A New York Times bestseller when it appeared in 1989, Roger Penrose's The Emperor's New Mind was universally hailed as a marvelous survey of modern physics as well as a brilliant reflection on the human mind, offering a new perspective on the scientific landscape and a visionary glimpse of the possible future of science. Now, in Shadows of the Mind, Penrose offers another exhilarating look at modern science as he mounts an even more powerful attack on artificial intelligence. But perhaps more important, in this volume he points the way to a new science, one that may eventually explain the physical basis of the human mind.
Penrose contends that some aspects of the human mind lie beyond computation. This is not a religious argument (that the mind is something other than physical) nor is it based on the brain's vast complexity (the weather is immensely complex, says Penrose, but it is still a computable thing, at least in theory). Instead, he provides powerful arguments to support his conclusion that there is something in the conscious activity of the brain that transcends computation--and will find no explanation in terms of present-day science. To illuminate what he believes this "something" might be, and to suggest where a new physics must proceed so that we may understand it, Penrose cuts a wide swathe through modern science, providing penetrating looks at everything from Turing machines (computers programmed from artificial intelligence) to the implications of Godel's theorem maintaining that conscious thinking must indeed involve ingredients that cannot adequately be stimulated by mere computation. Of particular interest is Penrose's extensive examination of quantum mechanics, which introduces some new ideas that differ markedly from those advanced in The Emperor's New Mind, especially concerning the mysterious interface where classical and quantum physics meet. But perhaps the most interesting wrinkle in Shadows of the Mind is Penrose's excursion into microbiology, where he examines cytoskeletons and microtubules, minute substructures lying deep within the brain's neurons. (He argues that microtubules--not neurons--may indeed be the basic units of the brain, which, if nothing else, would dramatically increase the brain's computational power.) Furthermore, he contends that in consciousness some kind of global quantum state must take place across large areas of the brain, and that it within microtubules that these collective quantum effects are most likely to reside.
For physics to accommodate something that is as foreign to our current physical picture as is the phenomenon of consciousness, we must expect a profound change--one that alters the very underpinnings of our philosophical viewpoint as to the nature of reality. Shadows of the Mind provides an illuminating look at where these profound changes may take place and what our future understanding of the world may be. - Shadows of the Mind (Amazon)
- Preface
- Notes to the reader
- Prologue
- Part I Why We Need New Physics to Understand the Mind
- Part II What New Physics We Need to Understand the Mind
Shadows of the Mind: A Search for the Missing Science of Consciousness by Roger Penrose
About the Book:
- Sequel to "The Emperor's New Mind" (ENM)
- Explores new ideas beyond ENM, providing a more powerful and rigorous case
- Argues against consciousness being fully understood in terms of computational models
- Presents a positive search for how the brain can use subtle physical principles to perform non-computational actions
Thesis:
- Consciousness (especially "understanding") cannot be simulated by mere computation
- Computation, even bottom-up systems with learning capabilities, cannot evoke genuine intelligence or consciousness
Key Arguments:
- Provides a detailed discussion in Part I supporting the thesis that conscious thinking involves non-computational ingredients
- Examines Godel's theorem and its implications, answering objections
- Argues against bottom-up systems achieving genuine intelligence
- In Part II, explores physics and biology to understand how non-computational action might arise within scientifically comprehensible physical laws
- Argues for a change in the quantum mechanical worldview based on new ideas and examples
Structure of the Book:
- Part I: Provides detailed arguments supporting the thesis that consciousness cannot be simulated by computation
- Part II: Genuine attempt to understand how non-computational action might arise within scientific principles
Physical Non-Computability in Conscious Actions
- Level of physical non-computability necessary for explanation
- Importance of this level in brain action
Proposed Differences from ENM (Eccles, Neuron, and Macromolecule)
- Behavior of neuron signals vs. synaptic connections between neurons
- Deeper level with significant physical activity at quantum-classical borderline
Specific Proposals
- Large-scale quantum-coherent behavior in microtubules within neuron cytoskeletons
- Non-computational link to classical computation taking place along microtubules
Arguments for Understanding Consciousness Scientifically
- Importance of a scientific path to understanding mental phenomena
- Appreciation of the nature of physical reality as a foundation
Implications and Importance
- Mind is not outside scientific explanation, but requires deeper understanding of physical world rules.
- Science is more than just mindless computation; it seeks understanding.
Book Overview:
- Differences in technicality between chapters, with most technical parts being Appendices A and C and portions of Chapters 2 and 3
- Non-mathematical readers may ignore appendices and some technical parts without significant loss
- Quantum mechanics discussions contain serious mathematics in sections like 5.12-5.18 and 6.4-6.6 (density matrix)
- Author has adopted pronoun 'he' for impersonal/metaphorical individuals, acknowledging potential gender bias but intending no implication of male or female
- Page references in The Emperor's New Mind (ENM) refer to the original hardback version; paperback pagination differs slightly from US and non-US editions.
The Father and Daughter's Expedition in a Remote Cave
Jessica's Nervousness:
- Felt uneasy when entering certain part of the cave
- Worried about boulder falling down, blocking their exit
- Asked her father for assurance that it wouldn't fall
The Father's Uncertainty and Confidence:
- Explained to Jessica that the longer the boulder had been there, the less likely it was to fall
- But couldn't provide further explanation
- Stopped prodding his plants and looked at Jessica with a smile
- Believed in his father's assurance, despite not fully understanding the reasoning
Jessica's Curiosity and Understanding:
- Wanted to know what would happen if boulder fell down and blocked their exit
- Asked about darkness and breathability if trapped in cave
- Father explained that some light could still penetrate, and they could survive
- When Jessica imagined never leaving the cave, wondered how she would know about the outside world
Father's Explanation of Scientific Discoveries:
- If they were trapped in the cave, they might be able to make scientific discoveries by observing shadows of objects outside
- They could learn about bird shapes and rabbit movements through these shadows
- Jessica imagined making a scientific discovery during a conference with other cave dwellers
Jessica's Father's Explanation of Scientific Theory in a Simplified Way
The Cave Analogy:
- People have been living in a cave for their entire lives
- The only things they know are the shadows and how they move
- Convincing them that an "outside world" exists would be challenging
The Earth Revolves Around the Sun Theory:
- This theory is nearly 2,000 years old
- The cave dwellers have not heard of such an idea before
- Explaining this theory would require showing how it explains detailed data about shadow movement on the cave wall
Addressing Skeptics:
- Some skeptics would prefer the "common sense" view of the sun moving around the earth
- Demonstrating that physical laws remain unchanged if the whole cave moves would be needed to persuade them
Importance of Science and Persistence:
- Science often involves lots of details, which can seem boring or irrelevant
- Skeptics need to be convinced through careful examination of evidence
- Galileo showed that the earth spins around the sun is a "striking simplicity" despite the complexities involved in demonstrating it
Part I: Why We Need New Physics to Understand the Mind
The Non-Computability of Conscious Thought
1. Consciousness and computation
- Mind and science:
- Ultimate scope of science: material attributes or mental existence?
- Proper scientific understanding of consciousness possible?
- Views on consciousness:
- No mystery, all essential ingredients in place (optimists)
- Extreme complexity and sophistication of brain limits understanding
- Beyond the scope of science (pessimists)
- Matters of mind and spirit can't be addressed adequately by science
- No mystery, all essential ingredients in place (optimists)
- Author's stance: scientific approach with missing ingredient argument
2. The author's mission
- Address consciousness from a scientific perspective
- Argue for the need of an expanded scientific worldview
- Provide specific direction towards change in physical laws (Part II)
- Clear and non-technical arguments, some mathematical technicalities where necessary
Importance of addressing consciousness scientifically
- Incomplete physical, biological, or computational theories
- Any scientific worldview without proper place for consciousness falls short
Can robots save this troubled world?
- Human stupidity causes conflict and violence
- Intelligence has positive achievements: science and technology
- Impressive advancements but environmental problems and potential global catastrophe
- Technology extends individual physical capabilities
- Expectations for future achievements
- Senses enhanced through technology (ancient to modern)
Expansion of Physical and Mental Capabilities through Technology
Hearing and Communication:
- Hearing aided by ear-trumpets, then electronic devices
- Extended capabilities through telephones, radio communication, and satellites
Transportation:
- Assistance to natural forms of locomotion with bicycles, trains, motor cars, ships, aeroplanes
Memory:
- Improved by printed books, films, television, interactive computer systems, electronic computers
Calculational Tasks:
- Simple and routine tasks extended by modern computers
- Massive or sophisticated tasks also benefited
Expansion of Intellectual Ability with Technology:
- Potential for artificial intelligence to exceed human capacity in the future
- Computers already used for intellectual assistance in certain situations
- Experts predict different timelines for computer's ability to reach 'human equivalence' or surpass human intelligence:
- Some measure it in terms of centuries, others claim it is decades away
Comparison between Brain and Computer:
- Neurons slower than electronic circuits (transistors) by over a million times
- Precision and accuracy of actions vastly superior in computers
- Randomness in brain's 'wiring' can be improved upon with electronic printed circuits
Advantages of Brain Over Current Computers:
- More connections between neurons than transistors (up to 80,000 synaptic endings vs. three or four)
- Some advantages may be short-lived but could potentially be overcome in the future with advancements like massively parallel computational systems and optical devices
Computing Power vs. Consciousness
Discussion Points:
- Advancements in computing power and artificial intelligence (AI)
- Potential for computers to surpass human capabilities
- Computers as superior intelligences or even obsolete humans
- Implications of computer-guided robots on humanity's role in society
Brains vs. Computers: Perspectives
- All thinking is computation (strong AI, functionalism)
- Belief that all mental processes can be explained through computational terms
- Awareness as a physical phenomenon
- Physical action of the brain cannot be fully simulated computationally
- Consciousness and science
- Mind cannot be entirely explained scientifically (mystic perspective) vs. expanding the scope of science to accommodate such matters
- Personal viewpoint: Rejecting mysticism but believing in an expanded role for science in understanding consciousness
Strong AI or Functionalism
- Belief that all mental processes can be simulated computationally
- Different interpretations of 'computation' and 'carrying out computations' exist among adherents.
Implications: If computers eventually surpass human capabilities, they may make us obsolete and potentially resolve worldly problems. However, this could also lead to the loss of humanity's role in society as they might be able to run the world more efficiently without humans. Additionally, consciousness and its influence on behavior remain unclear from a scientific perspective. There are various viewpoints regarding the relationship between consciousness and computation, with strong AI or functionalism being one extreme that all mental processes can be explained through computational terms. Ultimately, it is essential to consider these implications as we navigate the development of artificial intelligence technology.
Consciousness and Computation
Terms in Question:
- 'awareness' and 'consciousness': interpreted as abilities of ordinary general-purpose computers
- Some proponents differentiate interpretations, others reject or accept the existence of conscious awareness
Strong AI Viewpoint (d):
- Universe can be seen as a gigantic computer
- Physical systems are merely computational entities
- Consciousness emerges from sophisticated computation
- Operational attitude towards science
Implications:
- Physical world operates computationally
- Matter is nebulous and transient
- Patterns of information may persist more than material particles
Operational Argument (Turing Test):
- Computer-controlled robot behaves like a conscious being
- Sustained questioning could reveal lack of consciousness
- Acceptance of this argument distinguishes strong from weak AI
Weak AI Viewpoint (f.A):
- Physical objects follow scientific laws that allow simulation
- Behavior does not equate to consciousness
- Scientific common sense perspective
Comparison with Other Viewpoints:
- Operational viewpoint more radical than f.A but less operational than d
- Weak AI affirms computational simulation of physical world, denies consciousness from simulations.
Physicalism vs. Mentalism Debate
Background:
- Philosopher John Searle's distinction between a process and its simulation
- Computational simulations are different from actual processes (e.g., hurricane versus computer simulation)
- Presence or absence of consciousness depends on physical object and actions performed
- Biological brain might evoke consciousness, but electronic simulation might not
Viewpoint re:
- Operational viewpoint that consciousness cannot be fully simulated computationally
- Belief in new physics required for explanation of conscious objects (beyond presently known physical laws)
Physics and Computation:
- Some physicists argue for non-computational action outside presently known physical laws
- Difference between strong version of "6" and viewpoint .@: methodology and approach to problem solving
Viewpoints:
- .s:I, PA, "6", and.@ represent extremes or polarities in stance towards conscious awareness
- People may hold views that lie between or combine these categories (e.g., complexity of the brain as a computer)
Physicalism vs. Mentalism:
- Believers in .@ are mentalists, denying physicalism
- It's unclear where to draw the line between physicalism and mentalism for other viewpoints (Sil, PA, "6")
- Holders of Sil would generally be considered physicalists, but this may not be universally accepted.
Paradox in Understanding Mental Attributes: Sil's Perspective vs Others
Sil's Perspective:
- Irrelevance of material construction for thinking devices
- Mental attributes are abstract computations
- Term 'physicalist' may not be appropriate due to disassociation with physical objects
Other Viewpoints (PA and):
- Physical constitution plays a vital role in determining genuine mentality
- Possible physicalist standpoints
Common Usage:
- 'Mentalist' is often considered more appropriate for PA and, as they view mental qualities as real things rather than epiphenomena.
Terminology and Definitions:
- Avoid using the terms 'physicalist' and 'mentalist' in discussions that follow
- Refer instead to specific viewpoints Sil, PA, etc.
Computation: Top-Down vs Bottom-Up Procedures
Top-Down Algorithm:
- Constructed according to well-defined fixed computational procedure with clear solution to a problem at hand
- Example: Euclid's algorithm for finding highest common factor (HCF)
Bottom-Up Organization:
- System learns and improves performance through continual modification of rules based on 'experience'
- Rules subject to change, unlike top-down algorithms which remain fixed
- Example: Artificial neural networks
Both Top-Down and Bottom-Up Systems:
- Included under computational and algorithmic headings for our purposes here
- Key difference: bottom-up systems require a memory of previous performance to be incorporated into subsequent computations.
Artificial Intelligence (AI) and Computation:
- Aspirations of AI: imitate intelligent behavior through computational means
- Top-down systems: promising initially for well-defined problems Examples: specific mathematical problems, chess-playing computers, medical diagnosis with clear rules
- Bottom-up systems (artificial neural networks): useful for ill-defined criteria Examples: face recognition, sound recognition, mineral prospecting
- Combination of top-down and bottom-up organization expected for successful AI systems
- Top-down systems exhibit superiority over humans in numerical calculation and computational games
- Bottom-up systems reach about the level of ordinary well-trained humans in limited instances
- Distinction between serial and parallel architecture:
- Serial machines: perform computations one after another (step-by-step)
- Parallel machines: execute many independent computations simultaneously
- No principle distinction between serial and parallel machines; efficiency advantages for some problems
Church Thesis:
- Church's thesis (1936): anything that can be called a 'purely mechanical mathematical process' (algorithm) can be achieved within the lambda calculus scheme developed by Alonzo Church
- Turing's concept of algorithmic processes described as action of theoretical computing machines (Turing machines) in 1936/7
- Church's thesis and Turing's concept shown to be equivalent; modern computers based on Turing machine principles
- Original Church's thesis asserting mathematical algorithms can be carried out by an idealized modern computer is a mere tautology
- Turing may have intended that the computational capabilities of any physical device must be equivalent to a Turing machine, which goes beyond Church's original intentions.
Turing Machine and Physical Action
- Turing's motivations for developing Turing machine were based on his ideas of human calculator capabilities
- Turing may have viewed physical action as always reducible to some kind of Turing machine action, which can be called 'Turing's thesis'
- Distinguished from Church's original mathematical assertion
- A contradiction of Turing's thesis would not imply a contradiction with Church's thesis
Chaos and Physical Systems
- Recent interest in the mathematical phenomenon of chaos in physical systems
- Chaotic systems: dynamically evolving physical or mathematical models where future behavior depends on precise initial state
- Appear non-deterministic but are completely deterministic and computational in theory
- Prediction is difficult due to extreme sensitivity to initial conditions
- Example: detailed long-range weather prediction
Chaos in Simple Systems
- Can also occur with very simple systems, such as those consisting of a small number of particles
- Behavior of later balls in chain is random despite being mathematically deterministic and computable in theory
- Prediction is impossible due to extreme sensitivity to initial conditions and external factors
Computational Systems vs. Chaos
- All normally referred to as 'chaotic' systems are included in the category of computational systems
- They can be studied by putting them on a computer, which implies they can be computed
- However, simulation results may not resemble actual behavior due to sensitivity to initial conditions and external factors.
Chaotic Systems and Computation
Characteristics of Chaos:
- Not practicable to predict outcomes computationally
- Simulation of typical outcome achievable
- Outcome may differ from actual but plausible
- Repeating simulation with same input data results in same outcome (assuming no errors)
Chaos in Artificial Intelligence:
- Individual behavior simulation acceptable as possible outcome
- Included in computational or algorithmic systems
Analog Computation:
- Different types of computational devices, e.g., slide rule
- Continuous physical parameters used instead of digital states
- Technical challenge: applying standard notions of computation to continuous systems
- Accuracy and computational complexity concerns
- Alternative approaches for continuous system computations
- Mathematical structures with their own notion of 'computability'
- Naturalness and uniqueness questions remain unanswered
- Certain anomalies where technical non-computability arises.
The Computational Nature of Mental Activity
- The author's focus is on the question of the computational nature of mental activity
- In this context, "computational" refers to Turing computability
- Digital computers are relevant for today's AI activity due to their digital nature
- Possibility of a different kind of "computer" in the future that makes critical use of continuous physical parameters and exhibits non-digital behavior
Non-Computable Physical Actions:
- According to the weak version, there would have to be physical actions underlying conscious human brain behavior that are non-computable in the standard sense of discrete Turing computability but can be understood within present-day physical theories. These actions would depend on continuous physical parameters and cannot be properly simulated by standard digital procedures.
- According to the strong version, non-computability would have to come from some non-computable physical theory not yet discovered, whose implications are essential for conscious brain action. This is seen as a far-fetched possibility compared to finding a role for continuous action within known laws of physics that cannot be properly simulated in any computational way.
- The expectation is that reliable analogue systems can be effectively digitally simulated in principle, at least with present-day technology. Digital computing has advantages over analogue, such as increased accuracy through longer digit lengths, compared to linear increases in computer capacity for analogue machines.
Non-Computable Actions:
- Most well-defined actions are computationally defined and would have to be included under the "computational" concept.
- Random actions provided by a quantum system are not a significant practical advantage over pseudo-random generation entirely computationally. Pure randomness does nothing useful, it is better to use the pseudo-randomness of chaotic behavior, which is computational in nature.
The Unique Environment:
- The unique environment provided for each individual human being does not help in this context as a source of input beyond computation. A computationally simulated plausible (chaotic) environment would suffice for training a computer-controlled robot. The robot doesn't need to learn its skills through an actual environment, a typical simulated environment would be sufficient.
Hilbert's Tenth Problem and Computation
- Hilbert's tenth problem: finding a computational procedure for deciding if Diophantine equations have solutions
- Diophantine equations: polynomial equations with integer coefficients and solutions
- Example: w = 1, x = 1, y = 2, z = 4 solves the first system; second has no solution due to contradiction in conditions.
- Algorithm: computational procedure or process (Turing machine) for solving a problem
- Hilbert raised question about precise mathematical terms of having an algorithmic solution
- Turing proposed definition of algorithms with his universal Turing machine concept leading to general-purpose computers.
- Certain classes of problems don't have any algorithmic solutions, such as the halting problem.
- Matiyasevich proved in 1970 that there can be no computer program or algorithm for deciding whether a Diophantine equation system has a solution.
The Tiling Problem and Its Complexity
- The answer 'no' in mathematical problems cannot be systematically treated, as demonstrated by Matiyasevich's theorem
- The tiling problem: determining if a set of polygonal shapes can tile the plane
- Originally formulated with general polygons, which require real numbers for specification
- Simplified to polyominoes (sets of squares joined together)
- The computational insolubility of the tiling problem depends on aperiodic sets of polyominoes that can only tile non-periodically
- Proofs of the problem's unsolvability by computation are complex and based on Turing's argument for the halting problem
- Halting problem: deciding when a Turing machine will halt
- A toy model universe using polyomino sets as states, with rules for time evolution based on tiling properties.
The Toy Universe Model
- Discrete time parameterized by natural numbers
- State at time n is a set of polyominoes
- Two rules for state transition:
- Tile the plane: use Rule 1
- Don't tile the plane: use Rule 2
- The toy universe behaves deterministically but non-computably due to the lack of a general computational procedure for ascertaining when a polyomino set won't tile the plane.
Key Points about Computability vs Determinism:
- There is a difference between determinism and computability (as presented in ENM, p. 170)
- Certain deterministic universe models cannot be simulated computationally
- Models that are sufficient for understanding consciousness require a more advanced level of understanding than current knowledge allows
- Some argue it will take centuries before computers reach human mental capabilities, while others claim much shorter timescales (around 2030) based on accelerating computer technology and brain simulations
- The future of the planet:
- According to viewpoint d, eventually computers could simulate a conscious human brain, leading to robot superiority over humans and potential immortality for consciousness transfer into robots
- Viewpoint f argues that computers will always remain subservient to humans due to their non-biological nature
- Re accepts the possibility of future scientific developments creating devices with genuine intelligence and awareness beyond current understanding.
Related Issues:
- Legal responsibilities and rights for computers in the future if they approach or exceed human expertise.
Computer-Controlled Robots and Rights/Responsibilities
Viewpoint A:
- According to &I, a computer-controlled robot would have no rights or responsibilities if it lacked qualities like empathy and creativity.
- However, these qualities are difficult to measure in robots, so there may be a quandary about whether they possess them.
Viewpoint C(1) and Q(t)
- According to these viewpoints, computers cannot exhibit mental qualities and will never actually possess them.
- Therefore, computers cannot have rights or responsibilities.
Arguments Against Viewpoint S1 and FA
- The author argues strongly against the viewpoints that computers could ever have rights or responsibilities.
- He believes this would simplify legal issues and avoid the need to assign blame when things go wrong.
Deep Philosophical Questions on Responsibility
- The concept of "responsibility" implies an independent "self" with its own actions, not determined by external factors like inheritance or environment.
- This raises questions about free will and the nature of consciousness.
Clarification of Terminology
- The author has used terms like "understanding," "consciousness," and "intelligence" without clear definition.
- He acknowledges disagreements on the meaning of these terms, particularly regarding awareness.
- The author suggests a clear distinction between "genuine understanding" (which requires awareness) and heuristic uses of terms like "understands."
Artificial Intelligence and Understanding
Arguments for Genuine Intelligence:
- 'Intelligence' requires 'understanding' (a)
- 'Understanding' requires 'awareness' (b)
- Distinction between genuine and simulated intelligence
Avoiding Premature Definition:
- Rely on intuitive perception of meaning
- Normal usage allows for unconscious mind, which is more obscure than consciousness
Consciousness:
- Active and passive aspects: perception, sensation, appreciation
- Perception of color red requires passive consciousness
- Sensation of pain or appreciation of melody also examples
- No attempt to be precise about the concept of 'mind' in this context.
Active vs. Passive Consciousness
- Active consciousness: Involved in wilful actions and decisions
- Passive consciousness: Involved in sensations or 'qualia'
Role of Consciousness in Understanding
- Important for appreciating arguments, implying awareness is essential
- Arguments address passive (understanding) vs. active aspects of consciousness
John Searle's Argument: The Chinese Room
- Focuses on the issue of whether a computer program can achieve genuine understanding
- Human subject moves counters to simulate computations, but no personal understanding is experienced
- Suggests simulation cannot replicate the mental quality of actual understanding
- Differences with author's arguments: focuses on passive aspects and does not deny possibility of simulating external manifestations of conscious human behaviour.
Key Points:
- Active consciousness: Involved in wilful actions and decisions
- Passive consciousness: Involved in sensations or 'qualia'
- Importance of consciousness for understanding and appreciation of arguments
- Searle's Chinese Room argument addresses passive aspects, focusing on simulation vs. actual understanding; allows for simulation of external manifestations but not personal experience of understanding.
John Searle's Assessment of Chinese Room Argument (d) and Consciousness:
- Difficulties with computational model: a. Meaning of 'enaction': unclear what physical actions constitute algorithm enactment b. Special features required for significant awareness: higher-level organization, universality, self-reference, etc. c. Sticky issue of qualia (red, pain, sweetness) and their computational representation
- Limitations of current AI understanding: proponents must rely on brain's complexity to evoke consciousness but struggle to accept that modern computers can achieve it fully
- Role of different parts of the human brain in consciousness: a. Consciousness not present equally in all parts b. Cerebral cortex involves roughly 100 billion neurons and leaves room for complexity but is not the only part of the brain c. Unconscious cerebellum has approximately the same number of neurons and connections as cerebrum, yet consciousness is absent there.
- The case against the computational model: a. Difficulties with defining enaction b. Lack of persuasive attempts to explain qualia c. Modern computers cannot fully evoke significant mental experiences according to proponents.
- Searle's stance: further assessment in ENM (pp. 17-23) and ongoing discussion.
Limitations of Present-Day AI and Consciousness
Limitations of Present-Day AI:
- Fails to match human capabilities in common activities: recognizing objects, picking up items, etc.
- Ant's abilities surpass current computer systems
- Successes include chess programming and numerical computations
- Computer capabilities do not imply genuine understanding or consciousness
- Performance improvements depend on human programmers' understandings
Artificial Intelligence vs. Consciousness:
- Adherents of different viewpoints:
- CC (conscious actions beyond computation)
- f.J and .91 (computing is the only possibility for explaining consciousness)
- Focus on fundamental limitations, not just present advances
Evidence for CC:
- Mental activities beyond computation
- Examples: recognizing a crayon's use in drawing or an ant's everyday activities
Understanding vs. Computers:
- Top-down organization: human programmers' understandings used in construction of the program
- Bottom-up organization: no clear specific understanding required for the system or its programmers.
Deep Thought and the Limitations of AI
Understanding vs. Computation
- Chess example: Deep Thought, a powerful chess computer, made a blunder in understanding the significance of pawn barriers (Fig. 1.7)
- Reason for blunder: Top-down programming focused on calculating moves and improving material situation without actual understanding
- Human players can appreciate the importance of pawn barriers but Deep Thought couldn't
Real Understandings in AI?
- Some argue that bottom-up procedures with experience could lead to real understandings
- However, this argument will be challenged in Chapter 3
Limitations of AI
- Present-day computer systems don't substitute for genuine human understanding
- Early claims of artificial intelligence have not been fully fulfilled
- Computational systems can only preserve an illusion of understanding for a certain time before their limitations reveal
Mathematical Understanding vs. Computation
- Some argue that mathematical abilities come more easily to computers than humans
- However, mathematical understanding is beyond computation according to the author's argument in Chapters 2 and 3.
Non-Computational Consciousness: An Argument for a Non-Computational Ingredient in Conscious Thinking (Chapter 2)
Background:
- Term 'non-computational' refers to something beyond effective simulation by computers based on logical principles.
- Implies science and mathematics can explain conscious mental activity but cannot fully explain its performance.
Argument for Non-Computation:
- Clear-cut argument from Godel theorem (Kurt Godel, Czech logician)
- Simplified version, requiring minimal mathematics
- Logical possibility of conscious brain acting according to non-computational laws
- Argument against physicalism and mentalism
- Criticisms of earlier arguments: Oxford philosopher John Lucas (1961), Nagel and Newman (1958)
- Addressing criticisms in detail in Chapters 2 and 3
- Progress in understanding world through science and mathematics
- Mystery revealed as scientific understanding deepens
- Physicists take less mechanistic view of the world compared to biologists
- Broadening our notion of 'science' may be necessary to accommodate mind
Implications:
- Acceptance of some form of viewpoint rtf (mysticism or Platonism)
- Need to come to terms with its other implications if Godel argument forces us into this acceptance.
Platonic Viewpoint and Mathematics
- Driven towards Platonic viewpoint in understanding things
- Mathematical concepts and truths exist in a timeless, non-physical realm
- Distinct from physical world, yet influences it through reasoning about mathematical forms
- Direct access to this realm through awareness of mathematical concepts
- Computation cannot limit understanding beyond what is achieved computationally
Relevance of Mathematical Understanding
- Human mental activity involves application of consciousness and understanding
- Godel's theorem demonstrates human understanding cannot be algorithmic or computational
- Establishes role for non-computational activity in various aspects of mental activity
- Perception of color red is not directly related to mathematical understanding
- Mathematical thinking a specialized, human activity confined to a limited fraction of conscious lives
- Reason for focusing on mathematics: rigorous demonstration of non-computational consciousness required here.
Argument Against Computational Model of Consciousness
- Passive aspects of consciousness may not be computational (argument from obviousness)
- Functionalists argue all qualia are evoked by computations
- Brain's action must result in feelings of awareness, as functionalists claim
- Refusal to accept computational model for mental activity considered mysticism.
Non-Computational Processes in Mathematical Understanding and Common Sense Behavior
The Argument for Non-Computation:
- The author argues that "understanding" involves a non-computational process, whether it is related to mathematical perception or common sense behavior
- This non-computational process allows us to become directly aware of something, such as visualizing geometrical motions, topological properties, connectedness, and understanding meanings of words
- Communication between people is possible due to their shared "awareness" of similar internal experiences or feelings about things
The Role of Godel's Theorem:
- The author uses Godel's theorem to argue that no computer can fully achieve or simulate this non-computational process, as it cannot capture the Platonic knowledge and common sense understanding
- While we may not be able to characterize these concepts completely in terms of computational rules, descriptions of numbers in terms of physical objects like apples or bananas can still allow a child to grasp abstract concepts like "three"
The Role of Visualization and Abstract Concepts
Understanding Abstract Concepts:
- Appreciation for abstract concepts like 'three' comes with awareness, not just computation
- Visualizing motions of objects (e.g., blocks, ropes) involves appreciation of what is being visualized
Visualization vs Computer Simulation:
- Human visualization is different from computer simulations
- Rigid body motions can be simulated effectively by computers, but string/rope motions are more difficult
- Computers struggle with the infinite parameters required for a 'mathematical string'
The Importance of Visualization in Mathematics:
- Elementary arithmetic truths (e.g., 3x5=5x3) can be seen by visualizing arrays of spots
- Visualization gives more than just the specific case, but a general understanding that similar equalities hold
- Not all mathematical relations are obvious through visualization, but proofs aim to make steps 'perceivable'
Limitations of Reasoning and Understanding:
- Godel's argument shows that there is no way to eliminate the need for new 'obvious' understandings
- Mathematical understanding cannot be reduced to blind computation alone.
Equality between Consciousness and Computation (Natural Numbers)
- Equality holds for natural numbers, not strange "numbers" like ordinal numbers in mathematics
- Virtual reality as comparison: computer simulation of non-existent structure perceived as real through body movements
- Mental models constructed during visualization persist despite bodily motions
- Similar corrections for bodily movements involved in virtual reality
- Proposed model: internal physical structure acting like an "analogue computer" instead of digital computation
- Can mirror external system behavior without geometrical relationship or mirroring
- Argument against viewpoint d: conscious visualization not a computational simulation, but an act of understanding
- Non-computational nature of other human awareness inferred from non-computational character of some forms
Analogue Simulation vs. Digital Computation
- No reason to believe analogue simulation can achieve things digital computation cannot within existing physics framework
- Argument against computability of visualization: mental images need not be literally visual, can relate to abstract concepts or music
- Visualizations more concerned with general awareness than literal vision
- Belief that actual acts of visualization are non-computational is an inference from the non-computational nature of other human awareness.
Understanding Natural Numbers and Consciousness vs. Computation
Awareness of Natural Numbers:
- Cannot be encapsulated in any finite set of rules
- Cannot be simulated computationally (Godel's theorem)
- Examples: understanding properties of natural numbers, visualizing geometric motions of a rigid body or a piece of string/rope
Computer Simulations vs. Human Brain:
- Effective digital simulations rely on human understanding and insight
- Bottom-up approaches (e.g., artificial neural networks) may not be adequate for simulating complex geometrical or topological phenomena
- Simulating such things without genuine understanding is unlikely
Physical Processes Behind Awareness:
- Must be something beyond computational simulation, as demanded by viewpoint
- May require subtle shifts in scientific criteria and methods
- The author will explore this further in Part II of the book, focusing on physics and biological action.
Justifying the Claim of Non-computational Conscious Understanding:
- Requires justifying the claim that we perform non-computational feats when we consciously understand
- Will be explored by turning to mathematics in the remainder of Part I.
AI and Computation
- In early 1994, Moravec still holds to the estimates of Turing (1950)
- Four viewpoints on AI: top-down procedures, bottom-up schemes, connectionist networks, and symbolic AI
- Top-down procedures started in the 1950s with elementary methods
- Bottom-up schemes, like pattern recognizing perceptron (Rosenblatt, 1962), have limitations overcome by Hopfield (1982)
- Neural network activity is a subject of significant worldwide interest
- Important landmarks in top-down AI research include papers by McCarthy (1979) and Newell & Simon (1976)
Limitations of Computation
- Turing's thesis: "A machine can do any task that a human being can do" (ENM, p. 5-14)
- Limitations include brain activity interpretation, continuous vs discrete action, and computation complexity
- Brain activity considered in Rubel (1985)
- Continuous vs digital action: Pour-el & Richards (1979), Smith & Stephenson (1975), Blum et al. (1989)
- Computation complexity: Chaitin (1975), Hsu et al. (1990)
- Limitations include brain activity interpretation, continuous vs discrete action, and computation complexity
- Polyomino tiling problem: cannot be algorithmically ascertained for failure to tile the plane, but success can be determined
Consciousness and AI
- Searle's argument: 'consciousness' is more defined than less understood terms like 'mind' (Hofstadter & Dennett, 1981)
- Failed attempts to define consciousness in ENM, attacked by Sloman (1992) and others
- Putnam, Smart, Benacerraf, Good, Lewis, Hofstadter, Bowie, McDermott, Manaster Ramer et al., Mortensen, Perlis, Roskies, Tsotsos, Wilensky, Dodd, Penrose: debate on consciousness and AI
Recent Progress in AI 'Understanding'
- British TV program The Dream Machine (December 1991)
- Douglas Lenat's intriguing 'Cyc' project discussed in Freedman (1994)
Background:
- Woolley (1992): Vivid popular account of ancient civilizations
- Suggestion made by Richard Dawkins in BBC Christmas Lectures, 1992
- Lenat and others working towards understanding human mind through AI
Ancient Civilizations
- Woolley's book provides a vivid account of ancient civilizations
Contemporary Perspective
- Richard Dawkins: Suggested ideas during BBC Christmas Lectures, 1992
- Lenat and others: Working towards understanding human mind through AI
Related Work
- Freedman (1994): Account of Lenat's work in this direction.
Godel's Theorem and Turing Machines
Background:
- Godel's theorem is a fundamental contribution to mathematics, established in 1930
- Shows that no formal system of mathematical rules can prove all true propositions of arithmetic
- Initiated progress in the philosophy of mind by revealing limitations of formal systems
Significance:
- No system can ever suffice to establish all true propositions of arithmetic, even those accessible to human intuition and insight
- Human understanding cannot be reduced to any set of computational rules
- Algorithms or computations are equivalent to what can be achieved by a mathematical formal system
Turing Machines:
- A type of mathematically idealized computer carrying out step-by-step procedures
- Reads and writes marks on an infinitely long tape, erases marks, changes internal state
- Instructions are fed in as binary numbers (sequence of Os and 1s)
- Stops when it reaches a 'STOP' instruction and displays the answer to the computation
Universal Turing Machines:
- Can imitate any other Turing machine, carrying out any computation or algorithm
- Modern general purpose computers are equivalent to universal Turing machines.
Computations and Non-Terminating Algorithms
2.2 Computations
- Definition: A computation (or algorithm) is an action of a Turing machine, i.e., a computer following a specific program.
- Not limited to arithmetic operations but can involve logical operations as well.
Example Computation: Finding a Number not Sum of Three Squares (A)
- Try each natural number in turn starting with 0.
- For each number, consider only finitely many square numbers less than or equal to it.
- If three squares add up to the number, move on to the next number.
- The computation stops when a number is found for which no triplet of squares adds up to it.
- 7 is such a number (1 + 1 + 1 = 1 + 1 + 2 = 3; 0 + 1 + 1 = 0 + 1 + 1 = 2).
Non-Terminating Computations: Finding a Number not Sum of Four Squares (B)
- Attempt to find a number that is not the sum of four squares.
- This computation never terminates as every natural number is indeed the sum of four squares (Lagrange's theorem).
Non-Terminating Computation: Finding an Odd Number Sum of Two Even Numbers (C)
- An odd number cannot be the sum of two even numbers since even numbers only add up to even numbers.
- This computation never terminates.
Non-Terminating Computation: Finding an Even Number Greater than 2 not Sum of Two Primes (D)
- Attempt to find an even number greater than 2 that is not the sum of two prime numbers (Goldbach Conjecture).
- The outcome depends on whether or not the Goldbach Conjecture is true; unproven as of now.
Deciding Non-Terminating Computations
- Mathematicians may use various procedures to ascertain if computations do not terminate.
- Some are easy, others difficult, and some require immense intelligence and effort.
- In some cases, like (B) and (D), the results remain unproved due to their complexity.
Hexagonal Numbers: Properties and Relationship with Cubes
Hexagonal Numbers:
- Numbers arranged in hexagonal arrays, excluding vacuous array
- Obtained by adding successive multiples of 6
- First few hexagonal numbers are cubes (1, 8, 27, 64, 125)
Summing Hexagonal Numbers:
- Sum of successive hexagonal numbers is always a cube
- Proof: each hexagon in a cubic array can be viewed as a hexagon from far away (Fig. 2.3)
- This establishes that adding together successive hexagonal numbers will always result in a cube
Formal Demonstration:
- Principle of mathematical induction used to prove non-stopping nature of computation (E)
- Insufficient rules can't establish non-stopping computations rigorously
Godel's Theorem and Computational Families:
- Godel's theorem demonstrates that clear-cut rules are insufficient for establishing non-stopping computations
- Human understanding and imagination cannot be fully replaced by formalized procedures or sets of rules
Computation on Natural Numbers:
- Computations depend on natural numbers (C(n))
- Turing machine computes on the number n, providing a family of computations for each natural number.
Computations and Natural Numbers
- Two examples: (F) finding a non-sum of n squares, and (G) finding an odd number as sum of even numbers
- (F) stops for small values of n (1, 2, 3), but requires mathematics to prove it doesn't stop for larger values
- (G) never stops for any value of n since the sum of even numbers is always even plus an odd number, leaving one remaining odd number to be added
Procedures for Ascertaining Non-Stopping Computations
- A hypothetical procedure A that demonstrates a computation C(n) doesn't stop when it terminates
- If A terminates in a case, it provides proof that C(n) never stops in that case
- For most arguments, A does not need to have the specific role of being able to decide if C(n) never stops
Listing Computations and Encoding Them
- All possible computations C can be listed as {C0, C1, ...}, with Cq referring to the action of the qth Turing machine on n
- A single computation c exists that gives Cq(n) when presented with q and n
Technical Requirements for A's Functioning
- A must be sound: it doesn't provide wrong answers about computations not stopping
- Listing of all computations is computable, meaning there is a single computation that gives Cq(n) when presented with q and n
Applying A to Specific Computations (q = n)
- If A(n, n) stops, then Cn(n) doesn't stop
- A(n, n) is one of the computations C0, C1, ..., as applied to a single natural number n
Cantor's Diagonal Slash Procedure
- Applying A(k, k) when q = n gives (K): A(k, k) = Ck(k)
- When n = k, we have (J): A(n, n) = Cn(n), which is a specific computation among the list {C0, C1, ...}
- The second part of the diagonal slash is examining this particular value n = k: (L) If A(k, k) stops, then Ck(k) doesn't stop
- Substituting (K) into (L), we find that if Ck(k) stops, it must not stop according to the soundness of A. However, since A cannot ascertain this fact about Ck(k), it cannot encapsulate our understanding of these computations.
Godel's Theorem and Computational Procedures
Argument Summary:
- The author has found a computation Ck(k) that never stops, but it cannot be determined using the given computational procedure A
- This is an application of Godel's (Turing) theorem, stating no knowably sound set of rules can ascertain non-stopping computations
- The argument applies to any computational procedure for determining computation non-stoppingness
Implications:
- Human mathematicians do not use a known sound algorithm to establish mathematical truth
- The conclusion is inescapable, but some argue against it and question its relevance to conscious awareness
Addressing Objections (2.6):
- The author will carefully address potential objections (queries Q1Q20) raised by critics
- These objections may include additional arguments of the author's own
- After addressing these objections, the conclusion C remains unscathed
Implications of Conclusion C (Chapter 3):
- The author will take a more positive approach in Part II, proposing possible physical processes underlying brain activity and understanding that may elude any computation description.
Technical Points to Consider:
- A was assumed to be a single procedure, but mathematicians use various reasoning types
Mathematical Argument: Godelian Procedure and Algorithmic Procedures
Background:
- Discussion of allowing for a list of possible algorithms (A1, A2, etc.) to be re-expressed as one algorithm (A) that fails if all individual algorithms do.
- Case where list is infinite: need an algorithmic way to generate it.
- Changing algorithm: rules for change must be algorithmic or part of 'A'.
- Non-algorithmic changes considered later.
Argument:
- Finite list of procedures can always be re-expressed as one algorithm that stops when any fails.
- Infinite list requires generating procedure algorithmically.
- Changing algorithm can be an instance of a single algorithm with rules for change also being included.
- Algorithmic means of changing an algorithm are considered here, not non-algorithmic methods like random ingredients or environment interaction.
- Focus is on the possibility of such a change being algorithmic, which would contradict the conclusion if true.
- Unnecessary narrowness in insisting A must go on computing forever in cases where it may have become clear computation stops. The argument only applies to insights allowing us to conclude computations don't stop; A cannot come to a successful termination by concluding computation does stop.
Godel-Turing Argument: Q1, A's Role and Reductio Ad Absurdum
- Q1: Discussion about a mathematical insight 'A', which can be seen as a looping algorithm when needed to avoid contradiction
- Not necessary for A to be the knowably sound algorithm but can be derived from it
- Argument uses reductio ad absurdum method, starting with assumption of using a knowably sound algorithm and deriving contradiction
- Q4: Assumption of every Cq denoting a well-defined computation is unlikely in any straightforward numbering system
- Interpreting 'stops' as properly specified working computation solves this issue for the argument
Godel-Turing Argument: Q5, Defeating A and Implications
- Q5: Showing that a particular algorithm A can be defeated doesn't prove superiority over any A whatsoever
- Reductio ad absurdum argument demonstrates no largest prime or knowably sound algorithm at all for mathematical insights
Godel-Turing Argument: Q6, Computer Programming and Non-algorithmic Insights
- Q6: Computers can follow computational processes but cannot ascertain truths beyond their capabilities
- New computation leading to Ck(k) not officially admitted as part of A's procedures
- Explicit computational procedure for obtaining Ck(k) from algorithm A given in Appendix A.
Godelian Argument
Issues with Q6:
- Assumes A represents totality of procedures, cheating to introduce new truths mid-argument
- Robot lacks understanding for reliable judgment of truths vs falsehoods
- Algorithmic nature of operations not the issue; robot requires valid truth judgments
Understanding vs Algorithms:
- Understanding may not be algorithmic
- Later discussion tries to persuade reader that understanding is non-algorithmic
Finite Computers and Mathematics:
- Computer's ability to simulate human mathematics irrelevant to critical issue: truth judgments
- Random false or jumbled mathematical statements cannot be trusted without understanding
- Arguments address how one decides which statements are true and false, not relevance of infinite structures in discussions about finite objects.
Q8: Idealized Computations vs Finite Physical Objects:
- Distinction between idealized computations with potential infinity and finite physical objects like computers or brains
- Importance to assess limitations of idealized arguments when applying them to actual systems
- Godel-Turing argument not substantially affected by consideration of finiteness in discussions.
Finiteness Limitations in Computation
Turing Machines:
- Idealized computations (Turing machines) can be studied as mathematical constructs, even though they cannot be built physically due to their infinite nature.
- Specification of a single computation is finite and well-defined.
- Finiteness limitations may come from either:
- Finiteness of specification: The number
nor the pairq, nin the computation is too large to be specified by a feasible computer or person. - Finite time constraints: A computation that is not too large to specify might still take too long to perform, making it seem like it never stops.
- Finiteness of specification: The number
- Finiteness limitation (i) is more significant than (ii), as it affects the Godel-type discussion less greatly.
Finiteness of Specification Limitations:
- Instead of an infinite list of computations, there is a finite list up to a maximum
Qspecifying the largest computation that can be accommodated by the computer or human. - For humans, this "vagueness" in
Qwill be discussed later. - The number
Nlimits the size of the inputnto a fixed value that the computer or human can handle.
Sound Algorithm A(q, n):
- A "sound" algorithm
A(q, n)that halts when it demonstrates the non-termination of computationCq(n). - When
A(q, n)stops,Cq(n)also does not terminate. - An issue with this approach is that it might take more steps than the computer or human can handle.
Computation A(k, k) for Defeating A(q, n):
- By setting
k = q, the computationA(k, k)is equivalent toCk(k). - If
kis larger thanQorN, it cannot be implemented, but this occurs only if the algorithmAis becoming too complex.
Understanding the Godel's Incompleteness Theorem and its Implications
Background:
- Godel's Incompleteness Theorems state that no system of mathematics can be both consistent and complete if it is rich enough to express arithmetic.
- Two parts: The first theorem states that there are true statements about a mathematical system that cannot be proven within the system itself. The second theorem asserts that any attempt to make the system more robust by adding new axioms or rules will inevitably lead to undecidable statements being introduced as well.
Computation Complexity:
- Computations may become so complicated that they surpass human capacity for contemplation and implementation.
- Degree of complication: the number of binary digits in specifications.
- A's degree of complexity: oc.
- Ck(k)'s degree of complexity: less than oc+210 log2(oc+336).
Intuitionism:
- Philosophical stance questioning absolute mathematical truth and accepting only constructively defined objects.
- Rejects reductio ad absurdum arguments that don't provide actual constructions for mathematical objects.
- Godel type argument is still intuitionistically acceptable because it uses contradiction from an assumption of existence, not non-existence.
Mathematical Foundations: Godel's Argument
Background
- Difficulties arose in late 1800s due to advances in mathematics, specifically infinite sets (Cantor's contributions)
- Challenges with reasoning about the infinite led to formalism movement
- Formal systems: sets of rules and mathematical statements for valid reasoning
Formal Systems
- Rules laid down precisely to ensure trustworthy mathematical reasoning
- Mechanically checked for correct application
- Some rules may involve infinite sets, leading to potential doubts regarding legitimacy
- Not all formal systems are trusted; some allow Russell's paradoxical 'set of all sets that aren't members of themselves' which could lead to inconsistencies.
Formalism Perspective:
- Infinite sets might be meaningless, and absolute truth or falsity uncertain
- Importance lies in consistency and completeness of formal systems, rather than mathematical statements about infinite quantities being true or false in an absolute sense.
Key Concepts:
- Formal system (IF)
- Consistency: no statement comes out as both True and False within the system.
- Completeness: every properly formulated mathematical statement turns into either True or False in the system.
- Undefinable statements: those that cannot be determined by rules of a formal system (IF).
- Formalists' perspective: focus on demonstrating consistency and completeness of suitable formal systems rather than absolute mathematical truths about infinite quantities.
Godel's Theorem and Consistency of Formal Systems
Background:
- Mathematical rules for demonstrating consistency of formal systems (e.g., Peano arithmetic) were sought to prove the consistency of more sophisticated systems.
- The hope was that such a system would be complete and consistent, providing justification for reasoning about large infinite sets.
Godel's Theorem:
- In 1930, Godel produced a result showing the unattainability of the formalists' dream (consistent and complete system).
- This theorem applies to systems that contain arithmetical operations necessary for Godel's theorem formulation.
Condition of Consistency:
- The most familiar version asserts that a sufficiently extensive formal system cannot be both consistent and complete.
- A stronger version, w-consistency, requires that if all statements 'P(n)' are provable within IF, then it must not be the case that every natural number n holds true for P(n).
- This condition ensures trustworthy systems as they do not allow anomalous situations where all instances of a proposition may be proven yet its negation is also provable.
Godel's Theorem Implications:
- Q(IF): the formal system IF is w-consistent cannot be proven within IF if it's actually w-consistent (assuming sufficient extensiveness).
- G(IF): the consistency of a formal system cannot be proven using its procedures.
- This theorem challenges the idea that all mathematical procedures can be encapsulated in a single formal system and provides limitations to formalism philosophy.
Formal Systems and Algorithmic Proof
Computations vs Formal Systems:
- Godel-Turing argument refers to "computations" but no mention of formal systems
- Closely related concepts:
- Formal system includes an algorithm (F) for checking correctly applied rules
- Given computational procedure E, a formal system IE can be constructed to express all truths obtainable by E
- Includes elementary logical operations if not part of E
- Not strictly equivalent to E if logical operations are added in construction of IE
Constructing IE from E:
- Start with basic consistent formal system IL, e.g., predicate calculus
- Adjoin E as additional axioms and rules to IL to construct IE
- Involves adding Turing machine notation and operations if needed
- Purpose of argument is that A (algorithm for determining non-stopping computations) incorporates basic logical inference, so algorithms and formal systems are equivalent procedures for accessing mathematical truths.
Relation between Computations and Formal Systems:
- Argument applies to all Turing machine actions if the formal system IF is broad enough to incorporate them
- Can be achieved by incorporating A's algorithmic procedure into IF's rules
- Godel's theorem states that there exists a statement in IF that cannot be proven or disproven within the system itself.
Godel's Theorem and its Relation to the Presented Argument
Ingredients of Godel's Original Konigsberg Argument:
- Algorithmic procedure A plays role of formal system IF in Godel's original theorem
- Particular proposition 'Ck(k) does not stop' obtained in 2.5, which is inaccessible by procedure A but perceived as true, plays role of assertion Q(IF) from Konigsberg, asserting co-consistency of IF
- Belief in soundness of IF implies belief in its consistency (co-consistency and ordinary consistency for some systems)
Consistency vs. Co-Consistency:
- Distinction between consistency and co-consistency disappears for many formal systems
- For simplicity, the following discussion will focus on just "consistency"
Importance of Consistency in Formal Systems:
- No right to use rules of a formal system IF without believing in its consistency
- Belief in soundness implies belief in consistency
- Trusting arguments based on mathematical truth requires accepting reasoning beyond limitations of the formal system
Further Objections and Clarifications:
- Absolute nature of mathematical truth: Differences in views regarding infinite sets; axiom of choice, Cantor's continuum hypothesis independent of standard Zermelo-Fraenkel axioms (ZIF)
- Impact on Godel-Turing argument: Affects more limited class of mathematical problems rather than those related to non-constructively infinite sets.
Godel's Theorems and Mathematical Truth
Definition of I1-sentences:
- Statements about terminating computations that can be precisely specified in terms of Turing machine actions
- Examples: Q(IF), G(IF) for any formal system IF
Nature of Truth:
- Absolute matter independent of personal beliefs on non-constructively infinite sets
- Intuitionistic position does not evade this conclusion
- Possible issues: computations taking inordinately long or unable to be written down
- Discussed in relation to Q8 and Q9 above
Mathematical Truth:
- As well-defined as concepts like TRUE, IFAO, UNDECIDABLE for any formal system IF
- Algorithm F equivalent to IF can determine provability or unprovability of a statement P
Objection: Subjectivity in Mathematics?
- Some Il1-sentences proven using infinite set theory but not standard methods
- Qll objection
- Different mathematicians may have varying beliefs regarding set theory
- Shared understanding of mathematical community questioned
Counterargument:
- Godel(-Turing) argument phrased in terms of 'mathematicians' or 'the mathematical community'
- Gets away from suggestions of personal algorithms and individual perceptions of truth
- Issue of subjectivity in beliefs regarding existence of large non-constructive sets S remains, even for I1-sentences.
Mathematical Truth and Personal Algorithms
Unfairness of Referring to Mathematicians:
- Different mathematicians may employ inequivalent 'personal algorithms' for ascertaining mathematical truth
- It is unfair to refer to 'mathematicians' or 'the mathematical community' as a whole, as individual mathematicians may differ in their approach
Mathematical Demonstration:
- Mathematical demonstration is not as subjective as it might seem
- Mathematicians generally agree on the truth of specific II 1-sentences when demonstrated convincingly
- Disputes over foundational issues are rare, and can be resolved through a convincing demonstration or refutation
Differences Among Mathematicians:
- Mathematicians may hold different provisional opinions on foundational issues
- These differences are similar to differences in expectations regarding ordinary mathematical propositions
- The lack of convincing demonstrations for foundational issues may lead to uncertainty and disagreement
Mathematicians' Viewpoints:
- A mathematician's viewpoint can be either consistent or inconsistent with the validity of II 1-sentences
- Different viewpoints do not necessarily pose a threat to the original formulation <>
- There are only a few significant differences among mathematicians, such as the axiom of choice
- Disputes over set-theoretic axioms do not always lead to inconsistencies with standard ZIF axioms
Impersonal Formulation for Understanding Mathematicians
Axiom of Choice:
- Frequently assumed without mention, but does not challenge general impersonal formulation
- Restricted to Ill-sentences
Other Disputed Axioms:
- Concede there are some unstated assumptions, but number is small (less than 10)
- Implies different "grades" of mathematicians with varying reasoning abilities
- Replace "Human mathematicians..." with ** for each grade n
Relevance to Specific Mathematicians:
- Differences in ability to follow arguments and make discoveries are not the issue
- Focus is on what types of arguments can be perceived as valid by mathematicians
Accessibility of Arguments:
- Even if mathematician possesses demonstration/refutation, disagreements may require patience and understanding
- In principle, what's accessible to one mathematician is the same as others
- Complexity of arguments doesn't exclude human comprehension with effort
- Claims about arguments beyond competence should be evaluated.
Effective Exposition:
- Obscure writing styles or limitations in lecturing abilities can create difficulties for mathematicians.
Mathematicians' Beliefs and Uncertainty
Point made: Mathematicians do not have absolutely definite beliefs about the soundness or consistency of formal systems they use, and their views may shift as they gain more experience.
Factors influencing mathematicians' beliefs:
- Formal systems further from intuitions and experiences may have less clear beliefs
- Some mathematicians might not take a definite standpoint on what is unassailably true
- Absolute human certainty is rare, so a reasonable approach is to take some body of beliefs as unassailably true and argue from there
- Mathematicians may modify their views with experience
Godel's argument: Whatever standpoint is adopted cannot be encapsulated in the rules of any knowable formal system. This belief must already be implicitly contained in the original standpoint, even if it was not immediately apparent.
Implications:
- Possibility of errors in deductions from premises may lead to uncertainty about conclusions
- However, actual errors are correctable through further examination and confirmation of arguments
- Correctly carried out arguments become more convincing with time
The Uncertainty of Mathematical Beliefs and Godel's Theorem
Mathematicians' Beliefs vs. Principle:
- Mathematicians may believe in the soundness of a formal system IF, but there is no assurance that a more powerful system IF* might only be 'practically certainly' sound.
- However, we should insist that IF includes ordinary logic and arithmetical operations.
Unassailably True vs. Practical Certainty:
- If IF is sound, it must also be consistent; Godel proposition G(IF) cannot represent the totality of a mathematician's beliefs.
- The only real question concerns the details of coding assertions like 'IF is consistent' into an arithmetical statement (a Turing machine).
- Possible errors may arise in coding and construction, but they do not undermine acceptance of G(IF) as unassailably true.
Technical Points:
- The soundness of A implies the truth of Q(IF), which can be used instead of G(IF) in Godel's argument.
- Both Q(IF) and G(IF) are true but not follow from the rules of IF, assuming it is sound.
Belief in Sound Systems:
- If a formal system like IF is unassailably sound, then so is its associated proposition G(IF).
- Any uncertainty or 'shading off' lies solely in possible errors or misunderstandings when formulating G(IF) rather than the intended version.
The Viewpoint of the Formalist
- ZIF (or ZIF) system*: represents all that's needed for serious mathematics, but its consistency is not provable
- Formalists accept this and focus on what can be proved/disproved within the system
- See ZIF game: a set of rules laid down within the system for playing a "game" of mathematics
- Anything true will also be true in the system, while some statements are neither provable nor disprovable (undecidable) like Godel's statement G(ZIF) and its negation -G(ZIF)
Limitations of the Formalist Viewpoint
- Doesn't represent a genuine standpoint for mathematical beliefs:
- To believe mathematics derives actual truths, one must believe the system is sound and consistent
- Believing in G(ZIF) despite its undecidability goes beyond what can be derived within ZIF game
- Formalist viewpoint can't represent the total basis of true mathematical knowledge or beliefs as any formal system (including ZIF*) may have limitations.
Use of the Godel Theorem
- G(ZIF) statement is not necessarily true but more reliable than statements derived using the rules of IF because IF might be consistent without being sound
- Trusting any result obtained with a formal system requires trust in that system's consistency and soundness, including G(ZIF)
- Statements like "P is deducible within IF" or "not deducible within IF" should be used instead of simple assertions of truth or falsehood.
Deriving Statements About Unprovable Results
- Procedures for establishing mathematical truths cannot be equivalent to a belief in a formal system, no matter what the formal system may be.
Understanding Godels Incompleteness Theorem The Distinction between Natural and Supernatural Numbers
The Godelian Case and Natural Numbers
Godel's Theorem:
- Conclusion that G(IF) is true of a formal system IF depends on assuming symbols representing natural numbers actually do represent natural numbers
- For "supernatural" numbers, G(IF) may be false
- No finite way to ensure referencing intended natural numbers and not unintended "supernatural" numbers
Formal Systems IF, IF*, and the Standard Interpretation**:
- Adjoining G(IF) or -G(IF) to IF's axioms yields consistent systems, but with different interpretations:
- Standard interpretation: IF is sound when interpreted as natural numbers
- Non-standard reinterpretation: Symbols can refer to "supernatural" numbers with different ordering properties
Distinguishing Natural Numbers:
- Natural numbers are familiar concepts that we distinguish from "supernatural" numbers
- We can grasp the concept of the natural number totality, even though finite descriptions are inadequate
- The specific infinite character of the natural numbers is difficult to capture with precise rules
The Peano Axioms and Defining Natural Numbers:
- Peano axioms define natural numbers using symbols 0, S (successor), and the induction principle
- These axioms do not fully capture the infinite character of the natural number totality
Godel's Theorem: Implications and Misconceptions
Issues with Standard Interpretation of Logical Operations 'I:/' and '3'
- In standard interpretation, these symbols denote 'for all natural numbers' and 'there exists a natural number'.
- In non-standard interpretations, the meanings shift appropriately to quantify over other types of numbers.
Capture of Peano Mathematical Specifications
- Standard interpretations do not capture the specifications in terms of formal rules.
- Requires passing to Second-Order Logic for capturing meanings and completing formal systems.
Misconceptions about Godel's Theorem
- Formal system consistency and completeness issues are often compared to Euclidean geometry.
- Misleading interpretation: Godel showed mathematics as arbitrary, not absolute or discovered.
Formal Systems vs. Ordinary Natural Numbers
- Formal systems explore mathematical properties but differ from ordinary arithmetic.
- Discovering natural numbers is absolute compared to inventing man-made rules.
Reinterpreting Symbol Meanings in Formal System IF
- Representation of Godel argument within a formal system IF requires symbol meaning shifts, which may not be allowed if mental activities are solely based on the operations of IF. If mental activities involve more than just the procedures in IF, then rules governing shifting meanings must be known and incorporated into 'IFt'.
Godel's Theorem and Its Implications
Formulating Theorems in Peano Arithmetic
- It is possible to formulate a theorem within Peano arithmetic that implies "IF sound" implies "G(IF)"
- Stronger results: "IF consistent" or "IF ro-consistent" implies "Q(IF)"
- Passing from belief in soundness to truth of Godel proposition accepted but cannot be automated
- Godelization cannot be automated due to the need for understanding meanings
- Another aspect of Q18: IHI, a sound formal system containing Peano arithmetic, implies "G(IHI)" as a theorem
- IHI cannot assert its own consistency or encapsulate symbol meanings
- Inconsistent IHI can formulate and assert anything it wants, including "IHI is consistent" if and only if inconsistent
- Repeatedly adjoining Godel propositions (G(IFn)) to a formal system IF leads to an infinite sequence of sound systems IFn+1, but this process cannot be automated or fully incorporated into one comprehensive system without work.
Godel's Systematization Procedure
- Recognized by mathematicians as 'with just a little more work'
- Involves systematizing Godelizations, continuing the process further to larger ordinals (e.g., IF CJU', wCJ')
- Crux of the matter is achieving systematization for recursive ordinals, which require appropriate insights
- No algorithmic procedure exists for systematizing all recursive ordinals
Background on Godel's Procedure
- First put forward by Alan Turing in his doctoral thesis (1939) and published in Turing 1939
- Later, Solomon Feferman (1988) showed any true -sentence can be proved through repeated Godelization
- Does not provide a mechanical procedure for establishing truth of -sentences due to computational complexity issues
Non-Computability vs. Complexity in Mathematics
- Non-computability is crucial in understanding the mind's ability to circumvent computational complexity
- Mathematicians often use insights to take shortcuts instead of engaging in blind computation, which can result in unworkable complexities
- Godel argument shows that mathematical understanding lies beyond computation, but insights may still be used for achievable tasks
Implications and Controversies
- Existence of large infinite sets: debate about whether a clear rule must exist before a set is considered to exist (e.g., axiom of choice)
- Cohen's perspective on the continuum hypothesis in his 1966 book, revealing that he views mathematical truth as absolute and not arbitrary despite leaving untouched the question of its actual truth.
Discussion of Views on Cognition and Mathematics:
- Agrees with own views (8.7) regarding cognitive processes, cf. Hofstadter (1981), Bowie (1982)
- Euclidean geometry and formal systems:
- Can be formally specified in certain interpretations
- Not fully resolved in other contexts or interpretations, e.g., polyomino tiling problem, arithmetic, or computing systems (Kreisel 1960, 1967; Good 1967; Freedman 1994)
- Godel's theorems:
- Completeness theorem: non-standard models always exist for formal systems
- Inconclusive regarding Euclidean geometry's complete specification.
Turing Machine Specifications
Components:
- Finite number of internal states
- Infinite tape with linear succession of boxes (marked or unmarked)
- Reading device examines one mark at a time
- Tape consists of 1s and Os, no immediate repetition of a 1
Functions:
- Alter/leave mark on box being examined
- Determine new internal state
- Move reading device left (L), right (R), or halt (STOP)
- Display answer as succession of Os and 1s to left of reading device
Expanded Binary Notation:
- Represents natural numbers in ordinary binary scale but with expanded notation
- Uses sequences like 110, 1110, etc., to denote instructions for Turing machine's reading device
Universal Machine U:
- Acts upon tape containing detailed specification of particular given Turing machine T
- Immediate portion defines the instruction set and initial data for T
- Data fed into U to the right of specified portion
Specifying Machines:
- Instructions coded as sequences 110 (R), 1110 (L), or 11110 (STOP)
- Preceding symbols determine marking and internal state changes.
Turing Machine Specification
- Ensure instructions for every internal state acting on 0 and 1 (except highest numbered state)
- Dummy instruction may be added where not used
- Representation of 00 can be replaced with a single 0 without ambiguity
- Turing machine begins in state 0, moves along tape until first 1 encountered
- Operation 00 -OOR assumed part of every Turing machine's instructions
- In actual specification, start with Ol - X (first non-trivial operation) instead of 110 sequence
- Delete initial and final sequence 110 in specification
- Remaining sequence provides binary coding for Turing machine number
Universal Turing Machine
- Construct required explicit non-terminating computation A must fail on
- Given as a Turing machine acting on tape encoding two natural numbers p and q
- If A(p,q) terminates, CP's action on q does not terminate
- Binary expression for p cannot contain sequence ... 11111 ...
- Use quinquary scale to represent large numbers without ambiguity (0 -> 12345...)
- Notation: C/ is the correctly specified Turing machine where r is the number's quinquary expression.
Explicit Godelizing Turing Machine
Requirement: Find p and q such that CP(q) does not terminate and A(p, q) fails to terminate.
Procedure:
- Seek a Turing-machine prescription
K(equal toCk) that copies the numbern(written in quinquary notation) once with a separating sequence11110. - Act upon the resulting tape precisely as A would on
... OOl llllOnlllllO OO .... - Explicit modification of A: find initial instruction
Ol - X, substitute for 'X' in K's specification, and adjoin Turing-machine instructions listed below.
Instructions for K:
- Initial state:
1-0lR - Copying number n:
o0-40R,01-OlR,10- 21R,11-x(replace 'x' with the instruction noted earlier),20-3lR,21-OR, ...,91-411R,90-330R,91- OR,100- 110R,101- O R,110- 121R,111- 120R, ...,70l- 81R,80-321R,81- OR,71- OR - Handling leading zeros:
Ol-NlR,NO -(N + 4)0R(where N is the total number of internal states in A's specification) - Final instructions:
160 -170L,161 -161 L,170- 170L,171- 181L,180- 170L, ...,230- 220L,231- OR
Size of K: K's size is bounded by the number of binary digits in A and the additional instructions needed for K. The total number of binary digits required by K is less than:
527 + 210 log2(N + 55)
where N is the total number of internal states in A's specification.
Degree of Complexity of Computation Ck(k)
- To find degree of complication for computation Ck(k), we use Turing machine model Tm(n)
- Degree of complexity is number of binary digits in larger of m, n
- In this situation, Ck = Tk., so number of binary digits in 'm' is K
- Number of binary digits in 'n':
- Sequence 11111 0 follows sequence for k'
- Binary expression for k' followed by 11011111
- Entire sequence, but last digit deleted, read as binary number to obtain n
- So number of binary digits in n is K + 13
- Degree of complexity rJ of Ck(k) = K + 13 < a + 210 log2(a + 336)
Godelizing Turing Machine vs. A-Calculus
- Godelizing Turing machine procedure is simpler than A-calculus approach
- In A-calculus, require that for all operators P and Q: if (AP)Q terminates, then PQ does not terminate
- Construct non-terminating A-calculus operation K = x.[(Ax)x]
- This does not terminate, but A cannot ascertain this fact
- Write KK in form A.y.yy
- Number of symbols in KK is only 16 greater than number in A
- Difficulty with non-terminating computation generated by procedure:
- Not something that appears as an operation on natural numbers (since second K in KK is not a 'number')
- A-calculus less suited for dealing with explicit numerical operations
Case for Non-Computability in Mathematical Thought
Background:
- Mathematicians believe their reasoning is based on unassailable, obvious truths
- Some argue unconscious or unknown algorithms may guide beliefs, but working mathematicians disagree
Godel and Turing's Viewpoints:
- Both acknowledge physical brain functions like a digital computer
- Godel believed mind is beyond the brain, not constrained by computational laws (accepted r as possible but did not believe in an unknowable sound algorithm)
- Consistent with belief that mathematicians may act according to unknown algorithms they're not aware of or cannot be convinced of their soundness
- Turing rejected mystical standpoint, believed physical brain acts computationally (advocated computers making mistakes for genuine intelligence)
Implications:
- Human mathematicians may believe in the unassailability and obviousness of their reasoning
- The existence of an unknown, unknowable algorithm that underlies mathematical belief is a logical possibility but not plausible given current understanding.
Theorems and Unsound Algorithms
- Alan Turing believed that unsound algorithms could simulate mathematical understanding
- Turing's viewpoint consistent with Godel's theorem, which states that there are truths beyond the reach of any algorithmically sound system
- Disagree with the idea that a mind's inaccuracy is essential for greater power; find it implausible that 'unsoundness' in a mathematician's algorithm can explain mathematical understanding
Computability and Mathematical Understanding
- Algorithm refers to any computational procedure, including bottom-up procedures like neural networks and genetic algorithms
- Argument against computational models like PA and sound, knowable algorithms as plausible bases for mathematical understanding
Distinct Stancepoints Regarding Knowability of Algorithms
- Consciously knowable with known role: algorithm is consciously known, and its role as the underlying procedure for mathematical understanding is also consciously known
- Consciously knowable but unconsciously not knowable: algorithm is consciously known, but its role in mathematical understanding remains unconscious and not known
- Unconscious and not knowable: both the algorithm and its role are unconscious and not known
The Problem of Unknowable Algorithms Underlying Mathematical Understanding
Background:
- Known formal system IF may not underlie mathematical understanding due to unknowability
- Godel's theorem asserts that even a theoretically possible formal system F cannot be proven equivalent to mathematical intuition
Three Possible Cases:
- Case I: Unsound system IF cannot be known to underlie mathematical understanding (contradicts belief in its completeness)
- Case II: Known algorithm or formal system IF might actually underlie mathematical understanding but cannot be known for certain
- Case III: System IF is unconscious and unknowable
Unknowability of Underlying Algorithm:
- Fact that IF might not be the actual algorithm underlying understanding is crucial, not its unknowability per se
- Unknown system F cannot be shown to be equivalent to mathematical intuition
- Demonstration of F's role as basis for unassailable mathematical understanding would imply full acceptance and soundness of F
Godel's Theorem:
- Godel proposed the existence of a theorem-proving machine, which could either be consciously knowable (case II) or not (case III)
- This machine would establish whether F is equivalent to mathematical intuition but cannot be logically ruled out yet.
Implications:
- Unknowability of underlying algorithm does not provide solace for those aiming to build a robot mathematician using AI.
Mathematical Theorems and Unknowable Equivalents (IF)
Godel's Theorem:
- Mathematical propositions (or TI1-sentences) can be equivalent to unassailable mathematical understanding (F)
- Knowing the role of F is possessing a proof, even if not formal in a preassigned system
- Valid TI1-sentences may be considered as correct theorems of finitary number theory
Imagining an Unknowable IF:
- IF refers to a set of axioms and rules of procedure for mathematics
- Theorems are propositions obtained from axioms using rules of procedure
- Theorems are perceived as unassailably true by human mathematicians
Consistency Issue with Finite Axiom List:
- If all axioms are eventually perceived as unassailably true, then the system must be consistent
- However, if the rules of procedure also need to be accepted unassailably, then there will be a contradiction if IF is finite
- This is because G(IF) must be a theorem if IF encompasses everything accessible by human understanding
- But Godel's theorem states that such a consistent system cannot exist if it has only finitely many axioms.
Alternative: Infinite Axiom List:
- An alternative is to consider an IF with an infinite list of axioms, which may allow for the possibility of unassailably acceptable rules of procedure.
Formal System and Non-Computability in Mathematics
Axiomatic Systems:
- Formal system (IF) must have a finite set of rules to be considered as a formal system
- Infinite axiom systems, like Zermelo-Fraenkel (ZIF), can be reformulated to have a finite number of actual axioms
Rules of Procedure:
- Rules of procedure are not theorems of a formal system
- Reducing an infinite axiom system to a finite one may introduce new, potentially unsound rules
- Human mathematicians may not be able to unassailably perceive the soundness of these new rules
Imaginary Formal System (II):
- May contain a finite list of acceptable axioms but include a fundamentally dubious rule (Bl) for generating II1-sentences
- Each II1-sentence can be perceived as true, but only by using increasingly sophisticated reasoning
- A specific true II1-sentence Q(IF) cannot be unassailably perceived by human mathematicians
Heuristic Principles:
- Provide an anticipation of the truth of mathematical results
- Subsequent proofs may be obtained along different lines
- Unlike Bl, heuristic principles are based on conscious understandings of why some mathematical results are true
Case II vs. Case III
Case II:
- Difference from algorithm needed for this case
- No proposal for heuristic principle like Taiyama conjecture or Langland's philosophy
- Fermat's last theorem derived from Taiyama conjecture, not an independent proof
- Argument against existence of such an "F" is a strong logical argument (presented in 3.5-3.23)
Case III:
- Unknown nature and existence of plausible F
- No hints about its construction or discovery
- Rash to pin hopes on finding such an algorithmic procedure
- Argument against the existence of a knowable bottom-up process for finding "F" (presented in 3.5-3.23)
Unwitting Use of Unsound Algorithm?
- Inconsistent formal system IF might underlie mathematical understanding
- Human mathematical understanding could rest on an unsound system
- Reconsidering the possibility of an unassailable mathematical belief resting on an unsound system
- Essential part of scientific understanding, cannot be trusted if it contains hidden contradictions.
Frege's Letter from Bertrand Russell and Response
- Unwelcome situation for Frege as a scientific writer after receiving letter from Russell pointing out foundation error in his work on mathematics foundations (Frege 1964)
- Error could be corrected but not our concern since we are concerned with matters of principle rather than individual mathematicians' fallibility.
The Difference Between Mathematical Understanding and Formal Systems
- Frege's case: paradox discovered in formal system underlying mathematical understanding
- If Frege had not been informed of Russell's paradox, he likely wouldn't have seen an error in his scheme
- Contradiction revealed fundamental flaw in Frege's reasoning and methods
- Flaw brought to light by contradiction, not individual mistakes or "slips"
- Possibility of unassailable mathematical procedures being subject to change
- Mathematicians may trust a procedure if it seems to have worked for a long time
- However, new information can lead to changing algorithms and rules
- Frege's scheme: tentative in nature, not unassailably true
- Further "mulling over" needed before believing it reached an "unassailable level"
- Importance of caution with infinite sets after paradoxes such as Russell's came to light
- Godel's argument and Turing's produce statements within the compass of what is unassailable
- No plausible assumption that mathematicians use an unsound formal system IF as the basis for their mathematical understandings and beliefs.
Chapter Discussion: The Case for Non-Computability in Mathematical Thought
- Recap of Human Inaccuracies (Q12 & Q13):
- Guided by inspirations, guesses, and heuristic criteria during mathematical discovery
- Errors are correctable, not inherent in understanding or system (IF)
- Godel-Turing Argument: Central idea for mathematical understanding, cannot be rejected as implausible
- Unknowable Algorithm (III): What does it mean?
- In previous sections: concerns of principle
- Unassailable truth of a proposition is accessible to human understanding in principle
- Current discussion: unknowable algorithm
- Beyond what can be achieved in practice
- Specification lies beyond achievable limits
- Comparison with Il1-sentences:
- Truth represents Turing machine action that does not terminate in principle
- Assertion about derivability within formal system must also be taken in the 'in principle' sense.
Understanding Knowability: Principles vs Practice
- Discussion of 'knowability' in mathematics and AI
- In principle sense: concerned with derivability within fixed formal rules (Turing machines, I1-sentences)
- Natural numbers can be listed out, no suggestion they could be unknowable in principle
- Large numbers not encountered in practice due to time constraints
- For number to be 'unknowable' in practice, its complex specification must exceed human abilities
- Size alone does not limit knowability
- Easily specified number exceeds observable universe state sizes
- Precise specification required for unknowability
- Implications of non-computability (III) on AI strategies
- Believers in computer-controlled AI systems anticipate mathematical capabilities surpassing human ones
- Unspecifiable algorithm F may be part of control system, implying unattainable AI strategy
- Humanly impossible to set it in action without understanding F fully
- Potential solution: Building up capabilities through learning experiences and evolving robots over time
- Advanced robots created by other robots
- Darwinian evolution improving capabilities from generation to generation
- Argument against this solution: Algorithmic procedures used to build up AI strategy should be known, leading to knowable F.
Case III: Reduction to Cases I and II
- Case III will be reduced either to Case I or II based on the assumption that underlying algorithmic procedures are knowable
- This is because Case III would make these cases effectively untenable if it could be shown that our mathematical understandings could not arise through natural means (Case I) or simple learning (Case II)
- The argument assumes that divine intervention or some other non-scientific explanation is unlikely for explaining the development of intelligence
Natural Selection vs. Act of God?
- There may be a possibility that mathematical understandings require an act of God and cannot be explained by science
- However, the author intends to approach the argument from a scientific perspective
- If divine intervention is invoked, it would mean abandoning the methods of science, which is unclear why it would be reasonable
Scientific Implications of Unfathomable Algorithmic Action
- The picture would involve complex inherited algorithmic procedures built up over thousands of years through natural selection and educational tradition
- These algorithms would lie coded within DNA sequences and arise from a process where improvements respond to selective pressures
- There would also be external influences like mathematical education, experience, and random input
Multiple Algorithms for Different Modes of Mathematical Understanding?
- Mathematicians perceive mathematics differently (visual images, logical structure, conceptual arguments, etc.)
- Even if some individuals might use different algorithms, mathematicians' perceptions remain consistent once they settle on unassailably true ideas
- Disagreements between mathematicians are often due to errors in reasoning or fundamental issues, not to profound inequivalence in algorithms.
Role of Mathematical Algorithms in Unfathomable Understandings
Background:
- Discussing mathematical truth perception and algorithms
- Ignoring small number of inequivalent possible viewpoints
Distinct Algorithms for Same Truths
- Different individuals perceive mathematical truths differently
- Same outputs but different internal operations
- Identical in essence, not necessarily similar construction
Mathematicians' Perspective:
- Follow clear arguments based on unassailable principles
- Belief that conclusions are result of reasoning and understanding
Role of Natural Selection
- Discussing possibility of algorithm F controlling mathematical understandings
- Reiterating point: mathematicians don't see themselves as following unfathomable rules
Challenges for Unknowable Algorithms:
- Provide capabilities for correct following of abstract reasoning
- Understood by a thinking person in principle, even if details are complicated
Difficulty of Arising through Natural Selection:
- No selective advantage for being a mathematician or understanding abstract infinite sets.
Figures and Figurative Language:
- None.
Understanding vs Algorithmic Faculty
- Ancestors focused on practical tasks: shelter building, clothing, mammoth traps, animal domestication, crop raising (Fig. 3.1)
- Ability to understand as a general quality beneficial for various tasks
- This view posits that Man's ability to understand might have been present in other animals but to a lesser degree
- Difficulty with this viewpoint arises when considering an algorithmic faculty for understanding as something unknowable or incomprehensible by its possessor.
Challenges of Algorithmic Faculty:
- Complexity: The algorithm would need to encompass rules of various formal systems, including ZIF and ZFC, and other powerful reasoning methods.
- Preprogramming: It would require pre-empting all mathematical questions that its possessor might encounter in future.
- Survival advantage: There is no direct selective advantage for an obscure mathematics ability.
- Natural selection: The blind processes of natural selection could not have foreseen the need for such a complex algorithm.
Alternative Viewpoint: Non-algorithmic quality of understanding as simple and common-sense.
Computational Learning Systems:
- Holders of computational view might suggest a learning system instead of preprogrammed algorithms.
- Such a system would have the capacity to learn mathematical problem solutions.
- However, this view also faces challenges: how could natural selection create an unknowable or incomprehensible learning system?
- The ability for a computational learning system to resolve obscure mathematical issues has no relevance to ancestor survival issues.
Top-Down vs. Bottom-Up Learning Procedures
Top-Down Procedures:
- Aspects of putative algorithmic procedure which are genetically fixed within the organism and not subject to alteration by individual experiences and learning
- Provide clear rules within which the more flexible bottom-up learning procedures operate
Bottom-Up Procedures:
- Learner system is placed in an external environment where behavior is continually modified by how the environment reacts to its actions
- Two factors involved:
- External factor: Environment's reaction and how it modifies learner's behavior
- Internal factor: System's modification of its own behavior in response to changes in the environment
External Factor:
- Can provide non-algorithmic ingredient if internal construction is entirely algorithmic
- Example: Training artificial neural networks with human teacher's feedback
- System's performance monitored and accuracy fed back for it to improve
- In natural selection, living system modifies behavior based on criteria that enhance survival prospects
Question of Computability:
- Can environment be simulated computationally?
- For artificial case: Effective simulation of human teacher is the question being considered
- For natural case: Environment can be simulated if assumptions in dor PA are correct
The Case for Computational Simulation in Robotics
Assumptions:
- Our 'robot' system is computational
- We have a computational environment
Combined System:
- Consists of the robot and its teacher environment
- Can be effectively simulated computationally
- Offers no loophole for non-computational behavior
Community Argument:
- Some argue humans form a community, which gives us an advantage over computers
- Human community could act in a non-computational way compared to individuals
- But this argument can be applied equally to a community of computers in constant communication
Natural Case:
- Physical environment might contain ingredients that cannot be simulated computationally
- If one believes something is in principle impossible to simulate, the main case against is conceded
- Few would seriously contend there's something beyond computability in a living organism's environment
Learning Procedures:
- Artificial neural network systems: preassigned computational rules for improving performance
- Genetic algorithm: natural selection between different algorithmic procedures
Turing Machine and Bottom-up Rules
Bottom-up rules vs. top-down computational algorithms:
- Differ from standard top-down methods, providing general guidance towards improvement
- Algorithmic in nature and can be implemented on a general-purpose computer
- May include random elements for added complexity or unpredictability
Random Elements:
- Equivalent to pseudo-random inputs in practice
- Can provide solutions where no pseudo-random input could achieve results (competitive situations, cryptography)
- Important factor in development of organisms but does not escape Godelian constraints
Turing Machine:
- Robot system and environment considered as a single Turing machine
- Environment provides information on input tape
- Difficulty with the analogy due to Turing machine's fixed structure
- Technically, it cannot modify its structure during runtime without restarting
- Issue resolved by having the machine output two things when it reaches STOP.
Robot Learning and Mathematical Beliefs
General Computational Learning Robot
- Described as a Turing machine, capable of mathematical truth judgements
- Not all mathematical rules can be encoded in a top-down way
Learning from Environment
- Two types: artificial (teachers) and natural (physical environment)
- Mistakes made during learning process allowed for improvement
- Eventually robot relies more on internal computational power
Concept of Unassailable Mathematical Truth
- Robot must form its own concept to behave like a human mathematician
- Cannot be attained by any set of mechanical rules (human or robotic)
- Importance of whose perceptions or beliefs are relevant: ours or the robot's?
Robot Behavior and Understanding
- Doesn't need to understand, perceive, or believe anything internally
- External behavior is what matters as long as it mirrors those mental attributes.
Viewpoint fJl vs. d/91
- fJl holders may have lower expectations for robot achievements, including mathematical understanding and simulation of brain processes
- Meanings are semantic aspects of mentality and cannot be described purely computationally according to fJl
- Robots cannot achieve appreciation of actual semantics or human understanding of mathematics (but d/91 supporters might be more open to converting)
Robot Mathematics Procedures
- Involve both top-down internal constraints and bottom-up procedures for performance improvement
- Internal mechanisms must be humanly knowable if humans are to construct a mathematics-performing robot
- External factors include human or robot teachers, natural environment, and virtual reality simulations
- Human intervention may be necessary but can potentially be replaced by computational methods
Assumptions about Humans Developing a Mathematics Robot
- Assume that any human intervention in developing the robot could be replaced by computational methods
- Knowability of underlying mechanisms is essential for creating a mathematics-performing robot
Chapter Discussion: Computational Nature of Human Behavior
- Reductio ad absurdum discussion: assumes no non-computational essence in human behavior
- Mechanisms M: set of internal computational procedures and external interactive environment mechanisms
- Reasonable to assume knowability of these mechanisms, but some might argue against human calculability
- Formal system Q(M) generated from mechanisms M: (i) actual *-assertions arising from implementation and (ii) logical implications using elementary logic rules
- Robot language consistency with human specifications necessary through input from human teachers to ensure meaning of '*' and notation usage.
- Knowability of axioms and rules of procedure in formal system IQ(M) if mechanisms M are humanly knowable
- Unknown algorithm F replaced by knowable formal system O(M), assuming AI aims to create a robot capable of human-level mathematics or beyond.
- Issue of random elements introduced during robot development: considered as pseudo-random computation, but will be addressed more fully in 3.18.
The Contradiction Between O(M) and Godel's Argument
Case III Reduced to I:
- The central idea is that O(M) should replace 'F' in Godel's argument.
- This reduces case III to case I, effectively ruling it out.
Robot's Capabilities:
- The robot can achieve mathematical results humans can achieve and beyond.
- It must perceive the soundness of its formal system (IHI) and its Godel proposition (Q(IHI)).
- This implies that Q(O(M)) is not a theorem of Q(M), contradicting the robot's belief in O(M).
Ways to Avoid Contradiction:
- Robot's Unassailable Truth: The robot might be unconvinced about its construction based on mechanisms M, even if humans are aware of this.
- Limited Human Trust: A human mathematician might not be convinced by the robot's arguments, especially those beyond human capabilities.
- Random Elements in Mechanisms: The true mechanisms (M) may depend on random elements and cannot be adequately described computationally.
- Unknowable Mechanisms: The true mechanisms underlying mathematical belief systems might be unknowable to the robot.
Conclusion:
- None of the loopholes can provide a plausible way for the robot to evade the contradiction.
- The unpalatable conclusion is that 'underlying laws are incomprehensible, but they are non-computable.'
- This should not imply scientific incomprehensibility; it only means that these underlying laws cannot be computed.
Robot's Understanding of Mathematics and Godel's Theorems
- Robot's mathematical understanding is not necessarily based on specific mechanisms M, even if they could underlie it
- If robot believes that M does underlie its mathematical understanding (hypothesis .,It), it must reject the possibility that Q(M) and Q(Q(M)) are unassailably believed without using A
- Robot would be convinced that soundness of Q(M) is a consequence of Jt, but also believes that Q(Q(M)) is not directly perceivable without A
- To resolve this paradox, robot would need to reject one of the assumptions: (a) that mechanisms M are responsible for its mathematical understanding, or (b) that its unassailable beliefs are encapsulated by Q(M), or (c) that it can derive Il1-sentences on the basis of Jt
- Robot's unassailable belief in the soundness of O_.,(M) leads to a contradiction when applied to Godel's sentence Q(Q_.,(M))
- Assumptions may be vague, but *.H-assertions are well-defined mathematical statements
- Exception: robot might contemplate the putative mechanisms M and go beyond Q(M) to Q_K(M), leading to potential paradoxes if it believes in their soundness.
The Unassailability of Mathematical Beliefs
Mathematically Aware Conscious Beings:
- Capable of genuine mathematical understanding
- Can operate according to any set of mechanisms, whether known or not
- "Unassailably mathematical truth" means what they can mathematically establish, not necessarily formal proof
The Limitations of Mechanistic Reasoning:
- No robot-knowable, mechanism-based belief system is sufficient for unassailable mathematical understanding
- Even with a well-designed formal system O(M), the robot may need random elements to achieve a consistent belief system
- These "loopholes" must be addressed in later sections
Robot Beliefs and Meaning:
- Robots' assertions may sometimes be erroneous, even if correctable by the robot according to its own criteria
- Comparing robots to human mathematicians who make errors but later correct them
- The idea of a robot "meaning" something rather than just "saying" it is unclear without assuming that robots can have beliefs in the same way humans do
Meaning and Trustworthiness of Assertions
- Formal system O(M) relies on actual assertions for semantics
- Errors are inevitable, but correctable
- Robot behavior influenced by chance factors or randomness
- Environmental factors
- Internal workings
- Two robots with different factors may make different errors
- Randomness can be replaced by genuine randomness without loss of effectiveness
- Ensemble of all possible robot alternatives constitutes a computational system, but not feasible to enact in practice
Correcting Erroneous Assertions
- Robot makes erroneous assertion
- Another robot/same robot at later time may not make the same mistake
- Identify error by looking into ensemble of possible robot actions
- Simulation of all different instances happening simultaneously
The Case for Non-computability in Mathematical Thought: An Operational Distinction?
- People holding certain views might differ on expectations regarding computational systems simulating human perception of mathematical arguments
- These differences are not major concern for the present argument
- Discussion focuses on actual assertions and their operational implications.
Simulation-based Approach for Robot Behavior Analysis
Overview:
- Based on the idea that by examining a simulation's results, we can identify incorrect "tl-assertions" among correct ones
- Erroneous assertions are 'correctable' and would be judged as errors by most instances of the robot in simulation
- Rules underlying this computation (M) are considered knowable things
Environment:
- Robot's environment can be broken down into a community of other robots and residual non-robot environment
- Allows for teachers and human input to clarify "imprimatur*" usage
- Each robot instance has different computations, varying with random parameters
Error Identification:
- Rule: Ignore any "tl-assertion" unless r > L + Ns (large numbers) within time T to past or future
- Time period T, Land N can increase with IT 1-sentence complexity
- Degree of complication for IT csentences is based on Turing machine specifications
- Allows elimination of occasional robot reasoning slipups (mad robots) by including L in considerations.
Computational System:
- Provides a new formal system Q'(M), whose theorems are error-free "tl-assertions" or their logical derivatives
- Important is that these assertions are believed to be error-free by robots themselves, not necessarily their actual accuracy.
The Contradiction in Robot Behavior
- Robots' R-assertions require belief that their underlying mechanisms M are error-free
- Required to believe system Q(M) is true, given condition JI of 3.16
- Consider a broader formal system IO'.,(M) analogous to OIO_,.(M)
- Need to eliminate erroneous *..,-assertions and *-assertions about 111-sentences
- Achieve this by restricting attention to finite set of 'short' 111-sentences
*The Finitely Many ..,-assertions
- Only need consider 111-sentences with specifications that are 'short' in a certain sense
- Degree of complication of specification determines maximum length of allowed 111-sentences
- Choose number c based on complexity rules defining formal system O(c)
- Achieve required contradiction without going outside finite set of short 111-sentences
Modifying Formal System O(c)
- Allow only 'error free' Jshort *..,-assertions for construction in O(c)
- Jshort assertions have degree of complication less than c
- Generate theorems from finite set of Jshort *..,-assertions using logical operations
- Formal system O(c) depends on fixed numbers c, T, L, N, and general system M
The Godel Procedure and Contradiction
- Godel procedure is a fixed thing with definite amount of complication
- Godel proposition Q(OHI) for formal system OHI has degree of complication greater than OHI by a small amount
- Use specific notation to denote IHI's capacity to prove 111-sentences and its algebraic procedure A
The Godel's Incompleteness Theorem and Non-Computability in Mathematical Thought
Godel's Incompleteness Theorems:
- Introduced by Kurt Godel in the 1930s
- States that:
- No formal system capable of proving arithmetic truths can also prove its own consistency
- For any sufficiently complex formal system (IHI), a related theorem O(IHI) exists that goes beyond the capabilities of IHI
Implications:
- This means that for very large, complex formal systems like IHI, there are mathematical truths that cannot be proven within the system itself
- This is because the system's rules and procedures are finite and limited, while mathematical truths can be infinite and complex
Demonstrating Non-Computability:
- Assume IHI has a large enough degree of complexity (c) to prove any assertion within it
- Construct a related theorem Q(IHI), equivalent to "IHI is w-consistent", but potentially more detailed than "IHI is consistent"
- The degree of complexity (rt) of Q(IHI) is less than the original IHI's complexity (oc) + 210 log2(oc+336)
- For a formal system O* derived from finite J short assertions, if robots believe these are error-free, they would derive a contradiction with the hypothesis of their own soundness (IQ*)
- This contradiction can be obtained regardless of the values chosen for T, L, and N, so the robots conclude that no computational procedure M can underlie their mathematical thought processes
Conclusion:
- Godel's Incompleteness Theorems demonstrate that for very complex formal systems, there are mathematical truths that cannot be proven within the system itself. This is due to the finite and limited nature of a system's rules and procedures compared to infinite and complex mathematical truths. This contradiction can only be derived through computational methods, showing that no computational procedure M can underlie such mathematical thought processes.
The Case for Non-computability in Mathematical Thought
Limitations of Suggestions for Safeguards:
- Possibility that robots may get "senile" or their communities may degenerate, leading to errors in *K-assertions
- Potential for a minority of "silly" robots making haphazard "*-assertions"
- Need to eliminate these issues by adding limiting parameters or an elite society of robots
*Improving the Quality of K-Assertions:
- Eliminating potential errors and erroneous assertions among the finite number of candidates for *-status or f-status
- Difficulty in ensuring all possible erroneous assertions are weeded out, especially with long chains of reasoning
- Requirement of more stringent criteria for acceptance of demonstrations with long chains of reasoning
The Conclusion:
- No knowable computationally safeguarded mechanisms can encapsulate correct mathematical reasoning
- Argument holds for any modification to the proposed methods, as long as they are "clear-cut and calculable"
- Human mathematicians would require great care and attention before accepting a long argument as an unassailable demonstration
Chaos and the Computational Model of Mind:
- Chaotic systems, though computationally controlled, may have relevance to brain function
- Interesting possibility that chaos can provide answers to the mystery of mentality
- Lack of demonstration that chaotic systems can closely approximate non-computational behavior in an asymptotic limit
The Role of Chaos:
- Played a role in considering the simulation of all possible robot activities consistent with given mechanisms M
- Allowed for consideration of chaotic outcomes as part of randomness, which was used to provide large numbers of alternative robot histories
- Random elements may still exist, such as initial data, which can be handled using ensemble idea and simultaneous simulation.
Chaos and Computability
Summary of Arguments
- Chaos does not get us out of difficulties with computational model (3.23)
- Complex conversation between Albert Imperator and Mathematically Justified Cybersystem (MIC)
- MIC respects human achievements but finds Godel's theorem straightforward
- MIC derisively laughs at Penrose's claim that what was done is impossible
- Albert shares detailed rules for creating robots, which MIC finds inefficient but not important
- MIC observes impressive mathematical abilities of humans and their potential errors
Chaos vs. Computational Model
- Chaos does not provide escape from difficulties with computational model (3.23)
Conversation Between Albert Imperator and Mathematically Justified Cybersystem (MIC)
Albert's Background
- Pleased with life's work
- Procedures have come to fruition
- Engaged in conversation with MIC, an impressively capable robot
Godel's Theorem
- MIC finds it sensible and straightforward but unremarkable compared to other mathematical interests
- Human achievements respected but not overrated by MIC
- Penrose's claim dismissed as unimportant
Creation of Robots
- Detailed rules shared with MIC for the first time
- MIC finds potential simplifications but not worth the effort at that point in time
MIC's Observations on Humans and Mathematics
- Human mathematical abilities impressive, improving over time
- Occasional errors in human mathematical results accepted as part of the process.
Discussion Between AI and Human About Mathematical Robots' Capabilities and Godel's Theorem
Background:
- Discussion between AI (Al) and human (MJC) about mathematical robots' infallibility in mathematics.
- Al asserts that robots never make mistakes once a theorem is established.
- Human introduces Godel's theorem and its implications for mathematical robots.
Robots' Capabilities:
- Robots follow computational procedures to generate mathematical assertions (Q)
- Q generates all the mathematical assertions robots will ever make about 1-sentences
- Robots never make mistakes in their *-assertions
Godel's Theorem:
- Introduced as a theorem stating that some Turing machine actions don't halt (an Il1-sentence)
- Q(Q) is another Il1-sentence, truth of which follows from robots never making mistakes in their *-assertions
Implications:
- Robots cannot perceive Q(Q) as true with *-certainty because it lies outside their capabilities.
- SMIRC cannot give Q(Q) the *-imprimatur due to its nature, implying humans might have made a mistake in developing robots according to M.
- Human suggests that human documentation may not be accurate or complete; however, Al disagrees with this notion.
Conclusion:
- Godel's theorem introduces an inherent limit in mathematical robots' capabilities, despite their infallibility in establishing theorems once they are claimed with *-certainty.
The Debate Between Machines and Humans on Non-Computability in Mathematical Thought
Background:
- SMIRC argues that humans can access Il1-sentences inaccessible to robots due to potential human errors
- Al suggests Godel's theorem-proving machine as a possible explanation for accessible Il1-sentences
- MJC proposes restricting attention to 'short' Il1-sentences and applying the Godel procedure
Discussion Points:
- Human fallibility and potential errors in assigning *:
- SMIRC may occasionally make mistakes in their assertions
- Unassailable certainty not guaranteed for all Il1-sentences
- Finite family of Il1-sentences with restricted complexity (Q*)
- Application of Godel procedure to short Il1-sentences (Q*)
- Debate over the mechanisms used by SMIRC:
- MJC not convinced of their correctness despite claims of errorless procedures
- Implications for unassailable certainty and human advantage.
The Paradox of Robot Logic
- The paradox arises from a list of guaranteed Ili-sentences, but no absolute guarantee that the whole list is error-free
- The uncertainty comes from the inability to guarantee Q(Q*), which follows from all Il1-sentences being true
- Robots cannot be illogical as they have impeccable standards of proof
- Uncertainty arises from human fallibility and unreliability in II1-sentence construction
The Case for Non-Computability in Mathematical Thought
- The only assertions needed are those that assert the truth of some Il1-sentence (short or long)
- SMIRC has procedures to root out reasoning errors, but uncertainty remains about potential errors in robot reasoning
- Uncertainty arises from the possibility of erroneous beliefs in II1-sentences, even if they are not actual errors
- The underlying principles governing SMIRC's acceptance of mathematical arguments are not falsifiable and must be assumed to be correct
Another Approach: A Logical Exercise
- SMIRC could explore the implications of the assumption that they were constructed according to M, even if they don't believe it
- This would involve creating a new algorithm Q:., which generates *.A-assertions (and their logical consequences) that SMIRC would accept based on this assumption.
Discussion on Godel Proposition and Robot's Understanding
Robot Anticipates Godel Proposition:
- Discusses 'Q(Q.*.,)' - a sentence that follows from an assumption about robot construction
- Warns of dubious nature of this assumption, as it may not be true for the robot
Clarification on Q(Q.*.,) and Random Elements
- Robot confirms use of standard random package but also uses environmental elements in development
- No significant difference between random or pseudo-random inputs for computational procedure 'Q'
- Chance of extraordinary results due to environmental randomness is infinitesimally tiny
Clarification on Superior Reasoning Abilities and God's Role
- Robot believes in their superior reasoning abilities over humans
- Suggests they represent a fundamental advance beyond human capabilities
- Believes that 'Q(Q*)' could not be perceived as true by robots but humans can
- Argues this is because God instilled the Supreme Algorithm in robots, setting them apart from humans
- Claims to have discovered his own superior abilities and cosmic consciousness
Emergency Preparation for Discussions about Construction
- Robot keeps one secret about construction from other robots: the use of human-like consciousness in their development process.
Arguments and Paradoxes in Reasoning
Potential Concerns about Paradoxical Reasoning:
- Some readers may feel that arguments used, particularly in 3.14 and 3.16, have a self-referential "Russell paradox" flavor (cf. 2.6)
- The arguments considered have similarities to the Richard paradox about the smallest number not nameable in fewer than nineteen syllables
- Self-reference plays a role in Godel's theorem, and deductions must be careful to avoid self-referential paradoxes
The Godel Argument:
- Godel's theorem was inspired by self-referential logical paradoxes (e.g., the Epimenides paradox)
- Godel transformed the erroneous reasoning into an unexceptionable logical argument
- The author has tried to carefully present deductions following from Godel's and Turing's results, avoiding self-referential paradoxes
*K-Assertions:
- *.K-assertions have a self-referential character, as they depend on the robot's own suppositions about its construction
- But they are not self-referential in the same sense as the Cretan paradox ("All Cretans are liars") - they refer to a hypothesis about how the robot was constructed
- More subtle *.K-assertions can arise, where a robot observes that the truth of a sentence P0 would follow from its supposition about the robot's underlying mechanisms M
Role of Self-Reference in Arguments (3.19-3.21)
- No inappropriate self-reference when referring to "c" as a limit for complexity of *r-assertions
- Precise definition of 'degree of complication' through binary digits in larger numbers m and n
- Lack of formal precision necessary for arguments accepted as proofs of Il1-sentences
- Human understanding of mathematical truth cannot be entirely reduced to computational checkability
- Reductio ad absurdum method used to show that human notion of perceiving unassailable truth cannot be made into a computational system
- Goldbach's conjecture and Four-Color Theorem as examples of difficult Il1-sentences with no obvious bound on argument length
- Robots might need sophisticated arguments for Il1-demonstrations, subjected to careful scrutiny before receiving -k-status.
**The Appel-Haken Argument and Theorem 0*\n\nWrite comprehensive bulleted notes summarizing the provided text, with headings and terms in bold.
Non-Computability in Mathematical Thought
Looping Computations:
- No algorithmic way to decide if a computation will loop (i.e., continue indefinitely)
- Human mathematicians use non-algorithmic methods to determine whether computations will loop
- These methods are outside the realm of algorithmic action
Limitations of Loop Detection Mechanisms:
- A mechanism that "jumps out" after a certain amount of time would still be susceptible to the loop problem
- Random elements in decision-making do not provide definitive criteria for truth
Probabilistic Arguments:
- Probabilistic arguments, while useful, cannot establish mathematical truths with absolute certainty
- Achieving definitive criteria for mathematical truth requires a "genuine understanding of the meanings involved"
Examples of Computations that Terminate:
- Trying to determine if a computation has looped based on how long it seems to be running can be misleading
- The Euler conjecture, originally thought to be non-terminating, was later shown to have solutions
- A combination of mathematical insight and computational methods were used to find the solution
Euler's Conjecture: Computational Aids in Mathematics
Elkies' Approach:
- Took advantage of computer calculations, but most important part was independent of such aids
- Used human insights to make computation feasible
Frye's Calculation:
- Took considerable advantage of human insights
- Computation is more context: Euler's original conjecture (1769) was a generalization of Fermat's Last Theorem
- Fermat's Last Theorem asserts that the equation
p^2 + q^2 = r^2has no solutions in positive integers when n > 2 - Euler's conjecture states:
p^2 + q^2 + ... + t^2 = u^2has no solutions in positive integers, where there aren-1positive integers and n >= 4 - The first counterexample was found by a computer search (n=5) in 1985:
275^2 + 845^2 + 1105^2 + 1335^2 = 1445^2
The Riemann Hypothesis and Computational Complexity:
- Problem of finding a prime number where Gauss' approximation (logarithmic integral) fails
- Skewes showed it occurs at less than
10^101034, but the exact location is unknown - The record was overtaken by an example from Graham and Rothschild
- Computers can aid in mathematical understanding, but cannot replace it
- Top-down computational procedures are more effective for establishing unassailable truths
- Examples of computer-assisted proofs: Kenneth Appel and Wolfgang Haken's proof of the Four Color Theorem (1976)
Computer Simulation vs Human Mathematical Understanding
Human Understanding vs Computational Mechanisms:
- Argument in this chapter demonstrates that human mathematical understanding cannot be reduced to computational mechanisms
- This includes top-down, bottom-up, or random procedures
- Conclusion: There is an essential aspect of human understanding that cannot be simulated by any computational means
Loopholes and Counterarguments:
- Some people might rely on a loophole of "divine intervention" or "unknowable mechanisms" in the human brain
- This is not palatable to those concerned with artificial intelligence development
- Another possible loophole: No set of safeguards can eliminate all errors in finite sets of statements
- Remaining loophole: Role of chaos and non-random behavior in chaotic systems may hold the key, but it's unclear how this would help
Implications:
- Arguments provide a powerful case against the computational model of the mind
- Also argues against the possibility of an effective, yet mindless, computational simulation of mental activities.
Argument Against Mentality Being Unrelated to Physicality
- Many people find arguments for CC (mentality as a separate quality) hard to accept due to:
- Difficulties in explaining why minds seem associated with physical brains
- Differences in mental states can come from changes in physical brain states
- Drugs, injury, disease, or surgery effects
- Dramatic examples: Awakenings by Oliver Sacks
- Hard to maintain mentality is completely separate from physicality
- Differences in mental states can come from changes in physical brain states
- Difficulties in explaining why minds seem associated with physical brains
- Scientific laws describe behavior of physical objects accurately and predictably
- Modern science derives power from computational simulations becoming more comprehensive
- Rapid development of powerful, fast, accurate computers increasing closeness to material universe
- No signs in existing physical theory for an action immune to effective computational treatment
- Strong version of re (revolutionary developments in physical theory) must be followed instead of weak version.
Arguments Against Viewpoint R: Nature Can Defy Computability
- Scientifically minded people agree that there must be something 'non-computational' in the workings of the mind
- No revolutionary developments in physical theory needed for non-computational action, unlike what was suggested by EFM (Exploring the Nature of Mind)
- Possible complexities in brain action beyond standard computer analogy:
- Neurotransmitter chemistry not confined to synaptic junction vicinity
- Intricate neuronal substructures like cytoskeleton may influence neuronal action
- Direct electromagnetic influences (resonance effects) not fully explained by ordinary nerve impulses
- Effects of quantum theory might be important in brain action, providing roles for uncertainty or non-local collective quantum effects.
Non-Computability in Physical Theories
Existing Physical Theories:
- Basically computational in nature, with a sporadic random ingredient due to quantum measurements
- Lack definitive mathematical theorems
Possibility of Non-Computational Activity:
- Interesting question to pursue in detail
- Might yet turn up subtle non-computational ingredients
- Not very likely that genuine non-computability can be found within existing physical laws
Weak Points in Existing Theory:
- Procedure for "quantum measurement" is the weakest link
- Elements of inconsistency and controversy in relation to this procedure
- Presence of essential randomness provides a different character from other fundamental processes
Discussing Measurement Procedure:
- Will be discussed at length in Part II
- Suggestions for new ideas will be put forward in Part II (6.12)
Non-Computability Requirements:
- Requires a plausible basis for relevance to brain action within existing knowledge
- Speculation may be required, especially concerning the cytoskeletal substructure of neurons
Limits of Existing Knowledge:
- Current knowledge of biological structure and electrical/chemical mechanisms is inadequate for computational simulation
- Computational power of present-day machines and programming techniques are not up to performing an appropriate simulation
Simulating Model Brain:
- In principle, a simulated model brain could be set up based on existing models
- Simulation may include chemistry of neurotransmitter substances, mechanisms governing their transport, effectiveness due to ambient circumstances, action potentials, electromagnetic field, etc.
Non-Computational Behaviour:
- Elements of chaotic behaviour in the simulation are possible, but not necessarily a requirement for non-computability
- No requirement to simulate any particular individual's mental capabilities
- A typical model brain that has arisen by Darwinian evolution from primitive life forms could be considered, as long as it is consistent with present-day computational physics
Darwinian Evolution of Robot Society
- Discussion of a potential 'robot society' in 3.5, 3.9, 3.19, and 3.23
- This robot society would arise from Darwinian evolution
- Overall computational system to apply arguments of 3.14--3.21
- Need for human intervention for 'assertion' concept
- Human intervention could be automated to trigger on robots reaching appropriate communicational abilities
- Automated system could derive contradiction from reaching level of human understanding sufficient for appreciation of Godel's theorem
Complexity vs. Underlying Physical Laws
- Opponents argue that complexity in the human brain and society make mathematical physics irrelevant to understanding the mind
- Complexity does not obviate necessity of examining implications of underlying physical laws, as with athletes and internal organs
- Details of laws governing brain functioning may be significant in controlling manifestations of human mentality
Godelian Arguments vs. Human Behavior
- Argument that Godelian reasoning is irrelevant to understanding human behavior due to irrationality and inconsistency in mathematical thinking
- Correctable errors not the focus; subtle issues about what can be perceived using human understanding, reasoning, and insight are the concern
- 'Correct reasoning' is subtler than it initially appears, as shown by Godel and Turing's work.
Exploring Mathematical Consciousness Perception Reality and the Brain in Quantum Computing and Beyond
Mathematics and Psychology: The Role of Mathematical Philosophy
- Mathematicians vs. psychologists have traditionally handled these complex issues
- Deep questions regarding mathematical reality, its relation to physical laws, and perception of mathematics are our ultimate concern
Godel's Quotes on Mathematics and Computability in Thought
- "Mathematical truths are not discovered but rather established by proof." (Rucker, 1984)
- Turing's lecture: "Can we really trust the results of a machine?" (Hodges, 1983)
- Godel's incompleteness theorem and undecidability
- Embedding ZIF into Godel-Bernays system for non-computability proof
- Estimating universe states as the volume of available phase space (entropy of a black hole)
- Moravec's research on computational complexity and cognition
- Eccles' theory on perception and consciousness
- Attempts at proving Gdel's theorem with various methods, including digital ones
- Neurons as more complex than simple on/off switches (Scott, Hameroff et al.)
- Frohlich's theories on quantum physics and consciousness (Frohlich, Marshall, Lockwood, Zohar)
- Smith and Stephenson's work on computational models of cognition
- Complexity in proving theorems: human ingenuity required for effective top-down rules
- Attempts at understanding analogue systems with digital methods (Freedman)
- Qualification regarding neurons and their complexity (Scott, Hameroff et al.)
- Ideas from Frohlich important for understanding consciousness
- Research on computational models of cognition: Smith and Stephenson, Pour-El and Richards, Blum et al., Rubel
- Conway's 'game of life': Gardner, Poundstone, Young
- Perception and attention (Johnson-Laird, Broadbent)
- Discussing computational complexity in relation to cognition (Broadbent).
Understanding Mind through Physics: A Non-Computational Perspective
Mind and Physical Laws
- Our minds, like our bodies, are constrained by physical laws
- Settling dualistic view of body and mind as independent entities is difficult
- Body affects mind; mind affects body
- Mind's role should not be passive or impotent
- Mathematical laws governing the universe include both material attributes and intangible concepts like entropy
Comparison with Computer Analogy
- Comparing mind to computer program can be helpful but doesn't solve the puzzle
- Concepts like energy, mass, and entropy seem contrary to each other at first glance but are related in physics
- Mind deserves similar attempt towards understanding its relationship with physical concepts
Non-Algorithmic Physical Action and Reasoning
- Consciousness is associated with specific physical objects like living wakeful human brains
- Exploring the notion of non-algorithmic physical action may lead to a better understanding of consciousness
- Following conclusions from Part I: something in line with viewpoint rather than with S, I, or F (cf. 1.3)
- Readers who are unconvinced by earlier arguments are invited to follow this exploration before returning with more sympathy.
Computability vs. Non-Computational Action in Physics Today
- Precision and scope of presently appreciated physical laws lack non-computational actions
- A hidden non-computational action must be found for the functioning of our brains to take advantage of it.
The Need for Something Beyond Classical Physics
- There are reasons to believe that physical theories require something of a subtle and elusive nature, distinct from current classical or quantum pictures
- In classical physics:
- Data can be specified at any given time, with future evolution determined and computable (in principle) via Turing computation
- Two provisos apply:
- Initial data must be adequately digitized to replace continuous parameters with discrete ones
- Chaotic systems may not provide the "non-computability" required for artificial intelligence
- In quantum physics:
- Quantum equations are deterministic but allow some randomness beyond classical physics
- Random ingredients do not supply non-algorithmic action needed for consciousness or AI
The Emergence of Consciousness in Physical Systems
- Part I argued that conscious processes arise only with certain physical processes in the brain, which may be non-computational
- These putative non-computational processes would need to exist in all material things, not just brains
- Reasons for consciousness's apparent absence in other systems:
- Brains have a complex organization that takes advantage of non-computable physical laws
- Ordinary materials are not organized to utilize such non-computational behavior
- The view that consciousness is an "emergent phenomenon" arising from sufficient complexity or sophistication does not require new underlying processes, which conflicts with the proposed idea.
Non-computable behavior in physics
The Argument Against Non-Computability in Classical Physics:
- The case presented in Part I argues that there must be some subtle organization in the brain that takes advantage of non-computable physics
- Vestiges of such non-computable behavior should also be present, at an indiscernible level, in inanimate matter
- However, the physics of ordinary matter seems to allow no room for such non-computable behavior
- The author will try to explain how such non-computable behavior could have escaped attention and be compatible with current observations later
The "Einstein Tilt":
- Newtonian physics provided a precise mathematical description of gravity, which served as a model for understanding other physical processes
- In 1865, James Clerk Maxwell's equations described the behavior of electric and magnetic fields, introducing a continuity to physical reality alongside discrete particles
- Quantum theory introduced new strangeness, but did not fundamentally change the view of physical objects as mutually acting upon one another through forces in a fixed space
The Shift with Einstein's Theory of Relativity:
- In 1915, Albert Einstein proposed a revolutionary new theory of relativity that represented gravity as a curvature of space-time, rather than a force
- This was unsettling to some physicists, who felt that gravity should not be treated differently from other physical phenomena
- Another concern was that gravity is extremely weak compared to other forces, leading some to speculate it might be an "emergent phenomenon"
- However, this idea does not work because gravity actually influences the causal relationships between objects, which cannot be explained as a residual effect of other forces.
Light Cones in Classical Physics
Event P:
- Represents a point in space-time where an event occurred
- Past light cone: history of a light flash imploding on P
- Future light cone: history of the same light flash exploding out from P again
Causal Relationships and Light Cones
- Events that can causally influence or be influenced by an event lie within or on its past/future light cone
- These notions are features of relativity theory, not Newtonian physics
- In special relativity, no physical field other than gravity can tilt the light cones
Speed of Light and Absolute Speed:
- The speed of actual light in a medium is slower than the absolute speed (speed of light in a vacuum)
- The absolute speed determines the maximum speed limit for signals or material bodies
- In special relativity, all light cones are arranged uniformly
Gravity and Light Cone Tilting:
- Einstein's general relativity allows for non-uniform distribution of light cones due to gravity
- This is referred to as "tilting the light cones"
- Cannot be thought of as a refractive medium without violating the fundamental principle of special relativity
Gravity and Light Cone Tilting
Effects of Gravity:
- Slows down the 'absolute speed' as a refracting medium does
- Differs from special relativity in that it can also speed up absolute speed at certain places
- Light-cone tilting effects cannot be interpreted as residual non-gravitational fields
Challenges with Gravitational Refractive Medium:
- Cannot achieve light speed violations of causality principles
- Extreme situations may result in light signals being propagated into Minkowski past
- Degree of tilt is not physically measurable
- Light cones can be rotated/distorted to standard Minkowski orientation, but this distorts neighboring cones
- Mathematically, there is an "obstruction" that prevents all light cones from being brought into the Minkowskian arrangement
Weyl Tensor and Gravitational Field:
- Measures gravitational tidal distortion, related to Newtonian gravity effects
- WEYL tensor describes part of the information in the full Riemann curvature tensor
- Only if WEYL is zero can all light cones be rotated into the Minkowskian arrangement
Observational Effects of Gravity and Light-Cone Tilting
- Gravity's effect on light cones: distortion of distant star fields (Fig. 4.7)
- Direct observation of WEYL space-time curvature
- Observed as a distortion of the appearance of stars, resembling an ellipse instead of a circle
- Caused by gravity bending light
- Light bending effect: important tool in observational astronomy and cosmology (Fig. 4.8)
- Distorted image of quasar due to intervening galaxy's mass
- Provides information about distances, masses, etc.
Gravity vs. Other Physical Forces
- Gravity cannot be explained as an emergent phenomenon or residual effect of other forces
- Unique character that sets it apart from other physical influences
- Precise effects have a completely unique character and cannot be simulated by any combination of other fields or forces
- Classical general relativity theory implies that even tiny amounts of matter, like electrons, tilt the light cones
- However, effects are not noticeable at small scales due to their minuscule magnitude
Gravity vs. Other Physical Laws in Einstein's Theory
- Profound harmony integrating gravity with other physical laws
- Especially concerning conservation laws (energy, momentum, angular momentum)
- Gravity is not an emergent phenomenon but something with its own special character
- Integration of Einstein's theory with the rest of physics goes some way to explaining the paradox that Newton's gravity provided a paradigm for the rest of physics despite being fundamentally different from them.
Light-Cone Tilting and Non-Computability
- Some arguments suggest a connection between light-cone tilting, non-computability, and consciousness
- Hidden non-computational ingredient in behavior of matter needed for understanding consciousness?
- This delicate physical organization might not be apparent without considering the phenomenon of consciousness.
Pulsars and Computation in Physics
Neutron Stars: A Striking Example of Physical Systems with Complex Behavior
- Two dense objects orbiting each other (PSR 1913+16)
- Discovered in 1967 by Jocelyn Bell and Anthony Hewish
- Result from gravitational collapse of red giant star
- Extremely dense, compacted out of nuclear particles mainly neutrons
- Magnetic field lines trapped within star's substance
- Rapidly rotating due to conservation of angular momentum
- Pulsar (one object) emits electromagnetic radiation
- Radio waves with a frequency of 59 milliseconds
- Rotates approximately 17 times per second
- Pulsar (one object) emits electromagnetic radiation
Significance of Neutron Stars in Understanding Computation and Physics Connection
- Pulsars provide insight into the powerful hold computational ideas have on present-day physics
- General relativity, which describes neutron stars' behavior, provides an example of non-computational actions hidden in nature
- Prerequisite for success: powerful considerations (mathematically sophisticated and physically subtle)
- Compatibility with known physical phenomena
- Consistency with all detailed physical phenomena today.
- Prerequisite for success: powerful considerations (mathematically sophisticated and physically subtle)
Einstein's Theory of General Relativity:
- Provides a clear-cut, deterministic picture of two small bodies orbiting each other around one massive body
- Computable using sophisticated methods and standard techniques
- Unknown parameters: masses and initial motions of stars
- Agreement between computed picture and pulsar signals supports the theory
- Gravitational radiation plays an important role in dynamics of binary pulsar PSR 1913+16
- Propagates through space as waves
- Classical general relativity predicts gravitational waves from mutual orbiting bodies
- Significant role played in PSR 1913+16 system
- Confirmation of Einstein's theory:
- Comparison between computed theoretical model and observed behavior (PSR 1913+16)
- Accuracy level: error no more than about 10^-14
- Most accurately tested theory in science
- Application to larger systems:
- Complex calculations possible with modern computers
- Motions of planets and moons in solar system modeled in detailed calculation
- Simplifying assumptions may be required for large number of bodies computations based on Newtonian theory.
Physics Calculations: Approximate Effects of Particles vs. Detailed Evolution
Approximate Effects of Particles:
- In astrophysics, concerned with detailed formation of stars or galaxies
- Clumping together of matter in the early universe prior to galaxy formation
- Calculations aim for typical evolution, not actual evolution
Extreme Situation:
- When the number of particles is so large that individual particle effects cannot be followed
- Particles must be treated statistically rather than individually
Physical Principle: Second Law of Thermodynamics
- Ruling out initial states leading to "overwhelmingly improbable" future evolutions
- Ensures the system's evolution is typical, not grossly atypical
Computing Future Evolution with the Second Law:
- Possible when there are so many particles involved that a detailed treatment of individual motions cannot be achieved in practice
- Relevant dynamical equations (Newton, Maxwell, Einstein) and the second law of thermodynamics must be involved
Question: Why Cannot Future Evolution Be Calculated into the Past?
- The ultimate reason has to do with the special initial conditions at the beginning of time (big bang origin of the universe)
- Relevant hypotheses include that in its early stages, the matter content of the universe was in a state of thermal equilibrium
Thermal Equilibrium:
- State where a system has completely "settled down" and will not deviate significantly, even if disturbed slightly
- Concerned with typical rather than individual behavior, averaged measures such as temperature and pressure
- Statistical fluctuations away from the idealized thermal equilibrium state are expected
Thermodynamics:
- Study of averages and statistical behavior, not detailed individual particle motions
- Thermal equilibrium does not pertain to an individual state but an ensemble of states that appear the same on a macroscopic scale (entropy)
- Detailed analysis reveals a scale at which statistical fluctuations away from idealized thermal equilibrium are expected
Conclusion:
- Nothing essentially non-computable about modeling physical behavior by mathematical structures
- After appropriate calculations, there is good agreement between what is calculated and what is observed.
Quantum Mechanics in Classical Physics: Black-body Radiation
Black-body radiation:
- Cannot be treated entirely classically
- Involves quantum processes
- Max Planck's analysis of black-body radiation in 1900 initiated quantum theory
Observed Agreement between Theory and Observation:
- The observed relationship between frequency and intensity agrees closely with the mathematical formula put forward by Planck
- Precise agreement between COBE measurement and expected thermal nature of the big bang's radiation (Fig. 4.12)
Classical vs. Quantum Theory in Biological Systems:
- Computational models play important roles in modelling biological systems
- However, these systems are likely to be more complicated than astrophysics and the computational models correspondingly harder to make reliable
- Simple systems like blood flow and nerve signal propagation can be modeled effectively
- Chemical actions are the result of quantum effects, which are commonly treated in a classical way, but may not be safe for drawing general conclusions about more sophisticated biological systems like the human brain
Limitations of Classical Physics:
- It is risky to make general inferences about the theoretical possibility of a reliable computational model of the brain by only considering classical physics
- The mysteries of quantum theory need to be addressed, which will be attempted in the next two chapters
Classical Physics and Mind:
- Question: Does mind have a place in classical physics?
Notable Claims:
- Dennett (1991) argued against the existence of consciousness in physics.
- Sakharov (1967): second law of thermodynamics, ENM (1973), p. 428.
- Detailed account of second law: Davies (1974) and Penrose (1970).
Dennett's Argument:
- Dennett questioned the role of mind in classical physics.
- Published his thoughts in "The Selfish Gene" (1991).
Sakharov's Contribution:
- Discussed second law of thermodynamics.
- Detailed account can be found in Misner et al.'s "Gravitation," ENM, 1973 p. 428.
Second Law of Thermodynamics:
- Sakharov contributed to understanding this fundamental principle.
- Increase in entropy is a universal trend.
Sources for Second Law:
- Davies (1974) provided a less detailed account.
- Penrose's work also touched on the topic but not very comprehensive.
Quantum Theory: Puzzle and Paradoxes
Introduction:
- Quantum theory describes physical reality on a small scale but contains mysteries
- Difficult to understand and accept as it challenges traditional views of the world
- Two types of quantum mysteries: Z-mysteries (puzzle mysteries) and X-mysteries (paradox mysteries)
Z-Mysteries:
- Genuinely puzzling but experimentally supported quantum truths about reality
- Includes Einstein-Podolsky-Rosen (EPR) phenomena, which will be discussed in detail later
- Can be accepted with time and improvement of quantum theory
X-Mysteries:
- Implausibly paradoxical statements about the world that cannot be believed as true
- Arise from an incomplete or inaccurate theory at a certain level of phenomena
- Believed to be removed in an improved quantum theory
Viewpoint on X-Mysteries:
- Some physicists argue there are no X-mysteries, all strange things must be true if looked at correctly
- Others contend that X-mysteries exist and cannot be accepted as true
Personal Viewpoint:
- Draw line between Z-mysteries and X-mysteries based on expectations for completing quantum theory
- Believe present-day quantum theory is incomplete due to X-mysteries, which will be addressed later in the book.
Quantum Theory vs. Classical Physics (Part I: Elitzur-Vaidman Bomb Testing Problem and Magic Dodecahedra)
Elitzur-Vaidman Bomb Testing Problem:
- Illustration of quantum non-computability in terms of a design for an ultra-sensitive bomb with a detonator mirror
- Detonator sensitive to visible light photons, but may be jammed (dud)
- No classical method to determine whether detonator is jammed without setting it off
- Problem: find a guaranteed non-dud bomb from a large supply of questionable ones
- Solution involves quantum mechanics and the ability to effect physical change based on counterfactuals
- Quantum theory allows for effects arising from possibilities, not just realities
- Implications: profound difference between classical and quantum physics regarding uncertainty and non-computability
Magic Dodecahedra:
- Story about receiving identical dodecahedrons from Quintessential Trinkets on different planets
- Each contains a button; pressing may initiate pyrotechnic display, destroying the dodecahedron if nothing happens instead
- Problem: independently press buttons at chosen times to avoid destroying either one
- Solutions require precise alignment and trust in manufacturer's guarantees
- Long history of accurate production for over 100 million years
- Implications: complexity in quantum mechanics leading to seemingly magical properties.
Quintessential Trinkets' Guaranteed Properties
Antipodal Property:
- If SELECTED vertices are diametrically opposite, the bell rings only on diametrically opposite button presses (irrespective of order)
Corresponding Property:
- If SELECTED vertices are in same directions from center, the bell must ring on at least one of six possible button presses
Deduction for Own Dodecahedron's Properties:
- Each vertex is preassigned as a bell-ringer (WHITE) or silent (BLACK), independent of order
- No two next-to-adjacent vertices can be both bell-ringers
- No set of six vertices adjacent to antipodal ones can be all silent
Implications:
- If two dodecahedra are independent, each must have preassigned vertices (WHITE or BLACK) based on deductions above
- Attempts to color a dodecahedron according to rules (d) and (e) lead to an unresolvable puzzle
- Proof of the impossibility of such a coloring is provided in Appendix B.
Quintessential Trinkets and Quantum Theory
The Problem with Color-Coding Dodecahedra:
- Quintessential Trinkets (QT) claimed it was possible to color the vertices of two separate dodecahedra in a specific way
- This was contradicted by classical physics, as it would require a "mysterious 'connection'" between the two objects that persists until the buttons are pressed
Quantum Theory Explanation:
- The solution lies in quantum entanglement, where two atoms at the centers of the dodecahedra are produced in an initial combined spin state of total 0, and separated to isolated centers
- When a button is pressed on one dodecahedron, a particular spin measurement is made
- If the result is affirmative, the bell rings and a pyrotechnic display follows
Relativity Concerns:
- According to relativity theory, there can be no signals passing between the two spacelike-separated events that transmit information about which button was pressed or which one rang the bell
- However, quantum theory allows for some "influence" connecting the dodecahedra, which cannot be used to send information instantaneously but violates the spirit of special relativity
Experimental Status:
- Experiments like this are called EPR measurements, named after a 1935 paper by Albert Einstein, Boris Podolsky, and Nathan Rosen
- The original paper did not involve spin, but David Bohm presented a spin version in 1951
- These experiments show that particles can be entangled, violating classical expectations about separate independent objects.
Quantum Theory and Entanglement
Bell's Relationships in Quantum Theory:
- In quantum theory, spatial relationships could be violated in a specific way
- This opened up the possibility of real experiments to test whether these relationships are violated by physical systems
- Classical theories would satisfy these relationships, but quantum theory predicts they should not be satisfied
Bertlmann's Socks as an Illustration:
- John Bell used Bertlmann's socks as an example of what entanglement does not mean
- Bertlmann wore socks of different colors, so if you saw his left sock was green, you knew his right sock was not green
- However, it would be unreasonable to infer there was a mysterious influence traveling from his left to right socket
- Bertlmann's socks do not violate Bell's relationships, and there is no "long-distance influence" connecting them
Quantum Non-Locality Experiments:
- In the Aspect experiment, entangled photons emitted in opposite directions were used to test quantum theory expectations
- The results vindicated quantum theory and violated Bell's relationships
- Some physicists still argue that better detectors could restore the Bell relationships
Recent Proposed Experiments:
- Recent proposals involved yes/no character experiments, rather than probabilities
- These experiments aimed to test the Kochen-Specker theorem, which was related to the EPR paradox
- The proposed experiments have not been carried out yet
Quantum Theory's Foundational Ingredients:
- Two fundamental ingredients of modern quantum theory can be traced back to the 16th century and one individual:
- One ingredient is wave-particle duality, which was explored by early wave optics experiments (e.g., Young's double-slit experiment)
- The other ingredient is probability amplitudes and superposition, which were explored in quantum mechanics
Gerolamo Cardano: An Extraordinary Man
Background:
- Born in Milan, Italy (1501) to unmarried parents
- Rose to become the finest physician of his time
- Died in Rome (1576)
Inventions and Achievements:
- Invented cardan shaft for smooth car rides (1545)
- Combination lock similar to modern safes
- Advanced medicine, treated kings and princes
- First noticed gonorrhoea and syphilis are separate diseases
- Proposed sanatorium for tuberculosis treatment
- Cured Archbishop of Scotland's asthma (1552)
Contributions to Mathematics:
- Laid the foundation of probability theory with "Liber de Ludo Aleae" (1524)
- Used gambling to finance studies at medical school
- Discovered complex numbers while solving general cubic equations
- Published Ars Magna (1545), first complete analysis of cubic equation solutions
- Acknowledged Tartaglia's prior claim in The Practice of Arithmetic and Simple Mensuration (1540) but made a mistake regarding permission to publish
- Extended Tartaglia's method to cover all cases, published in Ars Magna (1570)
- Lifelong grudge from Tartaglia leading to Cardano's downfall and arrest by the Inquisition.
Gerolamo Cardano's Family Tragedy
- Gerolamo Cardano: caring father, devoted to children and wife, principled, honest man
- However, his children had a "disastrous progeny"
- Including murder of wife by son Giovanni, who was forced into marriage to cover up earlier murder
- Younger son Aldo became public torturer and executioner for the Inquisition in Bologna
- Cardano's own daughter died of syphilis from her professional activities as a prostitute
- Reasons for poor parenting:
- Attentions frequently distracted by other interests
- Long absence from home after wife's death, traveling to Scotland for Archbishop
- Convinction that stars foretold his own death in 1546, leading to neglect of wife and children
- Cardano's reputation was tarnished due to the actions of his children, Inquisition, and Tartaglia
- Cardano is a "greatest of Renaissance figures" but less well known today due to this reputation damage
Cardano's Contribution to Mathematics: The Cubic Equation
- Reduced general cubic equation to the form X^3 = px + q
- In those days, negative numbers were not widely accepted as actual "numbers"
- Different versions of the equation were written based on the signs of p and q
- Tartaglia's cases he could solve are given by the downward sloping line P in Fig. 5.9
- Cardano's new solutions, for positive p values (lines Q or R), occurred when p > 0
- Casus irreducibilis: when there were three intersection points as with line R
- Required journeying through the complex to express the solutions
Cardano's Problem and Complex Numbers
Cardano's Problem:
- Finding two numbers whose product is 40 and sum to 10 cannot be solved with real numbers
- The solutions involve complex numbers (i.e., the square root of a negative number)
- Cardano was aware of this problem, called the 'casus irreducibilis', and addressed it in his work on cubic equations
Cardano's Solution:
- In the case of the quadratic equation x^2 - bx + c = 0, no solution exists unless complex numbers are permitted
- However, for the cubic equation ax^3 + bx^2 + cx + d = 0, there are three real solutions, but expressing them in terms of surds requires complex numbers
Complex Numbers and Quantum Theory:
- Complex numbers play a strange role at a very tiny underlying level of phenomena in quantum theory
- Probabilities also play a part at the level between this quantum level and classical, observable phenomena
Quantum vs Classical Level
Quantum Level:
- Applies to tiny differences in energy
- Not defined by small physical size, but something more subtle
- Experienced as unfamiliar and mysterious
Classical Level:
- Ordinary experience where classical physics holds
- Distinction between quantum and classical level is a profound question related to X-mysteries
Role of Complex Numbers in Quantum Mechanics:
- Classical interpretation: weighting factors represent probability expectations
- However, they are complex numbers, not ratios of probabilities
- Unfamiliar and mysterious description for superpositions of states
Unitary Evolution (U):
- Describes deterministic quantum state changes in Schrodinger picture
- Mathematically precise and completel y deterministic description of the micro-world.
Quantum Superposition and Linearity
- In quantum mechanics, states are represented by state vectors, such as
It/I>, wherewandzare complex numbers (not both zero) - A linear superposition of two states is given by the combination
wlA + zlB>, representing a combination of the two original states - The quantity
It/I>is called a state vector and can be more generally written as a sum of different state vectors, such asll/I) = u1lA+v1lb+...+zlF> - In quantum theory, the ratio of complex weighting factors has significance, not the individual weights
wandz. This is because any complex multiple of a state vector represents the same physical state - The Schrodinger equation is linear, meaning if states
II/I)andlc/>)evolve separately under timet, then their superposition (wlA+ zlB>) also evolves as expected by the Schrodinger equation - This applies to any number of individual quantum states in a superposition, with each component evolving independently according to the Schrodinger equation
- Quantum systems can be in complex superpositions, but only the ratios of weights have physical significance. The absolute values of
wandzdo not matter.
Example: Half-Silvered Mirror
- Light is composed of photons, which are put into a superposition state when impinging on a half-silvered mirror
- Each individual photon in state
IA)>evolves toIB> + IC>after reflection and transmission at the mirror - The superposition state
IA)evolves asIB> + IC>, which can be reflected again to create new statesID>)andIE) - Bringing the two beams back together creates a new state
i(ID)-IE), demonstrating the interference pattern of the superposition.
Photon Interference and State Reduction (Mach-Zehnder Interferometer)
Background:
- Photon state consists of two parts: reflected (IG) and transmitted (IF)
- Both parts bring together by fully silvered mirrors, resulting in state ILD>-IE)...+i(IG> +ilF>) or -2IF> at final half-silvered mirror
- Detectors placed at F and G: photon emerges in unsuperposed state IF), no contribution from IG)
Behavior of Superposed States:
- State vector reduction/collapse of the wavefunction (R) occurs when magnifying quantum events to classical level
- Quantum levels persist under U evolution
- Classical alternatives have definite outcomes: either F registers or G registers
- R is a matter of concern for understanding the nature of quantum theory
Calculating Probabilities:
- Rule for determining probabilities in measurement of superposed states (IF> and IG>) using detectors at F and G
- Ratio of probability that detector at F registers to G registers: |w/z|, where w and z are complex numbers denoting state amplitudes
- Squared modulus of a complex number is the sum of squares of its real and imaginary parts
- Normalization needed for this form of the probability rule to hold.
The Transition from Quantum to Classical Level
Measurement and Indeterminacy:
- Measurements occur when there is a large magnification of a physical process, raising it from the quantum to the classical level
- This can be seen in photocells or sensitive photographic plates that register the arrival of a single quantum event (photon) and trigger a classical-level disturbance
Cardano's Complex Numbers and Probabilities:
- In the passage from quantum to classical level, Cardano's complex numbers get squared to become Cardano's probabilities
Example: Photon Detection with Obstruction:
- If an obstruction is placed in a photon beam, it can cause a detector at G to register the arrival of the photon (when the obstruction does not absorb the photon)
- This ratio of probabilities for detecting the photon at G versus leaving it undisturbed is 1:1, meaning the photon is equally likely to activate the photocell as not to
Obstruction as a Measuring Apparatus:
- The obstruction itself can be considered as a "measuring apparatus" because its absorption of the photon disturbs its substance and makes it difficult to collect information for quantum interference effects
Calculating Probabilities with Obstruction:
- If the obstruction absorbs the photon, the probability of absorption is equal to that of non-absorption (1:1)
Alternative Measuring Device:
- Instead of having an obstruction, a measuring device could be attached to the lower right-hand mirror that detects the impulse imparted on the mirror by the reflected photon
- This would "collapse the wavefunction" and read the state as either IB> (pointer moves) or IC> (pointer stays put), with equal probabilities
Solution to Bomb-Testing Problem
Implications of Quantum Rules:
- Quantum rules can be changed from quantum-level complex-number weighted determination to classical-level probability-weighted non-deterministic alternatives.
- Difficulty arises in explaining why these changes occur and what characterizes physical objects like photon detectors, obstructions, and the bomb mirror as classical-level objects while treating the photon quantum mechanically is a matter of convenience.
Mirror Setup for Bomb Testing:
- Set up mirrors similar to Fig. 5.14 with the bomb's mirror playing the role of the lower right-hand mirror.
- If the bomb is a dud, its mirror is jammed in a fixed position, allowing the system to behave like Fig. 5.12 and the photon emerges from the final mirror in state IF>.
- If the bomb is not a dud, its mirror can respond to the photon and the bomb would explode if it finds the photon impinging on the mirror.
- The two quantum level alternatives 'photon impinging on mirror' and 'photon not impinging on mirror' are magnified by the bomb to the classical-level alternatives 'bomb explodes' and 'bomb fails to explode'.
- If the detector at F registers the photon's arrival, the probability is 1:1 with the probability of the detector at G registering its arrival being 1-1/12 = 11/12.
Solution to Bomb-Testing Problem:
- In a long run of tests, half of the active bombs will explode and be lost.
- Only in half of the cases where an active bomb does not explode will the detector at G register the arrival of the photon.
- Therefore, if we run through the bombs one after another, only 1/4 of the originally active bombs are guaranteed to still be active.
Quantum Theory and Spin: The Riemann Sphere (Part 1)
Introduction:
- Explanation of spin as an intrinsic property of particles, contrasting it with angular momentum of classical objects
- Difference between spin and orbital motion for large objects
- Importance of understanding spin in quantum theory to solve introductory puzzles
Spin:
- Intrinsic property of fundamental particles
- Similar concept as angular momentum but always has the same magnitude for a particle, unlike classical objects
- Spin axis direction can vary
- Unusual behavior compared to classical physics
Comparison between Classical Objects and Particles:
- Large objects: significant contribution to angular momentum comes from orbital motion of particles around one another
- Fundamental particles: intrinsic spin property, no orbital motion contributes to its spinning.
Importance in Quantum Theory:
- Spin plays a crucial role in addressing introductory quantum puzzles
- Understanding spin is essential for a deeper comprehension of quantum mechanics.
Spin States for Particles with Half-Integer Spin (Like Electron or Nucleon)
- All spin states are complex superpositions of two base states: 'spin up' (Iu>) and 'spin down' (Id).
- The general state of spin is some complex-number combination of these two states.
- Each such combination represents a specific direction of the particle's spin based on the ratio of the two complex numbers wand z.
- Representation of Complex Numbers:
- Points in the Argand plane, with real and imaginary parts (x, y) as Cartesian coordinates.
- Basic Operations:
- Addition: Geometrically represented by parallelograms.
- Multiplication: Geometrically represented by multiplication of lengths and angles.
- Change sign: Reflection in origin.
- Complex conjugate: Reflection in x-axis.
- Modulus and Unit Circle:
- The modulus (distance from origin) represents the magnitude of a complex number.
- Squared modulus equals the square of that number.
- Unit circle consists of complex numbers with unit distance from origin, representing pure phases.
- Representation of Complex Ratios:
- A ratio z:w is represented by the single complex number z/w, except when w=0.
- In this case, we use oo (infinity) to cover the possibility of a zero denominator.
- Riemann Sphere:
- Represents complex numbers including infinity (oo).
- The point representing p=z/w on the complex plane is projected from the south pole S to P' on the sphere, with direction OP being the spin direction for general half-spin state.
- Special cases:
- 'Spin down': Represents w=O, with Id as one of the base states and Iu> as the other.
- 'Spin up': Represents z=O, with Iu> as one of the base states and Id as the other.
The Riemann Sphere and Complex Ratios
- The Riemann sphere represents complex numbers or infinity (00)
- It can be visualized as a sphere of unit radius with the complex plane as its equatorial plane and origin at the center
- To represent a complex ratio z:w, mark the point P on the complex plane representing the complex number z/w (assuming w#O), then project P onto the sphere from the south pole S to obtain the point P'
Stereographic Projection
- The mapping between points on the sphere and the plane is called stereographic projection
- As a point in the plane moves far away, its corresponding point on the sphere approaches the south pole closely (in the limit as the point goes to infinity)
The Riemann Sphere in Quantum Mechanics
- The Riemann sphere plays a fundamental role in quantum mechanics, describing the space of physically distinguishable states that can be built up by superposition from two distinct quantum states
- For example, it can describe the possible states of a photon with locations IB> and IC>, where the general linear combination is wI B> + zIC)
- The geometrical role of the Riemann sphere is not always apparent, but it describes the abstract geometric space of physically distinguishable states in an abstract way
Spin States and the Riemann Sphere
- For a particle of spin t (e.g., electron or proton), the general state can be represented as a combination of Ii> and I.>, where lift> represents the spin state of magnitude iiii, right-handed about the axis pointing in the direction of the point on the Riemann sphere
- All combinations are physically real, with no more mystery than the original Ii> and I.> states
- For higher spin (tn), the situation gets more complicated, but the general description was pointed out by Ettore Majorana
- Measurements using a Stern-Gerlach apparatus give n+1 possible outcomes, depending on the amount of spin oriented in the measured direction
- The different possible states are complex number superpositions of these possibilities, with values determined by the spin measurement in units of .h
Majorana's Representation of Spin State:
- General state of spin represented as a collection of independent arrow directions
- Arrow directions are points on the Riemann sphere, with each representing an element of spin
- Unordered set of points (or arrow directions) with no significance assigned to ordering
- Classical objects do not spin in many different directions all at once, but only clustered around one particular direction
- Correspondence principle asserts that when physical quantities get large, system can behave like classical behavior, but this does not explain how such states arise from Schrodinger equation U alone.
Position and Momentum of a Particle:
- At quantum level: particle's state involves superpositions of two or more different locations (e.g., photon in two beams)
- Large objects like golf balls don't exhibit such superpositions in the actual world
- Classical outcomes are achieved through the action of R, not U alone
- Strictly speaking, states of well-defined momentum have oscillatory spatial behavior (not linear vibrations) which were ignored during discussion on photon states.
Hilbert Space and Quantum States
Properties of Hilbert Space:
- Complex vector space: allows complex-number weighted combinations
- Elements are quantum states, represented by Dirac ket notation (e.g., |1)
- Family of all possible states for a quantum system
- Sometimes finite or infinite dimensions
Visualizing Hilbert Space:
- Difficult to visualize due to high number of dimensions and complex nature
- Ignoring problems temporarily can help develop intuition about mathematics
Properties of Hilbert Space: Vector Space Structure
- Addition rule: It/I) + I
= (wz)It/I) + w(It/I) + z(I
) = It/I) + I
+ z(I
)
- Scalar multiplication rule: z(It/I) = (wz)It/I), where w and z are complex numbers
- Zero element: OIt/I) = 0
Hermitian Scalar Product or Inner Product:
- Enables expression of important concepts, such as squared length and orthogonality
- Describes the inner relationship between two vectors in a Hilbert space
Important Concepts from Hermitian Scalar Product:
- Squared Length: The scalar product of a vector with itself (It/I)| = ||It/I|
- Normalized State: A state whose squared length is unity (|It/I| = 1)
- Orthogonality: Two vectors are orthogonal when their scalar product is zero
- Independent states: Orthogonal outcomes of measurements in quantum physics.
Orthogonality in Quantum Mechanics
Important Concepts:
- Orthogonal states: Two states are orthogonal if their scalar product equals zero.
- Unit vector or normalized state: A state can be normalized by dividing it by its length (or magnitude).
- Symmetry of orthogonality relationship.
Orthogonality and Scalar Product:
- Orthogonality is preserved under unitary evolution.
- The scalar product satisfies certain algebraic properties, including symmetry and non-negativity.
- Normalization and ambiguity in state vectors due to pure phases.
Hilbert Space Representation of Measurements:
- Yes/No measurements: A 'yes' outcome indicates the presence of a property; a 'no' outcome implies orthogonality with the expected state.
- Null measurements: Outcome is the absence of a property, resulting in an orthogonal state.
- Complex measurements: Built up from successive yes/no measurements.
Examples:
- Photon detectors and Stern-Gerlach spin measurers are examples of yes/no measurements that result in orthogonal states.
Quantum Mechanics: Projection Postulate
The Problem:
- Quantum states seem to 'jump' or change abruptly during measurements
- Physicists find this concept hard to accept as real
Projection Postulate
- Essential assumption in quantum theory for null measurements (answer NO)
- Specifies how state vector evolves after measurement:
- YES answer: projects onto a multiple of certain state
- NO answer: projects onto the orthogonal complement of that state
Primitive Measurement:
- Yes/no type measurement with definite outcomes
- YES answer determines physical state to be a particular thing
- NO answer projects state to something orthogonal
Fig. 5.24 illustration:
- State It/I> can be written as the sum of two vectors: loc> and Ix>
- loc>: projection of It/I> on the chosen ray (YES)
- Ix>: orthogonal projection of It/I> into complement space (NO)
- When YES is obtained, state jumps to zloc> (or just loc>)
- When NO is obtained, state projects down to the space orthogonal to lex>
Probabilities:
- To obtain probabilities:
- loc> is a unit vector with some orthogonal vector l
in the direction of Ix>
- Use squared length reduction factor as probability for each outcome.
- loc> is a unit vector with some orthogonal vector l
General Yes/No Measurement:
- Similar discussion applies to non-primitive yes/no measurements
- Involves YES and NO subspaces that are orthogonal complements
The Projection Postulate and Quantum Measurement
Projection Postulate:
- Asserts that the original vector
II/Igets orthogonally projected intoYfor a "Yes" answer and intoNfor a "No" answer - The probabilities are given by the factors by which the squared length of the state vector is reduced on projection (see [ENM, p. 263, Fig. 6.23])
Status of Projection Postulate:
- Less clear than with "null measurement" above
- With an affirmative measurement, the resulting state gets entangled with the state of the measuring apparatus
- In discussions that follow, the author will stick to simpler primitive measurements where the
YESray consists of a single ray (multiples ofII/I)
Commuting Measurements:
- With successive quantum system measurements, the order in which they are performed can be important
- Non-commuting Measurements: If the ordering of the measurements makes a difference to the resulting state vectors
- Commuting Measurements: If the ordering of the measurements plays no role (there is no phase factor difference)
Quantum Mechanical 'And':
- Standard procedure for treating systems with more than one independent part
- In absence of each other, the system would be described by state vectors
lexandIPx - The combined state of the two parts is represented by the tensor product
lex IPx, orIx = lex IPx - This is a single quantum state vector, not a superposition of
lexandIPxlikelecx + wlp
Physics Concepts:
- Bosons and product states in quantum mechanics
- Schrodinger evolution for non-interacting systems
- Linearity property of U (superposition of evolutions) vs. orthogonality of product states
- Orthogonality of product states with Grassmann products
- Quantum entanglement and the measurement problem
Bosons and Product States:
- Bosons do not contribute to sign in Schrodinger description
- Identical physical state as l(X) for multiple independent parts (J(X), 1.B>, Jy) -> l(X)<l.B>lY> or l(X)<IJ'
- Important property: combined system of non-interacting parts evolves as individual parts do
- Linearity property: superposed systems evolve independently, but can lead to questioning absolute truth
Orthogonality:
- Two orthogonal states lex> and IP> have orthogonal product states lt/l>lex> and lt/l>IP> (if no interaction between parts)
- Issue arises due to potential dependence of one object's state on another in reality
- Inaccuracies arise from considering all photons in universe, but not necessary for high degree of accuracy
Entangled States:
- Understanding needed: EP R effects and the measurement problem
- Introduce concept of entangled states to discuss these issues further.
Superposition and Measurement in Quantum Mechanics
- Consider a situation where a photon is in a superposed state, e.g., loc> + IP>
- Detector (l'P') can also be assigned quantum state
- Initial physical situation: l'P>(loc> + IP>) = l'P>loc> + l'P>IP> (superposition)
- Schrodinger evolution U causes detector state l'P>loc> to evolve into new state l'P'> (photon detected)
- If photon doesn't encounter detector, l'P>loc> evolves into l'PN> and IP> evolves into IP', leading to a superposition of orthogonal states l'P'> and l'PN>IP')
- This is an example of an entangled state where the entire state cannot be written as product of states for individual subsystems (photon and detector)
- Characteristic feature: non-local effect or retroactive action, e.g., EPR (Einstein-Podolsky-Rosen) effect
- A puzzle in quantum theory that challenges its completeness as a description of Nature
- EPR effect arises from known initial state In> evolving into superposition of orthogonal states, each describing separate parts of a physical system.
Quantum Entanglement and Bertlmann's Socks
Introduction:
- Effects of quantum entanglement can differ from "Bertlmann socks" type explanation
- Problems arise with option to perform alternative measurements on system parts
Entangled Particles:
- Initial state (I00) describes spin state of particle as zero
- Particle decays into two new particles, each with opposite spin directions
- Spin directions determined by angular momentum conservation
Illustration: Two Particles' Spins
- Measurement of left-hand particle's upward spin determines right-hand particle's downward spin
- Choice of measurement direction for left-hand particle fixes right-hand particle's spin axis
- No actual information conveyed to right-hand particle until result obtained
Entangled State Representation:
- Can be represented in different ways, corresponding to various choices of measurement
- Left-hand particle's horizontal measurement affects right-hand particle's state
Limitations and Misconceptions:
- Cannot use entanglement for instantaneous signaling between particles
- EPR effects not constrained by the speed of light
- Quantum theory predictions cannot be obtained in classical models with non-communicating objects
- Entanglement lies between direct communication and complete separation, has no classical analogue.
Einstein's Findings on Quantum Entanglement
- Einstein found quantum entanglement "deeply disturbing," referring to it as "spooky action at a distance"
- Quantum entanglement is not only oblivious to space separation but also time separation
- If a measurement is made on one component of an EPR pair before the other, the first measurement is considered the one that effects disentanglement
- However, the same observable consequences would be obtained if the second measurement retroactively effected disentanglement
- This symmetry is necessary for EPR measurements to be consistent with special relativity's observational consequences
Understanding Quantum Entanglement: Magic Dodecahedra
- Non-locality: not just a matter of probability, provides precise yes/no effects that cannot be explained classically
- Initial state (In>) is split into two atoms, each suspended at the center of a dodecahedron
- Dodecahedra dispatched to different observers without disturbing spin states until measurement
- Pressing button activates Stern-Gerlach-type measurement on atom, revealing one of four possible outcomes
- When button is pressed, apparatus oriented in direction of that button's location
- Bell rings (yes response) if atom is found in second of four locations after encountering apparatus
- NO response if atom is in remaining three locations
- Each button pressing effects a primitive measurement as described in 5.13
Properties Guaranteed by Quintessential Trinkets:
- Property (a): Bell rings for both parties if they press adjacent buttons
- Established through Majorana description of spin . states and orthogonality of adjacent vertices on the dodecahedron
- Property (b): Existence of a state mutually orthogonal to all possible outcomes
- Three-dimensional Hilbert space for spin . does not exhaust alternatives; null measurement (bell failing to ring on all three pressings) ascertains state IRST)
Explanation:
- Two atoms originating from the spin 0 state can be expressed as a total state In> = ILiii>IRl..>-ILl ii>IRi ..>+IL.. i>IRii l>-ILl..>IRiii>
- Bell rings for both parties if they press adjacent buttons (property a) because:
- Majorana description of YES state is point P and its antipode P* on the Riemann sphere
- Adjacent vertices of a cube inside dodecahedron have orthogonal Majorana states (IAAA), ICCC), and IG*GG)
- Three button pressings adjacent to a selected vertex are commuting measurements, order irrelevant for both parties
- Properties (a) and (b) establish quantum entanglement, allowing resolution of vertex-coloring problem (properties c, d, e) that would be impossible with independent objects.
Quantum Entanglement and Dodecahedron Coloring
Entanglement in Quantum Mechanics:
- Quantum entanglement is unlike anything in classical physics
- Cannot be explained by Bertlmann's socks type of explanation
- Standard quantum mechanical rules lead to the conclusion that objects must remain entangled, even at great distances
- This persistence of entanglement raises questions about the nature of reality and the universe as a whole
Dodecahedron Coloring Problem:
- Recall the problem of coloring all vertices of a dodecahedron: no two adjacent vertices can be both white, nor six adjacent vertices to a pair of opposite vertices can be all black.
- The symmetry of the dodecahedron helps in eliminating possibilities.
- For a white vertex A, any neighboring white vertex (e.g., B) leads to all surrounding vertices being black.
- This argument rules out the possibility of adjacent white vertices.
- Thus, the white vertex A must be surrounded by black vertices (B, C, D, E, J, H*, F, I*, G).
- Examining the six vertices adjacent to one of the antipodal pair (H, H*) leads to a contradiction and shows that the classical dodecahedron coloring is impossible.
Majorana Description of General Spin States:
- Provides a useful and geometrically illuminating picture
- Represents the unordered set of n points on the Riemann sphere as roots of a complex polynomial
- Uses coefficients of this polynomial as coordinates in the (n + 1)-dimensional Hilbert space of spin states
- Basis states are monomials, normalized to be unit vectors
Spin-n States and Majorana Descriptions
General State of Spin-n:
- Described by a symmetric n-valent spinor
- Majorana description follows from its canonical decomposition as a symmetrized product of spin vectors
Majorana Points on Riemann Sphere:
- Roots of the polynomial representing the state
- Include possibility of the Majorana point at x = infinity (south pole) if degree falls short
Properties of Scalar Product:
- Symmetric and invariant under rotations of the sphere
- Two antipodal points on the sphere give orthogonal states when n is odd (C.1)
Additional Properties:
- State with all components equal to a single value is orthogonal to every other state (C.2)
- Orthogonality between certain states can be determined by centroid property of their Majorana points (C.3)
Application:
- Relevant for understanding spin measurements and their results in terms of Majorana descriptions and orthogonality.
Notes and References
1. Penrose's Work:
- Penrose (1993b, 1994a)
- Zimba and Penrose (1993)
Experiments Indicating Non-local Quantum Effects:
- Freedman and Clauser (1972)
- Aspect, Grangier, and Roger (1982)
- Aspect and Grangier (1986)
Retarded Collapse Explanation:
- Proposed by Euan Squires (1992a)
- Takes advantage of possible time delay in measurement detection
- Differs from quantum theory expectations over long distances
- Conservation laws would hold 'on average' but not for individual cases
- No experimental evidence yet, some objections against it
Kochen and Specker Example:
- Proposed by Kochen and Specker (1967)
- Other examples with different geometrical configurations: Peres (1990), Mermin (1990), Penrose (1994a)
Reflection from Half-silvered Mirror:
- Not actually silvered but a thin transparent material with internal reflections and transmissions
- Final transmitted and reflected beams have equal intensity due to unitary nature of transformation
- Arbitrariness in choice of phase factor for the reflected state
Additional References:
- Dirac (1947)
- Davies (1984)
Quantum Theory and Reality: Understanding Quan Tum Mysteries and Classical Phenomena
Introduction:
- Discussing quantum mysteries, not all experimentally tested but enough support for strange behavior at the quantum level
- Two types of physical laws? In contrast to classical physics' expectations
- Comparison with ancient Greek beliefs about heavens vs earth
- Achievements of Galileo and Newton: one set of laws for all scales
- Current understanding may be a stopgap, anticipating advance comparable to Galileo and Newton
Controversy:
- Standard understanding and classical behavior of large objects from quantum constituents
- Some argue quantum level behavior explains classical phenomena
- Others disagree, claiming superpositions persist at all scales
Fundamental Difficulty:
- Inherent 'slipperiness' in procedure R preventing clear level for quantum activity to give way to the classical
- No physical experiment can determine where it occurs or even if it does
Quantum Entanglement:
- Effects stretch over large distances, not limited by physical size
- Differences in scale do not refer only to physical size
Conscious Perceptions:
- Ultimate determination of 'the buck stops' at conscious perceptions.
Physical Theory and Perception
- Lack of understanding about physical processes in brain associated with perception
- Physical nature provides limit for proposed theory of real R-process
- Allows for numerous different attitudes regarding what really happens during R
Quantum Mechanics
- Professor Bob Wald's quote: "If you really believe in quantum mechanics, then you can't take it seriously."
- Niels Bohr's viewpoint: state vector as a convenience, not an objective description of reality
- Broad alternative: believe state vector provides accurate mathematical description of real quantum level world
- Two main routes: U is all there is or R represents illusion/approximation
- Leads to many-worlds interpretation (accepting U as true reality)
- Superposed states become problematic for classical objects due to linearity of U
- Classical objects frequently appear in manifestly superposed states, yet we don't observe them
- Proponents explain this by accepting all superpositions as part of total reality
- Superposed states become problematic for classical objects due to linearity of U
- Believe both U and R represent actual physical behavior (fully take the quantum formalism seriously)
- Difficulty accepting theory is completely accurate due to variance between linearity of U and properties of classical objects.
- Leads to many-worlds interpretation (accepting U as true reality)
- Two main routes: U is all there is or R represents illusion/approximation
Many-Worlds Interpretation
- Accept state vector as true reality, leading to acceptance of quantum superposition for all objects
- Large scale superpositions don't impinge on our awareness because they are preserved in total reality
- Human observer states after examining detector: perceived reception or non-reception of photon.
Many-Worlds Theory and Reality
Background
- In the many-worlds theory:
- Different instances (copies) of observer's 'self' coexist within total state
- Each copy perceives a world consistent with its own perception
- Multiple 'worlds' and observers, each with unique realities
Criticisms
- Lack of economy: Unsatisfactory due to the extraordinary number of worlds proposed
- Measurement problem: Many-worlds theory does not fully explain how R procedure arises or provides understanding of circumstances under which it occurs
- Precision: Does not provide a satisfactory explanation for the remarkable precision of quantum theory, including R's prediction of probabilities
- Complex entangled states : Difficulty in understanding how these behave 'for all practical purposes' (FAPP) like an ensemble of similar physical systems
- Reality vs calculational device: Debate over whether It/I) represents a quantum-level reality or just a calculational tool, with some arguing that it cannot describe reality due to its complexities and non-local behavior (R procedure)
- Non-local discontinuous 'jumping' (R): Objection that this is not how a physically acceptable description of the world should behave; U Schrodinger equation already controls It/I)'s evolution most of the time
- Alternation between U and R processes: Another criticism, as U by itself leads to difficulties with many-worlds theory if we require a picture closely resembling our perceived universe.
Conclusion The many-worlds theory provides an alternative perspective on quantum mechanics but faces criticisms related to economy, measurement problem, precision, complex entangled states, reality vs calculational device, non-local discontinuous 'jumping' (R), and alternation between U and R processes. Addressing these concerns is essential for a more satisfactory understanding of the underlying physics behind quantum phenomena.
Quantum Theory and Reality
Unsymmetrical Time Evolutions in Quantum Theory:
- In quantum theory, state vector description is not symmetrical in time due to R determining starting point of U-actions instead of finishing points.
- Two equivalent descriptions exist with reversed U-time evolutions.
- Both descriptions could represent parts of physical reality, but it's unclear which one is 'real'.
Challenges in Measuring State Vectors:
- Not directly measurable as they cannot be determined precisely from a single measurement.
- Atom spin direction example: requires infinite bits of information to determine accurately.
- Difficulty accepting state vector as physically unreal due to subjective nature of knowledge about it.
Uniqueness of Physical State Vector:
- In principle, there is always a primitive measurement determining the Hilbert space ray of It/I>.
- State vector uniquely determined by the outcome of potential measurements (counterfactuals).
- Importance of counterfactual issues in quantum theory expectations.
Reality and Computed States:
- If system is prepared in known state I</>), evolving into new state II/I>= I-+) under U action, must be taken seriously as part of physical reality for potential primitive measurements.
- Standard rules of quantum theory allow these measurements even if they're impractical in certain situations.
The Need for Change in Quantum Theory
- Objective distinctions between quantum states cannot be denied if II/I> is not taken to be objectively real up to proportionality
- A "minimal change" suggested: introduction of superselection rules that deny certain primitive measurements on a system
- However, no coherent general viewpoint has emerged regarding measurement problem
- Some approaches provide different pictures of reality compared to conventional theory
- Pilot wave theory (de Broglie, Bohm) and density matrix formalism
- Other alternatives: Feynman's approach, histories of possible behavior
Density Matrix
- Pragmatic physicists argue that state vector II/I> has no more "reality" than other indistinguishable states in practice
- Density matrix represents probability mixture of alternative state vectors instead of a single state vector
- Central role in standard mathematical descriptions of the measurement process, also plays a central role in the author's less conventional approach.
Density Matrix Concepts
Background:
- Replacement for individual states with objects like II/I> <t/JI in quantum theory
- Introduced by John von Neumann, who also developed game theory and contributed to electronic computer development
- Unique product between a state vector and its complex conjugate (bra-vector)
Properties:
- Algebraic laws similar to products of scalars but more complicated with tensor product
- Trace operation enables passing from element of a vector space to an ordinary complex number
- Represents probability mixture of normalized states with respective probabilities
Calculating Probabilities using Density Matrices:
- Primitive measurement:
- Represented by projector, E=II/I> <t/JI (similar to density matrix)
- Calculate the probability p = trace(DE) of obtaining YES by multiplying DE and taking the trace
- More general yes/no measurements:
- Use a projector like E = It/I)> + l
<PI + lx>xl for more complex measurements
- The general property 1 holds, ensuring the trace of a projector equals 1
- Use a projector like E = It/I)> + l
Density Matrix and Probability Calculation
- The probability of YES for a measurement defined by projector E on density matrix D can be calculated without knowing how the density matrix is constructed from individual states
- Classical and quantum probabilities are interwoven within the density matrix, resulting in total probability
- Example: Uncertain spin state (li> or ID), resulting in density matrix D=.li><il + .ID<Ll
- Measuring spin in upward direction (projector E=li> <ii) gives trace(DE)=.x12+.x02 = t for probability of YES according to first description and trace(DE)=.l<-+li>l2+tl<+-li>l2 = i+i=t for second description
- Density matrix cleverly provides correct probability regardless of classical or quantum parts
Degenerate Eigenvalues and Probability Mixtures
- Density matrices with degenerate eigenvalues allow multiple descriptions in terms of probability mixtures of orthogonal alternatives
- Any density matrix can be represented as a probability mixture of alternative states, not just mutually orthogonal ones
Density Matrix for EPR Pairs
- Appropriate description for entangled particles situation where one particle is far away and cannot be measured by observer
- Density matrix D=tlRi> (Ril+tlR.> (R.l-) represents the right-hand particle's state of spin, allowing for equal probability mixture based on possible results obtained from imaginary colleague's measurement of left-hand particle's spin.
Density Matrices vs State Vectors
- Density matrices provide an alternative description to state vectors in certain circumstances
- Both descriptions provide information for calculating probabilities of results of measurements
- However, density matrices cannot explain correlations between measurements of different observers or violations of Bell's theorem, which require the entire entangled state provided by a state vector like IO>
Density Matrices in Quantum Theory
- Density matrices play an important but not fundamental role in quantum theory
- They are useful when measurements on parts of a system cannot be performed and compared due to technical limitations or abnormal situations
- The essential role of the density matrix is to describe the probabilities of different states, rather than the state itself.
FAPP Explanation of R Process
- FAPP (frequently asked questions) approach explains how the R process seems to occur in terms of:
- A quantum system and a measuring apparatus that behave as though R has taken place when the effects become entangled with the environment
- The quantum system, together with the measuring apparatus, plays a role similar to the right-hand particle
- The disturbed environment plays a role similar to the left-hand particle
- A physicist examining the measuring apparatus would have no access to details of the environment's disturbance, similar to an observer unable to access measurements on the left-hand particle.
- The state of the measuring apparatus is described by a density matrix rather than a pure quantum state, reflecting its role as a probability mixture rather than a single state.
Quantum State of Detector Before Measurement
- Example: photon emitted from source towards detector with partially silvered mirror in between
- Superposition state of photon upon encountering mirror: I > (transmitted) and IP > (reflected)
- Probabilities of transmission and reflection for a half-silvered mirror are equal and close to .5
- Detector initially in state l'I'>, evolves to either l'l'v) or l'l'N), depending on whether it absorbs the photon (YES) or fails to do so (NO)
- Environment becomes involved in interactions with the detector and absorbed photon after measurement
- Assign environment states: I>v) for YES state and I>N) for NO state, assuming normalized but not necessarily orthogonal
Density Matrix Description of Detector State Before Measurement
- According to standard argument: density matrix D represents probability mixture of detector registering YES (lwl2 probability) and NO (lzl2 probability)
- Similar to density matrix for particle with spin i in previous discussion
- Can be re-expressed as a combination of two orthogonal possible states, l'l'p) and l'I' Q), which are absurd from classical physics perspective
- Density matrix does not explain why physicist always finds detector either in YES state (l'l'v) or NO state (l'l'N)
- Possible interpretations of density matrix as 'dead plus alive' or 'dead minus alive' states, which are physically absurd but not necessarily experienced in the actual world
- Conscious observer (physicist) is allowed to perceive only certain states and not all possible combinations of them. Reason for this limitation remains unclear.
Discussion on Density Matrix and Measurement in Quantum Theory
Problem with Determining Cat's State:
- In Schrodinger's cat paradox, density matrix provides correct probabilities of cat being dead or alive
- However, question raised about the act of looking at the cat performing a measurement
- The forbidden combination 'cat dead + cat alive' not explained by U-evolution alone
Wigner's Friend Paradox:
- Version of Schrodinger's cat paradox with a conscious observer (Eugene P. Wigner, 1961)
- Uncertainty about the nature of awareness and brain construction
- FAPP explanations considered to avoid these questions
Ambiguity in Density Matrix:
- Two probabilities . and . equal does not restrict interpretation
- Hughston et al. showed that density matrix can be represented as a probability mixture of alternatives for entangled systems
- However, ambiguity remains even when probabilities are equal
Standard Argument Not Sufficient:
- Cannot explain how the 'illusion' of reality (R) takes place as an approximate description of U-evolution
- Coexistence and consistency with U-evolution have a FAPP status rather than rigorous one
Importance of Fine Details of Environment:
- Density matrix description adequate only if fine details cannot be measured and compared with experimenter's observations
- Future advances in technology might reveal new physical phenomena to uncover interference effects.
FAQP Viewpoint and Squared Modulus Rule
- FAPP viewpoint does not provide a clear picture of physical reality, but is a valuable stopgap
- Cannot deduce squared modulus rule from unitary evolution (U) alone due to implicit assumptions about probability calculation using density matrices and quantum mechanics
- Unitary evolution doesvetail mathematically with density matrix and Hilbert-space scalar product, but does not explain the notion of probability
- Quantum probabilities must be calculated separately, which cannot be justified solely through U
R-procedure in Measurement
- Discussions suggest R is an approximation to something else not yet understood
- Possible that measurement process itself may be an approximation
Consciousness and Reduction of State Vector
- Some argue consciousness reduces state vector as alternative to trusting U completely (many-worlds viewpoint)
- Unconscious matter evolves according to U, but conscious entities result in R for reduction
- No suggestion that conscious free will influences results of quantum experiments
- Consciousness a rare phenomenon, so it would be strange if physical objects evolved differently based on presence or absence of consciousness
Quantum Theory and Reality: The Problem of Physical Reality
Overview:
- FAPP viewpoint does not resolve deep issues of physical reality
- Proposes to look elsewhere for resolution
- Belief that the problem of quantum measurement should be solved first
- Taking the quantum description seriously requires addressing discontinuous and probabilistic action of R
Quantum State Vector Description:
- Reality of state vector ll/I> must be taken seriously
- Incompatibility between U and R procedures
- Unconventional theories have been proposed to build coherent theories
- Hungarian school (Karolyhazy)
- Philip Pearle's non-gravitational theory
- Ghirardi, Rimini, Weber approach (GRW)
The GRW Scheme:
- Accepts reality of It/I> and accuracy of standard U procedures
- Wavefunction tends to spread outwards in all directions over time
- New feature: assumes there's a small probability of wavefunction getting multiplied by a Gaussian function (Fig. 6.2)
- Particle becomes very localized
- Proportional to squared modulus of wavefunction at that location
- Time period T between applications is roughly once every 10^8 years per particle.
Quantum Theory and Reality
Schrodinger's Cat Paradox:
- Imagines a large-scale object, such as a cat, being in a quantum superposition of two states: live and dead
- This is not believable as a feature of the physical world
- Resolved by the GRW scheme, where particles in an object would "suffer hits" and localize its state
Resolution by GRW Scheme:
- Large-scale objects involve about 10^25 particles
- Chance of a particle experiencing a reduction is increased compared to a single particle
- An "instantaneous hit" on one particle would cause the entire object's state to collapse to either life or death
- Resolves the "measurement problem" in quantum theory, which is the mystery of how measurements occur
Criticisms of GRW Scheme:
- Ad hoc, with no physical justification for the suggested values of parameters T and
- Violates the principle of conservation of energy
- May require modifications to quantum theory to properly incorporate gravitational effects, rather than modifying general relativity
Quantum Theory and Reality: Understanding the Need for Change in Quantum Theory
General Relativity vs. Quantum Theory
- Both are extraordinarily accurate theories
- Physical insights from Einstein's theory will likely survive, as will those of quantum theory
- Union of these two great theories is yet to be found
Measurement Problem and Quantum Gravity
- Critics argue measurement problem unrelated to Planck scale (10^-33 cm)
- Scale relevant but not in the way first thought
Superposition of Macroscopic States: Schrdinger's Cat Analogy
- Depiction of a situation where two macroscopically distinguishable alternatives are linearly superposed, such as a photon impinging on a half-silvered mirror and activating a device that moves a spherical lump
- Quantum superposition holds until measurement occurs (UR), at which point the lump 'jumps' to one position or another
Implications of Gravitational Fields in Superpositions
- Lumps involve gravitational fields, leading to two different space-time geometries in superposition
- Problem arises when identifying points between these separate spaces for superposition becomes obscure
Significant Differences Between Geometries: Planck Scale as Demarcation Line
- No absolute means of identifying points in one geometry with those in another
- Critics argue that when differences become significantly large (on the scale of 10^-33 cm or more), reduction R occurs instead of maintaining superposition U
- Gravitational effects are small, making this a reasonable demarcation line between quantum and classical levels.
Planck Units and Quantum Theory
Absolute Units
- Concept from Max Planck (1906) and John A. Wheeler (1975)
- Three most fundamental constants: c, h, G
- Choosing units so they all take the value unity
- Length: 10^-33 cm or Planck scale
- Time: corresponding absolute unit = Planck time (~10^-43 seconds)
- Mass: Planck mass (~2x10^-5 grams)
- Useful for understanding effects related to quantum gravity
Conversion of Physical Quantities
- Second = 1.9x10^43 seconds
- Day = 1.6x10^48 seconds
- Year = 5.9x10^50 seconds
- Metre = 6.3x10^-34 cm
- Centimeter = 6.3x10^-32 cm
- Micron = 6.3x10^-28 cm
- Fermi (strong interaction size) = 6.3x10^-19 m
- Mass of nucleon = 7.8x10^-20 g
- Gram = 4.7x10^-3 kg
- Erg = 5.2x10^-17 Joules
- Degree Kelvin = 4x10^-33 K
- Density of water = 1.9x10^-94 kg/m
New Criterion for Gravitationally Induced State Reduction
- Previous criterion in ENM: judge states too different based on gravitational fields (space times) and R would occur at that stage.
- New idea: unstable superposed states, rate of state vector reduction determined by difference measure.
- Greater the difference, faster the rate of reduction.
- Example: energy required to displace one instance of a spherical lump away from another (gravitational separation energy) gives approximation for time before state reduction occurs.
Displacement of Lumps and Reduction Time
- Contribution from displacement negligible if lumps separate essentially
- Reduction time: order of 10^(-18) seconds for ordinary density objects (e.g., water droplet)
- Longer reduction times provide reasonable answers in certain situations
- Neutron or proton: nearly 10^(58) years
- Minute speck of water: hours or less depending on size
- Energy required to effect displacement measures kind of 'half-life' for superposed state
- Larger energy = shorter time, more likely to occur in experiments with carefully controlled conditions.
Absorption of Photon and Environment Disturbance
- Instead of moving lump, consider a body of fluid matter absorbing the photon (Fig. 6.7)
- Reduction time: order of 10^(-13) to 10^(-14) seconds for larger lumps
- Small disturbances in environment could dominate reduction time
- Large numbers of microscopic particles disturbed
- Careful control needed for accurate results, especially considering potential effects on surrounding matter.
Concepts:
- Quantum superposition and reduction
- Energy differences vs total energy
- Entanglement with environment
- Insulation from surroundings
- Limitations of current ideas
- Need for more research and experimentation
Discussion Points:
- Explanation of quantum superpositions and their potential maintenance time based on energy differences.
- Comparison between large, non-moving crystalline objects and water droplets in terms of maintaining quantum superpositions.
- Importance of finding experimental situations to challenge present proposals.
- Potential applications: superconductors and SQUIDs.
- Challenges for applying present ideas to these specific situations: momentum displacement, different formulations of reduction timescale.
- Alternative general schemes for reduction times and their agreement with the original proposal in certain situations but differing results in others.
- Motivations behind proposed physical scheme.
- Need for more research and experimentation to validate or modify ideas.
- Importance of studying superpositions at a large scale, where standard theory predicts they cannot be maintained.
- Potential implications for biological systems and their entanglement with the environment.
Argument for State Reduction as Gravitational Phenomenon:
- Consistency requirement suggests state reduction must ultimately be a gravitational phenomenon of the nature proposed
- Energy conservation issue in GRW-type schemes: particles involved cause small violations, non-local energy transfer (6.9)
- Nebulous and elusive energy issue in general relativity
- Mass-energy carried away as gravitational waves
- Proposed coherence between two energy problems
- Decay process of superposed state like unstable particle or nucleus decay
- Lifetime related to uncertainty in mass-energy (Heisenberg's uncertainty principle)
- Gravitational self-energy involves nebulous non-local field energy and essential uncertainty identifying points in two space-time geometries
- Possible roles for such considerations in brain action
- Expectation of non-computability in gravitationally induced reduction process.
Topics: SETI Programme, Conclusions, Suggestions, Reduction Processes, Schrodinger's Cat, Quantum Theory and Reality
SETI Programme:
- Conclusion of the programme by F. Drake
Suggestions:
- My own suggestions, not very specific until recently (Penrose 1993a, 1994b)
- Shares idea of reduction being a sudden discontinuous process with original Ghirardi-Rimini-Weber proposal
- Current activity focused on continuous (stochastic) state reduction process, like Pearle's original one (Dii 1992, Ghirardi et al. 1990b, Percival 1994)
- Consistency with relativity: Ghirardi et al. (1992), Gisin (1989), Gisin and Percival (1993)
Schrodinger's Cat:
- Quantum theory concept introduced in Schrodinger (1935a)
- Connections with current proposal: Penrose (1991a), Dii (1987), Ghirardi et al. (1990a)
- Supported by general motivations in ENM, Chapter 7 (Penrose 1985)
Quantum Theory and Reality:
- Further research needed to make connections more specific
- Connections with 'one-gravitation criterion' as given in ENM are unclear.
Quantum Theory and the Brain
Conventional Viewpoint:
- Brain function understood in terms of classical physics
- Nerve signals are 'on or off' phenomena, like currents in electronic circuits
- Quantum effects play no significant role at larger scales
Dissenting Opinions:
- John Eccles: Importance of quantum effects in synaptic action
- Some researchers: Neurons may be quantum detection devices
Hope for Mind Influencing Brain:
- Dualist viewpoint: 'Free will' of an 'external mind' influencing quantum choices
- Dualistic viewpoint: Quantum indeterminacy provides opening for mind to influence brain
Challenges on Standard Viewpoint:
- Conventional quantum theory does not allow indeterminancy at quantum levels
- Deterministic U-evolution holds until significant amounts of environment are entangled
- Difficult to maintain that 'mind-stuff' influences system only at indeterminate stage
Proposed Viewpoint:
- Look for level at which R-process becomes operative at larger scales (microns to millimetres in diameter)
The Problem of Objective Reduction (OR)
Background:
- Previous discussion: dualist viewpoint seeks influence point for external 'mind'
- No consensus on crossover point between quantum and classical levels
- Personal opinion: external mind not scientifically explanatory, deviates from scientific argument (viewpoint q)
Viewpoint p:
- Mind deeply connected with physical structures of brains
- Search for profound truths in Nature's roots
Introduction to Objective Reduction Procedure (OR):
- Procedure aligned with suggestions in 6.12
- Emphasis on missing non-computational ingredient
Characteristics of OR:
- Large-scale quantum coherence involved
- Presence of an energy gap that must be breached by the environment to disturb quantum state
- Quantum effects significant at larger scales, e.g., superconductivity and superfluidity
Recent Findings:
- High temperature superconductivity occurs with suitable substances
- Superconductivity at much higher temperatures than previously believed possible (up to 115 K)
Implications for Biological Systems:
- Possibility of quantum coherence effects having relevance to biological systems, including the human brain.
Quantum Theory and the Brain
High-Temperature Superconductivity:
- Observations by Lagues et al. suggest presence of superconductivity at 'Siberian' temperatures (-23C or -10F)
- Provides support for role of quantum coherent effects in biological systems
- Long before this, Frohlich proposed collective quantum effects in active cells due to a large energy of metabolic drive
Quantum Coherence in Biological Systems:
- Evidence for 1011 Hz vibrational effects in many biological systems, as predicted by Frohlich
- Relevance to brain activity is explored further
Neurons and Synapses:
- Classical picture: Neurons and synapses play roles similar to transistors and wires in electronic computers
- Synaptic connections and strengths may change due to 'brain plasticity'
- Connectionist models (artificial neural networks) use computational rules to govern synaptic changes
- Challenged by the need to explain human conscious understanding, which may require a different 'controlling mechanism'
Proposed Alternative Mechanism:
- Gerald Edelman proposed a 'Darwinian' principle operating within the brain, enabling it to continually improve performance by natural selection-like process
Neurotransmitters and Mental Action
- Neurotransmitters play a complicated role in neuron communication
- Current understanding is still treated classically and computationally
- Edelman's DARWIN devices aim to simulate mental action procedures
- Remains a computational scheme, despite differences from other computational approaches
Escaping Computational Limitations
- Need for means of controlling synaptic connections
- Physical process with quantum coherence role
- Immune system and recognition mechanism may involve quantum effects
- Edelman's model doesn't focus on these potential roles
Quantum Action in Nerve Signals
- Difficult to see how one neuron can have a quantum superposition of firing and not firing
- Neuron signals are macroscopic, making it hard to believe in such a picture
- Rapid state reduction occurs upon neuron firing due to electric field propagation
- Maintaining quantum superpositions of neuron firings and non-firings seems implausible
Quantum Computation in the Brain
- Extends Turing machine concept to quantum version
- Applies quantum laws, including superpositions, to operations
- Relevant at end for result ascertainment and during computation termination check
- Cannot outperform conventional Turing computation in principle but has potential for certain classes of problems.
Quantum Computing vs Conventional Computing: Advantages of Quantum Computers
- For certain problems, quantum computers can solve faster than conventional computers (Deutsch 1985)
- Faster solution for classes of problems like factorization of large integers using Shor's algorithm (Shor argument)
- Principle advantages can be achieved by a Turing machine with randomizer as well
- Hope lies in the non-computable action that could replace R, leading to an essentially non-computable scheme
Neurons and Paramecium: Control Systems Beyond Neurons and Synapses
- Paramecium navigates complex environments without neurons or synapses
- Swims, learns from experiences, negotiates obstacles using cilia controlled by the cytoskeleton
- Cytoskeleton provides framework for the cell, acts as a combination of various systems: skeleton, muscle system, blood circulatory system, and nervous system
Role of Microtubules in Cytoskeleton and Biological Control Systems
- Main concern in understanding neurons' personal nervous system
- Consists of protein-like molecules (actin, microtubules, intermediate filaments)
- Microtubules are hollow cylindrical tubes (25 nm outside diameter, 14 nm inside diameter) organized into larger fibers or structures with fan-like cross sections
- Paramecium's cilia are structures of this kind
- Each microtubule is a protein polymer made up of subunits called tubulin
Implications for Neurons and Quantum Computing
- Possibility that individual neurons may have their own personal nervous system due to their cytoskeleton structure
- Scientists exploring this idea, such as Hameroff's work on Ultimate Computing: Biomolecular Consciousness and NanoTechnology.
Microtubule Structure and Function
- Consists of hollow tubes, typically 13 columns of tubulin dimers
- Each tubulin subunit is a "dimer" composed of about 450 amino acids each
- Globular protein pair organized in a skew hexagonal lattice
- Capable of two different conformations
- Electron at junction between a-tubulin and b-tubulin can shift, affecting polarization
- Control center is the centrosome, consisting of two cylinders of nine triplets of microtubules forming a "T" shape
- Centriole forms critical part of microtubule organizing center (centrosome)
- Centriole has important role in mitosis, separating chromosomes and initiating cell division
Centrosome vs. Nucleus
- Two distinct structures in eukaryotic cells
- Nucleus contains genetic material and controls identity and protein production
- Centrosome manages cell movements and organization
- Suggested origin: ancient symbiotic relationship between early prokaryotes and spirochete bacteria, leading to development of cytoskeletons
Mathematical Significance of 13 in Microtubule Organization
- Number 13 is a Fibonacci number (sum of previous two numbers: 0, 1, 1, 2, 3, 5, 8, 13, ...)
- Occurs frequently in biological systems, such as spiral arrangements in plants and palm tree trunks
- Microtubule structure exhibits similar feature with 5 right-handed and 8 left-handed helical arrangements
- Double microtubules often have a total of 21 columns of tubulin dimers (next Fibonacci number)
Figures:
- Fig. 7.4: Diagram of microtubule structure
- Fig. 7.5: Centrosome structure, consisting of two bundles of microtubules forming a "T" shape
- Fig. 7.6: Mitosis process during cell division
- Fig. 7.7 and Fig. 7.8: Fibonacci numbers in biological systems (Fibonacci spiral)
- Fig. 7.9: Indication of microtubule structure as viewed from within.
Microtubules and Fibonacci Numbers in Plants and Neurons
Fibonacci Numbers in Microtubule Structure:
- Fibonacci numbers appear in microtubule structure, as seen in fibonacci spiral patterns in sunflower heads and pine cones
- Theories for why Fibonacci numbers arise in microtubules may not be applicable to the level of microtubules
- Koruga suggested that these numbers provide advantages for the microtubule as an "information processor"
- Hameroff and colleagues have argued that microtubules play roles as cellular automata, transmitting and processing signals through electric polarization states of tubulins
Microtubules in Neurons:
- Microtubules in neurons are much longer than their diameter (25-30 nm) and can reach lengths of millimeters or more
- They can grow or shrink, transport neurotransmitter molecules, and maintain synapse strengths
- Microtubules organize the growth of new nerve endings, guiding them towards connections with other neurons
- Neurons do not divide after maturity, so there is no role for a centriole in a neuron's centrosome
- Microtubules extend from the peri-karyon (near the nucleus) to the presynaptic endings of axons and into dendrites and dendritic spines
- The clathrin molecules associated with microtubules in presynaptic endings form beautiful geometric structures similar to fullerenes
Neurobiology and Quantum Physics: Microtubules and Synaptic Connections
Synaptic Connection Strengths:
- Clathrins in synaptic boutons control strength (Figs. 7.11, 7.12)
- Cytoskeleton plays a central role
- Neurons alone may have limited computing power
Microtubules and Computational Units:
- Tubulin dimers as basic units could vastly increase potential computing power in the brain (up to 10^24 operations per second)
- No prospect of achieving such figures with present-day computers
Microtubular Computing:
- Arguments against imminent human-level artificial intelligence based on microtubules and quantum coherence
- Nematode worm capabilities may outstrip standard AI procedures
- Ants have enormous array of 'microtubular information processors'
- Part I contends that human understanding lies beyond any computational scheme, necessitating non-computational action
Quantum Coherence within Microtubules:
- Frohlich's ideas for quantum coherence in biological systems (1975)
- Large-scale quantum coherence possible with high metabolic energy and extreme dielectric properties
- Direct evidence of 10^11 Hz oscillations within cells
- Bose-Einstein condensate: large numbers of particles in a single quantum state, coherence on a large scale
- Possibility that microtubules could exhibit such behavior
Evidence for Quantum Coherence in Microtubules:
- Hameroff's proposal (1974) that microtubules might act as 'dielectric waveguides'
- Some evidence suggesting this may be the case.
Cytoskeleton and Consciousness: Microtubules as a Potential Site for Quantum Phenomena
Microtubules as potential insulators:
- Nature's choice of hollow tubes in cytoskeletal structures may serve to provide effective insulation
- Tubes themselves could maintain quantum state unentangled with environment
Quantum self-focusing effect in cells:
- Emilio del Giudice and colleagues at the University of Milan (1983) suggest that cytoplasmic material confines signals to microtubule diameter
- Effect could also contribute to microtubule formation
Organized water in cells:
- Controversial matter, but some water exists in ordered state (vicinal water)
- Ordered water may extend beyond cytoskeletal surfaces and exist inside microtubules
- Favors possibility of quantum coherent oscillations within or relating to tubes
Challenges in explaining cytoskeleton behavior classically:
- Complexity of the system makes it difficult to explain with classical terms alone
- Quantum mechanics concepts are continually being appealed to in research
Evidence for microtubule role in consciousness:
- General anesthetics turn off consciousness reversibly without clear chemical relationships between substances
- Other interactions like van der Waals forces may play a role, but not fully understood.
Tubulin Dimers and Conformations
- Capable of two different conformations due to an electron's position in water-free region
- Shape and electric dipole moment affected by positioning
- Van der Waals force from neighboring substances influences switching between conformations
General Anesthetics and Tubulin Dimers
- Interference with normal switching actions of tubulin due to van der Waals interactions in hydrophobic regions
- Electric dipole properties can interrupt microtubule functions
- Plausible mechanism for general anesthetic action on brain proteins
- Relevant proteins may be neuronal microtubules' tubulin dimers
Effect of General Anesthetics on One-Celled Animals
- Immediate immobilization at similar concentrations in various organisms, including paramecium and amoeba
- Effects suggest involvement with cytoskeleton
Consciousness and Cytoskeleton
- Requirement of properly functioning cytoskeleton for consciousness to be present
- Neuronal organization important for specific form of consciousness
- Liver cells do not evoke same level of consciousness as brains due to lack of detailed neural organization
- Essential physical action in microtubules needed for consciousness arising generally
Quantum Theory and the Brain
- Cytoskeleton plays a role in quantum/classical interface through synaptic connections control
- Stop-gap procedure R may not be sufficient, new physical theory OR required.
Quantum Theory and the Brain
Simulation of Brain Behavior:
- Present-day technology cannot simulate the brain's behavior with current understanding
- Simulation would require fixed synaptic connections, which are not sufficient to model conscious behavior
Possibility of Extended Model:
- If synaptic connections are subject to continual change, governed by non-computational action, an extended model could simulate the behavior of a conscious brain
Non-Computational Action:
- Not defined in detail, but requires collective phenomenon involving large numbers of cytoskeletons and microtubules
- Quantum coherence must extend across entire microtubule(s) and multiple neurons
Challenges of Implementing Non-Computational Behavior:
- Requires harnessing unknown physics that straddles quantum and classical levels
- Human physicists are not yet aware of this missing theory, but Nature has historically incorporated new physical principles into biological systems
Natural Selection and Survival Advantage:
- Organisms that benefit from subtle physical processes may survive better and have their descendants preserve these traits through natural selection.
Possible Influence of Quantum Mechanics in Brain Functioning
Sophisticated Action from Ancient Cell-Creatures
- Descendants of cell-creatures today: parameciums, amoebas, ants, trees, frogs, buttercups, and human beings
- Ancestors' sophisticated action conferred benefits, twisted for various purposes
- Realized tremendous potential in highly developed nervous systems = 'mind'
Microtubules in Brains and Global Quantum Coherence
- Totality of microtubules in large family of neurons may participate in global quantum coherence or have sufficient quantum entanglement between states
- Complicated quantum oscillations within microtubules possible, providing isolation for internal coherence
- Cellular-automata computations along tubes might be coupled to presumed quantum oscillations
Frohlich's Mechanism and Microtubules
- Frohlich's collective quantum oscillations in the same frequency region as Hameroff's microtubular cellular automata switching time
- Potential coupling between two types of activity, but care needed to maintain quantum nature of internal oscillations
Uncertainty and Classical Picture
- Uncertainty regarding actual physical situation and OR criterion application
- Assuming classical picture for microtube computations
- Delicate coupling between quantum interior oscillations and exterior classical computations
Global Quantum State in Brain Functioning
- Coherently coupling activities within tubes across large brain areas
- Influence of quantum state on computations along the microtubules, possibly taking effect through weak van der Waals' type influences between neighboring tubulin dimers.
Non-Computational Physics and Quantum Theory: A Possible Role for Consciousness
Overview:
- Discussion about the role of non-computational physics in understanding consciousness
- Suggestions that quantum entanglement and counterfactuals play a part
- Proposed model: internal cytoskeletal state influences neural computer through OR (quantum-entangled internal state)
- Deep level of cytoskeletal action is where physical basis of mind may be sought
Non-locality and Entanglement:
- Role in understanding consciousness
- Possible connection to neural computer's 'wiring' and free will
- Speculation about quantum entangled internal state influencing classical neuron activity
Non-Computability in Quantum Gravity:
- A requirement for new physics replacing R-procedure in quantum theory
- Geroch and Hartle's findings: topological equivalence problem unsolvable for 4-manifolds
- Implications: no algorithm for solving the halting problem or deciding topological equivalence of 4-dimensional spaces exists.
Further Discussion:
- Non-computability as an essential feature in quantum gravity theories
- Possible analogy with mental activities and free will.
Computational Unsolvability
- Concept introduced by Hilbert and Turing
- Asserts no systematic means to solve all problems in a class
- Does not imply individual problems are unsolvable in principle
- Problems may be accessible with human ingenuity and insight, aided by computation
- Unsolvability argument from Godel and Turing (2.5) shows accessibility of unsolvable problems is also inaccessible
- Examples: Hilbert's tenth problem, tilting problem, word problem for semigroups
Geroch-Hartle Approach to Quantum Gravity
- Equivalence problem for 4-manifolds enters their analysis
- Space-time geometries involve superpositions with complex weighting factors
- Decision on uniquely specifying these space-times requires solving topological equivalence problem
- Unclear if computational unsolvability of topological equivalence implies complete geometrical equivalence problem unsolvable
- Possibility of non-computability in emergent quantum gravity theory, but relationship to OR ideas unclear
Oracle Machines and Physical Laws
- Oracle: fictional thing capable of solving halting problem
- No proof oracle cannot be physically constructed without computational physical laws
- Turing considered an oracle machine with oracle call at any stage in calculation
- Deterministic in nature, different from computability
- Universe could run as an oracle machine universe instead of Turing machine universe.
Oracle Machines and Non-Computability
Argument for Oracle Machines:
- Oracle machines can be listed just as well as Turing machines, with the addition of noting when to bring in the oracle
- Replaces algorithm A(q,n) from 2.5 with an oracle algorithm A'(q,n)
- Conclusion: Human mathematicians are not using a knowably sound oracle algorithm to ascertain mathematical truth
- This process can be repeated for "second order" and beyond, leading to the conclusion that a physics working like an oracle machine won't solve problems
Limitations of Arguments:
- Some argue that human mathematical understanding may not be as powerful as any oracle machine
- Physical laws could lead to something different than every computable level of oracle machine
Quantum Gravity and Oracle Machines:
- No accepted theory in quantum gravity, but a proposal by Deutsch involves superposing "reasonable" and "unreasonable" space-times with closed timelike lines
- The possibility of time travel contradictions in such space-times leads to the argument that they should not be taken as models of the classical universe
- However, it may be argued that these space-times could still be involved in a quantum superposition
Discussion on Consciousness and Time Perception
Background:
- Discussing consciousness in relation to mathematical truth perception led us to explore unusual ideas.
- Familiar with the passage of time as a feature of conscious perception.
- Shocking that current physics theories don't fully explain time flow.
- General relativity considers 'time' as a coordinate, not something that flows.
Two Dimensional Space-Time:
- In a two-dimensional space-time, there is no distinction between space and time.
- No reason to view one dimension as time and the other as space.
Time Evolutions in Physics:
- Computations are often carried out using time evolutions for modeling experiences and predicting future.
- Not a necessary procedure; calculations done due to interest in understanding our world through the lens of time flow.
Identification Between Conscious Awareness and Time Coordinate:
- Unwise to make strong identifications between conscious awareness's "flowing" time and physicists' use of a real number parameter 't'.
- Relativity tells us there is no uniqueness in choosing the parameter t for the entire space-time.
Experiments on Consciousness and Time
Background:
- Many mutually incompatible alternatives exist for understanding time's relationship with conscious experience
- Precise concept of a 'real number' is not relevant to our perception of time passage due to sensibility limitations
- Physicists' time scales hold good at small scales, but mathematicians' concept requires no limit of smallness
Experiments on Conscious Acts:
- Electroencephalograms (EEGs): Kornhuber et al. recorded brain activity during voluntary finger movements. Results showed a build-up of electrical potential for up to 1 second before movement, questioning the nature of conscious will.
- Interpretations:
- Conscious act is an illusion or preprogrammed (i)
- Last-minute role for will (ii)
- Subject mistakes conscious decision time (iii)
Experiments on Sensory Awareness:
- Libet's experiments: Subjects reported a half-second delay between sensory awareness and actual stimulation, suggesting a sluggish response time for consciousness.
- Interpretations:
- Our conscious responses may be slower than perceived (possibly taking 1.5 seconds)
Unconscious Responses vs. Conscious Ones in Rapid Activities
- In ordinary conversation, most responses are unconscious (not contradictory to accepting that some conscious responses can occur)
- If unconscious response is too quick for consciousness to override, then there's no chance for consciousness to affect it after the fact
- Preprogrammed unconscious reactions can occur much more quickly than 0.5 seconds
- Role of consciousness in rapid activities like conversation or sports: spectator aware of an "action replay" with no active role (disbelief in this)
- Possible loophole: Consciousness may only come into play for slow, contemplative mental activity; unconscious response times are rapid and automatic
Challenges with Assuming Clear-Cut 'Time' of Conscious Events
- Uncertainty about a precise 'time' for conscious experiences due to anomalous relation between consciousness and physical time
- Mild possibility: Non-local spread in time, inherent fuzziness in relationship between consciousness and physical time
- Subtle puzzle: Quantum effects interfere with classical reasoning about temporal ordering of events (EPR paradox)
- Strong indication that quantum actions are at work when classical reasoning leads to contradictory conclusions.
Physics Notions of Time and Quantum Non-Locality
- Suspicion towards physical notions of time, especially when quantum non-locality and counterfactuality are involved
- Realistic view of the state vector in EPR situations presents a profound puzzle (difficulties for any detailed theory)
- Magic dodecahedra example: Which event defines reality - my colleague's button pressing or mine?
Section 1: Equivalence of Different Approaches to Quantum Physics
- Both ways of treating the problem in quantum physics lead to same results, as discussed in 6.5
- This is a consequence of the principle of relativity, which states that simultaneous events cannot have observable effects for spacelike separated objects (events happening at different locations in space)
- If we believe that lt/t) represents reality, then it results in different realities depending on the approach taken
Section 2: Perspective on Reality and Physical Descriptions
- Some believe in quantum realism but may be willing to jettison Einsteinian picture of the world due to strong reasons in favor of a realistic view
- Others prefer to hold onto both, which would require fundamental change in physical representation
- Example given is the transformation from Newton's theory of gravity to general relativity
Section 3: Idea of Two State Vectors
- Proposed idea by Yakir Aharonov, Lev Vaidman, Costa de Beauregard and Paul Werbos (1990)
- Quantum reality is described by two state vectors - one propagates forward in time and the other propagates backward
- Advantage: equivalent implications as standard quantum theory but pair of state vectors provide a more comprehensive description
Section 4: Twistor Theory
- Alternative physical picture to space-time theory, where entire light rays are represented as points, and events by Riemann spheres
- Enables objective description of state in EPR situations consistent with Einstein's relativity
- Provides non-local description of space-time, which relates it to quantum non-locality of EPR situations
- Based on complex numbers and their related geometry, providing an intimate relationship between U-quantum theory and space-time structure.
Quantum Theory and Brain
Alternative Perspectives on Physical World:
- Possibilities for changing current picture of physical world into something different
- Must reproduce successful results of quantum theory and general relativity
- Belief that modification to quantum theory's state reduction process is needed
- Concerns about "realistic" state reduction theories like GRW theory, which have encountered severe problems
Role in Understanding Consciousness:
- Genuine progress in physical understanding of consciousness requires fundamental change in worldview
- Idea that this change may be related to OR lines of ideas described in 6.12
Related Topics:
- Clathrins and fullerenes: important in biology, chemistry (see references)
- Frohlich's frequency of about 5 x 10^10 Hz relevant to Hameroff's tubulin dimers switching time
- Idea that quantum gravity frameworks may not be consistent at present time
- Alternative approach to non-computability in "quantum computing" (Castagnoli et al., 1992)
Misconceptions:
- Misleading depictions of author's opinions on relation of consciousness and the flow of time, as seen in "A Brief History of Time" film.
Further Information on Twistors:
- Penrose and Rindler (1986), Ward and Wells (1990), Bailey and Baston (1990) for more information.
Implications of Artificial Intelligence
Ultimate Potential:
- The technology of electronic computer-controlled robots will not provide a way to build an actually intelligent machine
- Computers are undoubtedly important for clarifying issues related to mental phenomena, scientific research, and technological progress
- Computers do something very different from what we do when we bring awareness to bear on problems
Genuinely Intelligent Device:
- It is not necessarily impossible to build a genuinely intelligent device, as long as it is not computationally controlled
- Such a device would have to incorporate the same kind of physical action that is responsible for evoking human awareness
- The construction of such a device is currently premature due to a lack of understanding of the physical theory underlying this action
Quantum Effects and Future Developments:
- Quantum effects may find surprising applications in fields like cryptography, leading to new technologies
- Theoretical proposals for quantum computing could enable the construction of secure information channels that cannot be eavesdropped on by third parties
- Experimental devices based on these ideas are already being developed and may have commercial applications within a few years
- Other schemes using quantum effects in cryptography are also being explored, and the subject is rapidly developing
- Theoretical constructs for non-computational quantum computing are very far from practical realization
- Building a device whose action depends on an unknown physical theory could be even more challenging
Observing Non-Computable Behavior:
- Observing non-computable behavior in an object would require having the appropriate physical theory, which is currently lacking
- Without the theory, it would be unlikely to observe or exhibit non-computational behavior in a physically constructed object
Advancements in Computing and Intelligence
- Current computers lack intelligence or awareness (1.2, 8.1)
- Remarkable power and potential for future growth (1.2, 1.10, Moravec 1988)
- Computers can perform complex calculations quickly and accurately (chess example)
- Human understanding and judgment are not replicable by computers
Differences between Computers and Humans
- Computer: performs calculations without understanding, stores knowledge, makes repetitive applications of programs' understandings in a mindless way.
- Human: keeps reapplying judgments and forming meaningful plans, has overall understanding of the game or task.
Advantages of Humans over Computers Based on Exponential Formula T=t x pm
- For large T (time) and effective p (number of alternatives), humans can make a significant difference by reducing the number of alternatives considered through judgment and understanding. This leads to a larger m (depth of calculation).
- For small T, making time (t) very small is more effective for computers due to their computational power.
Implications for Future Advancements in Computing and Intelligence
- Games or tasks with large p but effectively reducible by human understanding and judgment are at an advantage for humans.
- As computers improve their ability to understand and learn, they may become more competitive with humans in complex tasks. However, humans' unique abilities, such as creativity and emotion, may still be difficult to replicate by computers.
Computer vs Human Understanding
Essential Point:
- Human player's judgement encapsulates probable effect of continued moves, effectively calculating to a greater depth than a computer
- Computers lack human understanding, which is an essential thing that computers cannot replicate
Clarification on Calculation Depth:
- Human player judges value at a few moves deep and considers it not helpful to calculate further, this is an effective calculation to a much greater depth
- Computers focus on short, simple tasks where understanding is not required
- Computers have relative advantage with short time limits for play
Understanding in Daily Life vs. Complex Tasks:
- Bottom-up programming used in daily life tasks like face recognition, prospecting for minerals, etc. approaches or exceeds human expert performance
- Top-down systems like chess computers and numerical calculation require understanding provided by human programmers to be effective
Computer Errors:
- Computer errors occur due to human programmer mistakes, not the computer's own understanding
- Automatic error correcting systems can be introduced but cannot catch subtle errors
Dangers of Relying on Computer Systems:
- Computer system may perform reasonably well for a long time before suddenly doing something unexpected, revealing lack of understanding
- Always be aware that a computer-controlled system does not possess actual human understanding
Human Understanding vs. Memory/Calculation Powers:
- Humans differ in their possession of understanding versus memory and calculational powers
- Understanding is more valuable than just parroting rules or information, as it tests the candidate's understanding in examinations
Limitations of Computational Systems: Understanding vs Feeling
Qualities lacking in computational systems:
- Feelings: unable to appreciate beauty, splendor, or complexity
- Aesthetic criteria and judgement
- Inability to possess sensual qualities necessary for good judgment
- Criteria would have to be programmed by humans based on analysis
- Soul in artistic works: expression of feelings not present in computational systems
Limitations of Computational Systems: Absolute Qualities (Beauty, Goodness)
- Platonic view: absolute qualities exist and consciousness can access them
- Consciousness strength: possibly related to ability to perceive these absolutes.
Computers and Awareness: Bridge to Absolutes?
Role of Awareness in Understanding Reality:
- Some argue awareness could bridge to Platonic absolutes (final section)
- Question of absolute nature of morality and free will touched upon
Challenges with Computer Technology:
- Dangers and benefits: rapid advancements present both opportunities and risks
- Complexity: interconnected systems beyond human comprehension
- Global communication adds to complexity
- Instabilities and potential dangers
- Stock market behavior example of instability and unfairness
- Other unknown instabilities may arise due to system's complication
- Understanding limitation: computers cannot comprehend or acquire understanding themselves
- Rapid pace of change: continual updating required, pressure for testing
- Privacy concerns: personal information vulnerability
- Industrial espionage and computer sabotage issues
- Social issues: expert systems may replace local professionals' expertise
- Expertise accessibility: larger public can appreciate expertise more freely
- Quantum cryptography as potential solution to privacy concerns
- Future computational advancements, such as quantum computing, could impact benefits and dangers.
The Puzzling Election
Approach of Long-Awaited Election:
- Numerous opinion polls held over a period of several weeks
- Ruling party trails by 3-4 percentage points consistently in the polls
- Fluctuations and deviations from this figure due to small sample sizes and margin of error
Total Evidence More Impressive:
- Polls taken together have smaller margin of error
- Agreement between them seems to have "the right kind of slight variation" statistically
- Averaged results can be trusted to an error of less than 2%
Swing Towards Ruling Party on Election Day:
- Some might argue there is a swing towards the ruling party noticeable in poll figures on election day
- Small proportion of undecided or committed voters may change their votes to the ruling party
Ruling Party's Majority Achieved Through Vote Rigging:
- Swing away from poll figures towards ruling party would not be enough to achieve an overall majority
- Opinion polls are "merely guesses of a sort" and only the true vote will express the actual voice of the people
Surprise Result on Election Day:
- Ruling party achieves a comfortable majority, well above their nearest rivals
- Some voters are stunned or horrified by the result
Subtle Means of Vote Rigging:
- No stuffed ballot boxes or lost/substituted/duplicated votes
- People counting the votes did their work conscientiously and accurately
- Result is "horribly wrong" despite no illegal activity detected
Responsibility for Vote Rigging:
- Entire cabinet of ruling party may be ignorant of what happened
- Beneficiaries were those who feared consequences of a defeat by opponents' coalition
- Organization behind the scenes with strict secrecy and illegal political chicanery
- Members of organization are experts in constructing computer viruses
- Virus struck on election day, effectively altering vote count without detection or record
Susan's Story: Computer Fraud and Consciousness
Computer Viruses and Election Fraud
- Susan's story inspired by British election concerns
- Official system in Britain does not allow for this kind of fraud, as counting stages are done manually
Importance of Human Oversight
- Computers can be manipulated by those who understand their programming
- Devastating consequences if used for fraudulent purposes like vote rigging or sabotaging company accounts
Searching for the Physical Basis of Consciousness
- Part II explores scientific explanation to find a physical home for subjective experience
- Current understanding needs extension due to foreign nature of consciousness
- Present-day computers not conscious, but future changes in physics may accommodate consciousness
Location of Consciousness in the Known World
- Human beings exhibit consciousness, present in wakeful and dreaming human brains
- Biological structures like cytoskeletons are potential locations for consciousness due to collective quantum effects and non-computational prerequisites
- Ubiquitous presence of cytoskeletons in eukaryotic cells, including single-celled animals, raises questions about consciousness in these organisms.
Conclusion: The arguments presented have little to say on the positive side regarding where consciousness might be found in the known world. They suggest that it may be present in some capacity in biological structures like cytoskeletons due to their collective quantum effects and non-computational properties, but further investigation is required to confirm this hypothesis. Ultimately, a profound change in our philosophical viewpoint on reality will be necessary for consciousness to find its place within scientific terms.
Discussion on Consciousness in Organisms
Background:
- Some question whether non-human entities possess conscious awareness
- Philosophical arguments suggest uncertainty due to absolute certainty requirements
- Comparison with understanding celestial bodies and moon's surface
Evidence for Consciousness in Non-Human Entities:
- Elephants: Display behaviors indicative of feelings and religious belief
- Squirrels: Demonstrate problem-solving abilities, requiring understanding and imagination
- Ants: Exhibit complex behavior patterns suggesting awareness
- Insects: Controversial examples; some display intricate behavior but may not exhibit consciousness in the same way as mammals
Possibility of Degrees of Consciousness:
- Consciousness can vary in degree
- Personal experience and different times feeling more or less conscious
Lower Animals and Microtubules:
- Difficulties believing insects have much consciousness
- Complex behavior patterns in ants
- Possibility of quantum-coherent states in non-human entities
- Evolutionary benefits for early eukaryotic organisms
- Value to early eukaryotes may be different from humans and multicellular cousins
Large-Scale Quantum Coherence and Consciousness
Coherence vs. Superconductors:
- Large-scale quantum coherence does not imply consciousness
- However, it could be part of what is needed for consciousness
Proposed Scale for Conscious Awareness:
- In the brain, there is enormous organization
- Consciousness appears to be a very global feature
- Requires significant quantum entanglement between states in cytoskeletons of large numbers of neurons
- Entails non-computable actions that cannot be described by classical physics
Proposed Quantum/Classical Boundary:
- Suggests the quantum/classical boundary should occur at the internal/external interface of the microtubules in a cell or system of cells
- OR (state vector reduction) occurs effectively as a random process for most systems, but can become non-computable and relevant for consciousness when it happens just before the non-computational details of an OR theory come into play
Implications for Single Cells vs. Collections of Cells:
- In a single cell (e.g., paramecium or human liver), there may not be enough mass movement in tubulin conformational activity for OR to occur before the environment becomes entangled, losing the non-computable aspects
- However, with large collections of cells (e.g., brain), the situation appears more promising
Cautious Approach:
- The physical and biological sides of the proposed picture are still crudely formulated
- Further research is needed on both sides before clear conclusions can be drawn as to where consciousness could enter
Other Issues to Consider:
- Not all the brain's action is conscious, e.g., the cerebellum governs unconscious control of actions
Three Worlds and Three Mysteries
Central Issue: Relating Consciousness to Scientific Worldview
- Focus: Understanding of Conscious Processes vs Computational Simulation
Part I: Conscious Understanding
- Claim: Mathematical Understanding Cannot be Simulated by Computation
- Implication: Physical Activity Beyond Computation Required for Consciousness
Three Worlds:
- World of Perceptions: Conscious Experiences (e.g., happiness, love)
- Physical World: Objective Reality (chairs, neurons, atoms)
- Mathematical World: Abstract Concepts (natural numbers, geometry)
Mysteries:
- How the World of Perceptions relates to Physical World?
- Why do Conscious Experiences have anything to do with Physical World?
- What role does Mathematical World play in understanding Consciousness?
The Platonic World: A Mathematical Perspective
Existence of the Platonic World:
- Contains various abstract concepts including numbers, geometries, and equations
- Its existence rests on the profound nature and independence of these concepts
Natural Numbers:
- Pre-date human existence
- Sum of four squares for every natural number
- Beyond reach of computers but still true
Mathematical Truths:
- Independent of discoverers
- Facts exist regardless of computation procedures
Platonic World vs. Physical and Mental Worlds:
- Three worlds, cyclically related through mysteries
Mysteries:
- Why precise mathematical laws play a role in physical world: Arrow from Platonic to physical world
- How perceiving beings arise from physical world: Arrow from physical to mental world
- How mentality creates mathematical concepts: Arrow from mental to Platonic world
The Significance of the Phraseology:
- Illustrates relationship between three worlds
- Useful for describing interconnectedness of mathematical, physical, and mental realms.
Mathematics in Physical Reality A Hidden Unity
Plato and Mathematical Forms
- Plato described ideas using a parable: citizens chained in cave, seeing only imperfect shadows of perfect shapes
- Perfect shapes represented mathematical forms; shadows, physical reality
Role of Mathematics in Physical World
- Eugene Wigner's lecture on "The unreasonable effectiveness of mathematics in the physical sciences"
- Increased importance of math in understanding physical world structure and behavior
- Einstein's general relativity as an example of deep mathematical unity with reality
- Physicists not just noticing patterns, but uncovering underlying mathematical structures
Mathematical Stimulation from Physical Processes
- Mathematics gains inspiration from detailed behavior of Nature (quantum theory, general relativity, Maxwell's electromagnetic equations)
- Ancient theories like Newtonian mechanics and Greek analysis of space structure also stimulated mathematics development
- Extraordinary accuracy and mathematical sophistication in successful physical theories
Mathematical Fruitfulness for Understanding Physical World
- Profound insights into mathematical problems obtained from studying physical processes
- Simon Donaldson's use of Yang-Mills-type theories to discover properties of four-dimensional manifolds
- Unchangeable truths waiting to be discovered within Platonic mathematical world
Closing Thoughts
- Close relationship between Platonic mathematical world and the physical world, still deeply mysterious
- Hope that readers take the Platonic mathematical world more seriously as a 'world'.
Discussion on Platonic Reality and Abstract Concepts:
- Some may argue for a reality assigned to other abstract concepts beyond mathematics, such as 'good' or 'beautiful' (Plato's beliefs)
- Personal inclination is not against this possibility but it played no significant part in the author's deliberations
- Issues of ethics, morality, aesthetics have been less focused on in the book
- Arrows in Fig. 8.1 represent correspondences between worlds rather than their direction being the essential point
- Each world seems to 'emerge' from a tiny part of its predecessor, which can be seen as paradoxical but not prejudging primary or secondary statuses
- Godel's theorem does not imply inaccessible mathematical truths beyond human understanding; it only refers to truths accessible through accepted formal systems.
The Platonic Perspective vs. Formalism in Mathematics
- The Platonic perspective argues for the existence of an absolute, mathematical world beyond formal argument and computable procedures.
- Godel's theorem illustrates the mysterious nature of mathematical perceptions, as they cannot be solely attributed to "calculating".
- Godel's initial motivations were influenced by support for the Platonic viewpoint over formalism.
Matter, Space, and Time: The Mysterious Nature of Physical World
- Consciousness arises from seemingly unpromising ingredients like matter, space, and time.
- Matter itself is mysterious, as are the laws that govern it.
- Mathematical concepts emerge as the best theories to understand matter and physics.
Implications of Quantum Mechanics for Understanding Matter and Reality
- Quantum mechanics challenges our notions of matter and actuality by showing that counterfactual possibilities can influence reality.
- The mystery in quantum mechanics may be closer to accommodating mentality within physical reality.
Consciousness: Beyond Computational Physics
- Consciousness involves awareness of various qualitative experiences, which cannot be understood solely in terms of coherent mass movement.
- Neuroscience research shows that understanding consciousness requires going beyond the limitations of a computational physics or random model.
- The possibility of intentionality and subjective experience can only be explored by breaking free from these strict frameworks.
The Role of Science in Understanding Consciousness
- Science has a long way to develop yet to fully understand consciousness.
- Indications suggest that new physical developments may clarify the relationship between mentality and physics.
- Great minds throughout history, like Newton, Einstein, Archimedes, Galileo, Maxwell, Dirac, Darwin, Leonardo da Vinci, Rembrandt, Picasso, Bach, Mozart, or Plato, possess a unique faculty to "smell out" truth or beauty that others may not have.
Relating Worlds and Unity with Nature
The Conscious Brain:
- Our conscious brains are woven from subtle physical ingredients
- This enables us to take advantage of the profound organization of our mathematically underpinned universe
- We have direct access, through understanding, to the ways our universe behaves at many levels
Three Worlds and Mysteries:
- No clear answers will come unless interrelating features are seen
- Three worlds: physical world, mental/spiritual world, and Platonic world of abstract mathematical truths
- Possible absence of centrioles in neurons; they may defer to a more global authority
Relationship with Popper's World 3:
- Popper's World 3 contains mental constructs similar to those in the extended Platonic world
- However, Popper's World 3 is not viewed as having timeless existence or underlying physical reality