CHAPTER 5: Mathematical and Physical Reality


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…the question has regained interest among physicists, mathematicians, and philosophers, not to mention science fiction writers.

E.g., John D. Barrow Theories of Everything: the quest for ultimate explanation Fawcett/Balantine 1991, p246: “Reality is not intrinsically mathematical. Rather, it is only those aspects of reality amenable to mathematical description that we are good at elucidating.” Cf. also Nancy Cartwright How the Laws of Nature Lie Oxford UP 1983, p19, p199-200: “We are misled if we take mathematics too seriously. Not every significant mathematical distinction marks a physical distinction in the things represented.” Finally, cf. David Deutsch [“It from Qubit” Sept 2002 online archive p13]: “The misconception is that the set of computable functions (or the set of quantum-computational tasks) has some a priori privileged status within mathematics. But it does not… It is only through our knowledge of physics that we know of the distinction between computable and non-computable… or between simple and complex.” Barrow is an astronomer, Cartwright a philosopher, and Deutsch a mathematician.


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…it abstracts the world’s most general properties in the first place.

Effectively, this was also Aristotle’s view. Cf. Deason, in Lindberg & Numbers God and Nature, p167-8: “Aristotle rejected Plato’s mathematicism, believing that mathematics and physics study separate kinds of objects. For Aristotle, the quantities and shapes of mathematics were abstractions from physical entities. They captured certain qualities of material things but left unexplained the true natures, which could not be reduced to mathematics.”


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Idealizations are chosen that do correspond closely enough to reality…

Consider, for example the famous engineering joke that begins: “Given a spherical cow…” See also Wigner “The Unreasonable Effectiveness of Mathematics in the Natural Sciences, 1960: “…the construction of machines, the functioning of which he can foresee, constitutes the most spectacular accomplishment of the physicist. In these machines, the physicist creates a situation in which all the relevant coordinates are known so that the behavior of the machine can be predicted.” Also R.W. Hamming “The Unreasonable Effectiveness of Mathematics” The American Mathematical Monthly Volume 87 Number 2 February 1980: “…we approach the situations with an intellectual apparatus so that we can only find what we do in many cases… We select the kind of mathematics to use. Mathematics does not always work… Science in fact answers comparatively few problems.” Finally, see Frank Wilczek “Reasonably Effective: 1. Deconstructing a Miracle” Physics Today November 2006: “Part of the explanation for the success of mathematics in natural science is that we select what we regard as the scientifically interesting aspects of nature largely for their ability to allow mathematical treatment.”


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…computation is simply the latest metaphor by which to understand the appearances we call nature.

On the other hand, physical laws, expressed as equations, are computable functions by definition. “It is only through our knowledge of the physical world that we know of the difference between computable and not computable. So it’s only through our laws of physics that the nature of computation can be understood.” [Deutsch “Physics, Philosophy and Quantum Technology”, p4]


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…Einstein… saw no necessary connection between experience and reality…

“How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” [From ‘Geometry and Experience’, an expanded form of an Address by Albert Einstein to the Prussian Academy of Sciences in Berlin (27 Jan 1921). In: Albert Einstein Sidelights on Relativity, translated by G. B. Jeffery and W. Perrett (1923).] However, thought and its products are not independent of experience.


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An evolutionary general theory…one development of a broader capacity to model, abstract, and generalize.

Eric Baum has proposed such a theory, a key point of which is the mind’s ability to generalize problem solving to accommodate many variations. Generalization is exactly what mathematics is about. According to this theory, effective mental procedures arise by first discovering a compact program that works on a large sample of real-world situations. Such a core program is enshrined in the genome, and guides the development of an expanded version in the brain. This, in turn, facilitates the construction of appropriate problem solving techniques—such as found in mathematics. [Eric Baum “A Working Hypothesis for General Intelligence” Proceeding of the 2007 conference on Advances in Artificial General Intelligence. See also: Baum What is Thought? MIT Press, 2004.]


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…aspects of physical reality…require future developments in mathematics.

An example might be the problem of modeling protein folding. The mathematical problem seems intractable, but obviously the physical problem is not, for nature does it—we don’t yet know how. [Thomas Homer-Dixon The Ingenuity Gap Alfred A. Knopf, 2000 p118]


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…the universe is “really” nothing other than a mathematical structure…

Plato rejected the immanent reality of the natural world while Aristotle embraced it. From a Platonist viewpoint, only number, space, and geometrical relationship—i.e. strictly mathematical properties—are real. Plato took up the Pythagorean program to reduce the four elements to geometric solids. Hence, his basic ontology consisted of geometry, not matter. Descartes similarly tried to reduce physics to geometry.

In recent times, Max Tegmark explores the idea of physical reality as mathematical structure [“Is the ‘theory of everything’ merely the ultimate ensemble theory?” Annals of Physics, 270, 1-51 (1998)]: “In other words, some subset of all mathematical structures… is endowed with an elusive quality that we call physical existence, or PE for brevity… Since there are three disjoint possibilities (none, some or all mathematical structures have PE), we obtain the following classification scheme: 1. The physical world is completely mathematical. (a) Everything that exists mathematically exists physically. (b) Some things that exist mathematically exist physically, others do not. (c) Nothing that exists mathematically exists physically. 2. The physical world is not completely mathematical.” (Just for completeness, I would add a third possibility: that the physical world is not mathematical at all!) Tegmark assumes the Platonic idealism he hopes to establish, beginning with the notion that physical existence is a property or quality that may or may not be attributed to some more primary (mathematical) form of being.


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…early Christian dogma, which expected a fit between reason and the world as a creation of the divine Mind.

Renaissance mathematics encompassed both a quantitative approach to nature and sheer number mysticism. The latter included numerological interpretation of scripture and supposed theological implications of magic squares and harmonic ratios, in the Pythagorean tradition. The ancients impressed upon the early scientists the merit of deductionism that became the ideal of classical science, so that Kepler followed Plato in trying to assimilate the planetary orbits to the five regular solids, and Galileo insisted on exactly circular orbits. Today, Kepler’s three laws and Galileo’s laws of dynamics are fundamental, but their erroneous idealist beliefss are dismissed as metaphysical. Yet, the relationship of mathematics to physics has hardly outgrown the attempt to assimilate physical reality to idealized mathematical schemes.


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science as we know it could not have developed without such restrictions…

According to Wigner, symmetry principles relate to laws as laws relate to events. Laws facilitate prediction; if we knew everything in advance, laws would be superfluous. Similarly, symmetry principles facilitate the formulation of laws; if we knew all the laws, symmetry arguments would be superfluous. [E. P. Wigner Symmetries and Reflections Indiana University Press 1967]


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…natural situations, which are generally neither static nor symmetric.

“In short, [spontaneous symmetry breaking] provides a way of understanding the complexity of nature without renouncing fundamental symmetries. But why should we prefer symmetric to asymmetric fundamental laws? In other words, why assume that an observed asymmetry requires a cause… What the real nature of this principle is remains an open issue, at the centre of a developing debate…” [Stanford Encycopedia of Philosophy: symmetry and symmetry breaking, sec4.2]


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…there are far broader ways in which mathematics makes over the world.

An obvious example is the cultural influence of computers on modern life. A more subtle and pervasive influence derives from the proposition that any mathematical function that can be computed by a person using an “effective procedure” can also be computed by a digital computer—known as the Church-Turing Thesis, named after two of its early proponents, Alonzo Church and Alan Turing. This notion must be distinguished from the idea that any physical process can be simulated by a digital computer. For, if a physical process has already been redefined as some algorithm, then it is computable simply because it corresponds by definition to an effective procedure.


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Galileo famously called mathematics the language in which nature is “written.”

Under the spell of Platonism, Galileo believed that planetary orbits should be perfect circles, not ellipses. (For similar reasons, he made the absurd statement that all human or animal movements are circular.) [Holton EHP p101] He also thought that nature is composed of such geometric forms as triangles and squares. Galileo formulated gravitational acceleration not as distance proportional to t2, but as the difference in distance covered in succeeding one-second intervals, which follows the sequence of odd integers. His Platonism thus confined him to number lore in place of the mathematical expression eventually due to Newton.


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There is no more guarantee…than there is that ordinary language can capture all of human experience.

Cf. Richard W. Hamming “The Unreasonable Effectiveness of Mathematics” in The American Mathematical Monthly Vol 87 No. 2, Feb 1980: “Indeed, to generalize, almost all of our experiences in this world do not fall under the domain of science or mathematics… When you consider how much science has not answered then you see that our successes are not so impressive as they might otherwise appear.”


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A Platonic interpretation of mathematics regards the totality of mathematical possibility to be a finished and timeless structure.

Like many mathematicians, Roger Penrose [The Road to Reality Alfred A. Knopf 2004, p63-4] assumes the platonic existence of mathematical truths. He supports this contention with the fact that Cantor derived the set of natural numbers purely from the notion of ‘set.’ But this begs the question, for the concept of set—though more abstract—hinges as much on experience of objects as the concept of integer. He also argues that alien beings, in an amorphous environment lacking definable “objects” (including perhaps their own bodies!), might nevertheless come upon an idea of integers—by carefully disregarding assumptions that seemed to them self-evident in their physical situation. (This is parallel to how various alternative geometries and algebras are discovered by human mathematicians.) I find this rather to be a reason for not assuming a fixed platonic world, since it involves deliberately transcending intuitions. Because no argument can actually prove or disprove platonism, he argues additionally that it has been useful, in both mathematics and science, to treat mathematics platonically. That may be so, and yet what is useful in one context may not be in another. It might prove useful as well to consider mathematics a genetically and culturally constructed map of nature, reflecting both the physical world and the adaptive human ability to fit abstractions to it. To reify mathematics as a platonic realm is to deny both its inbuilt relation to physical reality and its significance as a human construct.


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the whole of math cannot be contained in formal definitions.

According to Gödel, true statements can be made that cannot be proven within a given formal system. Some mathematical Platonists seem to take this to mean that there pre-exists a body of truths outside what can be proven in any formal system whatever. However, transcending a given idea or point of view does not mean transcending all possible points of view.

Hilbert had hoped to compress all of mathematics—an endless number of mathematical truths—into a finite set of axioms. Gödel showed this was not possible, simply by demonstrating one mathematical truth that would elude any such expression. Chaitin showed that there are an infinite number of such elusive truths—an infinite number of true but unprovable mathematical assertions. (This follows from the fact that their algorithmic information content is greater than that of the axioms and rules of inference employed to prove them.) [See Gregory Chaitin “Algorithmic Information and Evolution” 1988/1991] “The only way to ‘prove’ such facts is to assume them directly as new axioms, without using reasoning at all.” [Gregory Chaitin “The Limits of Reason” Scientific American 294 no3 (March 2006) pp74-81]


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The corpus of mathematics has a history and is a work in progress, subject to indefinite expansion and refinement.

Gregory Chaitin envisions a future mathematics that evolves, modeled on biology. [Chaitin “Algorithmic Information Theory: some recollections” 25 May 2007 sec: “challenges for the future”] Also: “…in a certain sense pure math is closer to biology than it is to theoretical physics, because pure mathematics provably contains infinite irreducible complexity.” [Gregory Chaitin “How Real are Real Numbers?”]

Decades before, Dirac had remarked: “…modern physical developments have required a mathematics that continually shifts its foundations and gets more abstract… It seems likely that this process… will continue in the future and that advance in physics is to be associated with a continual modification and generalization of the axioms at the base of the mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation.” [P.A.M. Dirac “Quantized Singularities in the Electromagnetic Field” Proc of Royal Soc (A) 133 (1931)]


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But there is little justification…when it is not possible to exhaustively specify either.

Such arguments may not be convincing to idealists and have not prevented some mathematicians from proposing that nature, in some sense, simply is mathematics, or some part of it. Max Tegmark, for example, believes that a complete theory of everything is possible because the natural reality it describes is a “mathematical structure.” He argues this on the grounds that such a theory would have to be truly objective, in the sense of being free from human “baggage”; an alien race of scientists would come to an equivalent version of the same ultimate theory. The only such universal description, he reasons, is mathematics. Therefore, the universe itself must be a mathematical structure. [Max Tegmark “The Mathematical Universe” 2007]. However, this strikes me as a rather circular non-sequitor and wishful thinking. What is cogent in it is simply that a theory of everything would describe a mathematical structure (for example, a model), not necessarily that it would describe physical reality itself.

Tegmark has developed a “Computable Universe Hypothesis” to argue that physical reality is but a subset of mathematical reality. [Tegmark ibid, sec VII F] This asserts that only computable functions “really exist”—in other words, that the rest of mathematics is not mathematics at all! He identifies the continuum as the main obstacle to his scheme, and proposes to overcome it by restricting mathematics to computable functions. However, this proposal seems to run counter to the premise that mathematics is transcendent. Why isn’t this restriction simply a form of “baggage”?

In contrast, David Deutsch rejects any special status for computable numbers within mathematics: “The only thing that privileges that set of operations is that it is instantiated in the computationally universal laws of physics. It is only through our knowledge of physics that we know of the distinction between computable and non-computable…” [David Deutsch “It from Qubit” Sept 2002 online archive, p13]

According to Roger Penrose, “The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, Mandelbrot set is just there!” [Penrose, The Emperor’s New Mind, p95] But the same can be said of any algorithm that produces a geometrical representation, indeed any mathematical construct. Any formal system can be imagined to exist prior to and apart from thought, and to produce results that “existed” prior to their demonstration. Similarly, how a particular chess game plays out may be imagined to exist potentially, before the game is actually played. A good Platonist, Penrose also holds that “When mathematicians communicate, this is made possible by each one having a direct route to the truth, the consciousness of each being in a position to perceive mathematical truths directly.” [p428] I would say rather that this “direct perception” is made possible because some mathematical truths are generalizations that other minds would naturally arrive at. Very particular mathematical truths might be less obvious to many mathematicians. Many proofs of contemporary theorems would be quite complex and hardly a matter of direct perception.


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How does the world happen to have three spatial dimensions when any number is theoretically possible?

Cf. Craig Callender “An Answer in Search of a Question: ‘Proof’ of the Tri-dimensionality of Space” Sec 1: “The idea that there might be more than three spatial dimensions is not very startling to the contemporary reader. Programs in quantum gravity and the popular science literature [not to mention science fiction and New Age literature] abound with speculation that the number of spatial dimensions may be anywhere between three and twenty-five. Some have even suggested the number is a fraction or even complex, and still others that this number evolves with time.”


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Is the (three-)dimensionality of space an illusion?

See Smolin Time Reborn, p175: “We believe that space is an illusion and that the real relationships that form the world are a dynamical network a bit like the Internet or cell-phone networks… The idea is that space emerges, just as thermodynamics emerges from the physics of atoms.” But what can it mean that “space is an illusion,” except that it is a cognitive construction of the brain, serving the needs of the organism? Yet, that is equally true of ‘time’, which Smolin holds to be “real.” While it is meaningful to speak of space and time “arising” in cognition, is it meaningful to speak of it arising into existence, out of something more fundamental? Thermodynamics is a level of physical description, which emerges from a description of atoms; but it is not an illusion in contrast to the reality of atoms.


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…the fact that such questions can be posed does not mean they make sense or have meaningful answers.

The first question assumes that it is sensible to generalize the concept of ‘spatial dimension’ to some arbitrary number. However, the geometric or topological meaning of ‘dimension’ must be distinguished from the algebraic meaning employed in physics: as a parameter of some purely conceptual space. The second question assumes that laws of physics exist logically prior to and independent of the physical universe, with the power to somehow select or imply a world of a given dimensionality. The third involves a logical recursion, since one may then ask about the number of dimensions of the meta-space (or meta-time) in which physical space may take on varying dimensions.


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The question of why we live in a world of three dimensions… is a matter of convention and definition.

Barrow and Tipler [The Anthropic Cosmological Principle Oxford UP 1988, p261-69] review arguments that have been given for why there must be three dimensions. Of these arguments, three are directly physical, though based on anthropic reasoning; the remainder might best be called mathematical or conventional, concerning the general relationship of mathematics to physical reality. The three physical arguments are: (1) The planetary orbits are stable only with the inverse square law, which presumably requires three dimensions. (2) Atomic electron orbits are stable only in 3D (a premise contested by some). (3) Reverberation-free and distortionless wave propagation is possible only in 3D. The stable orbit arguments—(1 & 2)—basically assume what is to be shown: that power laws are to be identified with dimensionality. Kant had argued from the inverse square law (of gravity or electromagnetism) to three dimensions. But the connection between power laws and dimensionality could be argued the other way around. Assuming four dimensions, for example, would imply an inverse cube law. [Craig Callender “An Answer in Search of a Question: ‘Proof’ of the Tri-dimensionality of Space”, sec2]


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To consider this convention as real, however, is a metaphysical assertion.

Smolin is particularly critical of the block universe: “The block universe is the culmination of the movement begun by Galileo and Descartes to treat time as if it were another dimension of space. It gives a description of the whole history of the universe as a mathematical object…” [Lee Smolin Time Reborn, p59] Also: “The great attraction of the concept of a deeper, mathematical reality is that it is timeless, in contrast to the fleeting succession of experiences. By succumbing to the temptation to conflate the representation with the reality and identify the graph of the records of the motion with the motion itself, these scientists have taken a big step toward the expulsion of time from our conception of nature.” [p34] And also: “The fact that we can make this correspondence between a mathematical object and a record of past motion does not imply… that behind the real evolution in time of the real world there exists a complete correspondence to a timeless mathematical object. To posit this further relation is a pure metaphysical fantasy…” [Smolin “The Unique Universe”, June 2, 2009]


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…they also required eventual revisions and posed limits on physical knowledge.

In addition to physical limits to observation or measurement, such as posed by finite c and h, there are computational limits to what can be practically calculated. Physical resources needed to explore the extremes of the very small and very large grow asymptotically in both directions away from the human scale. Computational resources expand from a zero point of complexity, and may maximize at the human scale—for instance, the resources needed to understand the human brain. [Paul Benioff “Resource Limited Theories and their Extensions” arxiv quant-ph 030308v3, 2008]


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Particles are indistinguishable when we cannot point to them individually…

The usual interpretation of the Bose-Einstein statistics is that the particles concerned are indistinguishable, and so must be tallied differently than in classical statistics, where individuals can be identified by their state-dependent properties. But the B-E statistics could also result if the particles were distinguishable but connected by some force that tends to put them in the same state. In contrast, the Fermi-Dirac statistics usually assumes particles that can appear to have a repelling force acting between them, such that no two can occupy the same state within an atom. [M. L. Dalla Chiara, R. Giuntini, D. Krause “Quasiset Theories for Microobjects: a comparison” in Elena Castellani, editor, Interpreting Bodies: classical and quantum objects in modern physics, Princeton University Press, 1998, p149] From a certain point of view, it turns out that other statistics are theoretically possible between the extremes of Bose and Fermi statistics. However, these seem to be defined only in two-dimensional space.


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…it makes no sense to speak of identifiable individual quanta of energy.

But even in this analogy, identifiable properties of individual dollar bills can make a difference under certain circumstances—for example, if they are counterfeit or collector’s items. Whether a “dollar” is an object with individuating properties beyond a set of defining properties depends on how we do the accounting. This aspect of the shell game of quantum physics begs the question of what one expects to find under the shell. Are waves and eigenstates “things,” potentially with individuating properties, or are they sums in energy accounts?

By definition, photons cannot be “observed” as objects. Rather, their existence is simply inferred from the fact that light energy is absorbed or emitted in discrete amounts. This effect could be a property of discrete matter without implying discrete light. It was for this reason that Planck initially opposed Einstein’s “discovery” of photons.


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entropy measures randomness or disorder, while information informs.

Shannon information measures a decrease in entropy or uncertainty, which still must make a difference to some agent. Entropy, like probability, depends on how a system or situation is partitioned. While the ratio of heat to temperature in a given situation is quantifiable and definite, the quality or usability of energy is relative to purpose. The interpretation of entropy is unclear in the exotic realms of general relativity and quantum theory, particularly in the physics of black holes. For example, it is unclear whether the relevant degrees of freedom should be associated with the thermal atmosphere, with the horizon, or with the singularity. [Robert M. Wald “The Thermodynamics of Black Holes” Living Reviews in Relativity Volume4/2001 (, sec 6.2.]


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…ignored the caveats of E.T. Jaynes regarding the ambiguous nature of entropy.

See: E. T. Jaynes “Gibbs vs Boltzmann Entropies” Amer. J. of Physics, Vol 33 No. 5, May 1965, p396-98. See also: S.F. Gull, “Some Misconceptions about Entropy”, 1989. Jaynes cautioned that there is no such thing as the entropy of a physical system, since any given physical system corresponds to many possible thermodynamic systems. Consider, for example, a pot of purple paint, made from mixing blue and red paint. It should have a higher entropy than either of its constituents; yet as a new constituent to mix with another color, the purple paint should have a lower entropy than the new mixture.

Jaynes reminds us, on the one hand, that entropy measures our degree of ignorance about the microstate of a system, when we know only its macroscopic thermodynamic parameters. On the other hand, entropy measures the degree of experimental control over the microstate, when only macroscopic parameters can be manipulated. It’s not so much a property of the physical system as of the experiments performed on it, reflecting an agent’s participation. While it is unclear how experiments can be performed on the universe as a whole, Jaynes’ admonitions only served to spur on efforts by cosmologists to transcend the relative or anthropomorphic aspect of the information/entropy concept.


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It is nonsense when applied to real things.

Consider, for comparison, trying to delete the “path information” of a planet. Does the planet leave a wake like a passing ship? Would erasing the path information mean to (1) retrace the path itself, (2) undo the physical influences it had, or (3) destroy a record of measurements? The first could only mean reversing the history of a reversible system; but such systems are fictional idealizations. The second could only mean undoing all traces or effects of the motion (the wake), but these cannot even be fully specified. The third possibility is simply to eliminate an instrumental artifact. But, surely, even in the quantum realm, deleting information from an artificial record is not the same as deleting some part of nature—unless one assumes from the outset that nature at that level is nothing more than a record!


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Such outlandish notions follow from adopting an idealist platform to begin with.

It is fair enough to liken a scientific theory to a computer program that predicts observations [e.g. Gregory Chaitin “The Limits of Reason” Scientific American, March 2006, p79]. It is a great leap from there to the notion that physical reality itself “registers information,” or is a program or a computer that processes information. Some do not hesitate to make this leap. Yet, physical reality is not the same kind of thing as a theory about it, nor the same kind of thing as a computer program that simulates some aspect of it. It is certainly not the same kind of thing as a literal computer. Biological systems do indeed register information. And digital computers do indeed process information. However, to claim that the universe does these things is simply to project human meanings indiscriminately. Such abuse of metaphor ends in bad metaphysics and possibly bad science.


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Applied to the universe as a whole, it then implies a limit on the total of information that can exist.

The ‘Beckenstein bound’ says that the amount of information in any physical system is finite, which implies discreteness: “In physics, the Bekenstein bound is an upper limit on the entropy S, or information I, that can be contained within a given finite region of space which has a finite amount of energy—or conversely, the maximum amount of information required to perfectly describe a given physical system down to the quantum level.” [Wikipedia: Bekenstein bound] Bekenstein himself reminds us that information-theoretic entropy and thermodynamic entropy are two very different concepts, even quantitatively.


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What is the relationship between information (entropy) and real structure in black holes?

The term ‘black hole’ is used somewhat loosely, at least in literature for the general public. Sometimes it refers to the volume inside the event horizon, sometimes to the “surface” just outside the event horizon (stretched horizon), and sometimes to the singularity. The relevant concept of temperature is also ambiguous. In one context, temperature is considered an effect of Hawking radiation generated at the horizon. Yet, measured from a distant location outside the event horizon, temperature is thought to vary inversely with energy input; since in principle nothing can escape the black hole, the temperature of a large black hole must then be vanishingly small. Nevertheless, black holes according to string theory have a positive specific heat, varying directly with energy input; the “stretched horizon” is considered even to be indefinitely hot! Temperature as conceived in classical physics refers to the energy and motions of atoms or other particles. Such detail is undefined inside the event horizon, where it cannot be observed. Approaching the singularity, temperature seems to be meaningless. Cf. Lee Smolin “Final Letter” (in discussion with Leonard Susskind) Edge []: “The black holes that were successfully described by string theory have a property that typical astrophysical black holes do not have — they have positive specific heat. This means that when you put in energy the temperature goes up. But most gravitationally bound systems, and most black holes have the opposite property — you put in energy and they get colder. It appears that the methods used so far in string theory only apply to systems with positive specific heat, therefore no conclusions can be drawn for typical astrophysical blackholes.”


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Mass then becomes like thermodynamic varibles, such as temperature and pressure, which disregard internal microstates.

Whatever the state of the matter inside the black hole, its gravitational effect continues to be felt outside, defining the event horizon. (This raises the ugly question, incidentally, of why gravitation is different from electromagnetism, insofar as gravitons can escape the black hole while photons cannot.) Entropy continues to be proportional to mass, but can no longer be associated with structure—nor, therefore, with microstates or information. The argument over whether information can be absolutely lost in a black hole seems to pivot around the ambiguity between an internal and external view of black holes, which corresponds to the ambiguity regarding temperature. Susskind has labeled this ambiguity “complementarity” (after Bohr). At root, however, it seems to confuse a mind’s eye view with the actual situation of a real observer—which may be true of Bohr’s complementarity as well.