Essay Abstract

The ''unreasonable effectiveness of mathematics'' in describing the physics of empirical reality is simultaneously both trivial and profound. After all, the relevant mathematics was, (in the first instance), originally developed in order to be useful in describing empirical reality. On the other hand, certain aspects of the mathematical superstructure have now taken on a life of their own, with some features of the mathematics greatly exceeding anything that can be directly probed or verified by experiment. Specifically, I wish to raise the possibility that the real number system, (with its pragmatically very useful tools of real analysis, and mathematically rigorous notions of differentiation and integration), may nevertheless constitute a ''wrong turn'' when it comes to modelling empirical reality. I shall discuss several alternatives.

Author Bio

Professor Matt Visser is a Fellow of the American Physical Society, a Fellow of the Royal Society of New Zealand, and a Member of FQXi. He obtained his PhD at the University of California at Berkeley, and undertook postdoctoral research at the University of Southern California, Los Alamos, and Washington University in St Louis. He moved back to New Zealand 10 years ago. His research largely addresses the interface between quantum physics and general relativity, with particular emphasis on wormholes, analogue spacetimes, black holes, and cosmology. Trained as a physicist, he is currently based in a mathematics department.

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  • Post #2

Prof. Visser,

It would seem that this effectiveness breaks down when we get to certain processes where the mathematics become non-linear but the "real" world shows a finite outcome. Do you think we just do not understand the underlying mechanics or do you think mathematics might need new tools to help us describe these? I have in mind renormalization for one, but there are other aspects such as energy densities of particles in mind. What are your thoughts on how to handle non-linearities?

Regards,

Jeff Baugher

    Matt,

    In your summary, you ask "Exactly which particular aspect of mathematics is it that is so unreasonably effective?" in describing empirical reality.

    I would argue, that is not an aspect of mathematics at all, but rather, an aspect of physics. Specifically, some physical phenomenon are virtually devoid of information. That is, they can be completely described by a small number of symbols, such as mathematical symbols. Physics has merely "cherry picked" these sparse information-content phenomenon, as its subject matter, and left the job of describing high information-content phenomenon, to the other sciences. That is indeed both "trivial and profound", as noted in your abstract.

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      • Post #4

      Dear Dr Visser

      It so happened that my essay contains a reference to your article

      http://fqxi.org/community/forum/topic/1413

        Dear Jeff:

        There are certainly many well-understood situations where both mathematics and physics become non-linear, but do so in a controlled way. The occurrence of nonlinearity, by itself, is not necessarily a disaster for either mathematics or physics.

        Some nonlinearities however, are very difficult to deal with, (turbulence for example), and it is far from clear how to handle that...

        You also mention renormalization. There are two separate issues with renormalization that are often conflated: (1) renormalization as a tool for keeping perturbative infinities somewhat under control; and (2) renormalization as a way of re-expressing "bare" parameters in terms of physical measurements. Even if one has a finite field theory, without a single infinity anywhere in the system, one would still have to renormalize in the sense of (2).

        Perturbative infinities are certainly tricky to deal with. We have some pragmatic techniques which work most of the time, but certainly not everyone is happy with the situation.

        Regards

        Matt Visser

        Dear Robert:

        There are some tricky issues hiding in the phrase "virtually devoid of information".

        In a highly technical information-theoretic sense any efficient axiom scheme (Euclid's axioms, the ZF axioms) can be argued to be "virtually devoid of information", since you are encoding large swathes of mathematics in "just a few symbols".

        But very few mathematicians or physicists would agree that Euclidean geometry itself, or ZF set theory, is "virtually devoid of information".

        Similarly - though the precise details are still an open question - the Navier-Stokes equations, a very compact set of equations which can be written down with very limited number of symbols, seem to encode all of turbulence. And an understanding of turbulence would hardly be "virtually devoid of information".

        This suggests a need for a modified interpretation of the phrase "virtually devoid of information", one that not only considers the compactness of the axiom scheme itself, but also takes into account the size and complexity of the model one can deduce from the axiom scheme.

        Without somehow taking this into account, one could easily fool oneself as to the information content of a specific model by considering a sloppy redundant axiom scheme.

        While physics has by and large certainly picked problems that are relatively clean, and to some extent the relevant mathematics has been developed to address these clean problems, the phrase "virtually devoid of information" is perhaps overkill. Physicists certainly are interested in complex systems, and the techniques of physics are increasingly being used to address complex systems.

        Regards

        Matt Visser

        Dear Yuri:

        Thanks. From time to time I still periodically think about Sakharov's induced gravity, and how to get a little more out of Sakharov's proposal...

        Regards

        Matt Visser

        Matt,

        I understand your point. There are indeed "tricky issues hiding in the phrase "virtually devoid of information." That is precisely my point. I am using the term "information" as it is used in Information Theory, not Physics or Math. Physicists, have completely misunderstood just how "tricky" knowledgeable observers can be, when it comes to extracting information from observations. Consequently, attempting to use equations, "devoid of information", to describe what observers are doing is "THE" source of all the weirdness in interpretations of quantum theory. Observers behave "symbolically" towards observations, not "physically." They can (and often do, without even realizing it) treat a measurement not as a datum, but as a "serial number", used to label the set of "behaviors" that need to be "looked up" and performed whenever that symbol is observed. Such types of behavior are vastly more non-linear than anything dealt with by the equations of physics, or phenomenon like turbulence. For such observers, it is not the equations, but rather the initial conditions, in the observers memory, that dictates behaviors.

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        • Post #9

        Dear Professor Vissar,

        Although I am not a mathematician, I quite enjoyed reading your exceptionally well written essay. "The "unreasonable effectiveness" in describing the physics of empirical reality is simultaneously both trivial and profound" In my essay, Sequence Consequence, I mention the fact that the number 1 is often considered to be the most important number in a competitive list, yet it is always accorded the least whole value in currency systems. I believe that one real Universe having one real appearance can only be perpetually occurring once in real here and real now in one real dimension. For this reason, I believe that only 1 of anything can ever exist once. All real stuff has to be kept in one dimension. I suppose it would be more sensible of me to believe that there could be three abstract spatial dimensions, however, if that was the case, how is the abstract stuff in the three abstract dimensions distributed? Is the heavy abstract stuff in dimension A, the average abstract stuff in dimension B and the light abstraction stuff in dimension C? Does the abstract stuff stay in the relative dimensions, or does it intermingle in a measurably orderly fashion that you could attach an accurate number to?

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          • Post #10

          Hi Matt,

          An important essay, at the core of our models of physics, the "unreasonable e ectiveness of mathematics". Which number system to use is a question I would have not though of. You wrote a clear understandable and concise essay.

          Changing the subject a little, I find in my own work two areas where the effectiveness of mathematics breaks down. These are addition, and calculus. See: http://www.digitalwavetheory.com/DWT/5_Math-Physics_Connection.html and http://www.digitalwavetheory.com/DWT/9_Paradoxes.html

          If you should care to comment I would appreciate it very much.

          Thanks,

          Don Limuti

          Dear Joe:

          Thanks for the feedback.

          You mention: "I believe that only 1 of anything can ever exist [at] once".

          There is a sense in which (assuming classical physics) this is almost axiomatic - in classical physics, particles (and more generally, objects) are distinguishable with well-defined locations, and by adding enough qualifiers to your description of an object you can indeed in principle uniquely specify the object. So, essentially by definition, with a detailled enough description there's only 1 of any object you might wish to consider...

          In quantum physics this feature seems to go away: Quantum elementary particles of the same type are, as far as we can tell, utterly indistinguishable from each other, which is a key feature leading to (for instance) Bose-Einstein condensation, which we certainly do see experimentally...

          So there are very good reasons for being extremely careful and cautious when discussing issues of "uniqueness" in the physical universe we inhabit.

          Regards

          Matt Visser

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          • Post #12

          Which Number System Is ''Best'?

          For example Mp/Me=1836 is a true dimensionless constant. I found that it is a beautiful symmetric number because 1+8=3+6=9, after it is converted to numerological addition. In the binary code 1001 present mirror symmetry.

          See my essay

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          • Post #13

          Dear Professor Vissar,

          Thank you for your considerate answer. All real stuff is unique. All human abstract thoughts attempt to impose fixed systems of duplication on all physicality finding congenial mental security in commonly collective shared practicality. "Quantum elementary particles of the same type are, as far as we can tell indistinguishable, which we certainly do see experimentally." Naturally formed snowflakes are indistinguishable to the naked eye, yet experimentation has proven that no two snowflakes of the trillions that have fallen have ever been found to have been identical. One real Universe can only have one real law. If there have never been identical natural snowflakes, it is physically impossible for any particles of any type to be identical just because they were fabricated. Every one of the sparks created by CERN has to be unique.

          Dear Prof. Visser,

          This is a very intriguing essay, and it approaches both the theme of this edition of the FQXi essay contest, as well as the previous one. It is very catchy, and pleasant to read. I think the study of the various number systems may be useful for many reasons. It would be useful to have a monograph, or maybe a branch of mathematics, dealing with such number systems and what happens to them when apply various mathematical operations or functions - are they preserved, or we access another number system? You mentioned the square root, also one can add trigonometric functions, exponentials and logarithms, and so on. Also, what kind of coordinate transformations preserve a vector space over such a ring? I think there are many reasons to research these ideas, not only for mathematics and physics, but also for computer science, in computational geometry for example. It is known that there are numerical problems due to the finite amount of bits that can be represented in the memory of a computer, and such studies may provide wiser choices of the numerical representations. I am thinking for example how floating point numbers are customary represented exponentially. Computational geometers usually define a constant, representing the mathematical tolerance used for approximations. Many algorithms work with squared lengths, for example in comparing distances there's no need to take the square root and find the length, since the square function is monotone, and it is better to avoid this, for speed optimization. For this reason, it is customary to define a second constant, which is the square of the tolerance. But normally programmers stop here, when in fact they apply trigonometric functions, geometric transforms and other operations which may need also the tolerance to be transformed. This limitation leads sometimes to inconsistencies and numerical issues. Therefore, the mathematical libraries used by programmers may benefit too from a more careful way to represent various numbers, and to control the way they transform under various operations and functions.

          Best regards,

          Cristi Stoica

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          • Post #15

          Dear Matt Visser,

          Are all constructible numbers really countable? Didn't ancient mathematicians count {2,3,...}? Weren't the algebraic irrationals already treated by Cantor? Aren't computable reals just rational approximations of reals? Is there really something between countable and uncountable? Galileo did definitely not agree on that there are more hyper-reals than reals. So don't I.

          Maybe I mistook you. I largely agree with some of your intentions. While you are living in New Zealand and I in the old center of arbitrary mathematics, I have different antipodes of mine in mathematics: Weierstrass, Dedekind, Cantor, and Hilbert. Do you dare commenting on my essay?

          Regards,

          Eckard

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          • Post #16

          My vote for which number system that best models reality is that it depends on the reference frame of the observer (ie, the physicist or mathematician trying to model reality) relative to the reality that he/she's trying to model. For example, a finite observer within an infinite set of finite balls might view each ball as an integer. But, a hypothetical, infinite-sized observer outside this set would not be able to see the boundaries of each of these balls (because they're infinitely small relative to him) and so the set would appear to him to be smooth and continuous. Therefore, he/she might like to use the real numbers to describe the set. I put these thoughts into my entry for the last FQXi contest and they're also at the below if anyone is interested. Thanks!

          https://sites.google.com/site/ralphthewebsite/filecabinet/infinite-sets-ii

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            • Post #17

            Hi Roger,

            I cannot see who is the author of the linked statements on infinite-sets.

            What about your claim that what is the best model of reality depends on the reference frame of the observer, I strongly disagree. Does it matter for a model of the moon whether it is observed from the earth or from somewhere else?

            Also, I would like to object against sloppy use of "infinitely small relative to" something. Aren't Leibniz's infinitesimals strictly speaking finite?

            I never came across in serious literature with an infinite-sized observer.

            Regards,

            Eckard

            Matt

            An interesting resume on number systems. Can you comment on the suggestion by Smolin, and Ken Wharton here, that after abstracting to numbers from reality to predict evolution of a system (nature), when renormalised of 'mapped' back to reality, we have no guarantee that similarities found mean there is any real physical relationship of the algorithms to the natural processes.

            In other words, the maths used is 'unrealistically representative' of the renormalised mathematical model, but not necessarily also of reality itself. This then would be the reason for all the problems and anomalies.

            Also, do you consider that conceptual ontology must be correct before abstraction is used? And might limiting abstraction to, perhaps, rational numbers and rigorously applying the rules and structures of logic to matters of 'process' or 'mechanisms', then allow more precise quantitative modelling?

            I've explored the logic and ontology route, particularly regarding kinetics, and believe found some success. I'm not however competent to now evolve the mathematical structures to accompany it.

            If you have the time I'd be grateful if you'd also read my essay and comment.

            Thank you for yours, and very best of luck in the competition.

            Peter

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            • Post #19

            Eckard,

            Hi. The author of the link was me as indicated by the preceding sentence:

            "I put these thoughts into my entry for the last FQXi contest and they're also at the below if anyone is interested."

            Everything we observe in reality depends on the perspective of the observer relative to the thing being observed. In regard to your example of the moon, an observer on Earth would view it as an orbiting satellite. But, a hypothetical microbe living in the interior of the moon might view it as an almost infinitely big rock and see the Earth as the satellite.

            In regards to the comment that you've never seen anything in the "serious literature with an infinite-sized observer.": First, I used the word "hypothetical" to show that this was an imaginary observer and did this just to show what a set might look like from his viewpoint. Second, whenever mathematicians discuss infinite sets, the mathematician can be the observer describing what is going on in the set.

            Thanks.

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            • Post #20

            Dear Matt

            Is it time to read my essay more closely?