Essay Abstract

Eugene Wigner famously argued for the "unreasonable effectiveness of mathematics" for describing physics and other natural sciences in his 1960 essay. That essay has now led to some 55 years of (sometimes anguished) soul searching -- responses range from "So what? Why do you think we developed mathematics in the first place?", through to extremely speculative ruminations on the existence of the universe (multiverse) as a purely mathematical entity -- the Mathematical Universe Hypothesis. In the current essay I will steer an utterly prosaic middle course: Much of the mathematics we develop is informed by physics questions we are tying to solve; and those physics questions for which the most utilitarian mathematics has successfully been developed are typically those where the best physics progress has been made.

Author Bio

Matt Visser is a mathematical physicist based in the Mathematics Department at the Victoria University of Wellington in New Zealand. He obtained his PhD at UC Berkeley, working on supergravity field theories. Since then he has (among other things) worked on QFT under external conditions, quantum scattering, general relativity, cosmology, black holes, Lorentzian wormholes, and "analogue spacetimes". He has published over 200 scientific articles, one single-author research monograph on wormholes, and three edited volumes.

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Hi Matt,

this is a beautiful essay. I'm sympathetic to your middle road practical point of view on this subject, though I'm also somewhat sympathetic to the "So what? why do you think we developed mathematics in the first place?" point of view that you use as a straw man. (I also like that you managed to bring in one of my favorite science fiction novels: The Dispossessed).

You bring up the technical problem of making rigorous sense of quantum field theory. This leads me to wonder why there seems to be no middle ground in quantum field theory between the usual perturbative approach and the rigorous constructive quantum field theory approach. To make an analogy with general relativity, one can think of the post-Newtonian approach used by Cliff Will as analogous to the Feynman diagram approach to quantum field theory, and one can think of the Christodoulou-Klainerman type global existence theorem as analogous to constructive quantum field theory. But for those of us in GR in the vast middle ground between Will and Christodoulou, general relativity is a manifold with a metric satisfying the Einstein field equations. And we are free to tackle those field equations using whatever our favorite method is: exact solution, numerical, perturbation theory, PDE, etc. Is there any such middle ground in quantum field theory? and if so, what is it? and if not, why not?

You mention that sometimes physics gets ahead of mathematics and sometimes mathematics gets ahead of physics. But what seemed mysterious to Wigner was when mathematics developed completely separately from the physics of the time later turned out to be essential for new physical theories. In particular, he mentions the central role of complex numbers in quantum mechanics. In my essay in this category, I propose an explanation for why complex numbers were first found by mathematicians and then only much later found to be important for physics, but I would be interested to see if you have a different explanation.

    Dear Matt Visser

    I liked your essay and your collection of ``common sense'' points about quantum mechanics. And I believe that the topic of ``collapse and reification'' is key and indeed requires better physics more than mathematics. As fundamental mystery, perhaps entanglement might compete equality with this (the strangeness of writing a multi-particle wave-function in configuration space) and also require new physical thinking more than mathematics. Also, I just saw that Kastner's latest paper (arXiv:1502.03814, Haag's Theorem as a Reason to Reconsider Direct-Action Theories) also stressed the importance of Haag's theorem. I see the discipline of mathematical-physics as a porthole encouraging the transfer of possibly useful math into physics as needed and showing interesting physics problems to the mathematical community for cross fertilization.

    Thanks again for your interesting contribution. Dave.

      Hi David:

      Thanks for the comments. Regarding the quantum analogue of the "middle ground" between general relativity's PPN formalism (parameterized post-newtonian formalism) and rigorous Christodoulou-Klainerman-type global existence theorems; there is already an issue with quantum mechanics, let alone quantum field theory...

      Quantum mechanics (as currently formulated) has no PPC formalism (parameterized post-classical formalism); quantum mechanics is pretty much an "all or nothing" deal; there are no "tuneable parameters" that allow one to interpolate between classical and quantum. (I first heard this argument from Ian Redmount, it may have a longer history that I am unaware of.) This is what the PPN formalism in GR is designed to do --- set up a phenomenological framework in which one can experimentally/observationally test whether it is in fact GR, or some other phenomenological model, that describes empirical reality. In quantum physics, with no extant PPC formalism, this is much trickier to do --- the complex of ideas related to Bell's inequalities seems to be the closest one can get.

      When it comes to QFT, again there is no PPC formalism, and in some sense the Feynman diagram expansion is the "middle ground". What would PPC look like for QFT? My guess (and it is a guess) is that we should take another look at the CPT-spin-statistics theorems and embed them in a coherent framework that allows for controlled violations of CPT and spin-statistics. Experimentalists do look for violations of CPT, but can only do so in an ad hoc case-by-case fashion, (eg: check that the mass of an antiparticle equals that of its partner), there is no general phenomenological framework for them to work in.

      So I feel there is a middle ground in QFT, but it's largely Feynman diagram technology, (maybe add in lattice QFT, and soliton techniques). Something is clearly working very well with those approaches, even in the absence of formal existence proofs.

      Let me reply to your comments regarding the complex numbers in a separate posting...

      Regards

      Matt

      Dear Matt Visser,

      Your fine essay was a joy to read; your less-than-worshipful analysis of quantum mechanics and QFT, a breath of fresh air. As you note, math is simply a way of codifying regularities. Assuming Kronecker's dictum, then the question of the non-mystical source of integers would appear the key question, and I argue that these arise directly from the physical logic of AND and NOT 'gates' at all levels of physical reality. As you say, prosaic. Your observations on the obfuscations of mystics, and the horrible price paid by trees, is quite astute.

      I find your discussion of quantum pedagogy very well done and agree with all of your major points, including those concerning the uncertainty relation, 'tunneling', collapse of the wave function, and decoherence, and that these require new physical understanding, not new mathematics.

      As for 'collapse of the wave function' implying no physical basis of the notion of memory, history, or trajectory, I hope you will find time to read my current essay, The Nature of Bell's Hidden Constraints, which analyzes a local model of spin based on the physics of particles in the Stern-Gerlach apparatus, entirely overlooked by Bell's oversimplified assumptions, and precluded by his "hidden" constraints.

      Your observations on utility versus precision are extremely appropriate and those on the non-ontological, non-interpretive essence of "shut up and calculate" ring decidedly true.

      Thank you for entering this insightful essay,

      My best wishes,

      Edwin Eugene Klingman

        Dear Matt Visser,

        On page 4 of your essay you asked the question, 'So where are the "real" open questions in quantum physics?' I allege that, according to Alan Kogut's NASA science team that discovered the space roar, there are 3 independent empirical data sets that confirm the space roar. Do you agree or disagree with my allegation?

        -- D. B.

          Dear Mat Visser,

          A well written essay, I enjoyed reading it. I totally agree with your line of thought, "The problem is not with mysticism per se, but with excessive mysticism used as a tool to obfuscate....Heisenberg uncertainty relations (as commonly presented) suffer from their own level of excessive mysticism and obfuscation".

          However, your conclusion, " the close relationship between mathematics and physics is not at all surprising -- the reason for the close relationship is in fact utterly prosaic", denies that there can ever be any reason for that relation. Your kind attention is invited to my essay, 'A physicalist interpretation of the relation between physics and mathematics', wherein I propose a prosaic (not at all mystic) reason for the relation: Changes in the physical world happen entirely by way of motion; motion is a space- time relation that can be expressed mathematically; so all changes follow mathematical rules; so it is no wonder the patterns we observe in the world are mathematically explainable.

          Dear Prof. Visser:

          I enjoyed reading your clear and cogent essay on how math and physics are mutually supportive but distinct. I especially appreciated your analysis of "Quantum Conundrums", where you (or perhaps Dr. Shevek) question whether the accepted interpretation of QM really makes sense.

          The accepted view of QM is that the physics (and mathematics) of the microworld are fundamentally different from those of the macroworld, which of course creates an inevitable boundary problem. I take the radical (and heretical) view that the fundamental organization is the same on both scales, so that the boundary problem immediately disappears. Quantum indeterminacy, superposition, and entanglement are artifacts of the inappropriate mathematical formalism. QM is not a universal theory of matter; it is rather a mechanism for distributed vector fields to self-organize into spin-quantized coherent domains. This requires nonlinear mathematics that is not present in the standard formalism.

          My essay is entitled "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory", and presents a simple realistic picture that makes directly testable experimental predictions, based on little more than Stern-Gerlach measurements. Remarkably, these simple experiments have never been done.

          Alan Kadin

            Hi Matt,

            I enjoyed reading your essay. I especially like that you address some problem areas including the interpretation of wave function collapse. You paint mathematics and physics as being loosely allied for convenience- rather than in a fundamental inseparable relationship. Which is I should think true of the disciplines (you would know best being involved with both). Though perhaps it is not so within nature. Getting to the end you reiterate your prosaic view of mathematics. That is a view I used to hold. I used to argue that it is just a a language. However I confess to now holding the more romantic notion that: in a changing universe, rather than just the 'stuff' it is made of, it is at least as much the totality of unmeasured 'mathematical' relations between the elements of (Object)reality that bestows its character, and provides the specific forces for change. If we were to ask;' which is more important substance or relation?' it would be hard to promote one over the other. Thinking about chemistry it is the form of molecules, the internal and external relations that gives their characteristic properties and behaviour not just the constituent elements. There is of course a difference between mathematics 'in vivo', in the wild, just as the living organism in vivo is different from the one (however accurately) described on paper.Can there be such a thing as wild mathematics rather than imagined and written,belonging to different facets of reality- I'd like to think so.

            A very good read, good luck, Georgina

            Dear Matt Visser,

            I identify closely with much that you have professed in your essay. Moderation is not mean; it simply means the exclusion of extremes. Your points "Clarity is typically more important than precision", and "The key issue here is usability versus precision" are acknowledged.

            The essence of our problem concerning the relation between mathematics and physics is that what we seek (and often find) are utility values that are applicable to our personal needs. Relativity is more important (more useful) than precision (aka absolute truth).

            "So the close connection between mathematics and physics is dynamic not static" follows naturally. The term 'dynamic' in this context means variable. Could the connection work any other way if the intention is to generate useful information under changing circumstances?

            When we look at the very large we see an assemblage of many things, and their relatedness. When we look at the very small it is the same. In time we find ways of dividing even the smallest things into smaller, related things. We are unable to define the smallest units of existence or to say with any degree of certainty that they do not exist. What we have are relations, and relations to other relations, but no fixity. There is no absolute standard of motionless fixity except that to which individuals attach in their minds. Since such motionless fixity assumes a god-like lacking in confirmation; in the macrocosm of 'all-there-is'; all there are are relations. All there is for mankind to understand is our appropriate relationship to all there is.

            Gary Hansen

            Dear Matt,

            We have enjoyed reading your essay and broadly agree with you. Fully agree about the dynamic tension between physical theories and mathematics...what we call `frailty of the connection'.

            A few remarks on the middle ground between quantum and classical mechanics, in the light of your conversation above with David Garfinkle. We fully agree there is a big difference between

            Newtonian gravity - PPN - General Relativity

            on the one hand, and

            Classical Mechanics - ?? - Quantum Mechanics

            on the other. With hindsight, we feel the reason for this difference is apparent. GR, as we agree, has a built in structure which allows it to reduce to Newton's gravity for weak fields and small speeds. As is well-known, there is no analogous built in structure in quantum theory which permits the recovery of classical mechanics in the limit. We believe the biggest difference between quantum and classical mechanics is the absence of macroscopic position superpositions in the latter theory [this of course being the root of the measurement problem]. And quantum mechanics is unable to explain this, because it claims that even for large masses position superpositions must be seen (Schrodinger's cat). To us this is an indicator that quantum theory is incomplete and approximate.

            It is tempting to believe that the theory of Continuous Spontaneous Localization [CSL] which is the continuum version of the GRW you allude to in your essay, is a worthy phenomenological candidate for such a middle ground, taking a role somewhat analogous to PPN. There is a new constant of nature in the theory, the so-called rate constant, which is proportional to the mass of the object: it goes to an extremely small value for small masses, so that CSL reduces to quantum mechanics in this limit. For large masses, the rate constant is large enough that CSL (being a stochastic nonlinear theory) destroys macroscopic positions at a rapid rate, causing wave-function collapse, and hence explaining the measurement problem and the Born probability rule, as well as recovering classical mechanics. Thus CSL seems to provide a nice universal dynamics, with the quantum and classical as limiting cases.

            We agree with you that CSL has its own limitations [it will be interesting to know though, which limitations you regard as serious ones]. Nonetheless, the fact that CSL makes experimental predictions in the middle ground which are different from the predictions of quantum mechanics, and that these departures are testable in the laboratory and are being tested, makes it an attractive benchmark against which we may evaluate quantum mechanics.

            With best regards,

            Anshu, Tejinder

            Hi Matt,

            This was a pretty good essay. I had three main complaints, though. First, I found it a bit disjointed. Perhaps the ten page limit was a hindrance. For example, I'm not sure how the technical end notes were meant to connect to the rest of the essay.

            The second issue I had was with your characterization of the problems in quantum theory. While de Broglie's momentum-wavenumber relation and Einstein's energy frequency relation are certainly intriguing aspects of quantum physics, I'm not sure I would call them the "central mystery." Take the notion of contextuality, for example. It's a topic that has been central to a good number of papers on quantum foundations in the last decade or two, and yet, arguably, it has nothing to do with the concept of a "wavicle." Certainly within the quantum foundations community, contextuality is seen, these days, as a deeper issue.

            Honestly, I'm not really sure how the issues in quantum theory really had/have anything to do with the nature of mathematics at all. At least you didn't make a convincing argument that they do from my reading of the essay.

            Finally, I would say that "usability versus precision" is a false dichotomy. While I sympathize that many mathematical physicists over-extrapolate meaning from their results and that a healthy dose of operational realism ought to be included, I see no reason we must sacrifice one for the other. Certainly there is nothing that is a priori necessarily mutually exclusive about the two ideas.

            Cheers,

            Ian

              Dear Dr. Visser,

              You wrote: "Mathematics is simply a way of codifying, in an abstract manner, various regularities we observe in the physical universe around us."

              Accurate writing has enabled me to perfect a valid description of untangled unified reality: Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one's normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

              Warm regards,

              Joe Fisher

                Hi David:

                To follow up and respond to your comments regarding the unreasonable effectiveness of complex numbers in quantum mechanics. I interpret your essay as saying something along these lines: "Complex numbers are the minimal extension of the real numbers; the real numbers are known to be useful for classical physics; so when one goes beyond classical physics it is perhaps not all that surprising that complex numbers show up". I would agree with this as far as it goes, but would add a few more issues to the discussion.

                The complex numbers may *first* have shown up in finding roots of polynomials, but as you point out in your essay, they soon led to the concept of (what would now be called) an algebraic field, and are in some sense a minimal extension of the field of reals. But in addition, complex numbers are also an efficient way of handling 2-dimensional rotations, and via Euler's result

                exp(i theta) = cos(theta) i sin(theta),

                very quickly get tied into the notion of waves, ultimately leading to Fourier analysis. Part of the reason people spent so much time on developing the theory of the complex numbers is that they were very broadly useful, not just narrowly useful in defining roots of polynomials. In particular, the fact that complex numbers show up so naturally in Fourier analysis implies that they are going to be useful for any wave-based physical model --- such as, (1850-1900 and thereafter) classical Maxwell electromagnetism, and then, (1925 and thereafter) wave mechanics. As soon as one realizes that quantum physics is related to wave physics, (the Einstein and de Broglie hypotheses), one should not be at all surprised that complex numbers prove useful.

                So overall I'd say that I'm still happy with the central thesis of my essay: If a certain branch of mathematics is useful (for physics, engineering, geology, astronomy, biology, whatever) then more people will work on developing that branch of mathematics. If a well-developed branch of pure mathematics turns out to have some use in the natural sciences, then the natural scientists will quickly appropriate that strain of pure mathematics and turn it into applied mathematics...

                Regards

                Matt

                Thanks for the comments...

                It will take me a little while to digest what you have to say in your essay...

                Regards

                Matt

                Dear DB:

                There are really two questions here: (1) is the "space roar" real? (2) is the "space roar" important? (Since many people will not know what the "space roar" is, observationally there seems to be excess signal in the radio frequencies normally associated with radio galaxies, with the observed signal intensity being roughly six times what was naively expected.) But should we really get all that excited by a factor six discrepancy between observation and naive theory in a small part of the electromagnetic spectrum where we already know there are plenty of radio sources? Overall, at least for the time being, I think this is best left to the radio astronomers to worry about...

                For instance, mis-estimating the average distance between radio galaxies by a factor of two will mis-estimate their number density by a factor of eight, and more than adequately account for the "space roar"... There are also many other possibilities one might think of... So while the "space roar" appears to be a real signal, it may not be indicative of "fundamentally new physics". There is an old adage: "When you hear hoofbeats --- think horses, not zebras".

                Regards

                Matt

                Dear Joe:

                There is no gentle way to put this --- your essay is failing to usefully communicate with the intended audience. To save other people the time that might be spent in reading your essay, let me provide three salient quotes therefrom:

                "Had Isaac [Newton] had any concern for truth and a rudimentary grasp of reality..."

                "Had Albert [Einstein] shown any interest in truth and had he had a rudimentary grasp of reality..."

                "Had Hawking had even the most rudimentary grasp of reality..."

                Comments along these lines will certainly influence people --- but not in a positive manner. Some of the other phrases you use, such as "erroneous abstract zero" and "invisible illuminant" are certainly striking; but for all the wrong reasons. I strongly feel you should carefully re-assess your entire essay, and your serious misconceptions regarding both mathematics and physics.

                Regards

                Matt

                Dear Ian:

                Thanks for your comments.

                --- Technical note 1 was there to make sure I minimized the number of formulae in the main text of the essay itself.

                --- Technical note 2 was there of keep the essay more focussed, by allowing me to avoid bringing specific technical details of electro-magnetism and acoustics into the body of the essay.

                --- Regarding the difference between the "Einstein--de Broglie" relations on the one hand, and "contextuality" on the other; it seems to me a little odd that the inequalities coming from contextuality-related arguments never seem to involve Planck's constant, while the Einstein--de Broglie relations do very explicitly involve Planck's constant. Somehow the presence of Planck's constant screams "quantum" to me in a fundamental manner.

                --- Regarding the tension between usability and precision (more precisely, hyper-technical and excessive precision), consider for instance the Henstock-Kurzweil integral, which is even more powerful than the Lebesgue integral, which in turn is more powerful than the Riemann integral. There are purists who feel we should of course be teaching the Henstock-Kurzweil integral in freshman calculus; I have very strong reservations... Indeed I would be very leery of being a passenger in any vehicle whose design depended critically on the engineers using the Henstock-Kurzweil integral instead of the Riemann integral. (In fact, I'd probably prefer the engineers to stick to upper and lower Riemann sums to obtain explicit bounds on quantities of interest.) Now there are places where the Henstock-Kurzweil integral is useful, but just because you can develop such a mathematical formalism does not mean it it always desirable to do so... I am not saying that usability and precision are exclusive, I am saying that it is probably not worth setting up a formalism that is significantly more precise than what you really need to get the job done...

                Regards

                Matt

                Dear Dr. Visser,

                Thank you ever so much for reading my essay. Alas you are correct in that my unified explanation of reality will not appeal to the utterly ignorant abstractions addicted community at this site.

                Please either refute my contention that real light is inert and there is no physical space or accept it, but please do not leave me with the impression that you are also an abstraction addicted ignoramus.

                Imploringly,

                Joe Fisher