The utterly prosaic connection between physics and mathematics by Matt Visser

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

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

Dear Matt Visser,

Maybe, complex numbers are already the minimal extension of the rational numbers? Anyway, already the extension of the natural numbers to the integer numbers may be interpreted as a trick to allow convenient calculation without shift operations.

In principle, 19th century physics could also be formulated without negative and complex numbers. This is not just an idea that I adopted from other engineers; it was also admitted by Pauli (cf. Der Pauli-Jung-Dialog). Don't get me wrong, I don't deny the practical necessity of using complex calculus.

While your superb essay confirms your excellence, you and virtually all physicists might nonetheless have failed to get aware of a trivial trifle; the use of complex calculus in physics and technology has been based on Heaviside's trick to attribute zeros to the not yet existing future and split the continued. This implies fourfold redundancy due to omission. Well, one can easily avoid the dilemma by following Einstein and Hilbert and denying the distinction between past and future. However, this way the logic consistency gets lost. Instead I prefer using complex calculus properly, step by step.

So far, nobody could show in what I am wrong when I state that cosine transformation of measured data is as good as their Fourier transformation.

MPEG demonstrated, it actually works. Therefore, I doubt with all unbelievable consequences that ict and ih are absolutely indispensable for physical reasons. Again, don't get me wrong. I don't claim that they can actually be replaced.

Faithfully yours,

Eckard Blumschein

Hi Matt,

I tend to agree with you that math is efficient because we have developed it for areas in which it is efficient, so it's kind of a pointless question if you interpret it literally. It's like asking why email is efficient at delivering messages.

However, I think you're missing the more relevant question, that is, are there aspects for which math cannot be used, or it is "efficient" for everything? I see you're not very convinced by the mathematical universe hypothesis. Arguably you are right that saying that there are more mathematical structures to be discovered is hardly a useful prediction, but the question is there a mathematical structure for ANY observation that we can make (natural science or any other), or not? Maybe you have a chance to look at my essay, I think you might find we have some things in common.

-- Sophia

Dear Matt Visser,

Great essay. Your comment "unnecessary and excessive obfuscation" leading to the example of "distressingly common misconception that "uncertainty relations" are intrinsically a quantum phenomenon -- utterly ignoring the fact that engineers have by now some 60 years of experience with utterly classical timefrequency uncertainty relations in signal processing, and that mathematicians have by now over 80 years experience with utterly classical time-frequency uncertainty relations in Fourier transform theory." is something I agree completely with.

The second example you mention ""tunnelling"/"barrier penetration". Despite yet more common misconceptions, tunnelling is a simply wave phenomenon; it is not (intrinsically) a quantum physics phenomenon. Under the cognomen "frustrated total internal reflection", the classical tunnelling phenomenon has been studied and investigated for well over 300 years, with the wave aspects (the "evanescent wave") coming to the foreground approximately 150 years ago .." adds more certainty to your argument.

I enjoyed the discussion on the standard model, where no one doubts it's accuracy, but perhaps time will give us a better understanding of some of the technical aspects and issues you highlight.

Finally, I agree with your comments on "usability". In my essay here, I start with the usability issue and argue that it can be greatly enhanced by providing visual models of objects that match the properties of the fundamental particles of the standard model. I would be very interested in your comments on any of the specific models provided.

Your essay was a very enjoyable and insightful read.

Thank you and regards,

Ed Unverricht

Dear Matt,

Great essay and very clear. I particularly liked your section on Quantum conundrums.

I liked the way you dealt with quantum complementarity and I also have an instinctive reaction against such ideas which do not present a unified and simple picture of natural phenomena.

In talking about the measurement problem or the problem of the collapse of the wavefunction you may be interested to read my essay "Solving the mystery". I take the position that the apparent mystery is because of our lack of understanding.

If we adopt a different interpretation of reality in which we are dealing with real physical waves, then the notional collapse of the wavefunction is in fact an interaction of a real physical wave with an atom of the detector or measuring device. The measurement problem then is resolved.

Best regards

Richard