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

      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

      Dear Matt,

      Please find my attempt to better explain my point of view here .

      Thank you for your estimated effort to respond.

      Regards,

      Eckard

      Dear Matt,

      Thank you for your essay.

      I liked your point that the back-and-forth connections between mathematics and physics will continue to twist and strain not only due to technological limitations, but also due to the personalities involved (i.e. mathematics and physics are both human endeavours).

      I also liked your point that theorists pay heed to experimentalists who only adjust one aspect of an experiment at a time - similarly, theorists should not overlay speculation on speculation.

      Adding to your point that mathematics encodes the patterns and regularities in the data stream, I would argue that the crux of the tension between mathematics and physics is that the data stream does not include the arrow of time - and therefore mathematics is not able to model this empirical feature of nature. That is, the act of measurement and quantification (combined with the use of the continuum) eliminates from the data stream the arrow of time before mathematics can do its magic. This, in my view, is why we end up with time-symmetric theories and,for example,the measurement problem you discuss.

      Thank you for writing the essay.

      Kind regards

      Spencer Scoular

      11 days later

      Hi Matt--

      Your essay was a joy to read: passionately yet cogently argued and right to the point. To all of your points, I can only respond, "YES!" In particular, I subscribe to your thesis that what we do in physics is identity patterns and regularities in Nature, which we then codify using mathematics. There is no mystery here. Mathematics is so effective because we have spent the last 350 years industriously making it so.

      On a minor note, I also enjoyed your use of quotations. You've got to be the only guy in the history of FQXi who has managed to work in a cite to Ken Wilber!

      On a completely different note, I observe that your outstanding book, Lorentzian Wormholes, is 20 years old. To the best of my knowledge, you never put out a second edition. Don't you think that it is about time?

      Best regards,

      Bill.

      6 days later

      Dear Matt,

      I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

      All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

      Joe Fisher

      7 days later

      Dear Matt,

      This is a well argued and well thought out exposition. I enjoyed reading it as much as I enjoy reading your relativity papers; you always bring up interesting topics and back them up with solid research. While I can think of counterexamples to the idea that much of the mathematics we develop serves a well defined purpose, I do agree with and enjoy your no-nonsense approach to the central theme of the contest. I also found particularly satisfactory your witty critique on how speculation gets disconnected from reality. Your mention of some of the interpretations and non-interpretations of QM made me wonder (again) which is your approach to the problem. I hope one day you'll write a paper about it.

      Wish you good luck in the contest! Should you have the time and curiosity to read my essay, your comments are more than welcome.

      Warm regards,

      Alma