Essay Abstract

This paper argues against a strong philosophical interpretation of the leading role of mathematics in all of physics. To do so the paper focuses on a specific case study, that of the truly astonishing success of symmetry groups in modern particle physics. Specifically, I analyze the case of one local gauge symmetry, that of the strong nuclear interaction. I would say this is an especially pertinent case study, as gauge symmetry applies throughout most of our current best fundamental physics and the intimate relation with the physics it describes is particularly astonishing. The paper advocates for an understanding of mathematics only as an (especially appropriate) language which does nothing but describe patterns, a subset of which are instantiated in Nature. With such an understanding I argue that the effectiveness of mathematics is not unreasonable; on the contrary, it is to be expected. Such an explanation undermines the viewpoint that takes gauge symmetry principles as a priori reasonable or as some sort of necessary meta-laws. Likewise, such an explanation weakens the reasons to endorse a strong ontological commitment to the mathematical entities (as the diverse variants that suggest that the universe is fundamentally mathematical, like [Tegmark, 2014] or [French, 2014]).

Author Bio

From March 2015 Aldo Filomeno will join the UNAM at Mexico DF as a postdoctoral researcher in philosophy. He defended last October his PhD in Philosophy of science. His thesis dealt with the notion of law of nature and is entitled: "On the possibility of stable regularities without assuming fundamental laws". Previously he studied engineering of telecommunications and philosophy. More details of his research can be found on: https://unam.academia.edu/AldoFilomeno

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Dear Aldo Filomeno,

While many essays speak of Grothendieck's 'dessin d'enfants', or the Langlands program, or the Monster group, etc. in hopes that these will answer some questions, you address probably the most significant mathematics of the last 60 years, gauge theory - and you properly put it in its place.

As no other general field of math has had the impact on physics as has gauge theory, I'd say you have your priorities right. Also, you rightly include "(and vice versa)" when you note the relation of symmetry principles for every continuous global symmetry of the Lagrangians to which corresponds a conservation law.

It seems to me that if one had to classify these entities, the conservation laws would be ontological facts, while the symmetry principles would be epistemological effervescences.

The really amazing thing, to me, is that none of the basic symmetries are exact, from iso-spin to SUSY. You mention 'astonishing success' such as the prediction of the Omega minus. But have there really been that many such? Perhaps I'm jaded, but when it comes to trying various hierarchical layouts of the known particles, just how hard was it to notice "one seems to be missing here"?

I very much enjoyed your "Elegant but contingent..." analysis of gauge symmetry. I particularly liked that (B) assumes "a language full of non-actualized possibilities, and that the mathematics constituting our best physical theories is only one of the infinite possible mathematical descriptions of the regularities of the world."

My essay begins by 'extracting' regularities from measurement data to find a 'best' [in a max entropy sense] feature vector that represents the measured properties. Your analysis of the strong force leads again to the particular final choice being made among an extremely vast space of possibilities.

[I have suggested another possibility in The Chromodynamics War in which SU(3) is based on structure, not color. You might enjoy some of the ideas in this 'hard sci-fi' novel.]

It is not insignificant that the gauge fields are put in 'by hand' to a large extent.

But, to return to your "Elegant but contingent...", I have made the same argument against the Bell's basic model: it is 'Elegant but contingent' in the sense that it is contingent on a constant field, which leads to a null experimental result, and hence begins with a contradiction. There is much more to the story which is presented in my essay . I invite you to read it and would welcome feedback.

Thanks again for your very well written and quite on-target essay.

My best regards,

Edwin Eugene Klingman

    Dear Aldo Filomeno,

    Your essay is an interesting and great read. My interest started with your question "Why is group theory so central to describing the physical world?" You introduce symmetry and spend time on SU(3) - "The color invariance is represented by the symmetry group SU(3), the Special Unitary group of degree 3". I have always been troubled by the naming these 3 properties "colors" and calling them Red, Blue and Green. Doesn't the subject require a better description of something so important and fundamental to world of hadrons?

    My essay takes a very specific look at the SU(3) symmetry and extends the two properties (spin and handedness) of the electrons SU(2) symmetry to the world of hadrons and the SU(3) symmetry by adding a specific property of flow as the third property. I hope you get a chance to comment on it.

    Finally, your comment "The "unreasonable" success of symmetry principles in physics" begs an explanation and your essay makes a big contribution to the subject.

    Regards and good luck in the contest.

    Ed Unverricht

      Dear Eugene Klingman,

      I very much appreciate your compliments and especially the comments.

      Your point about the inexactitude of the symmetries seems to me particularly appealing: scientific or philosophical research should be carried out pursuing this line. So far I do not know which side I would bet on. I envisage arguments refusing the importance of such inexactitudes as well as arguments that might employ such inexactitudes against a too idealized view of the laws of nature.

      I also agree that one could interpret symmetries and conservation principles as you suggest (being the latter more fundamental). Then, we face questions regarding the conservation principles: Are they a plausible unexplained primitive of our ontology, or should the postulation of the conservation of some property be somehow explained?

      Finally, I'll read your essay which looks really interesting!

      Warm regards,

      Aldo Filomeno

      Dear Ed Unverricht,

      thank you very much for your comments!

      In principle, the property has been named 'color' as it could have been named in any other way. How we could have a better scientific description of such properties I do not know. Philosophers have historically struggled to better understand what are properties, but it is hard to attain an answer (as usual in philosophical matters). This reminds me of a paper from a great philosopher and physicist called Alyssa Ney entitled: "Are There Fundamental Intrinsic Properties?" (available online). Perhaps you enjoy it. (Let me know!)

      I'll read your essay, it looks very pertinent to what I say!

      Warm regards,

      Aldo Filomeno

      Dear Joe Fisher,

      thanks a lot for reading and criticizing the essay. I'm extremely eager of hearing criticisms to improve the paper.

      Regarding your additions of 'abstract' to my quote, I would agree with some of them, but crucially not when I say "deeply physical" (as it's just the opposite that I'm trying to convey).

      Your other observation of your untangled reality is interesting but complex to follow; this has raised my interest to read your essay!

      Warm regards,

      Aldo Filomeno

      a month later

      Dear Aldo,

      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

        Dear Joe,

        ok I'll read it in a pair of weeks.

        Thanks,

        Aldo

        20 days later

        Dear Aldo Filomeno,

        While I will not vote and not even comment on your essay because I seem to be at odds with mainstream. You wrote: "the acceptance of (A) is hardly disputable, (B) should be justified". If you had a look into my essays, you may hopefully understand why I consider from the perspective not of a mathematical model but of conjectured reality the acceptance of (A) hardly indisputable. Concerning symmetries, I observed that ideal symmetries tend to be mathematical artifacts that just reflect redundancies. Real symmetries are rarely or maybe never absolutely perfect. Isn't the harmonic oscillator an unphysical ideal? What is wrong with this view?

        I should add that I maintain that I consider it unjustified to integrate in case of frequency analysis over available past and not yet written future data. Cosine transformation is evidently as good as Fourier transformation except for it omits an arbitrarily chosen zero.

        Genuinely curious,

        Eckard Blumschein

          Dear Eckard,

          thanks for your comments. Your essay looks interesting and original; I'll certainly read it when I have the time. As to your comments, I'm curious to see how you argue that (A), i.e. that the world displays spatiotemporal patterns, is hardly indisputable. How could it be disputed? I took it as a premise of my argument (an obvious statement but it was important to make it explicit for the clarification of my conclusion).

          Your view of symmetries as informing of redundancies in the world is clearly one standard possible interpretation of them. I would say it is not incompatible at all with the moral I wanted to highlight in my paper (if any, it's in the same line), namely the lack of any special or necessary status of the laws of current physics.

          (By the way I don't understand your last observation!)

          When I read your paper I'll let you know, warm regards,

          Aldo

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