Mathematical Beauty, Physics, And Truth

Mathematical beauty can guide physics: this is what happened for the foundation of QED by Dirac. At least, so it looks at first sight, and so he said. However, Dirac was guided by one intuition deeper than “beauty”: finding an equation of maximum simplicity to describe the electron… while knowing the Klein Gordon relativistic equation didn’t describe the electron, finding a simpler (first order) PDE that would be “relativistic” guided his search. Then see what happened. He knew that the simple wave equation are first order (although conventional strings are second order PDEs). Doing so Dirac re-invented unknowingly part of Cartan Spinor theory, a pure mathematical theory invented 15 years earlier. The Dirac equation he found led to experimental predictions, which were found to be true.

General Relativity too had a mathematical origin: Riemann, in the 1860s, got the idea that force will manifest itself as a deviation of geodesics. The idea is actually even older, in 3 dimensions, going back to Buridan (1350). That’s how Buridan superseded Aristotelian physics with his “Impetus” theory (the first order of the mechanics we have now).

Special Relativity was invented differently: a number of equations were found to explain effects observed, until Poincaré built a coherent logical whole resting on the idea that the speed of light should always be measured to be c. In particular electromagnetism was found to the essence of Relativity.

The picture is from CERN. The waves are from beaches of Western North America. Ultimately, it seems likely to me that nonlinear phenomenon are needed to understand hydraulics in full. But present day hydraulics, like Quantum Physics (away from collapse), is linear…

So the opposition is not so much between mathematics and physics, it’s between shallow ideas and deeper ideas. Physicists had no deeply new ideas, ideas which can stand-under, understand, for generations. Much of that has to do with denying that the Foundations of Quantum Physics are worthy of consideration.

Mathematical beauty can guide physics: but who guides mathematical beauty? 23 centuries ago, mathematicians then in power decided that Euclidean mathematics was beautiful, and non-Euclidean mathematics (invented prior) was ugly. Let’s not talk of the ugly anymore, or, at least, too complicated, they opined. After a few generations of pounding that notion, it became a claim that nothing existed in geometry, but for the beauty of geometry in a plane. Mathematicians got so dumb they forgot that the axiom of parallels was just an axiom, not a theorem (they tried to demonstrate it for nearly 20 centuries, whereas it would take ten seconds to explain to them what idiots they were, had they a brain in that direction…)

Indeed, never mind that Pytheas of Marseilles and his successors had, thanks to spherical geometry, computed the size of the rounded Earth most precisely. So, clearly mathematics on a sphere was extremely useful! In particular, true, and in existence!

Some say equation are beautiful. Equations themselves are subjects to interpretations. For example Henri Poincaré’s E = mcc, rolled out at the Sorbonne in 1899, is not clear. Similarly Einstein GR equation, basically: Curvature – Mass-energy, is not clear, as Einstein pointed out: right side is ill defined. After Dirac discovered his equation he realized it had to live in “Spinor Space”. So interpreting an equation gave the space where it had meaning.


Right now the most fundamental problems in mathematics and physics are clear to yours truly:

First, mathematics use an infinity axiom, namely that there is infinity. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: There is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.

This, this Infinity Axiom, in my opinion supposes too much, relative to the physical and practical realms, just like Euclidean geometry supposed too much relative to the practical and physical realms. Indeed, in practice, flat geometry does not exist. Same for infinity: in practice infinity cannot exists (not enough particles to count all the numbers). The Infinity Axiom introduces infinities in physics which are a mathematical artefact. This philosophical point is too hard for most top theorists to understand, the ones the Wall Street Journal is in love with (because of there are leading minds officially sanctioned in physics, thus as higher principles, so it is in in economics, sociology, hence plutocracy is rightfully supreme; see below).

Second, Quantum Physics is about WAVES. This enormous conceptual breakthrough was from Louis de Broglie. Waves are beautiful, especially Quantum Waves. Yet, in practice, waves are NOT linear. They are often nearly linear, right, but not quite (just like Euclidean geometry doesn’t quite exist, except as a figment of the imagination, and even then… ). However, present day mathematics has not been focused on nonlinear waves, so we don’t have a notion of “mathematical beauty”of nonlinear waves.

And guess what? The formalism of quantum Physics itself says that the “collapse” it can’t do without is nonlinear.

And now for a word of wisdom from that rather tall little thief friend of ours, Richard P. Feynman: “Physics is to math what sex is to masturbation.” There has been too much self dealing in physics, too much nonsense at the highest level! Bohr’s philosophy, which underlays his satisfaction with the Copenhagen Interpretation of Quantum Physics, is a surrealistic horror: he thought that clarity contradicted truth (or idea to this effect… actually the exact contradiction of the beautiful idea of equation).

Want new physics? Do like Buridan, Oresme, and their friends and students, seven centuries ago: invent new mathematics (they invented the second page of calculus, the first one was from Archimedes himself, 16 centuries before). That’s done by working on the axioms, introducing new ones.

So when is a system of thought X deeper than another Y? When X implies Y, by under-standing it, namely introducing deeper (“under”) reasons for its standing.

String theory has been the equivalent of the crystal spheres and epicycles construction which replaced the evidence all could see, that Earth, the small thing turned around the Sun, the big thing (the Greeks knew from computations, looking at the Moon, and shadows, that the Sun was millions of kilometers away…) Right now the big thing is Quantum Collapse, that’s what needs to be understood. String Theory does a few things, like cancelling some infinities as a problem (my proposal above is much more radical… also, unavoidable…)

Meanwhile, while those self-esteemed super brains make super theories of supersymmetries of super strings (their concepts involve the word”super” very much…), to make a theory of Quantum Gravity, little Patrice has noticed this: there is NO experiment, and, a fortiori theory of gravity in the double slit… Why? Because the super minds, too busy being super, have not noticed that we lack experiments there (after they read this, they will steal the idea, and run to the closest physics journal edited by their friends to publish it as their great insight).

Patrice Ayme



Note 1: the preceding was inspired by the following WSJ article:

Einstein’s character was more like that of an artist than a scientist, his older son, Hans, said: The great physicist reserved his highest praise for theories that are beautiful, rather than ones that merely fit the facts. When, in the latter half of his career, Einstein spent most of his time trying to discover a unified theory of gravity and electromagnetism, he paid little attention to new experiments and focused mainly on trying to find the best mathematical structure. Alas, the strategy got him nowhere.

According to the physicist and prolific blogger Sabine Hossenfelder, Einstein and others who work in a similar way are “lost in math,” the title of her lively and provocative book. Until the early 1970s, few theoreticians fitted such a description—most of them were taking inspiration from the results of experiments. It was this strategy that led them to the so-called Standard Model, which describes the inner workings of atoms with remarkable success. Over the past four decades, however, theoretical physics has gone astray, in Ms. Hossenfelder’s view. Part of the problem, she feels, is that so many theoreticians have allowed themselves to be seduced by the aesthetic appeal of mathematical theories that are going nowhere.

As she explains, the use of beauty as a proxy for truth has an impressive pedigree: Not only was it espoused by Einstein, it also became the obsession of the almost comparably brilliant English quantum physicist Paul Dirac. In 1975 he wrote: “If you are receptive and humble, mathematics will lead you by the hand . . . along an unexpected path, path where new vistas open up . . . from which one can survey the surroundings and plan future progress.” Toward the end of his life, he declared that any theoretical physicist who disagreed with him should give up research and do something else.

As a result of this misguided focus on beauty, Ms. Hossenfelder says, her generation of theoretical physicists has been “stunningly unsuccessful.” The multiverse—the idea that our universe is only one of a vast number—is one of the fashionable concepts that she believes is a dud… 

Ms. Hossenfelder believes string theorists are deluded. “Nature doesn’t care” about mathematical beauty, she declares. Clever physicists have been led up the garden path before, she stresses, pointing to the once-fashionable theories of the ether that Einstein later demonstrated to be redundant.

Ms. Hossenfelder has paid a high price for her counter-orthodoxy…”

And the WSJ to conclude by discreetly celebrating the Fuhrerprinzip which Hossenfelder violated:

“The best string theorists are confident that they are heading in the right direction not only because of the theory’s mathematical beauty but because of its huge potential, despite its formidable challenges.

When Ms. Hossenfelder reiterates in her final chapter that many of the world’s most accomplished theorists are “lost in math,” we cannot help wondering whether it is she who is lost. Time will tell whether many of the world’s leading theoretical physicists have spent decades barking up the wrong tree. Meanwhile, it is pleasing to read that Ms. Hossenfelder now has a research grant and has resumed work on the subject she plainly cares deeply about, no doubt steering well clear of what she regards as bandwagons. In that respect, at least, Einstein would have been proud of her.”



After the plutocratic horror critique above, I must re-establish some justice to Sabine (and myself, indirectly). Here is Nature:

Lost in Math: How Beauty Leads Physics Astray Sabine HossenfelderBasic (2018)

“Why should the laws of nature care about what I find beautiful?” With that statement, theoretical physicist and prolific blogger Sabine Hossenfelder sets out to tell a tale both professional and personal in her new book, Lost in Math. It explores the morass in which modern physics finds itself, thanks to the proliferation of theories devised using aesthetic criteria, rather than guidance from experiments. It also charts Hossenfelder’s own struggles with this approach.

Hossenfelder — a research fellow specializing in quantum gravity and modifications to the general theory of relativity at the Frankfurt Institute for Advanced Studies in Germany — brings a trenchant new voice to concerns that have been rumbling in physics for at least two decades. In 2006, Lee Smolin’s The Trouble with Physics and Peter Woit’s Not Even Wrong fired the first salvos at the trend of valuing mathematical elegance over empirical evidence. Both books took on string theory, a ‘theory of everything’ in which the fundamental constituents of nature are strings vibrating in many more spatial dimensions than the familiar three. Since its entry into mainstream physics in the mid-1980s, the theory has failed to make predictions that would unambiguously verify or falsify it.

Hossenfelder, too, tackles string theory, but her broadsides are more basic. She points to the paucity of experimental data, exacerbated as the machines needed to probe ever higher energies and smaller distances become more costly to build. Given that, she is worried that too many theorists are using mathematical arguments and subjective aesthetics to judge a theory’s validity.”

By the way, my own theory of Quantum Foundations predicts Dark Matter and Dark Energy… It also predicts unpredicted, in contradiction-with Einstein, mass behavior in, say the 2-slit experiment… Namely a dispersion of mass during translation…

Here is more of Nature:

For example, Hossenfelder questions the desire for naturalness — the idea that a theory should not be contrived or have parameters that have to be fine-tuned to fit observations. The standard model of particle physics feels like such a contrivance to many physicists, despite its spectacular success in predicting particles such as the Higgs boson, discovered at the Large Hadron Collider (LHC) at CERN, Europe’s particle-physics laboratory near Geneva, Switzerland. In the theory, to prevent the mass of the Higgs from ballooning beyond reasonable bounds, certain parameters have to be set just so, rather than be derived from first principles. This smacks of unnaturalness.

To get rid of this ugliness, physicists developed supersymmetry — an elegant theory in which every known particle has a hypothetical partner particle. Supersymmetry made the Higgs mass natural. It also showed how three of the four fundamental forces of nature would have been one at energies that existed shortly after the Big Bang (an aesthetically pleasing scenario). It even unexpectedly provided a particle, the neutralino, that could explain dark matter — matter that is unseen, yet thought to exist because of its observed gravitational effect on galaxies and galactic clusters. Hossenfelder explains that in combining everything that theoretical physicists value (symmetry, naturalness, unification and unexpected insights), supersymmetry has become “what biologists fittingly call a ‘superstimulus’ — an artificial yet irresistible trigger”.

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5 Responses to “Mathematical Beauty, Physics, And Truth”

  1. G Max Says:

    Very interesting. So why is string theory not beautiful, and what’s wrong with it?


  2. Benign Says:

    tl;dr in entirety, but reminded me of previous discussion of quantum collapse, in which you opined that the gravitational trigger in the Penrose-Hameroff et al. theory is arbitrary, to which I forgot to agree, having had the thought myself.

    What is the nonlinear “resistance” that motivates this mechanism? Too many holes.

    There may be (is) stuff happening in dimensions we can’t even measure. I continue to conceptualize a space below/beyond/above/inside/outside (?) ours from which our reality bubbles into existence. Gets me through the night.

    cheers! back from toscana!



    • Patrice Ayme Says:

      The small dimension thing was started in early 1920s (Kaluza Klein), unifying gravitation and e-m. It’s the key idea of String theory with compact Calabi-Yau manifold …
      I am into DYNAMICAL dimension theory, where particles themselves are the dimensions… And generate collapses through nonlinear tail interactions/singularization…
      Hoping Tuscany was not boring…


  3. ianmillerblog Says:

    I suspect beauty is in the eye of the beholder. I do not see anything at all beautiful in the Lagrangian of the Standard Model, whereas you could say Newtons equations are Beautiful (or not, as it strikes you) but in the end they are not exactly right. MY view is compare with observation.

    As for Feynman’s quote, did he really say that? I would have though he would have put the sex part the other way around.


    • Patrice Ayme Says:

      You are very right about the Lagrangian of the SM. If anything it’s obviously as pretty as the Ptolemaic model… That means extraordinary ugly. It’s like this Higgs thingie: if I suppose there is an ubiquitous interaction all over, OK, I gather some inertia… And knowing what the masses are, I can predict how much…

      The BURIDAN-Newton laws are beautiful, because they are sparse. So is the RIEMANN-Einstein (Ricci) curvature = mass-energy… Or Poincare E = mcc, Or de broglie mcc = hf…

      Attributed to Feynman: “Physics is like sex. Sure you can get some interesting results, but that’s not why we do it.” Somewhere in Leonard Susskind’s “The Theoretical Minimum” online lecture series he says something to the effect of “as my friend Dick Feynman used to say, ‘physics is to sex as mathematics is to masturbation’ – but I think he meant it in the sense that with physics, you’re doing it with a partner”.

      Feynman like to emphasize he got on with the ladies, he and his tam tam…


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