Does the beauty and effectiveness of math in understanding the world prove anything?

May 1, 2016 • 10:30 am

One of the disadvantages of shopping for food early Sunday morning is that Krista Tippett’s “On Being” program is on National Public Radio at 7 a.m. And, of course, I have to listen, cursing to myself for an entire hour. Why do I do it, you ask? I could say that I need to keep my finger on the pulse of America, and that’s one reason, but it also serves as an Orwellian Sixty Minutes of Hate. (“Hate” is too strong; I think that Tippett and her followers are pitiable, though she’s very well compensated.)

Today’s show, actually, wasn’t so bad (I heard only about 40 minutes), as it featured a man who resisted all attempts to couch his thoughts as woo: Frank Wilczek, an MIT professor who, along with David Gross and H. D. Politzer, won the Nobel Prize in Physics in 2004 for work on the strong interaction.  Tippett being Tippett, the topic, of course, wasn’t really physics per se but, as you can see from the show’s title (“Why is the World So Beautiful?“), the “spiritual.” Wilczek has also written several popular books (I haven’t read them), one with the unfortunate title of A Beautiful Question: Finding Nature’s Deep Design. I doubt that it’s teleological or osculates faith, but I wouldn’t have used the word “design”, which of course implies a Designer.

At any rate, Wilczek tackled an interesting topic: the beauty of mathematics and how well “beautiful and simple equations” describe the structure of the cosmos through physical laws. Why are such simple and “beautiful” theories so useful in describing the laws of physics? The wonder that Wilczek evinced resembled that expressed by Eugene Wigner in his famous 1960 paper, “The unreasonable effectiveness of mathematics in the natural sciences.” Here’s a quote from that paper:

It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them. The observation which comes closest to an explanation for the mathematical concepts’ cropping up in physics which I know is Einstein’s statement that the only physical theories which we are willing to accept are the beautiful ones.

I doubt that most physicists would consider this a “miracle” (Wilczek didn’t come close to using that word); and, as I discuss in Faith versus Fact, there are anthropic reasons for the laws of physics being constant rather than variable (our bodies wouldn’t function, and no organism could evolve, if the laws varied wildly), as well as for mathematics being able to describe the laws of physics.

This leaves two questions: why are the laws of physics described with such simple—and, to physicists, beautiful—equations? And why, as Wilczek maintained, has beauty served physicists so well as a guide to truth? In fact, at one point Wilczek said that, when deriving an equation to explain physical phenomena, “It was so beautiful that I knew it had to be true.”

As a working (or ex-working) scientist, I recoil at such statements. To me, beauty cannot be evidence of truth, though it may be a guide to truth. If so, how does that work? But I even wonder how often mathematical beauty itself, which, after all, doesn’t come out of thin air but builds on previous equations known to describe reality, guides the search for truth completely independent of empiricism.  I’m not qualified to answer that question, nor the questions of whether even more beautiful equations could be wrong, or whether it’s all that surprising that the effectiveness of math in describing physics is “unreasonable.”

Of course Tippett tried to turn all this toward spirituality, and at one point asked Wilczek if this beauty was evidence for Something Bigger Out There that others have called God, but he batted away the question. The woman tries to force everything into her Procrustean Bed of Spirituality. But leaving that aside, I have three questions for readers to ponder, and—especially for readers with expertise in math and physics—to give their take in the comments:

  1. Aren’t there “ugly” theories that describe reality? What is a beautiful theory, anyway?
  2. Are there beautiful theories that physics has proposed that turned out to be wrong?
  3. Is it even worth pondering the question (if the proposition is true) about why physical reality is explained by such simple and “beautiful” equations? My own reaction would be “that’s just the way it is,” but clearly people like Tippett want to go “deeper.”

*********

By the way, at the end of the show, Tippett announced the major donors to the show, and the first of these was—surprise!—the John Templeton Foundation (JTF). I hadn’t heard that before, and the JTF isn’t listed among the “funding partners” on the show’s main page:

Screen Shot 2016-05-01 at 9.10.21 AM
WHERE IS TEMPLETON???

But if you search her site you can see the funder of several episodes, and Tippett’s connection with Templeton (she asked the JTF for, and was given, nearly $600,000 to fund nine episodes of “On Being”).

Screen Shot 2016-05-01 at 9.12.59 AM

There would be every reason, of course, for the JTF to sponsor a show that tries to connect science with something “deep”. Further, the blurb for the show on Google, below, jibes very well with Templeton’s aims, for both deal with the Big Questions:

Screen Shot 2016-05-01 at 9.44.03 AMTempleton:

Screen Shot 2016-05-01 at 10.00.04 AM

The “Big Questions” are usually those that have no satisfactory answer, like “What does it mean to be human?” or “Why are equations describing the world so beautiful?”, but they serve to reassure the public that science, as Sir John Templeton asserted, could point us to the divine. That was the explicit aim of the millions left by Sir John to the JTF.

Templeton and Tippett—a match made in Heaven!

124 thoughts on “Does the beauty and effectiveness of math in understanding the world prove anything?

  1. Max Tegmark has been pushing the idea that the Universe IS Mathematical, and has a book to that effect. Google “The Multiverse & You (& You & You & You…)” for an interview of Tegmark by Sam Harris.

    1. I just picked up Tegmark’s “Our Mathematical Universe: My Quest for the Ultimate Nature of Reality” after skimming tbrough it at a bookstore. Looks fascinating to me.

      1. The book unblinkingly follows the theories of modern physics to their logical conclusion: The Multiverse. Physicists are not in agreement on the multiverse idea, but Max points out that the existence of the Multiverse is not a theory or hypothesis, but rather a prediction of theories that are already accepted (quantum theory). I’m uncertain about the Multiverse, but the book is an interesting read and it suggests the Universe is not simply “described by” mathematics, it is in a very real sense a Mathematical Structure. If true, that would go a long way toward explaining “the unreasonable effectiveness of mathematics”.

        1. I have never taken to the version of the Multiverse where there are alternative versions of you and me and so on, since I just don’t get how other pocket universes could bear even an approximation of a duplicate of anything. So I am curious if such an idea is really to be taken seriously.

          1. I think it follows from the belief that the multiverse is infinite in size with random variations throughout. Thus you’d expect everything to happen eventually, just like if you generate an infinite number of randomly selected words you’d expect it to eventually generate the entire contents of the Library of Congress laid end to end.

            But we don’t actually know that the universe or the multiverse is infinite, and we don’t know that the variations are sufficiently random to generate every possibility. A biased word generator might not generate this sentence even with infinite time.

          2. Sometimes when you’re so “close” to something, i.e. our own existence, it’s easy to lose perspective: it may seem unlikely and unpalatable to consider the existence of other universes where there are similar, if not even identical, “versions” of you and me, until you realize how unlikely and unusual it is for us to be who we are in THIS one. If it happened once, why not again? If it happens again, why NOT in multiple variations?

            We see a similar phenomenon all around us albeit on a “smaller” scale: go to different places on the Earth, and you will see “versions” of different types of plants and animals suited to particular environmental niches, some more similar than others; go out into space, and you’ll find differing “versions” of planets, asteroids, suns, etc.- they may differ wildly, but they all exist within, and are determined by, the same basic laws of physics that hold true all across this universe (if there were a “Designer”, this would be it)- the underlying “theme” doesn’t change. So, in another universe, given that its basic laws of physics were at least close to ours, the chances of a universe with life at least similar to our own become increasingly likely.

            And then there’s infinity: even if the multiverse were proven to be true yet it were also proven that it only contained a finite amount of “data”, over an infinite period of time any conceivable arrangement of that data becomes possible. To say, “Even given an infinite amount of time it might never happen” is meaningless, as one is never going to reach the end of that infinite period of time to be sure that it DIDN’T.
            Another interesting thing to consider is that, in an infinite span of time, “time” actually loses its meaning: all material events take place in a “now”; what you really have is an endless collection of “nows”, in one (or more) of which someone identical to you is saying that THEY’RE not comfortable with the idea of YOU existing.

            Does your brain hurt, yet?

        2. “The problem with the multiverse of course is not that you can’t directly observe it, but that there’s no significant evidence of any kind for it: it’s functioning not as a testable scientific explanation, but as an excuse for the failure of ideas about unification via superstring theory. Siegfried makes this very clear, with his argument specifically aimed at those who deny the existence of “supertiny loops of energy known as superstrings”, putting such a denial in the same category as denying the existence of atoms. Those who deny the existence of superstrings don’t do so because they can’t see them, but because there’s no scientific evidence for them and no testable predictions that would provide any.”

          http://www.math.columbia.edu/~woit/wordpress/?p=6196

    2. The wet dream of platonists.

      There is enough results by Chaitin and computer scientists to suggest math is semi-empirical.

      And there is *a lot* of math that doesn’t make it as physics.

      1. A Bunge-style theory of reference shows that mathematics is used in the factual sciences to exactify our *ideas* about reality, not reality itself. (Not surprising, given the implausibility of Platonism.)

        Consequently, Wigner’s question is actually: “how do our ideas match reality”, which is a more fruitful question, IMO.

    1. Well, the standard model is just that — a model. It has yet to be elevated to the level of theory. I suspect it will not, but will be explained by some other theory. Don’t know if that affects its “beauty” or not.

      I also read Tegmark’s book and found it quite interesting. It’s similar to a book by Brian Greene but goes farther on the math idea. Otherwise, as DrDroid pointed out, it assumes quantum mechanics is true and goes on from there. Tegmark, as far as I know, gets no money from JTF.

      1. I still wanna know how you got a picture of *my* cat for your avatar. 😉

        But there’s an article in the latest Physics Today suggesting that the standard model can be simplified by subsuming it under another theory. (And I think it’s by Frank Wilczek. But I can’t check now, it’s at work and I’m at home.) So, right you are.

    2. If you ask physicists, it may have to do with that our particle sector is so “cold”, low energy.

      The simple fields like the inflaton and nothing else applies in a hot universe, but as it cools down the complexity increases due to seemingly random (or not) choices. C.f. homogeneous liquid description vs polycrystalline descriptions.

    3. Earth, Air, Fire, and Water. Ah, those were the days. Things were better back then — simpler, and more elegant.

        1. I think beauty is more an instinctive concept – we like complexity that has a consistent interrelationship – a pattern – and at other times we like simple patterns, and sometimes we like lots of diversity – even interspersed with apparent randomness. Music and art plays on this and different things appear beautiful at different times. I suspect a lot of bits of physics maths are simple and systematic, some complex and systematic, and some are adjustments to make things conform with evidence which happen to always give the right answer but not quite sure why. In the case of the standard model, and Feynman’s action, we know its right because his assumptions about the movement of electrons conform with evidence and explain many other things, whilst all the components of the standard model have now been observed – and besides numerous actual inventions depend on them.

          also on the Occams razor thing about the simplest explanation being more likely to be right. This is usually true – so long as it actually does explain the phenomenon. Sometimes a complex explanation really is needed.

          1. We find beauty (and solve problems) through identifying patterns – and sometimes we like a bit of diversity and shift in those patterns – and sometimes even a short interlude of randomness between patterns
            In science however, the patterns have to solve problems that is they have to be confirmed by replicated empirical evidence in a form that our senses can register

  2. human mind can string a thousand arguments together without getting itself into contradictions…

    There are few human minds (maybe none) that can do this by itself without help from others and/or writing it down on computers, etc.

    I reject this starting premise and therefore what follows, especially alleged miracles.

    1. Neurologists tell us that most humans can only hold together about 5 thoughts in their mind at a time

      1. When I was young I could barely keep three thoughts in my head before I’d drift off into my own little world. Then it seems they were always replaced with thoughts of girls, and the thoughts one gets when ones thoughts are of girls when one is a teenage boy.

        Eventually I was able to train myself to keep numerous thoughts in my head. I practised by designing wooden sailing yachts in my mind, as I was waiting for something, or when I went to bed. I would design the keel and then the hull planking (cedar strip epoxied, usually) and add more and more parts, trying to hold it all together until it either faded away into sleep, or the earlier parts became a single fuzzy conglomeration. I wish I had known of the memory palace back then.

        I’ve found as I get older the number of thoughts I can hold at a time decrease, although I think that’s probably also a side effect of arthritis (ankylosing spondylitis) and it’s effects and treatment. There is nothing like lack of sleep to really mess with a persons mind.

        I met a guy who does the same thing, except he designs houses in his mind.

        1. There’s a true story from years ago about a blind man who built his own house from scratch: not only did he have to design it all in his mind, at every stage of its construction he had to remember what he’d done and what he hadn’t (lest he step through a hole in an unfinished roof or floor).

      2. Im sure it varies between people – maybe this is wrong or it applies to sustained succession throughout the day

        maybe its more like in the course of the day for most of us we think of a certain amount of things at a time in succession and more things when we have relatively “quiet” moments when we aren’t interacting with others a lot.

    2. I don’t think he meant one person thinking all those thoughts at once or even over time. I think he did in fact mean “help from others”. I think “the human mind” here stands for “humanity’s capacity for logical thought”.

      (I do, however, dislike the phrase “unreasonable effectiveness of mathematics”. It strikes me as eminently reasonable that our descriptions of the world (and their extrapolations) should, well, describe the world.)

  3. a couple of answers perhaps?

    1. Aren’t there “ugly” theories that describe reality? Yes – String theory (whether it’s true or not is a different question)

    2. Are there beautiful theories that physics has proposed that turned out to be wrong? Yes – the phlogiston theory. With elegant mathemetics they could calculate exactly how much phlogiston was needed for a given combustion.

    3. Is it even worth pondering the question (if the proposition is true) about why physical reality is explained by such simple and “beautiful” equations? My own reaction would be “that’s just the way it is,” but clearly people like Tippett want to go “deeper.” Depends on the units chosen. For instance if you choose the speed of light for one of your units of measurement E=mC^2 is not very mysteriously elegant.

    1. Or maybe the math we invented is just damn good. It is our invention….if it made clunky equations, we’d modify it so that it didn’t make such clunky equations.

      As much as I dislike describing math as a language (because it really isn’t but I suppose it is metaphorically), I think we adjust our syntax in math just as we adjust our grammar in a living language.

      Tippet seems to suggest that math was given to us on high from God. It would fit though, God has given us so many plagues, he probably thought it was funny to give us math. 😀 I kid, I kid about the math.

      1. Well, what makes something beautiful? It seems to have something to do with neurotransmitters, right? Or maybe it’s resonance with something in our perceptive organs. I’m in a hurry, so I’m expressing this badly.

        1. In this context, to me, beauty is in numbers that relate to each other in some way so that there are no parts left out. There are myriad ways in which this is done, and even more ways in which relationships between numbers leave something left out. When we find no bits left out, we experience a feeling of completeness and we call this experience ‘beauty’.

          1. In fact quantum theory, with its discrete steps, is not a beautiful mathematical theory. Beautiful theories consist of smooth continuous relations, as you have in Newtonian physics and in relativity.

    2. String/M theory is actually simple (beautiful). It is the universes it creates that is ugly, but that happens in all applications of simple theories.

  4. Jerry, you have great fortitude and resilience. I would have turned that off in an uncontrolled motor action as soon as I heard the word, “miracle”.

    As an antidote, I think Douglas Adams is in order.

    This is rather as if you imagine a puddle waking up one morning and thinking, ‘This is an interesting world I find myself in — an interesting hole I find myself in — fits me rather neatly, doesn’t it? In fact it fits me staggeringly well, must have been made to have me in it!’ This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, frantically hanging on to the notion that everything’s going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for.

    1. We are also this puddle, but what i am speaking of will be experienced very slowly. Obviously global temperatures are rising and we are and will be experiencing the fallout from that. But on a longer scale our sun is slowly aging. A billion years from now it will be obviously hotter. Billions of years later life will be impossible on the surface. Later, our oceans will have boiled away, and so on.

  5. This leaves two questions: why are the laws of physics described with such simple—and, to physicists, beautiful—equations?

    The second question is about human psychology. My guess would be along the lines that finding simple equations underpinning things gives a lot of insight and understanding, and evolution has programmed us to like insight and understanding (since, of course, that is what evolutionarily expensive brains are for).

    On the first question, why is nature underpinned by physics that can be described by simple ideas and maths? I could only guess along the lines that PCC-E does, that we can only be present in a universe with underlying stability. Maybe one could only have very complex entities such as ourselves when built on a very ordered substrate.

    The “unreasonable effectiveness” of maths does not seem to me to be puzzling. All basic maths is arrived at as a distillation of real-world empirical experience.

    And all more-advanced maths is then logically entailed by those foundations, so is still “about” the real world.

    Finally, I agree that “beauty” is only a rough guide to truth, but it does seem to be a fairly reliable pointer.

    1. Whenever I read the phrase “the unreasonable effectiveness of math(s)” I try to visualize a universe where math isn’t the least bit effective in describing anything. The laws of physics, 1 + 1 = 2 — they all break down and can’t be applied or pinned to anything because reality is just so damned arbitrary.

      And that would be a reasonable world, one where math is ineffective. You know, the kind we expect and do not find.

      It seems to me that all this oohing and aawing over the astonishing success of math in describing regularities in Nature is suspiciously similar to those arguments for God which assume that the default of Nothing is ever so much more likely than the existence of Something. We have a crisis! We need an explanation for the extraordinary and astonishing fact that something exists instead of Nothing! Plus the universe is not Pure Chaos! Wheel in the Mind of God, making and noticing patterns.

      1. My take is that reality exists (and has always existed) “by default” for the simple fact that there is no such thing as “nothing”; it’s a word and concept demanded by our dualistic thought system as a counterpart to “something”. It’s as ridiculous to claim that “God made the world out of nothing” as it is to claim that “God made the world out of zero.”
        Michael Shermer, in his book, “How We Believe”, quotes someone as saying, “It seems impossible to think of nothing at all”. I quite disagree: when I “think of nothing”, what mental impressions do I get? Why, nothing, of course; if I DO find myself forming a mental impression, I can be certain that I’m NOT thinking of “nothing”- in this way, “nothing” acts as a “placeholder” much as “zero” does in mathematics- it shows that there COULD be something there, but there isn’t. “Space” is the dualistic counterpoint to “Matter”, by the way, although it actually IS “something”, being a part of reality- it is measurable (when figured from one piece of matter to another), can have qualities (warm spaces, hot spaces, tight spaces, dark spaces), etc.

        1. Yes, the minute “Nothing” is described the emptiness has qualities. A genuine nothing-nothing could not even be said to exist. I think the concept then is a bit of a trick, perhaps as you say a placeholder for what cannot be conceived of.

          1. This is why the dichotomy of something vs. nothing is only relevant to the realm of existence. The proper dichotomy to consider is existence vs. non-existence which leaves no wiggle room for those so inclined to essentially and wrongly claim that existence is caused by existence.

    2. I agree that when mathematicians and physicists talk about beautiful theories, they mean simple and powerful/comprehensive ones. And comprehensiveness can itself be viewed as a ticket to simplicity in the ultimate theory (“of everything”).

      But as to why simplicity is such a good truth, I’m surprised nobody has yet mentioned Solomonoff Induction. Imagine an infinite list of all possible theories, stated in some language, and ordered by their Kolmogorov complexity. At each higher level of complexity, there are more possible theories. Therefore at some point of complexity the probability of any theory of greater or equal to that level of complexity being true, has to start getting smaller and smaller, lest the probabilities sum to greater than one.

  6. 1. The equation of the Standard Normal Curve is far from beautiful, even though it accurately describes probability distributions in countless aspects of nature.

    2. Newton’s laws of motion and gravity are simple and beautiful, but Einstein showed that reality isn’t quite that simple.

    3. No matter how many questions are answered by science, those answers will always generate new questions. But that’s what makes science so fascinating. I’d prefer “we don’t know [b]yet[/b]” to “that’s the way it is.”

    1. Oops, wrong HTML. It should have come out as yet.

      I confess that I was in a hurry to get in the first post and even that short delay dropped me to eighth place. You gotta be quick on the website.

  7. 1. I’ve never heard anyone describe the Schrödinger equation as a beautiful equation.

    2. Newtonian Mechanics is beautiful, but wrong with respect to relativity.

    3. It is except when it isn’t. Just as we must accept “simple, but not simpler,” we must accept “beautiful, but not more beautiful.”

    1. Newton’s idealized two body equation may be simple and beautiful but for three or more bodies the necessary introduction of perturbation theory results in equations that are very unwieldy which required long and arduous calculations before computers.

  8. Diana-

    This topic reminded me of Douglas Adams also, but something slightly different:

    “”The Babel fish is small, yellow, leech-like, and probably the oddest thing in the universe. It feeds on brain wave energy, absorbing all unconscious frequencies and then excreting telepathically a matrix formed from the conscious frequencies and nerve signals picked up from the speech centres of the brain, the practical upshot of which is that if you stick one in your ear, you can instantly understand anything said to you in any form of language: the speech you hear decodes the brain wave matrix.”

    It is a universal translator that neatly crosses the language divide between any species. The book points out that the Babel fish could not possibly have developed naturally, and therefore it both proves and disproves the existence of God:

    Now it is such a bizarrely improbable coincidence that anything so mindbogglingly useful could evolve purely by chance that some thinkers have chosen to see it as a final and clinching proof of the non-existence of God. The argument goes something like this:

    “I refuse to prove that I exist,” says God, “for proof denies faith, and without faith I am nothing.”
    “But,” says Man, “the Babel fish is a dead giveaway, isn’t it? It could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don’t. QED.”
    “Oh dear,” says God, “I hadn’t thought of that,” and promptly vanishes in a puff of logic.
    “Oh, that was easy,” says Man, and for an encore goes on to prove that black is white, and gets killed on the next zebra crossing.
    Most leading theologians claim that this argument is a load of dingo’s kidneys. But this did not stop Oolon Colluphid making a small fortune when he used it as the central theme for his best selling book, Well That About Wraps It Up for God. Meanwhile the poor Babel fish, by effectively removing all barriers to communication between different cultures and races, has caused more and bloodier wars than anything else in the history of creation.”

    -The Hitchhiker’s Guide to the Galaxy (copied and pasted from Wikipedia)

    1. I’ve long enjoyed that bit about disappearing in a puff of logic. Adams had such a great way of using words.

  9. Notice that many of the mathematical laws are “ideal” laws. There are a lot of subtle interactions and friction and so on that demolishes the simplicity that others claim to be beauty.

    And we don’t have models that truly predict things like radioactive decay. Yes, we can predict the overall rate, half-life, and how unstable a radioactive isotope is. But separate a hundred radioactive atoms, and your odds of predicting the first to decay is 1%–no better than chance.

    Don’t some of the theories to reconcile the fundamental forces require dividing infinities… That’s hardly a beautiful solution.

    All-in-all, the idea of beauty in those physical laws is somewhere between picking-and-choosing, and subjective interpretation.

    1. “a lot of subtle interactions and friction and so on that demolishes the simplicity”

      reminds me of:

      “I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic.”

      – Horace Lamb

  10. In my limited experience, specialized professionals are prone to call anything relevant to their studies/interests as “beautiful”, whether it’s a slug, a mold, cuisine that by appearance could easily be pre- or post-stomach, or a suppurating wound. Endless flies most beautiful, I guess.

  11. As pointed out to me in my comment on the UK bills not being pretty:

    “Beauty is in the eye of the beholder.”

    Many find beauty in the things they create, be they beautiful or not. Others can’t find ever elusive perfection in their work, never satisfied no matter how much and how high it’s praised by others.

  12. “…there are anthropic reasons for the laws of physics being constant rather than variable…” The Anthropic Principle, while it makes sense, is not a very satisfying theory. It’s untestable, and tautological.

    1. It is eminently testable. I don’t understand why people insist with promoting an erroneous claim to the contrary.

      The weak anthropic principle predicts a distribution, which our world should not be in the tails of. For a prediction that was later tested, see the cosmological constant as predicted by Weinberg. And no, we weren’t located in the tails.

  13. As for mathematics being beautiful, I think it depends on your perspective on the problem. I recall from “Surely You’re Joking, Mr. Feynman” (pg. 86 paperback), an incident where he was calculating the trajectory that bullets would be fired out of a plane, when asked what if the observer was sitting in a different location in the plane than the gunner. He was shocked since, if he had been using (x,y) coordinates, he would just have to add a simple correction; but since he had been using polar coordinates, correcting for the change would turn into a big mess.

    1. I’ve had physicists and mathematicians and such tell me (or the audience) that they do a change of variables for aesthetic reasons sometimes. I think in these situations “beautiful” means “easily solvable”.

  14. Several commenters above noted that Newton’s equations are beautiful/simple but wrong.

    Yes, but if you compared Newton’s and Einstein’s equations side by side without knowing which is right, Einstein’s equation is vastly more surprising and beautiful and simple than Newton’s, in the sense that it explains and connects a much wider variety of seeming unrelated things than Newton’s equation.

    I think beauty is a guide to truth in this sense. If you can encapsulate a wide variety of apparently unrelated things in a single equation that is derived from first principles (possibly radically original first principles, like those Einstein used), and it gives a good match to reality in the various realms where it can be tested now, such a success is very unlikely to be an accident, because there was no wiggle-room in the derivation. The uglier and more complex the theory, the more likely that it contains ad hoc elements that lack deeper significance. These might well explain current phenomena but don’t give us much insight into deeper first principles, so we can’t really have confidence in their extension to new realms.

    The shocking success of Einstein’s theory reflects on the validity of his first principles, and this gives us confidence in their extension to new realms.

    1. Newton’s equations are NOT wrong. The are perfectly right in flat space and with low velocity relative to the speed of light. They are right enough to get us to Pluto with great accuracy.

        1. It would be wrong to predict Mercury’s perihelion using Newton’s equations, but Newton’s equations are not wrong. They are a special case. Einstein’s field equations reduce to Newton’s in flat spacetime and low velocity. Nor can the field equations apply at all energy and distance scales. That is why so many physicists waste their time on string theory.

          1. The numerical predictions of the equations are always wrong. There is no “gravitation-free” limit. There is no classical reality. There is no determinism. No force is ever instantaneously tranferred from one point to another distant point. Even just considering the special theory of relatity, and ignoring gravity, anything that moves violates Newton’s equations. The first principles on which they are based are false.

            For the range of physical parameters we live in, the errors are usually small. But our world runs on the “errors”. The whole structure of our universe would be completely different if there had been no relativistic big bang, no quantum tunnelling, etc, all of which are prohibited by Newton’s equations.

            1. Newtonian mechanics did not purport to answer questions such as action at a distance or quantum tunneling. This is like saying the Modern Evolutionary Synthesis proved Darwin wrong.

              Aristotlean physics was wrong. The miasmic theory of disease was wrong. The theories that superceded them contradicted them. GR contains Newton’s mechanics as a limiting case, just as whatever theory supercedes GR will contain it as a limiting case. When that happens, GR will not be proved “wrong”. It will be superceded by a more general theory.

              1. The flat earth theory is the limit of the spherical earth theory as the radius goes to infinity. Is the flat earth theory not wrong then?

                You make a good distinction between two kinds of being wrong, though. Yes, a theory that is not a limiting case of GR would be more wrong than one that was.

              2. I should add that Newton’s equations did purport to be universal laws, and did prohibit QM tunneling. And they were never used just in the limit of zero gravity and zero velocity. They always gave wrong numerical predictions.

              3. Since quantum tunneling was unheard of, unimaginable, in Newton’s time, why in the world would you think his theory would address the question? Did Darwin’s theory address epigenetics? Does the fact that it didn’t prove Darwin’s theory “wrong”?

              4. It expressly prohibits any low-kinetic-energy particle of any kind from escaping a sufficiently-deep potenial-energy well .

                You seem to argue that, because Newton could not have known this was wrong, then it wasn’t wrong.

                Epigenetics does prove that one aspect of Darwinian theory is wrong (though as you know, Darwin did hedge that part later). That’s not to say that the violation is important (especially since it only persists for a couple of generations), but it’s a violation nonetheless.

              5. Classical Newtonian mechanics cannot explain quantum tunneling, but as far as I know, neither can GR, which is a classical theory. Do you have some insight from some secret unified theory?

          2. Newton’s equations account for ninety percent of the precession of the perihelion of Mercury’s orbit. This was known in 1843. Einstein’s theory accounts for the remaining ten percent because Mercury being the closest planet to the Sun moves just fast enough to make the relevant relativistic term large enough to make an appreciable effect.

    2. This, and its responses, have lots of interest to me. There’s just one point I want to make about several of these, and it has to do with “wrong” or not, and theories being limiting cases.

      I think the latter is a too simplistic way of seeing relationships such as Newton to Einstein, and thence when it is appropriate to call a theory wrong. And let’s stick to really basic physics.

      It seems that the concept of emergence, of Newton out of Einstein, of Maxwell out of quantum electrodynamics, and in the reference below, of the so-called Many Worlds of Everett as a semi-classical emergence out of Everettian quantum mechanics (the latter briefly described in my lengthy thing way below). See

      David Wallace “The Emergent Multiverse”,

      probably very expensive Oxford Press, so you’ll want a good university library. Despite the title, this book is no popularization, but much of it can be read by laymen, sorry, laypersons.
      (To refer to a female as a layperson might get me in trouble with those “snowflakes”!!)
      An earlier companion compilation from 2 conferences with Wallace as one editor (look under Simon Saunders) is “Many Worlds”, giving all 99 sides to the story. These books are 2012 and 2010 resp.

      Actually people who scoff at Everett’s so-called Many Worlds interpretation should take a look and see if maybe they will appear later something like the scoffers at natural selection do now. These are very serious and knowledgable people making that kind of a multiverse very plausible.

      But I diverge. To get back, it would be easy for me to unoriginally make fun of taking limits, but of constants

      e.g. c —> infinity or h —> 0 or in this case, the radius of the earth —> infinity.

      I don’t think people who criticize in that facile way are grappling with the real issue.

      But these ideas of emergence of older theories out of more basic ones is not just vague babble, and is much more subtle. It is done in very precise ways, and gets away from silly arguments by philosophers of science who can never figure out some things other than to say science is always ‘eventually wrong’ isn’t it, yet Newton’s wrong theory can land something on a tiny asteroid about 200 million km away!

    3. Compared to the discussion just above, this will seem banal, but although I agree with your point (which I think basically reduces to Occam’s Razor), I wouldn’t use the word “beauty” the way you used it. There are words that will less poetically and more precisely convey what you meant about the characteristics of equations or theories that work well.

  15. Thinking about this has actually lead to a bit of sympathy with the religious. I’ve only done a BS in Physics, but the attitudes I’m about to express are, I think, shared by at least some professional physicists.

    You see, the mathematics of physics is… strange. We’ve all heard of quantum mechanics and relativity, and how they act differently than we’d think. Most people don’t understand just how weird it all is. It’s incredibly counter-intuitive.

    The ‘miracle’ is that the mathematics explains it just fine. What I struggle to even sorta-kinda comprehend, the equations handle without any difficulty.

    I understand the theory. I understand the equations. I’ve done experiments to confirm them both. I’ve seen them in action and they work. And yet I still, at some fundamental level, have no idea what the hell just happened.

    I told myself to trust the equations. They’re tried, tested and true, but it’s so unusual to have that be the ONLY thing to go on. Because it’s only the math that makes sense. Before I got to those classes, every equation made sense with regards to reality. Relativity doesn’t. The only place it makes any damn sense is on paper, and yet there it is, in reality.

    While trying to describe how these equations made me feel, I couldn’t help but fall into the language of the theologian. It’s knowing without knowing. Have faith in the equations. The fundamental rules of the universe are beyond our understanding.

    To be so confused, and yet to stand in front of the data confirming all these equations… I suspect this is what the religious feel like. To not understand, and yet to have experiences that validate that which you cannot comprehend.

    The key difference is that science is much better evidence than what religion provides… but I suspect it’s the same shared feeling between me staring agog at my first lab experiment confirming these incomprehensible ideas, and a Christian witnessing what they believe to be a miracle.

    1. I am not sure what you are saying. The DEs and PDEs so beloved by physicists are solvable (or not) in many other instances. They are not more or less mysterious or anything there.

      1. No, the equations are perfectly fine. That’s part of my point. The equations are all that is fine. The math works out, despite everything you thought you knew about how the universe works saying it shouldn’t.

        And even after you’ve come to trust your maths and your lab results, what the math represents is very difficult to understand.

        I understood the math, without always understanding the concepts.

        1. The underlying implication between the clear relationship between mathematics and physics is the necessary conclusion that if any phenomenon can be represented mathematically, then what must follow is that the implications of the mathematics itself must apply to, and become, the behaviour of the phenomenon itself. In other words we can DISCOVER and DERIVE what behaviours the phenomenon will exhibit purely by manipulation the equations and “seeing what must follow”. Why the physical world can be described mathematically at all, why there is such a connection, is in my opinion still a great mystery. We do NOT derive the basis of mathematics from what we see in reality, as some say. We derive a mathematical from a limited set of axioms that bear no relation to anything but themselves.

    2. We’ve all heard of quantum mechanics and relativity, and how they act differently than we’d think. Most people don’t understand just how weird it all is. It’s incredibly counter-intuitive.

      The ‘miracle’ is that the mathematics explains it just fine. What I struggle to even sorta-kinda comprehend, the equations handle without any difficulty.

      The fault is not in the stars (or the quanta) but in ourselves. Our intuitions are honed for “meso scale” reality, which makes them useful for survival. At greatly larger or smaller scales things are sufficiently different that our intuitions boggle.

      So it seems impressive, even miraculous, that the mathematical descriptions at these scales work so well while we are left in the dust. But this “unreasonable effectiveness” of the mathematics is actually a mirror of the reasonable ineffectiveness of our intuitions at very large or very small or very fast scales.

  16. We regard it as elgant which doesn’t have all the clutter and unnecessary details and we set out to create theories with that principle in mind. We leave away all of the messy things that we regard as unimportant for the aspects under scrutiny. They are however far from gone. We merely found out that some parts of reality can be neglected for now, because they don’t seem to have an effect.

    What we isolated then gets abstracted. Abstraction is quite marvellous and sems to arise ouf of analogy-making faculties, which seems to be the “core of cognition” as some cognitive psychologists argue. This gives us an intuition of properties and even design (rain is for watering plants and ruining parades). They already seem more elegant than the messy, lumpy things out there.

    Formulas seem to be quite elegant, but I am suspicious. They are often based on units that are not always neat. The complexity may get tugged away under a greek letter. It’s like shoving all the clutter of 3s, 1s, 4s and so on in a pie-shaped wardrope and pretending everything is neatly clean. Other things are defined to be neat and have meaning to us, humans. They seem elegant because it’s our own abstractions reflecting back on us. It’s a bit like wondering why it so happens that a pint of beer fits neatly into a glass.

    The elegant theory describes some small aspect of reality, and when it gets messy, we neatly cut it off there, and bring on another model that is taking it from there. Aren’t we cheating? One elegant theory of everything would be elegant. Due to this, the elegance may be an illusion.

    For more, see:
    1. Duhem-Quine thesis
    2. Model-dependent Realism

    1. Much of that was good, but some points:

      – Symbols is how our brains work, or we couldn’t learn much. (Symbolic clustering is the only way to avoid overtraining in neural nets.)

      – Units cam be elegant or avoided. Either as equations normalized in natural units, or as equations made unitless by taking ratios.

      – Reality isn’t what is described in theory, but the robust behavior of nature.

      – Hypothesis testing for robustness (i.e. observations, hypotheses and theories however incomplete the latter) can be made in isolation.

    2. I was thinking the same – we make short cuts of big long complicated strings of math and things look much cleaner and elegant. I think this is sort of what I was trying to express as well in my reply.

  17. #2 on the list above is easy. Early models of our solar system and the greater universe had explanatory power (which I take to mean beauty), but they were wrong. We are not in the center. The universe is not static, nor is it eternal.

  18. One can approach the observation from two directions. This is by the way an answer to the 3d question.

    One is to look at math. There is a lot of math that is never used by sciences. Not very efficient, and a selection bias that the “unreasonable” people, often platonists, doesn’t consider.

    The other is to look at physics. Generic laws comes from symmetries, Noether taught us that, and are hence simple and beautiful by their very nature.*

    Q: Aren’t there “ugly” theories that describe reality? What is a beautiful theory, anyway?

    A: Theories tend to be simple (beautiful), which means applications tend to be complex (ugly).

    Q: Are there beautiful theories that physics has proposed that turned out to be wrong?

    Many. Examples would include ladder of descent, 4 kinds of matter, 4 kinds of humours, the heliocentric theory (before orbits were considered), that nature abhorred vacuum, flogiston, Titius-Bode’s law, aether, and on and on.

    * One can dig deeper, but I am not sure it clarifies as much as muddies.

    The reason why Noether theorems work is that they encapsulate action.

    Action is based on a quantum particle field view of extrinsic and intrinsic properties. Action encodes how a particle behaves extrinsically as it follows conservation laws that conserves extrinsic properties, under the constraint of conserving internal properties such as spin, in analogy to how internal particle properties are conserved. (Say, conserve momentum due to symmetry under space translation.)

    I can’t say I see the simplicity on this level of detail.

  19. I finished reading and, being primed by Tippett and Templeton, … am I bad if I first read “Peabody Award” as Parody Award?

  20. I know nothing about physics, but from reading the Infinity puzzle, surely Feynman’s equations – which are key to quantum physics – are an example of ugly mathematics in physics. Feynman complained in a lecture 20 years after he came up with these equations – that no one had been able to come up with something more elegant. And no one has improved on his method since either. The technique involves numerous calculated adjustments and is the only way to avoid a result of infinity because actually an electron can samples all paths simultaneously.

    The strength of charge of an electron is relevant to this and it appears to be infinitely concentrated the closer you get to the apparent position of the electron because you can’t actually see it other than the blur of its orbit (I think that paragraph is right).

    Anyway, definitely though, In 1949 Richard Feynman devised a peterbation and renormalisation procedure to arrive at the actual path in mathematical terms. At quantum levels particles are able to travel in any pathway at all and even through time but they effectively always take the shortest path from A to B in current time unless some force holds them in a particular path. Many photon particles sent on converging paths are found to diffract and cancel each other out on some routes and in this reduced form are further reduced by a mirror, into a single ray proceeding from the mirror at the same angle as they approached the mirror. This reduction by interference to all paths is what statistically can be assumed even though the particle can take any path. This enabled Richard Feynment to calculate the probability that any quantum particle will travel from a spot A at a given time to B at a later time and to represent this diagrammatically. He calculated the likely path using an adaptation of the classical mechanics of Richard Lagrange, but involves lots of adjustments and is inelegant. Feynman won a nobel prize for his ground breaking work

    Also if there is a god why is quantum physics so counter intuitive and strange?

    1. “Also if there is a god why is quantum physics so counter intuitive and strange?”

      Because that’s the last gap he has in which to hide. 🙂

    2. Better question:

      If there is a god why do we have to discuss it after millennium of god proposals (religions)? It is hell on Earth.

  21. What is a beautiful theory, anyway?

    A beautiful theory is a theory which can describe a wide range of phenomena, and which has predictive power. A beautiful theory should also have a simple equation at its heart. The best example of a beautiful equation is the Dirac equation which was discovered by searching for a theory that encapsulated existing quantum mechanical ideas like unitarity and which could also accommodate the relativity principle, .. What this equation turned out to be, was not too difficult for Dirac to guess; he just had to look for the simplest equation that was consistent with the relativity principle, .. This relativistic quantum mechanical equation equation then made a surprising prediction that for every particle there was anti-particle. This prediction was confirmed by experiment. This is the hallmark of a beautiful theory / equation.

    Are there beautiful theories that physics has proposed that turned out to be wrong?

    Newtonian Mechanics, Classical Electrodynamics, .. Actually, these theories are not wrong they are rather an approximation or a reflection of a broader and more beautiful theory.

  22. Is it even worth pondering the question (if the proposition is true) about why physical reality is explained by such simple and “beautiful” equations?

    I think there is a definite mystery here. My guess is that this mystery is part of a wider mystery involving consciousness. It’s possible that were not even asking the right questions.

  23. I will put out there that the fact that the ratio of a circles’ diameter to its circumference is a damn irrational number (π ) is pretty ugly.

      1. Why do you say it’s an ugly number? It is 0.1 in base π arithmetic! You’re just a base 10 bigot! Think about all the oppressed people with 3.1419… fingers

  24. How about the mathematical theories of Pythagoras (ratios, proportions) regarding the structure of the universe? These theories were very influential during the Classical period and influenced such people as Plato, Pericles, and the architects of the Parthenon, Iktinos and Kallikrates. Pythagoras considered his mathematical theories as the essence of beauty. The PhD thesis by Anne Bulcken on the Analysis of the Proportions of the Parthenon & its Meaning argues that Pythagoras’ theories are at the very heart of the design of the Parthenon. She says that the Parthenon is a “musical sculpture.” Nonetheless, as beautiful as his theories were and how profoundly they influenced the Greek sense of beauty, his theories of the universe were wrong.

  25. Have not read the comments yet, but I’ll make the trite observation that ‘beauty is in the eye of the beholder.’

    Most people consider mathematics, physics, and a cosmos which has no explanatory need for God to be UGLY. Beauty is instead found in religion, spirituality, and woo. The universe is a story, and has a purpose, and cares about us.

    Astronomy? Ugly. Astrology? Beautiful. As above, so below: it’s all holistic and connected and hits the right emotional buttons. If all you’re going to do is describe and predict the behavior of inert matter then where’s the magic? Science unweaves the rainbow. “Math is hard.” Where do *I* fit into the picture, special snowflake that I am?

    Bottom line, I think whenever we understand, contemplate, or figure out something significant concerning what we care about, it will always seem beautiful to us. Mathematicians may overestimate how much other people care.

  26. Beuitiful but wrong:
    (1) orbits of planets are defined by Platonic solids
    (2) orbits are perfectly circular
    (3) Kaluza–Klein theory

  27. The fact is that most problems in the world cannot be solved with elegant mathematical equations. These include things like turbulent flow, climate, weather and the vast majority of engineering problems. Instead, solutions are approximated using various techniques. To say that the universe is mathematical is a gross overstatement.

    1. I have to agree. The equations governing fluid flow, for example, are ghastly. Viscosity and nonlinearity play merry hell with any simple model. James Gleick’s book ‘Chaos’ is a good primer on the limitations of mathematics to predict physical phenomena.

      But on the other hand, I marvel at how an almost trivially simple algorithm (it will fit in two lines of code on a home computer) can produce the infinite complexity of the Mandelbrot set.

      cr

    2. I agree too. There are some scientists who believe that ‘infinity’ does not exist – although it is *useful* in solving mathematical equations. The square root of -1 is useful too.

      And personally I find pi useful but ugly and disappointing. All those numbers going on and on, it’s not tidy and you can’t create a square with the same area of a circle through simple geometry. Huh.

  28. Knot theory is quite beautiful. It has some applications but the original idea, that different atoms were actually just different knots, turned out to be completely wrong.

  29. My biggest problem with Q1 is I am more inclined to classify theories as beautiful or mundane rather than beautiful or ugly.

    But a good candidate for the latter is the ungainly proof by computer of the 4-color map theorem found in the 1980s.

    There is a simple and elegant proof of a 7-color map theorem for doughnut/coffee cup surfaces, and one can fairly easily devise a map on a donut surface that requires 7 colors. Likewise, you can draw a map on a Mobius strip requiring six colors, and there is a simple proof that this is the upper limit of necessary colors.

    Likewise, there is a simple and elegant proof of a FIVE color map theorem for a flat surface. No map on a flat surface ever requires MORE THAN five colors.

    But for a long time mathematicians could not find any map that required 5 colors, nor prove that four was the upper limit.

    Finally, in the 1980s a proof of the four-color map theorem was found by a computer analysis that broke the whole thing down into multiple subcases (nearly 2000 of them). TO verify by hand takes hundreds of pages.

    Since then two simpler proofs have been found, but they too are computer-assisted, the most recent breaking it down into 633 cases instead of the old proof’s 1,936 separate cases.

    So why are the proofs of the actual upper limit for donuts and Mobius strips so elegant and beautiful, and the proof of the actual upper limit of a flat plain so ungainly, awkward, and top-heavy?

  30. Not much to say, but things fairly obvious:

    Are not two separate (but related) questions being sometimes mixed up here?

    1. Are they, and if so why are theories of fundamental physics beautiful?

    2. Are they, and if so why are theories of fundamental physics mathematical?

    Question 1 depends too much on the definition of “beautiful” to be very interesting to me. But if q.2. is answered yes and here’s why….., then the beauty of mathematics, as an answer, does it for q.1. But again, despite being a mathematician, an assertion that mathematics is beautiful is in the eye of the beholder i.e. the definition adopted of “beautiful”.

    It is amusing that Max Tegmark and Frank Wilczek are colleagues at MIT.

    My emotional answer to 2. is Tegmark’s, discussed above. I know of no other answers not superficial or misunderstanding Wigner (see below). But I don’t think anybody above pointed out that the phrases ‘mathematical equation’ and ‘mathematical system or model or structure’ are not obviously identical. Physicists like Wilczek usually stick to the word equation in popular stuff, rightly so. He has no need for what I just said. But Tegmark’s hypothesis is that the totality of physical reality is a mathematical structure, (probably very) far from being discovered yet, despite the extraordinary successes of quantum field theories and of general relativity. (And then all math structures surely exist in just as strong a sense as ‘our’ physical reality exists.)

    String theory and the standard model of particle physics resp. are a family of quantum theories and a quantum field theory resp. It is certain to most that the latter is ‘false’ or a special case of something more general, just because of dark matter’s existence for example. The LRC might just be in the process of discovering a particle outside the standard model. It seems to be 2 or 3 sigmas now (95 to 99.7% not a fluke), still far from the 5 sigmas for a genuinely accepted discovery.

    But falsity here, such as of Newtonian mechanics, is not all that simple. Would you say that if I could add something like error bars, it could then become ‘true as far as we know’?

    Finally, in the above discussion, the multiverse came up a bit, but I don’t think an important distinction was ever made. Tegmark’s book makes this very clear: if you reject it, are you rejecting
    (a) the logical consequence of these universe “bubbles” existing, following from most versions of inflation theory for some period before the observed big bang at about 300,000 years after the other standard model, of cosmology, would break down as a point? or
    (b) the seemingly logical consequence of Everett taking the basic evolution of the quantum state via a Schrodinger-like equation as everything in quantum theory (no collapse and no interpretation needed)?
    These two are apparently entirely different, on the surface at least.

    And finally, finally, I think nobody has mentioned that the more crucial aspect of Wigner’s famous question is not so much things like the fact that quantum electrodynamics has been experimentally verified in cases to an extraordinary 1 part in 10 billion or so. Rather, it is the fascinating fact that mathematical structures have been discovered beforehand, and others after but independently, by mathematicians (perhaps pursuing beauty!), which turned out to be exactly the right thing for physics: Riemann’s geometry for general relativity, Hilbert space for quantum mechanics, and less well known, distribution theory for Dirac’s delta ‘function’, and finally Elie Cartan’s work on the spin groups long before Dirac (unknowingly) made use of basically that.

    By the way, Dirac was married to Wigner’s sister. Graham Fermelo has written recently a terrific biography of Dirac: IIRC “The Strangest Man”. An old joke is the physics unit, invented in Cambridge, known as the dirac, and defined to be 1 word per hour.

    This did get long—sorry!

    1. No, I didn’t make the distinction between sundry equations and mathematical ideas of platonic objects, because it seems to me Tegmark et al are confusing them anyway.

      ” are you rejecting

      (a) the logical consequence of these universe “bubbles” existing, following from most versions of inflation theory for some period before the observed big bang at about 300,000 years after the other standard model, of cosmology, would break down as a point? or

      (b) the seemingly logical consequence of Everett taking the basic evolution of the quantum state via a Schrodinger-like equation as everything in quantum theory (no collapse and no interpretation needed)?

      These two are apparently entirely different, on the surface at least.”

      There are cosmologists like Ethan Siegel that notes that a) doesn’t imply that the string landscape, for example applies. The low energy physics could be the same in every universe of an inflationary universe. (In fact, they may better be, since the inflation potential and roll down would be the same.)

      And on b), which you seem to claim *is* entirely unrelated to cosmology, you always choose a theory (“make an interpretation” in old parlance) when you apply quantum mechanics. That the Everett theory is simpler (collapses some equations into one) doesn’t mean it is correct, that is part of this discussion of beauty and effectiveness. The theory that – as far as I understand – maps best to semiclassical physics is the relativistic one, where the decoherence process (the same process as in Everett’s theory) is observed with light cones. It seems to be the way quantum theorists treat and understand their subject in practice.

      “the more crucial aspect of Wigner’s famous question is not so much things like the fact that quantum electrodynamics has been experimentally verified in cases to an extraordinary 1 part in 10 billion or so. Rather, it is the fascinating fact that mathematical structures have been discovered beforehand, and others after but independently, by mathematicians (perhaps pursuing beauty!), which turned out to be exactly the right thing for physics:”.

      I did mention that when I noted how much of math (say, your “equations” above), an infinite set actually, turns out to be worthless for physics, That is what we would expect if math is unrelated to physics, except as it is used as a tool to understand physics.

      1. Thanks for the reply. I’m glad to know some have actually suffered through my turgid and over-lengthy prose!

        I’d forgotten to add at the end: If I wrote about 800 words and someone took 5 minutes to read it aloud (!), then the performance is rated at around 10,000 diracs.

  31. Then again, both maths and physics share some pretty similar principles in terms of quantity and magnitude, and inviolable laws that stuff has got to “add up” as it were. Which makes sense if our modern counting system developed as a sequence of names for quantities of analogous physical objects.

  32. I’m sure someone’s already made this point, but the question of beauty is an easy one. It’s a simple matter of definition. We often call things that work well “beautiful”. It’s not as though beauty is some kind of objective, absolute characteristic that simply exists alongside other characteristics like functionality; nor does beauty come first. If something first works well, then we call it beautiful.

    Furthermore, beauty is hopelessly subjective. How can “beauty” serve as a guide to truth? There are far more people who don’t find anything about physics or math beautiful than people who do.

    1. ‘Beauty’ or functionality is not the same as simplicity. I’m not sure whether it’s valid to compare nature to a computer interface, but building an interface for a friend’s stocktaking program has shown me that the simpler (more intuitive, foolproof and easy to use) the visual interface is, the more complicated and elaborate the code behind it has to be.

      cr

  33. I must say, that my interest in and my appreciation of, the Theory of Evolution itself is strongly influenced by my “wonder” at the implications of the mathematical models that describe it. Evolutionary Game Theory, the Price Equation, the Lotka-Volterra equation, the Quasispecies
    equation, and Fisher’s equation – which all describe the process and implications of the dynamics of Evolution. That most of these equations are inter-derivable and related I find rather astonishing. In applying this mathematics we get Evolutionary Algorithms which helps us engineer the “most adapted” solution to physical engineering problems. It is all, well, quite beautiful!

  34. I remember an watching an interview with Stephen Wolfram in which he pointed out that the idea of maths being unreasonable effective is overstated. He claimed that there are plenty of systems that maths has trouble modeling. I can’t find the video but here is a link to another one of his.

    https://goo.gl/bTaon8

    Around 3mins 30s, he points out that there might be a bit of a circular argument going on ie the stuff we understand well is exactly the stuff which maths can model.

    1. I would give Wolfram huge credit for cleverness in all the great things he’s done (e.g. Mathematika), but not so much for wisdom in general fundamental matters. Wilczek would be more my taste there.

  35. Very late to the party on this one, but I’ve been travelling and with only sporadic internet recently.

    As a statistician and mathematician, I’ve repeatedly encountered the “unreasonable effectiveness” or “so beautiful it must be true” tropes held by a surprising number of mathematicians. The older I get, the more these bother me, and I think the biggest reason is the following:

    The math is *never* beautiful. It’s almost always extremely complicated and intricate. But it’s the *notation* that disguises all the ugly machinery and generates a speciously “beautiful” equation or theory.

    As an example, take what I think could easily be argued as the most “beautiful” equation in all of mathematics, Euler’s formula: e^{i*pi} + 1 = 0. It relates arguably the five most common and important numbers in all of mathematics in a single, “perfect” equation. Beautiful, no?

    But the beauty is a fake. The constant “pi” is just notation for the ratio of a circle’s circumference to its diameter. The constant “i” is notation for the square root of -1, which only makes sense as notation for the algebraic solution to the equation x^2 + 1 = 0. And the natural exponential base “e” is either the limit of the “compound interest” expression (1 + 1/n)^n, or the sum of the reciprocal factorials of the positive integers.

    So, removing the notation and restating Euler’s beautiful formula, we get:

    One plus the sum of the reciprocal factorials of the positive integers to the power of the algebraic solution of x^2 + 1 = 0 times the ratio of a circle’s circumference to its diameter equals zero.

    That’s hardly beautiful, and hardly effective as well. It’s mathematical *notation* that is so effective. Mathematics itself is ugly and complicated.

    And to give my own answers to Jerry’s questions:

    (1) In line with the above, I would say that pretty much all mathematical theories are ugly, including the ones found in physics that describe reality. F = ma is a pretty looking formula, but again, it’s the notation that makes it look good. Think about what the notation represents, and even this simple theory is made a lot less elegant. The universe is a complicated place. Of course our models to describe it will reflect those complications!

    (2) The geocentric theory of the universe/solar system was a very simple and (speciously) mathematically beautiful theory of celestial mechanics. It was also very wrong.

    (3) Physical reality is not explained by simple or beautiful equations. Those who contend otherwise, I maintain, are mistakenly attributing the power of notation as some mythical ideal of mathematics.

  36. I think the “unreasonable” effectiveness of math is an overstatement. We don’t have successful mathematical modes for most complex natural phenomena. In fact, the number of natural phenomena we have failed to describe with math (brain, behavior, most biological systems) far outnumber those we can! There is a section bias when we say math is unreasonably effective.

    1. Are you not, analogous to “god of the gaps”, arguing for a ‘reasonable ineffectiveness of math of the gaps’?

      In any case, as I pointed out above, and applying to many other comments above, the “beauty” of mathematics, or of the applications of math to basic physics, is certainly not to do with the mathematical calculations that arise in concrete problems, to apply, or possibly falsify, a basic mathematical physics theory.

      You can argue about the beauty or not of basic Newtonian mechanics. But you are missing the point if you say ‘Look at that ugly calculation’ when I write out in detail the answer to some applied math question asking me to calculate how much snow she melted when an elephant slid down from the peak of my frozen slippery roof, given her weight, house dimensions, etc. And I’m sure Wilczek would say the same about calculations in Quantum Chromodynamics.

  37. P.V/T = constant.
    It is simple, in one form (the P1 P2 equivalence) symmetrical, and true over a wide range of circumstances.
    Sure it is behind the arch anti-capitalist – the Second Law Of Thermodynamics – and all sorts of diving problems. But that doesn’t make it less apt.

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