I will try to keep my answer relatively short for two reasons. One is that all this is very much off-topic on this forum. Secondly, because of your mode of argument is essentially of the kind "what I don't understand is not worth understanding", "what seems unenlightening to me is so for everyone" etc. Obviously there is no way to "disprove" arguments whose sole basis is "I think so", just like there is no way to disprove claims that Bach was a mediocre composer or Vermeer painter. These sort of statements tell us much more about the person making them then about the subject matter. One thing that they all share is that they dismiss the entire vast body of opinion and analysis by experts and scholars ("authorities") that constitute the basis of all "higher" culture (in which I include science and mathematics). Of course, "authorities" can be, on occasions, proved to be wrong. But when we hear opinions of the "I am telling you so" kind, we have no other choice but look at the credentials of the person making them. Thus when a Nobel prize winner or a Fields medal winner or someone like Arnold (who would have got a Fields medal were it not of the appalling weakness of Western mathematical establishment in the face of Soviet pressure) expresses eccentric views, they are worth listening to carefully. But of others we have a right to expect more humility.
> > It is also clear from history that mathematics developed from very concrete foundations in things like counting and measurement. It is incomprehensible to me that many mathematicians wish to deny this, preferring to believe in Platonic fairy tales. A nasty consequence of this denial was the 1960's "New Math" curriculum for American schoolchildren. Supposed to strengthen math comprehension, it did exactly the opposite.
Actually, this had nothing at all to do with Platonic "fairy tales" and most genuine mathematicians strongly opposed it. Most people behind it did not understand real mathematics. Mathematics is above all based on imagination and a certain kind of aesthetic judgement. For that reason it is much closer to what Bach or Mozart did than to what most physicists or engineers do. Formalism in mathematics is only finish and "filling in the details", its nice to have it when the price in time and energy spent is not to high but it is as inessential as is the lack of detail in Leonardo's "The last supper".
> > I cringe when I hear a mathematician talk about Fourier analysis as being about functions in L2. That notion ignores out a large part of the application space: "carrier waves", "flicker noise", delta functions, ... Here we see mathematicians wilfully avoiding *meaningful* infinity.
But this is an absurd misrepresentation. Mathematicians consider Fourier analysis (or harmonic analysis) in all kinds of contexts, e.g. combinatorics, additive number theory, boolean functions, etc, etc. Nobody ever speaks what harmonic analysis is about in complete generality because there would be very little that would be worth saying. The reason for restricting the definition to a certain class of objects (e.g. L2 functions on R^n or a semi-simple Lie group) is because then a certain definite body of theorems become valid (and others are not).
> >> In 1910 the mathematician Oswald Veblen and the physicist >> James Jeans were discussing the mathematics curriculum for physicists at >> Princeton university. "We can safely omit group theory" argued Jeans, >> "this theory will never have any significance for physics". Veblen >> resisted and it is well known that this fact had a certain influence on >> the future history of physics. >> >> This example is, in fact, an excellent illustration of the main point >> that people who argue like you do not get. > > Actually, this is a rather poor example for your argument. But first, to sh ow you that I'm partially on your side here, let me give you a better one. > > Non-Euclidean geometry was one of the great mathematical developments of the nineteenth century. It was driven entirely by the interests of mathematicians: it had no physical motivation. At the same time, the development of quaternions, their subsequent evolution into vector analysis, and the tremendously successful application of these developments to physics (especially electrodynamics) further entrenched three dimensional Euclidean space as *the* model for physical space. > > Then everything changed. Poincar=E9 and Minkowski reformulated Lorentz/Einstein special relativity as non-Euclidean geometry. Einstein then combined Minkowski's geometry with Riemann's, added some physics and came up with his general relativity. GR was such a huge intellectual leap that it seems inconceivable that he could have taken it without the foundation provided by "pure" mathematicians. Without that foundation, I don't think that even now, a century later, we'd have an adequate theory of gravity for astrophysics. > > But group theory? For half a century after the discussions you describe group theory had little influence on physics. Then, it made its big splash with Gell-Mann and Ne'eman's SU(3) theory of the hadron spectrum. But how much insight really emerged from group theory here? I recall Victor Weisskopf explaining the theory to a group of freshman (of which I was one). The gist was "the hadrons are the states of a spectroscopy, and they exhibit the patterns to be expected for a three particle spectroscopy". No group theory, all physics > > OK, you might say. The discovery passed through group theory to mechanism. Group theory was therefore important. The problem with this idea is that a number of other physicists were hot on the trail here, and there was no barrier to skipping straight to mechanism. Gell-Mann and Ne'eman got there first, and they happened to be unusually committed to mathematical abstraction, but a more concretely-minded physicist could have found the mechanism directly: it was not deeply hidden. > > The subsequent influence of the SU(3) abstraction on the development of this theory was negative. While Gell-Mann was certainly aware of the three-particle mechanism (he coined the term "quark"), he believed that mechanism was unnecessary. The trouble was that physical mechanisms have consequences beyond symmetry. In particular, if you hit a blob of particles with a probe of sufficiently small wavelength, you'll see that it's lumpy. And that's exactly what experiments revealed. Hadrons are not content-free consequences of SU(3) symmetry: they are composite objects, and the SU(3) symmetry is a consequence of their composition. > > This reveals the trouble with group theory here: it obfuscates the underlying physics. SU(3) could as easily represent the organizing principle behind somebody's stamp collection. The distinction between stamps and particles might not matter to mathematics, but it's a big deal in physics. > > But one good way to win a Nobel is to win the race. Gell-Mann and Ne'eman were the first ones to completely work out hadron spectroscopy: they won. A (to me unfortunate) consequence was that group theory has gained prominence in physics that goes far beyond its capacity for providing insight. For example, in place of Minkowski's clever geometry, the abstractionists now try to sell us the "Lorentz group". But the fact that Lorentz transforms form a group is trivial and unenlightening: it's the geometry that captures the physical essence here.
I will first refer to my introduction above. For mathematicians group theory is one of the most beautiful areas of mathematics - that alone provides all the justification needed, for mathematics like all arts is all about aesthetics and no more. But group theory is also the essence of symmetry, and hence underlies all science and art. Talking about it the way you seem to be doing, in just one single context, suggests very limited view point and understanding. Today the concept of group is ubiquitous in science and mathematics: ranging from discrete maths and number theory, to solving differential equations, the entire subject of topology (homotopy groups, homology groups etc), chemistry (crystallographic groups) and even biology.
>> There is also another, more direct reason. Knowing that >> there cannot be a general formula in radicals for the roots of a >> polynomial equation means that we no longer need to try to find one and >> instead can turn our attentions to other approaches. This is itself also >> useful in applications (just this of the number of people who post to >> this forum asking for "explicit" solutions of some equation or other). > > Indeed this is a very important result, but Galois theory itself is even less enlightening than other applications of group theory. Actually, it's one of the most enlightening discoveries in the history of mankind.
> >> Finally, where on earth did you get the idea that "philosophers have >> comprehensively demolished mathematical Platonism" or indeed that >> philosophers have "comprehensively demolished" any philosophical idea in >> the entire history of philosophy (including, of course, the idea of the >> Creator)? This is an astounding news to not only to me, but also news to >> my wife, who has been a professor of philosophy at one of the world's >> leading universities, has a PhD in the subject from Oxford University, >> etc, etc. It also would be of interest to physicists like Roger Penrose >> who, obviously in blissful ignorance of this great news, remain >> unabashedly "mathematical platonists". > > Penrose's Platonism is the source of his bizarre pseudophysical theory of how the mind works. To me, it is profoundly unscientific, based in faith in his subjective experience rather than objective evidence.
At last: "To me". > >> Could you please let us know the name of the philosophers who have >> performed this amazing feat? > > There's a *lot* of literature here: I'm surprised you are unfamiliar with it. Let's start with this paper: > > http://www.madepublic.com/getdata2.php?id=25357 > > You also might read "Philosophy of Mathematics (5 Questions)", edited by Hendricks and Leigeb, "18 Unconventional Essays on the Nature of Mathematics", edited by Hersh, and Hersh's book "What is Mathematics, Really?". > > There are a variety of good arguments against Platonism in the works above, but to me one seems especially unanswerable: mathematical Platonism requires that mathematicians possess a supernatural sense that connects them to an objective reality outside the physical world. There is neither any scientific evidence for this nor any explanation for what biological function such a sense would serve.
But the whole argument is just childish. Human beings are born unequal. Some have "absolute pitch" others doubt that such a thing is possible (it is). Some can appreciate the greatest of Bach and Mozart's music, other's don't. Some discover general relativity, others can't understand how to add fractions. What biological function does all this serve?
Mathematical Platonism is modern form is no more than a belief that the natural world is governed by "laws", which are discovered by human beings but exist independently of them and can be expressed in mathematical form. Like all metaphysics worth its salt, this belief can neither be validated nor refuted. Anybody who thinks that it can be "comprehensively demolished" is either using rhetorics more fitting to a political than a philosophical dispute or else should catch up on his Hume.
> > On the other hand, I'm very impressed by N=FA=F1ez and Lakoff's idea that mathematics is a phenomenon that emerges naturally from sufficiently sophisticated embodied cognition. Based in actual science (experimental psychology), this is a very plausible approach to understanding the true nature of mathematics.
I am not impressed. Philosophically I am close to Quine, and so I believe that ontologically there is no fundamental difference between the objects studied by mathematicians, such as groups or sets, and the ones studied by physicists such as atoms or electrons. They are all human posits which we use to "explain" the sense data which arise from some independent reality. But as the the actual nature of this reality we can only speculate and in doing so we can rely on nothing more then our aesthetic judgement. To me, the idea that that reality itself is describable by mathematics (an idea that, as I wrote above, can never be either established nor "demolished") is aesthetically the most satisfactory. I will not say that your view is wrong, for obviously I would be contradicting myself if I did, but I will say that I do have an opinion about your aesthetic taste.