I agree with your interpretation of Landau's letter but I also think your remarks about mathematics miss the point of what mathematicians do. Mathematicians do not concern themselves with the physical universe - if they did they would be something else. The results which they prove are meaningful within their own realm. The exact nature of this "meaning" is complicated, but it essentially relates to "procedures" (how arguments are conducted) than any physical reality. A great deal of mathematics (for example, almost all of probability theory) is concerned with "infinity", which arguably has no physical meaning at all.
Nevertheless, mathematics turns out to be important for other subjects, like physics, finance etc. This is not the right place for a discussion of why this is so, but in most cases it is a "side-effect" rather than something that mathematicians aim for. Nevertheless this "side effect" has shown itself to be of enormous importance. One of the many spectacular examples was the invention of group theory, which emerged in connection with an existence problem that seemed to have no relation to any physical reality (existence of solutions of polynomial equations in radicals). 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. Group theory was not created because mathematicians wanted to solve the "physical" problem of solving polynomial equations. It was known at least since Newton (and, in fact, before) that one could to so approximately. If mathematicians were physicists the matter would have stopped there - there was no physical reason to be interested in the problem of solving equations in radicals. However, the mathematics developed in order to solve this non-physical problem turned out to be among the most useful mathematics that has ever been invented. One reason was, of course, that working on a solution of a mathematical problem almost always leads to understanding that goes far beyond the narrow concerns of the problem.
So one reason why existence and non-existence theorems are important is that solving them leads to much deeper understanding of mathematics, which in turn turns out often to involve unexpected applications in other areas. 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).
There is a huge list of examples I could give of mathematics that has "no relation" to any physical reality and yet has had a big impact on applications. One example I have already mentioned - probability theory. Almost all important results in this field (in which I also include stochastic processes etc) involve limit theorems - which do not have a "physical reality" for rather obvious reasons. (A nice and easy to understand result is the very useful Kolmogorov's zero-one law <http://en.wikipedia.org/wiki/Kolmogorov's_zero-one_law> ). Probability theory is "useful" mainly because of the "inspiration" it provides to the practical subject of statistics, although in only a relatively small number of cases is this "relationship" an exact, "mathematical" one).
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". Could you please let us know the name of the philosophers who have performed this amazing feat?
On 14 Aug 2012, at 10:21, John Doty <email@example.com> wrote:
> On Wednesday, August 8, 2012 1:16:17 AM UTC-6, djmpark wrote: > >> Landau just expresses the frustration that non-mathematicians have when >> >> confronted with mathematical derivations and proofs that strive for rigor. >> >> They may need the results and material but, darn it, it's hard. > > I don't think so. Landau was much more mathematically sophisticated than you give him credit for. But mathematics is only meaningful to non-mathematicians to the extent that it relates to reality. > > Given that the philosophers have comprehensively demolished mathematical Platonism, it seems clear that the objects of mathematical study are products of the human imagination, not present in reality. It is indisputable that these objects are effective at modeling the behavior of real-world objects, but they also have odd properties of their own. > > Now, sometimes we can utilize these odd properties in applied mathematics. To generalize a statement by Hadamard, sometimes the path that leads to real world truth takes us through unreal realms of imagination. And then we may also need to be aware of physically unreasonable behaviors of mathematical objects, so as not to misapply them. > > But Landau singled out existence proofs. Non-constructive existence proofs are an example of mathematics that is unilluminating to a physicist. A notorious contemporary example is the Navier-Stokes equation: nobody knows if it generally has solutions or not. The Clay Mathematics Institute has offered a $1 million prize for a solution to the existence problem here. But to a physicist, this misses the point. The significance of the Navier-Stokes equation is that functions that satisfy it approximately also approximate real fluid flows. There is no need to know if an exact solution exists, since the Navier-Stokes equation describes the motion of a mathematical fiction: an "ideal fluid". An existence proof here would have no significance to physics. We're more interested in algorithms to efficiently find approximations. We have no doubt that in the real world the fluid will do whatever is inits nature to do, whether an exact Navier-Stokes solution exists or not. > > Generally, what Landau seems to object to is the tendency of mathematicians to study and teach meaningless properties of mathematical objects, rather than those properties that connect mathematical objects to reality. >