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Re: Landau letter, Re: Mathematica as a New Approach...
*To*: mathgroup at smc.vnet.net
*Subject*: [mg127887] Re: Landau letter, Re: Mathematica as a New Approach...
*From*: Andrzej Kozlowski <akozlowski at gmail.com>
*Date*: Thu, 30 Aug 2012 04:08:06 -0400 (EDT)
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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.
Andrzej Kozlowski
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