The Logic Off Topic list

GA Moore wrote:

>Just a side note - but Fourier was not a mathematician.
Yes he was.
For example, he taught mathematics at L'Ecole Polytechnique -- and other 
prestigious places. Why would they hire him as a mathematics lecturer if he 
wasn't a mathematician? To be honest, I can't find a single Internet site 
that says he wasn't.

>I think he was a physcist.
Does that mean you are not sure? Fourier could more strictly be described 
as a mathematical physicist. Not the same thing at all.

>He played fast and loose with the symbols to derive his
>results.
Not exactly, no.

>The mathematicians were shocked by the lack of rigor.
Not exactly, no. Fourier's work was later shown to be valid in all 
important respects.

>He was
>making all sorts of assumptions that were not on solid ground.
Not exactly, no. Depends who you're asking. Depends what you mean by 'solid 
ground'.

>However,
>over many decades there were able rigorously proove and further derive
>his ideas.
Not exactly, no.

Fourier claimed: "... but herein we have dealt with a single case only of a 
more general problem, which consists in developing any function whatever in 
an infinite series of sines of cosines of multiple arcs". He was extending 
the work of Lagrange. It was the basis of his investigation into heat 
flows. This necessarily involved the boundary value problem ... and 
therefore made it a problem within the calculus. The theory of boundary 
value problems is that of making solutions of differential equations fit a 
set of prescribed initial conditions. One way or another, most problems in 
mathematical physics reduce to boundary value problems. Usually, you solve 
for those, and you've solved your problem.

As to whether or not Fourier did actually solve his boundary value 
problems, this pretty much depends on your attitude to the distinction 
between pure and applied mathematics. Depends, in other words, what you are 
prepared to regard as 'a valid solution'. Very few pure mathematicians make 
a viable contribution to mathematical physics. The aims and standards of 
proof are somewhat different. Lagrange, Gauss, Archimedes, Euler, Riemann 
... some manage to straddle both camps. Fourier was like that. Far as I 
know, Jean Baptiste Fourier is generally regarded as a consummate 
mathematician.

A major difference between Fourier and others was the attitude to the word 
'function'. In that sense Fourier was of the lineage of Daniel Bernoulli 
who is generally regarded as the first really definable mathematical 
physicist. To Fourier, 'function' included not just those functions given 
by formulae, but also arbitrary representations -- i.e. sets of seemingly 
meaningless and disconnected points. This was what the stricter school of 
pure mathematicians did not seem to like.

Fourier initiated an enquiry into the nature of a function and what it took 
to sum them 'infinitesimally' given that not all functions, in Fourier's 
view, would be 'continuous' in what later became the 'accepted' sense. 
Roughly speaking, what Fourier provided was a method by which given 
functions with only a finite number of discontinuities in a given interval, 
and that therefore contained only a finite number of turning points within 
that interval, could nevertheless be summed by regarding them as an 
infinite sum of sines, cosines, or both. What Fourier pointed out was that 
sines and cosines are periodic, and that periodicity was a way to treat a 
function sufficiently well enough to allow outstanding and intractable 
physical problems -- such as the flow of heat through metal -- to be 
solved. Not only that, but any solution would be accurate enough for any 
reasonable purpose ... even though it might not be 'pin-point' accurate in 
the kind of way beloved by pure mathematicians. That was only ever really 
the issue. At least, that's what I always understood.

On the pure mathematicians' side, about 25 years later Dirichlet conducted 
the first rigorous (i.e. pure mathematical) study into Fourier series. He 
was the first to present a set of properly defined sufficiency conditions 
-- i.e. conditions that satisfied pure mathematicians. However, the 
Dirichlet definitions simply confirmed Fourier's original view about the 
generality of functions. But, against what Fourier had claimed, Dirichlet 
demonstrated that some of the 'inclusivenesses' and 'generalities' that 
Fourier proclaimed for his method, and thus for fully generalized sets of 
points, were false. This is probably what you are referring to. I do not 
really know, though.

Dirichlet achieved his aim with a classic and elegant reductio ab asurdum 
argument. One of the best in the canons of mathematical reasoning. But 
overall, what Dirichlet nonetheless did, in spite of this, was extend the 
function concept, within pure mathematics, in the way Fourier had 
originally suggested. The word 'function' therefore came to include very 
general and much broader correspondences of the Fourier kind.

Riemann later extended Fourier and Dirichlet's work yet further into the 
concept of the definite integral. Thus brought in yet more kinds of 
functions that failed to meet the 'piecewise continuous' requirements 
established principally by Cauchy. Riemann's demonstration that infinite 
sets of discontinuities did not necessarily remove the integrability 
property from the given set of points and/or functions was a big step 
forwards in helping to classify such sets. Riemann, though, limited his 
investigations to Fourier series. He did not go any further into the set 
theory itself. Later researchers, however, were able to establish the 
uniqueness of Fourier series. That's my understanding of the matter, anyway.

Therefore, what Fourier demonstrated was that there were indeed functions 
that although legally and properly expressible as functions, and that could 
therefore be integrated and differentiated, nevertheless did not have 
graphs that could be sketched. This was what the pure mathematicians found 
difficult to comprehend, what they initially had doubts about, and what 
they set about investigating. After all, they were forced to investigate 
it. They could hardly just dismiss it. Much like Heaviside did to later 
pure mathematicians with his somewhat eccentric method of solving the 
differential equations produced by wireless telegraphy, they simply had to 
check what Fourier did out ... simply because it worked. Same with 
Heaviside. They both produced results. If the pure mathematicians hadn't 
checked either of these guys out properly and then bent their theories to 
fit, they would have looked very stupid indeed.

Seems to me that the important point is that Fourier was quite justified in 
ignoring his (pure mathematical) critics. His, after all, was the first 
great step forward in boundary phenomena. As far as mathematical physicists 
were concerned the Fourier method worked just fine. It was plenty rigorous 
enough to produce the results they wanted, and to the kind of accuracy they 
desired in order to make problems in physics tractable. Fourier had made 
life a lot easier. Just find the appropriate Fourier coefficients.

While true there are theoretical restrictions, which Fourier admittedly 
bypassed, to the full and theoretical generality of his approach, those 
exceptions are of very little -- of almost no -- consequence to the fields 
of practice in which Fourier series are mostly used: the study of periodic 
phenomena in nature. Within that field, the exceptions to the Fourier 
method, although important to pure mathematicians, can on a physical level 
be regarded as anomalous cases with no real physical significance. And ... 
it is the physical significance that mathematical physicists and applied 
mathematicians are interested in. Mathematical physicists and applied 
mathematicians are both really mathematicians, though. As was Fourier.

In Fourier's defence, many say that he was quite right to ignore his 
critics ... but that he was probably mistaken in not conceding that those 
critics did maybe have a point when it came to discussing all possible 
curves in all possible mathematical universes .... even though the vast 
majority of those curves or points that might be dredged can make no 
conceivable appearance in the reality that mathematicians try to describe 
through their equations.

To be honest, Fourier published copious numbers of papers in pure 
mathematics. E.g. his work on determinate equations. Lagrange had shown how 
the roots of such an equation could be extracted by using a second equation 
whose roots were the squares of the differences of the roots of the 
original equation. Sturm gave the final solution. But ... this seems about 
as pure mathematical as you can get to me. Surely, anyone who can do that 
is a mathematician?

I am overall intrigued -- and a bit surprised -- by your statement that: 
'as a side note -- Fourier was not a mathematician'. As I said earlier, 
every book and Internet site I look at says that he was; and is. I have 
done my best to indicate why I am so puzzled by your claim, although I 
completely accept that everything I know is erroneous and therefore in need 
of correction.

Kool Musick
Keep Musick Kool


_________________________________________________________
Do You Yahoo!?
Get your free @... address at http://mail.yahoo.com