The Logic Off Topic list

Kool
You are obviously a clever guy, and you wrote a great article, but 
frankly, I don't think you fully know what you are talking about. I mean 
no disrespect, but your comments are misleading. I hope you won't mind if 
I set them straight - especially since everyone seems to take these 
comments as valid and insightful.

>>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? ..... Fourier could more strictly be described 
>as a mathematical physicist. Not the same thing at all.

Well a mathematical physicist is not a mathematician. Is a jazz pianist 
who plays something that sounds a bit classical, and actual classical 
pianist/composer? This is not a matter of snobbishness. Real 
mathematicians learn things in detail that scientists gloss over. 

>>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'.


Let me give you an example of the sort of fundamental mistake that 
Fourier made. 

If you have sequence of functions {  f_n(x) : n = 1, 2, .....} then it is 
>obivously< true that
the you can interchange the integral and the limit :

        integral( limit ( f_n(x), n -> infinty) , x = 0 to x = infinity) =
            limit ( integral(   f_n(x) , x = 0 to x = infinity)  , n -> 
infinty)

Right? Wrong. Consider this simple example :
   let  f_n(x) = 1/n for 0 <= x <= n, and f_n(x) = 0 elsewhere.

  Then the 
       integral(   f_n(x) , x = 0 to x = infinity)  = 1 for all n, 
and therefore
       limit ( integral(   f_n(x) , x = 0 to x = infinity)  , n -> 
infinty) = 1 

On the other hand, 
   limit ( f_n(x), n -> infinty)  = 0 (the zero function)
   integral( 0, n -> infinty) , x = 0 to x = infinity) = 0

Clearly 1 does not equal 0.

If you want a stronger example, consider the Heaviside function
   H_n(x) = 1 for x >= n, and 0 for x < n.

This function has a limit of the zero function also, but each of the 
integrals are infinity, so your equation above would look like
    infinity = 0
Clearly this is false.

This is the kind of thing Fourier did. He assumed these kinds of things 
could be done by 'pushing symbols around' as we say in mathematics.

>One way or another, most problems in 
>mathematical physics reduce to boundary value problems. 

I'm not sure if all of your readers will recognize that these are 
differential equations - which can be notoriously difficult to solve and 
are still studied to this day.


>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.

Exactly my point!!! However 'different' is a euphemism. Its like when 
children are asked what is 2 + 2 and someone says 4 but someone else says 
5 and we say that second answer is simply 'different' that the first. No! 
Its wrong.

Mathematical proof developed from the time of Gauss (in the early to mid 
1800's) because so many crap math was done leading to erroneous results 
.... which Fourier carried on the tradition of. 


> Lagrange, Gauss, Archimedes, Euler, Riemann 
>... some manage to straddle both camps. Fourier was like that. 

Excuse me while spit my coffee out at that statement. Guass, Archimedes, 
and Newton are considered the three greatest mathematical minds that 
lived, although Galois was clearly a genius of enormous stature who died 
prematurely. Riemann, Euler, Lagrange, along with Cauchy, Hilbert and 
others were clearly first rate also. But Fourier is not even a 
mathematician and in fact committed errors on the level of a C student.

>Far as I 
>know, Jean Baptiste Fourier is generally regarded as a consummate 
>mathematician.

Nonsense.

>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.

This is completely and totally untrue. It actually the reverse. 
Weirstrauss and others in the late 1800's lead the revolution in 
mathematics that considered functions to be defined on any metric space - 
very abstract indeed - I suspect this is too abstract for manuy 
physicists to  comprehend because they like to deal with real world 
things. (I mean to no disrespect to physicists, in fact my younger 
brother has a PhD in this.) Many functions were devised around that time 
which are can not be expressed as formula - some as limits or other 
things. Mathematics is nothing if not abstract, and it was not the 
mathematicians that didn't like the greater generality.


>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. 

i.e. the first valid study... Dirichlet was first rate.

>However, the 
>Dirichlet definitions simply confirmed Fourier's original view about the 
>generality of functions. 

You mean Dirichlet actually proved Fourier's wild conjectures happened to 
work out.

>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. 

uh huh!

>Riemann later extended Fourier and Dirichlet's work yet further into the 
>concept of the definite integral. 

That may be news to Sir Isaac Newton and Liebnitz who lived about 200 
years earlier, and who actually invented 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.

You could be right, but it was my understanding that another French 
mathematician Lebesgue was the one who discovered the most general form 
of integration which is called  the "Lebesgue Integral" and is based on 
measure theory - which is standard material for first year grad students 
in math, or advanced senior math majors.


>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. 

Again, I think this is wrong. Fourier is being credited with the 
discoveries of dozens of mathematicians who lived a generation or two 
before him.


>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 

I think it was the pure mathematicians who discovered these things in the 
first place. Maybe I should read up more on Fourier but the reason I 
don't know much of his work, is that I never saw any reference to any 
result of the kind you speak of - in fact the only time I ever saw his 
name mentioned was attached to the well known series. There were no 
theorems, lemmas, or corallaries named after him. There were no 
historicial footnotes about his discoveries - but there were plenty on 
many others. I heard more about Sonya Kobaleski, Weirstrauss' protege 
mathematician and young lover than I heard of Fourier.

>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. 

...at least in some ideal circumstances.... but how did he know they 
would work in all cases?

>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.

The mathematicians fleshed out the details to make these fellows look 
important while no one remembers their names.

>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. 

That true! Because mathematical phsycists are not mathematicians, and 
anything that looks reaonsable must surely be true, right? (See example 
above)


>While true there are theoretical restrictions, which Fourier admittedly 
>bypassed, to the full and theoretical generality of his approach, those 

Thats what i'm talkin' bout!

>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. 

Some of the most complicated stuff there is comes from nature. I remember 
one of my professors telling me 20 years ago that if they could simply 
solve non-linear differential equations, then the could unlock the keys 
to such things as earthquakes and weather prediction. To this day, those 
things have not been solved.

I think the proper restatement of the sentence above would be - they work 
out well in ideal conditions or of problems that you handpick to 
cooperate with these methods.

>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 ... 

his 'critics' ... you mean the people actually knew what they were doing?


Fourier had a good imagination for something but didn't have the 
mathematical horsepower to actually prove his ideas. That was left for 
mathematicians to do. In the end, his name is famous for having the 
original idea, but it would have been worthless if not on solid ground.