The Logic Off Topic list

I am sorry for the delay in responding properly but I have been very busy 
all day.

GA Moore wrote:

>Kool ... wrote a great article,
Heaven's sake ... it was just an email!!!

>  but
>frankly, I don't think you fully know what you are talking about.
Probably not.
I am happy to bow to your superior wisdom.

>  I mean
>no disrespect, but your comments are misleading.
OK.

>I hope you won't mind if
>I set them straight -
Not at all.

>especially since everyone seems to take these
>comments as valid and insightful.
Really? Then it really is best if you set them straight pretty smartish.

>Well a mathematical physicist is not a mathematician.
Depends who you ask, to be honest.

>Real mathematicians learn things in detail that scientists gloss over.
Depends on the historical epoch in which they are judged. This distinction 
between 'mathematician' and 'scientist', as also between 'pure' and 
'applied' mathematics is largely a modern contrivance and convention. 
Historically, they were not originally distinct. By your argument the 
Egyptians and Babylonians were not 'real' mathematicians either and 
therefore should not figure in books on the history of mathematics. Seems 
absurd to me.

>Let me give you an example of the sort of fundamental mistake that
>Fourier made.
Since these were not the 'errors' that Fourier made I have zero interest in 
your examples. The errors that Fourier ACTUALLY made I am very interested 
in. As was Dirichlet, actually.


>If you want a stronger example,
Not particularly, no thank you, because (a) I really don't see what point 
they are making; and (b) contrary to your assertion, they are simply NOT 
the kinds of errors that Fourier made

>This is the kind of thing Fourier did.
No.

>  He assumed these kinds of things
>could be done by 'pushing symbols around' as we say in mathematics.
Since you are not even close to what Fourier actually asserted and did, I 
really do not understand where this judgement comes from.

Anyway ... here's the kicker ... what exactly is wrong with that?
To put it another way ... if mathematics is going to increase, improve, and 
extend its boundaries, someone or other has to come up with exciting new 
ideas, things with new possibilities, new skeletons on which flesh and 
clothing can be put. So ... the people who come up with the skeletons ... 
by your criteria they are 'not really mathematicians'? Seems absurd to me. 
Nonetheless, you are entitled to your opinion. I am entitled to mine.

> >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
So what? I gave an accurate description of the nature of most problems in 
mathematical physics. They reduce to boundary value problems.

Kool said:
> >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!!!
My point also.

>However 'different' is a euphemism.
No it isn't.

>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.
This is plain ridiculous. 5 is known to be wrong at the moment that it is 
offered because we have a valid method AT THE PRESENT for coming by the 
correct answer. At the time that Fourier was working, there was no method 
for judging whether answers were right or wrong. That is what everybody was 
struggling with. It is all very well to sit in a comfy armchair now, some 
400 years later, and smirk and say 'blah blah was wrong'. At the time, 
there was no method.

The problem of the day and of the time was that, probably because there was 
no general method, mathematicians and scientists were nonetheless 
consistently able to get correct answers using Fourier's methods, although 
they had not the slightest idea why or how. You may know NOW, sitting in 
the 21st century, but they didn't know back then. So ... they set about 
investigating how and why they could sometimes get correct results, and 
they set about trying to find out when and how it wouldn't work either. 
They knew ALL ABOUT 2+2=5, so your example is just fatuous and plain 
ridiculous . And what they managed to do was come up with proper methods.

By the same token, the Ancient Egyptians did really know how to produce a 
right-angle triangle whenever they wanted to by using what we now call the 
Pythagorean Triangle. They did not, however, have what we would now regard 
as a valid proof. The Greeks caused a revolution by coming up with such 
things. So ... the Ancient Egyptians were not real mathematicians? Is that 
what we are to conclude? So ... when someone writes a book on the history 
of mathematics they should go straight to Ancient Greece and not bother at 
all with Ahmes and people like that? Because Ahmes and people like that 
could not possibly be real mathematicians because they were not Greeks? 
Strange book on the history of mathematics that would be IMHO.

>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.

Let's see ... Carl Friedrich Gauss, 1777-1855, Jean Baptiste Fourier, 
1768-1830. Pretty contemporaneous.

Gauss radically changed the nature of mathematics. It just became a 
completely different subject. He helped change it because of, admittedly, 
the problems caused by the 'intuitive' approach of people like Fourier. And 
... let's not forget that the 'Fourier approach' was pretty much the same 
approach adopted by Euler, 1707-1783, Lagrange, 1736-1813, and, really 
every other mathematician in history up to that point. Mathematics in that 
era was certainly in a mess. That mess was caused by calculus. And ... 
Gauss more than anyone showed the way out of it. Speaking generally, the 
Gauss method was WAY better than the Fourier method, and you've got no 
quarrel from me if that's all you're saying. However, your proposition that 
Fourier was not 'really a mathematician' is denied. It is denied not just 
by me but by every single book ever written on the history of mathematics, 
and every single Internet site I have examined. I invite anyone reading 
this to check out that fact.

Point is, Fourier lived at the very time that mathematics was changing to 
become what it is today; and changing in such a way as to make it clear 
that the real future of mathematics lay in the methods of Gauss and those 
who followed him and not those who worked like Fourier. It was, however, a 
transitional era. And ... Fourier belonged to that era. A member of the 
theoretical old guard of his day maybe, but nevertheless a real and working 
mathematician of that day, and one of the ones involved in demonstrating 
perhaps why there ought to be a change ... as well as demonstrating what 
was valuable about the old methods. Because what Fourier did was most 
valuable.

>Excuse me while spit my coffee out
Go ahead.

>But Fourier is not even a mathematician
You are entitled to your opinion. I do not share it.
And, to be honest, nor does anyone else I have read who has printed a book 
on the history of mathematics share that opinion. You cannot, in fact, 
write a history of mathematics and the calculus without mentioning Fourier. 
What he did was that important.

>and in fact committed errors on the level of a C student.
Depends where you're standing to judge him. At the time, it was not 
possible to give A's B's and C's because there were no standards and nobody 
to set an exam, and nobody to mark it. Nobody would even have known what to 
PUT in the exam, come to that. This was fringe and trend-setting stuff.

Anyway, Fourier was plenty smart enough to set people a challenge. A man 
must surely be pretty smart to set people a challenge that it took people 
like Dirichlet and Riemann and Lebsgue to resolve. Smart as Fermat, I'd 
say, who also set a challenge that it took aeons to resolve. Or are you 
going to say that Fermat wasn't a real mathematician either?

> >Far as I
> >know, Jean Baptiste Fourier is generally regarded as a consummate
> >mathematician.
>
>Nonsense.
Once again, you are entitled to your opinion. I do not share it. The 
majority opinion sides with me. I rest content with that.

>Mathematics is nothing if not abstract, and it was not the
>mathematicians that didn't like the greater generality.
This is not even remotely close to what I said.

>You mean Dirichlet actually proved Fourier's wild conjectures happened to
>work out.
If they worked out ... then they could not have been that wild, could they? 
Seems as though they were spot on to me.

> >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.
Sorry, but you are again misrepresenting what I said. I clearly said 
'extended ... yet further into'. I did NOT say 'invent'.

> >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 ...
Let's see ... Riemann, 1826-1866, Lebesgue, 1875-1941.
Strange .... I could SWEAR that my original sentence has a 'Later' in it. 
And ... what do I see when I check those dates? I see that Lebesgue was 
born 9 years after Riemann had died. Seems like a 'Later' to me.

>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.
Your point being?

> >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.
What Fourier is being credited with is demonstrating that there did in fact 
exist proper functions that could be expressed as functions and that worked 
as functions, but that nobody had yet invented a proper theory for. If he 
had not had that original idea, what on EARTH would there have been for 
those later 'dozens of mathematicians' to research into? Fourier stood at 
the head of a whole new branch of mathematics, and it is rightly named 
after him.

>Maybe I should read up more on Fourier
I think so, to be frank and honest. There's rather a lot about him you just 
don't seem to know. Which is probably why you have been reduced to making 
things that you then claim he simply must have done, and which he frankly 
did not do by any conceivable reading of history just to try to make your 
point. Me ... I just stick to what he actually did ... and I just also 
stick to readily admitting the things he did not do. My job is easy because 
I do not have to invent problems that Fourier never remotely tried to solve 
as examples of how stupid he was. He may have been stupid, but please stick 
to demonstrating this with problems that he did actually tackle. And ... if 
you stick to problems that he did actually tackle you will observe that he 
was right the bulk of the time ... which was the problem everyone else had 
with him. Please stick to what he actually did. Life's a lot easier that way.

>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.
I wonder why on earth the Fourier series is quite so well known? Let me see 
... could it be because it's INCREDIBLY useful, because it stimulated new 
mathematics, because it still provides wonderful research topics today ... 
stuff like that?

>  There were no
>theorems, lemmas, or corallaries named after him.

So what? He is named for what he did, and for the theoretical contribution 
he did actually make to the history of mathematical thought.

In about 1750 (please forgive me for not remembering the exact date) Euler 
defined a function as a variable quantity that is dependent on another 
quantity. This does, admittedly begin to approach today's definition. 
However, it is simply not good enough. It was FOURIER, and not anyone else, 
who stood up against that definition. He met great opposition in doing so, 
but NOT for the reasons that you personally have given. It was FOURIER, and 
not anyone else, who took the very important step of demonstrating that the 
proposed Euler definition was totally unsatisfactory. He did this -- as you 
know -- by introducing series with sines and cosines as terms. It was this, 
and only this, that led to the later concept that a given representation of 
a function might only be valid for a certain range of values. I think even 
you will surely agree that this is a vitally important part of what we call 
calculus today, no? Well ... Fourier did that. And it was based on 
FOURIER'S suggestions -- and not anybody else's suggestions for there were 
a lot doing the rounds at the time -- that Dirichlet proposed that a 
function is a correspondence that assigns a unique value of the dependent 
variable to every permitted value of an independent variable. Please not 
the importance of 'unique' and 'permitted'. Since you teach calculus you 
will appreciate how vital all of this is. That is what Fourier initiated. 
Nobody else initiated it. Everybody else wanted to trot merrily along with 
what Euler and Cauchy had proposed. Fourier just smiled and showed that it 
simply would not do. He had his idea. It worked. Dirichlet agreed with him. 
Just as well that Dirichlet did, actually.

>There were no
>historicial footnotes about his discoveries
This is because Fourier's discoveries were not footnotes. Seems plain and 
simple to me. Fourier's discoveries were absolutely vital to the theories 
and development of mathematics. He challenged the prevailing definitions. 
He showed that they were totally inadequate. Since you seem to like 
bringing in rather ridiculous analogies, then it's as if Fourier climbed in 
a boxing ring with Euler -- and many important others -- and knocked them 
down. He showed that they were all wrong and totally barking up the wrong 
tree in trying to define a function. They were all wrong. He was the one on 
the right track. He was not ALL right, but he was on a path that was much 
more fruitful than anyone else's. Therefore, he has a whole branch of 
mathematics named after him. And rightly so.

Seems to me that to deny that Fourier is a mathematician is a bit like 
denying that someone who can enter a boxing ring with Muhammad Ali and 
knock him down is not really a boxer but could only have done it by a 
fluke. Nice fluke. Maybe so ... but he'll go down in the history of boxing 
anyway ... and even though people will argue for ever afterwards if he 
really did knock down Muhammad Ali with skill, or whether Ali just kind of 
happened to fall over his own feet by accident while Fourier was 
simultaneously flinging his arm. Either way, the man fell. And ... Fourier 
was there swinging his arm. There's no denying that.

> >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?
He didn't. He was wrong there, as others proved, and as I have accepted. 
But he was right that they worked IN SOME CASES. The 'some cases' bit was 
absolutely vital at the time. It is simply that you are suffering from 
rather a large dose of hindsight, because the only thing you know about is 
the theories within calculus that actually worked. You seem to know zip 
about the many theories that didn't work. But then ... why should you? 
They're of scant interest to a working mathematician after all. 
Nevertheless, I strongly suggest you curl up with a good book on the 
history of calculus one day and read a bit more about the host of theories 
of that time that DIDN'T work. Lots of good people studied them and lots of 
good people couldn't get anywhere with any of them. Therefore (a bit like 
the phlogiston theory) they are known only to the few people like me who 
have an arcance kind of interest in the history of the subject.

Fourier is famous because out of all the truly crackpot theories of the day 
about limits and calculus and so on and so forth, HIS was the one that 
contained the seed that actually bore fruit. I think, actually, if you knew 
a bit more about all those bogus ideas that simply didn't pan out; and if 
you found out a bit more about how even people like Cauchy were locking 
themselves up in dark rooms and screaming and going almost mad (with Cauchy 
that really and nearly happened); and if you read a bit more about how they 
were clutching at every straw they could find to try to resolve these 
issues, then I think you'd have just a tad more respect for Fourier and for 
his achievements. I repeat, if you really think that Fourier's an idiot and 
a bogus mathematician then you really should catch sight of some of them 
others. You'd probably give up the subject in order not to be associated 
with them.

Nobody is denying that Fourier was wrong in certain respects with his 
ideas. But ... there were lots of others who were totally wrong in every 
way. And that's really the point here. Out of all the people who were 
totally wrong, he was the one person who had a really rather smart idea 
that could be worked on. None of the others were in the least workable. And 
that, surely, is a notable achievement. In my boo, that makes him a 
mathematician. Quite why in your book it doesn't is a bit beyond me, 
frankly, but everyone else who studies this subject sides with me so I feel 
quite OK about it all.

In any case ... what's so shameful, and what's there to turn up your nose 
at, at the errors of mathematicians. Euler was wrong about a fair few 
things as well. So ... are you now going to propose that Euler must have 
been a bad mathematician because he made a couple of boo-boos? Just where 
would that kind of madness end?

> >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.
This is out and out snobbery. Of the worst kind.

>That true! Because mathematical phsycists are not mathematicians, and
>anything that looks reaonsable must surely be true, right?
So also is this.

Paul Dirac is another example. A fine mathematician who found a method for 
solving equations that made a major contribution to theoretical physics. 
You, doubtless, would call him 'not really a mathematician'. Nevertheless, 
he has a whole algebra named after him on account of what he contributed to 
mathematical and algebraic theory whilst dealing with the mathematics of 
quanta. But then hay ... mathematical physicists are not mathematicians, 
right? Strange then, isn't it, that Dirac should have a whole system of 
algebra named after him. Probably, though, it's not a bona fide part of 
algebraic theory. How could it be? Dirac was 'only' a mathematical 
physicist, after all. Not really a REAL mathematician. Yet ... there his 
system of algebra stands. What ARE we going to do about that?

> >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!
I never denied them.

>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.
This is what applied mathematicians, mathematical physicists, and 
physicists do. They do not try to solve problems that do not help them 
understand better how nature works. They concentrate on solving those kinds 
of problems that are relevant to the solutions of practical problems. 
What's wrong with that, exactly?

> >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 knew what he was doing. It was them other guys who had a problem.

>Fourier had a good imagination for something but didn't have the
>mathematical horsepower to actually prove his ideas.
What you mean is, to prove his ideas to YOUR satisfaction, which is 
essentially to the satisfaction of a set of criteria that only entered 
mathematics through the work of Gauss and with whom he was admittedly 
contemporaneous.

As a whole, mathematics did very much better to follow the path of Gauss 
than of Fourier. I am not for one moment denying that. It would be rampant 
stupidity. Nevertheless, there is a role in mathematics for the kinds of 
creative insights that people like Fourier have. He set a challenging 
problem. He showed his methods would work in at least some circumstances. 
He set such a problem that it took generations before people could sort it 
out. He kept people like Dirichlet and Riemann well occupied. Not bad for 
"a mere hack". Congratulations, Mr. Fourier. And, in fact, this is a big 
issue in mathematical pedagogy. Why do so many mathematicians burn out at 
such a young age? Is it because the insistence on logical rigour gradually 
stifles their imagination? How are people like Ramanujan to be encouraged, 
without at the same time sacrificing the things that make mathematics so 
valuable and the subject it is. Mathematics will definitely change. It 
always has. It will change when more people of other cultures start doing 
it more intensively.

>That was left for mathematicians to do.
It was left for other mathematicians to do. Fourier was a mathematician.

>In the end, his name is famous for having the original idea,
... and ... what on earth ELSE could he be possibly be famous for??!!!

>but it would have been worthless if not on solid ground.
???!!!!!

I find this a truly astounding but ununderstandable remark. It would surely 
have been even MORE worthless if it had never been thought of. Where would 
we be today without Fourier's original demonstration that the definitions 
of an integral -- as proposed by people like Euler and Lagrange and so many 
others, and as so tamely accepted by everyone else in his day -- were 
false? This is why his name went forth into mathematics.

Fourier was a mathematician. You are entitled to your opinion. I am 
entitled to mine. Mine, though, has the singular virtue that it is accepted 
by every historian of mathematics I have ever come across and whose book I 
have read and whose Internet site I have looked at. This doesn't mean that 
you're wrong. Just means that there's more people like me than there are 
people like you. But hey ... I really am a very boring little guy. Just one 
of the crowd.

Feel free to say anything further that you wish on this subject. Me ... 
with these two emails I'm now writing, I'm done with it. I have way more 
important things in life than this.

Fourier was a mathematician.

Kool Musick
Keep Musick Kool


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