The Logic Off Topic list

>>    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, 
>
>I'm sorry but wouldn't the result of this be: limit( ((1/n)*x, x = 0 to x 
>= y), y -> infinity) which is not equal to 1 for all n?

I think its right as it was : 
  integral(   f_n(x) , x = 0 to x = infinity)  = (1/n)*n = 1 for all 
values of n > 0.

  the limit of a constant (1) is the constant (1).

  So RHS = 1 (right hand side)

while LHS = integral of the limit which is the integral of the zero 
function which is 0.

I suspect that you're mistake is the same one of Fourier, of assuming you 
can interchange the two infinite processes. So you're in good company.


> But I thought the equation you started with was ONLY true under certain 
>pre-conditions, the one about the limit of the integral is the same as the 
>integral of the limit. One of the pre-conditions being that the functions 
>must 'converge uniformly'( I hope this is the correct term). This means 
>that the limit( abs(f_n -f), n -> infinity) = 0, where abs is the absolute 
>function and f is a function. You introduced a function into the equation 
>that violates that pre-condition; both functions aren't uniformly convergent.
>Check your calculus books, man. At least you gave me a jolt to check my 
>college first year calculus stuff again. Thanks, :-).

I think you're confusing absolute convergence with uniform convergence. 
But I never stated that those preconditions were met by the example I 
gave. In fact, the reverse. My point was, that without suitable 
precautions 'common sense' leads astray.


>> 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.
>
>Wrong. Fourier's equations satisfy the precondition I mention above. 

I didn't say "Fourier did exactly this". I was trying to give a simple 
example to show what can go wrong. Even Kool's paper mentioned several 
times that Fourier did not consider the general case to suit 
mathematicians, and was making assumptions about pre-conditions as you 
call them without any basis. Sure his stuff works in 'nice cases' if you 
skip the step of showing any preconditions are met. 

Part of the job of mathematicians is to find which preconditions are 
needed and absolutely critical.

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

GA Moore wrote:

>I suspect that you're mistake is the same one of Fourier, of assuming you
>can interchange the two infinite processes. So you're in good company.
Does 'suspect" mean you're not sure if this was the error, or kind of 
error, that Fourier made?

>My point was, that without suitable
>precautions 'common sense' leads astray.
And ... often even with 'suitable precautions', such as those suggested for 
calculus, even the best thought-out calculations can also lead astray.

>I didn't say "Fourier did exactly this".
Good.

>I was trying to give a simple example to show what can go wrong.
If they are not examples of the kinds of errors that Fourier specifically 
made, then I really don't see the point of them. The debate here -- or so I 
thought -- is specifically about Fourier and Fourier's errors, and not more 
generally about the kinds of things that 'can go wrong' in mathematics.

>Even Kool's paper
I did not write a paper.

>Sure his stuff works in 'nice cases' if you
>skip the step of showing any preconditions are met.
Fourier had preconditions. He was talking about the theory of heat. There, 
it worked. His error was in suggesting that his ideas would work more 
generally.

What you don't seem to grasp is that people were truly baffled that 
Fourier's ideas worked at all and in any department. Please try to put 
yourself in that era and in that mindset. Fourier deserves credit -- and he 
gets it from everyone except you apparently -- for demonstrating that they 
worked at all. Nowadays, Fourier's ideas are totally ordinary because they 
are a part of everyday life. In its day, it was mind-boggling. Nobody could 
understand how or why it worked AT ALL, never mind try to find out what 
kinds of cases it did and did not work in. The major thrust of the day was 
to demonstrate that Fourier's ideas simply were not workable, period. It 
was not to find out their limits. The initial impetus was to disprove them 
totally. Once it was clear that they did work in at least some cases, then 
the hunt was on to find out which specific cases they worked in. Hindsight 
really is a wonderful thing. With hindsight, we can see that they will work 
in at least some properly defined cases. At the time, it was a mystery that 
they would work at all. How many different ways can I try to get this 
seemingly simple point across?

>Part of the job of mathematicians is to find which preconditions are
>needed and absolutely critical.

And ... a part of the job of mathematicians is to have new and exciting 
ideas to drive mathematics forwards. This is what Fourier did. Mathematics 
needs its dreamers and visionaries. Without them it would become a stagnant 
subject. Long may they live.

I, for my part, am quite happy to call dreamers and visionaries of this 
kind 'mathematicians', even if not all of their ideas prove to be fruitful. 
You, obviously, are not. From what you have been saying you are only 
prepared to restrict the term mathematician to those who come along to the 
new house afterwards and make sure that the rooms are all square and that 
the roof fits properly. Well ... I salute those who go out into new terrain 
and build tents in new areas, and then have the conviction to turn them 
into huts, and then into houses. True that lots of those tents, huts and 
houses will probably -- and rightly -- be blown down by the necessary wind 
of rigorous and logical thinking. I do not for one moment deny the 
importance of directing such a gale upon the house for without being tested 
by it the new idea really isn't worth much. Seems to me, though, that 
rigorous and logical thinking is not a lot of use for actually discovering 
things. It's only any use for making sure, after the fact, that the things 
discovered can be and are well proven. But first ... there has to be 
something worth proving. Mathematics desperately needs people like that for 
without them it has no life. But hey ... what do I know. I'm just an idiot, 
really.

Long live Fourier.

Fourier was a mathematician. ANY book -- and I do mean ANY -- on the 
history of mathematics (excepting only the one yet to be written by GA 
Moore) will tell you that. There has to be a reason why he is in all those 
books on the history of mathematics.

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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>Fourier was a mathematician. ....
>Fourier was a mathematician.

Keep repeating it until you make us believe.