[FOM] Mathematics and formalizability

[Vladik Kreinovich]

Martin Davis provided very good examples illustrating exactly the point that I 
was making. I think these example clarify, to the list readers, the difference 
between rigorous mathematics and heuristic methods that I was talking about in 
my email. Many thanks. 

However, judging by Martin's reply, I was probably not clear enough in stating 
my position. I personally have done both proofs and heuristics (when working 
with astronomers), I personally think both activities are useful, and I do not 
intend to pass any judgement on which of these activities is better.

My objective was not to provide new definitions or new approach, God forbid, 
but rather to state a sociological fact that, in my opinion, explains 
differences in understanding the term "mathematics". 

* many physicists would consider Heaviside calculus and Dirac's delta functions 
(equal to infinity when x=0 and to 0 otherwise) mathematics even before these 
notions were formalized (as distributions in the case of Dirac's delta 
functions).

* on the other hand, I do not think that many mathematicians at that time 
considered Heaviside calculus a mathematical result. 

If Dirac submitted a paper to a mathematical journal explaining that he 
introduces a function that is equal to infinity at 0, without explaining how to 
do it rigorously, I do not think he would have been published. 

I agree with Martin that this type of activity is considered, by many, 
mathematical practice. 

On the other hand, I understand the reluctance with which many working 
mathematicians consider such heuristics. Martin gave three examples when 
heristics later turned out to be correct, after proofs were provided. There are 
however, numerous similar examples when heuristic results turned out to be 
wrong. The existence of such example is one of the main reasons why there was a 
revolution of rigor in the 19 century mathematical analysis (Cauchy, 
Weierstrass, etc.). 

There are two different activities: proving theorems and heuritics analysis of 
mathematical concepts. Many working mathematicians with whom I talk consider 
proving theorems mathematics, but providing interesting heuristics (like some 
exciting un-proven ideas about chaos) not mathematics. 

This is not a question of judgement, these working mathematicians agree that 
the heuristic activity is useful, just like physics can be useful to 
mathematicians, and art can be.

For example, in his first paper, Mandelbrot, the author of the notion of a 
fractal, considered random processes for which the spectrum S(\omega) is 
described by a power law A\omega^{-\alpha}, and for which the overall energy 
\int_0^\infty S(\omega) is finite. Mathematically, it is impossible. Nice 
heuristics led to many practical applications and later were redone in a 
rigorous way. Were his original papers mathematics? If Mandelbrot "proved" 
something using his inconsistent assumptions would that be considered a 
mathematical result? Would his paper be accepted by a purely mathematical 
journal? Would he be give tenure at a Math department if this was his only 
activity? 

I can understand the logic behind both opinions, that heuristic activity is 
mathematics and that it is not, I merely wanted to state that there is a 
difference between the two definitions, and there is a quite large group of 
mathmaticians who would claim that heuristics are not mathematics.  

I probably overstated the percentage of mathematicians who would argue that 
non-rigorous "proofs" is not exactly mathematics, for which I apologize.

However, 

> This narrow approach flies in the face of history and mathematical 
> practice. Most mathematicians aim for "rigor" but give little if any 
> thought to formal systems. Rigor (and ultimately formalizability) can be 
> regarded as goals, but that's all. And as E.T. Bell once remarked 
> "Sufficient unto the day is the rigor thereof".
> 
> Here are some examples of mathematical activity that don't meet Vladik's 
> requirements;
> 
> 1. Euler sums 1/n^2 to pi^2/6 by a heuristic argument factoring the power 
> series for sine as though it were a polynomial and using the relation 
> between symmetric functions of the roots of a polynomial and its  
coefficients.
> 
> 2. Italian algebraic geometry
> 
> 3. Heaviside operator calculus
> 
> Martin
> 
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