[FOM] Re: Question on the Scope of Mathematics

[Vladik Kreinovich]

There are two different notions of what mathematics is. 

We mathematicians usually define mathematics by the level of rigor, while 
others define mathematics by objects of study: if it is about abstract 
mathematical objects, it is mathematics, eevn if there is no rigor. 

Mathematicians all (99% probably) agree that mathematics is something that is 
formal or at least formalizable. 

However, physicists and engineers have a completely different notion of what 
mathematics is. To them, heuristic arguments about mathematical notions, 
development of new heuristic methods of solving, say, differential equations, 
is clearly mathematics, even when nothing is proven and all the arguments are 
made on a heuristic physical level of rigor. In short, when a physicist 
develops a heuristic method of solving a specific equation (e.g., Schroeginer's 
equation), this is labeled as physics. If the same physicst proposes a general 
idea for solving different equations, physicists would call this activity 
mathematics. 

>From my experience, it is difficult to convince a physicist or an engineer to 
switch to our definition of mathematics, because with our definition, the above 
heuristic activity about mathematical objects becomes no-man's land: it is not 
physics, and it is not mathematics, so what is it? 

Vladik

P.S. The difference between the two definitions only appears when we have a 
rigorous result about a physically meaningful equation. A physicist would 
probably claim that this acticity is phyics, because it has direct physical 
applications; a mathematician would claim that it is mathematics, because 
theorems are proven. 

Dmytro Taranovsky wrote: 

> My preference is to define mathematics broadly; and use the word "formal
> mathematics" for the narrow notion of mathematics.  However, some believe 
> that
> all mathematics is formal mathematics, and in any case, to avoid uncertainty
> and controversy, as much as reasonable of one's mathematical work should be
> valid as formal mathematics.