[FOM] Re: Question on the Scope of Mathematics
vladik at cs.utep.edu
Fri Jul 30 15:13:17 EDT 2004
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
>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?
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
> 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.
More information about the FOM