[FOM] pure vs applied mathematics in general

[Kreinovich, Vladik]

We have to be careful to separate two different notions of "mathematics":

* Mathematics as professional mathematicians view it means, in a nutshell, using the highest standards of rigor, i.e., proving results; once a result is proven, it remains valid forever -- as opposed to, e.g., physics where a well-supported and universally accepted law may later turn out to be not always valid, as happened, e.g., with Newton's mechanics. This is the notion with which most people on the list are most familiar.

* On the other hand, if you ask a physicist or an engineer what is mathematics, to them any handling equations and formulas, even when done on a somewhat heuristic level, is mathematics -- especially if these formulas are not specific formulas about a specific physical object, but rather more general.

These two definitions have a big intersection, but they differ: e.g., if someone comes with a heuristic idea of how to solve a differential equations from a certain reasonably general class, then from the viewpoint of an engineer it is mathematics, while from the viewpoint of a professional mathematician, it is not, since nothing is proven. 

Such a distinction is important, since many ideas of "real" mathematics (with proofs) have their origin in such heuristics. For example, the delta-function started as physicists' heuristic and became a legitimate generalized function (distribution), Heavyside calculus was similarly formalized, etc. 
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From: fom-bounces at cs.nyu.edu <fom-bounces at cs.nyu.edu> on behalf of Harvey Friedman <hmflogic at gmail.com>
Sent: Sunday, May 7, 2017 10:24 PM
Very coincidentally, I think, I just contributed briefly to some
ongoing correspondence at mailing lists associated with the Ohio State
University mathematics community concerning related kinds of questions
for mathematics generally. I attach a copy of this very recent
correspondence. I have kept only my name on this correspondence
because the two other writers might not want this to be more public.