FOM: Defining mathematics

[Vladimir Sazonov]

Matt Insall wrote:

> In an earlier post, I suggested that I would argue that, according to one
> ``definition'' of Mathematics presented in this forum, Einstein's major
> contributions to twentieth century science are mathematical in nature.  In
> this post, I intend to present such an argument.  The definition to which I
> refer is the following(posted by Professor Mycielski, and referred to on
> 12/22/99 by Professor Sazonov):
>
>      Mathematics is a kind of *formal engineering*, that is engineering
>      of (or by means of, or in terms of) formal systems serving as
>      "mechanical devices" accelerating and making powerful the human
>      thought and intuition (about anything - abstract or real objects or
>      whatever we could imagine and discuss).

I am not sure that Professor Mycielski would agree
(however, I see no real and deep reason for that)
with this definition which could be written shortly as

Math. = FS (Formal Systems, possibly in a wide sense).

According to a posting from Mycielski, he defines

Math. = ZFC.

Recall also the definition of Professor Samuel Buss:

Math. = FOLS (FOL based formal Systems).

Note that ZFC \in  FOLS \subset FS.

I insist that more general definition Math. = FS is *really*
more general (say, ZFC or FOLS cannot properly imitate/interpret
arbitrary FS) and believe that it is philosophically more
"correct" or "realistic" because instead of Platonistic fictions
which all of us like so much it refers to reality (cf. more
detailed version of the definition above), Platonistic fictions
being only a part of our psychological real life related to
mathematical activity. I think that solid ground for mathematics
is better than sand.


Vladimir Sazonov