FOM: Definition of mathematics

Vaughan Pratt pratt at CS.Stanford.EDU
Thu Dec 30 21:11:30 EST 1999

From: Jan Mycielski <jmyciel at euclid.Colorado.EDU>
>Indeed if we regard the problem of defining mathematics as a
>problem of natural science (that is mathematics is viewed as a physical
>process just like other physical processes), then the answer is: 
>mathematics is the process of developing ZFC, i.e., the process of
>introducing definitions and proving theorems in ZFC.

ZFC is to mathematics as electrons are to electronics, or organic
molecules to biology: we can think of the one as constituting the other,
but this viewpoint omits much important intermediate structure.

>My prefered formalism (for ZFC) is not first-order logic, but
>logic without quantifiers but with Hilbert's epsilon symbols. In this
>formal language quantifiers can be defined as abbreviations. This has the
>advantage that the statements in such a language do not refer to any
>universes. So this does not suggest any existence of any Platonic (not
>individually imagined) objects. 

When I do mathematics, regardless of what might be happening in my brain
cells, I feel as though I am working in a world of mathematical objects.
The perception of a Platonic universe is very strong for me, independently
of its reality or lack thereof.   I'd find it hard if not impossible
to prove things if I had to work in a framework expressly designed to
eliminate that perception!

>[Category theorists tried to achieve a better definition of
>mathematics (other than ZFC). But their definition seems to be more
>complicated and hence inferior. Notice that the only primitive concept of
>ZFC is the membership relation, hence it is difficult to imagine a simpler
>theory in which mathematics can be formalised.] 

You're comparing apples and oranges when you say that ZFC is simpler than
category theory on the ground that membership is ZFC's only primitive
concept.  In that sense composition is the only primitive concept of
category theory.  Do you really find composition more complicated than

>But what are imagined sets? My answer is:
>They are imaginary containers intented to contain other
>imaginary containers (one of them, called the empty set, is to remains
>always empty).	

Spoken like an algebraist, or for that matter a category theorist.
I think your outlook is much closer in spirit to the abstract categorical
viewpoint than you allow.  If you were to view *structured* sets such
as groups, vector spaces, and lattices in the same abstract way you view
sets, pretty soon you'd be a rhinoceros yourself.

>Hilbert writes there that "general objects" can explain
>quantifiers (although he introduced his epsilon-symbols much later) and
>he writes that sets are mental objects which can be created prior to their

Category theory is about general objects understood abstractly in terms
of how they transform.

Vaughan Pratt

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