# [FOM] vagueness in mathematics?

Jacques Carette carette at mcmaster.ca
Mon Feb 27 20:47:47 EST 2017

There are quite a few terms in mathematics which are never defined.
"structure" and "forgetful functor" are two.

A few others are:
- variable, parameter, symbol
- the semantics of expressions
- ex: is x/x equal to 1, or is it equal to \lambda y. if y = 0 then
undefined else 1 ?
[both of those semantics is perfectly fine, just incompatible]
- solution
- what "axiom schemas" really range over
- ex: does the induction 'scheme' range over all predicates, or all
predicates-which-can-be-written-down?

There are also issues which are seriously under-discussed:
- (unlike in logic) syntax vs semantics.
- polynomials in expanded form, Horner form, "collected" form
[the point is that these are invisible wrt most definitions of what
a polynomial is]
- "sparse".  Sparsity is very 'basis' dependent.  For example, a
polynomial expressed
in the Chebyshev basis could be "sparse", but would not be in the
usual basis.
- when is it allowable to 'pattern match' on the shape of an object?
- ex: what are you really doing when you have that z = x * y, but in
some mathematical
process you take as 'input' z, and (somehow!) proceed with using x
and y.
One could say that this is sloppy (but never wrong).  There is, in
fact, a
reflection-reification explanation to this, but why is it never
discussed?

It is worth noting that there are similar issues elsewhere.  Most
prominently, in computer science, most people incorrectly identify
arrays and matrices; but arrays are a method of memory storage, while
matrices are representations of (finite dimensional) linear operators
with respect to a given basis.  Asking a computer scientist what a 0x0
matrix is, is an interesting exercise; I am frequently entertained by
the various 'answers' I get to such a query.

Jacques

On 2017-02-15 11:10 AM, Robin Adams wrote:
> [...]
> So, in my opinion, "structure" and "forgetful functor" are vague
> concepts in today's mathematics.
>
> --
> Robin