[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 

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 

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.


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

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