[FOM] Notations in mathematical practice

[Harvey Friedman]

On Sun, Oct 25, 2015 at 4:21 AM, Arnon Avron <aa at tau.ac.il> wrote:
> In his posting on free logic, Harvey Friedman made
> the following side remark:
>
> "Mathematicians just want to make sure that there is no practical
> ambiguity in what they write."
>
> Do they??

I only meant "practical" in the sense of "for professional
mathematical purposes". I.e., for one professional mathematician
reading another professional mathematician's claims and proofs. They
make obvious inferences as to what is meant, because what is not meant
is things that are patently false. So one starts with the idea that
what was intended is correct and reasonable, and then one works
backward. This is generally done very quickly and easily. If it is not
done very quickly and easily, then the author is asked to clarify, and
the matter is resolved.

Since the very idea of having "perfectly rigorous proofs" has only
recently penetrated the mathematical community, they are not going to
care about the fully legitimate issues that you raise. Of course, all
such issues need to be resolved nicely, preferably with a Gold
Standard, as or if "perfectly rigorous proofs" become of interest to
the mathematical community.
>
> Well, try the following experiment: ask an ordinary  mathematician
> whether or not the following is an identity:
>
>      f'(x) = f(x)'
>
> My experience is that when I ask this question some professor of math
> (with no much background in logic, and sometimes even with) he starred
> at me, starred at the formula, and then, after some hesitation, tells
> me :"well, it depends on what you mean."...

.......