[FOM] vagueness in mathematics?

W.Taylor at math.canterbury.ac.nz W.Taylor at math.canterbury.ac.nz
Fri Feb 10 23:12:08 EST 2017

Stewart Shapiro <shapiro.4 at osu.edu> raises an interesting point:

> Mathematics goes to great lengths to avoid any kind of vagueness or
> indeterminacy. In what sense has it succeeded or not succeeded?
> Doesn't vagueness enter in to almost every other subject?

To the first sentence:- yes, I'm sure we all agree.
To the third - yes, very much so.  Perhaps CS comes closest to math this way.

To the second sentence - almost every appearance of vagueness is rapidly
sorted out by appropriate distinguishing definitions, (as SS goes on to say).

But it seems to me, that there is at least one example of vagueness that
lurks behind a definition and axioms, one that is quite close to the concerns
of many here.  Specifically, just WHAT is included in the phrase,
"collection of ALL sets of natural numbers".

OC it is easy to reply, "All of them - anything you can think of or point to
which is clearly a set of natural numbers, is in the collection".

That's fine, but begs the question.  Specifically, what about those sets
of natural numbers (if any) that we CAN'T think of individually?  Whether
there are any such sets, or what properties they may have, is, I suspect,
a strong source of vagueness which continues to somewhat bedevil math
even to this day - or at least bedevil the philosophy of math.

> Perhaps the more important question here is the extent to which mathematics
> tolerates some sort of indeterminacy in its concepts.

So I would say, that here at least, considerable toleration has been ongoing.
And the indeterminacy is definitely only in the CONCEPT - not at all in
what is allowed to be done with them, i.e. the formalism, (after all,
we can always choose different axiom systems to cope with the latter.)

Perhaps this is a clear distinction between math and philosophy of math?

-- Bill Taylor

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