[FOM] vagueness in mathematics?

[Chris Scambler]

Isn't there a propensity for vagueness to enter into mathematics with
respect the question: what makes for a good axiom?

Perhaps 'indeterminacy' is better than 'vagueness' here, but the point
remains.

Would V = ultimate L be a good axiom for set theory? Even in the best case
I think this would be unclear.

On Feb 10, 2017 6:06 PM, "Stewart Shapiro" <shapiro.4 at osu.edu> wrote:

> Harvey suggested that some short pieces on philosophical topics be posted
> here, to see if we can generate discussion.  Here is a humble attempt to so.
>
> Here are some related questions, prompted by Harvey:
>
>     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?
>
> Philosophers and linguists mean different things by “vagueness”.
> Sometimes, the focus is on sorites series, the ancient “paradox of the
> heap”.  One might be hard put to come with a series of mathematical objects
> that slowly goes from those having a certain feature to those that don’t.
> Perhaps the more important question here is the extent to which mathematics
> tolerates some sort of indeterminacy in its concepts.
>
> In 1945, Friedrich Waismann introduced the notion of open-texture.  Let P
> be a predicate from natural language.  According to Waismann, P exhibits
> open-texture if there are possible objects p such that nothing in the
> established use of P, or the non-linguistic facts, determines that P holds
> of p or that P fails to hold of p.  In effect, Pp is left open by the use
> of the language, to date.
>
> Waismann explicitly limits focus to empirical predicates.  He notes that
> mathematics does not exhibit any open-texture.  I am not sure of this.  It
> is, of course, hard to imagine a borderline case of, say, “even natural
> number”.  But mathematics has traditionally dealt with other notions, less
> settled.
>
> The lovely Lakatos study, Proofs and refutations concerns the notion of a
> “polyhedron”, focusing on a supposed proof of a theorem, attributed to
> Euler.  The dialogue, which loosely follows history, focuses on strange
> cases, wondering whether they are indeed polyhedra.  One is a picture
> frame, another is a cube with a hollow interior.
>
> I would think that the notion of a polyhedron is as mathematical as it
> gets.  Of course, nowadays, we do not rely on inchoate intuitions, or
> paradigm examples, to indicate our concepts.  We insist on rigorous
> definitions, ultimately, perhaps, in a formal foundation, such as that of
> set theory.
>
>
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