[FOM] vagueness in mathematics?
cscambler at gmail.com
Fri Feb 17 01:41:11 EST 2017
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
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
> 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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