[FOM] vagueness in mathematics?
Stewart Shapiro
shapiro.4 at osu.edu
Fri Feb 10 12:50:28 EST 2017
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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