[FOM] vagueness in mathematics?
SCOTT RANDAL TIDWELL
srtidwell at ucla.edu
Sat Feb 11 11:40:40 EST 2017
Thanks to Stewart for starting a discussion.
Here’s a sketch of a view that I find attractive. Vague or fuzzy concepts
are not uncommon in mathematics. Polyhedron, set, and number are all
examples, at least in their early development. The need for algorithms and
proofs, however, has led to a concerted effort to replace vague concepts
with more precise surrogates.
Vague concepts are sometimes left alone in the empirical sciences, however.
Maybe some are left alone because there is no real gain to be had from
finding precise surrogates (ice is an example discussed by Mark Wilson).
Others might be left alone because of the prevalence of competing
idealizations in models of the same or related phenomena. The pressing
question is not whether two phenomena are instances of concept P, but
rather how well and in what circumstances model M applies to them.
On Fri, Feb 10, 2017 at 2:58 PM, <fom-request at cs.nyu.edu> wrote:
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> Today's Topics:
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> 1. vagueness in mathematics? (Stewart Shapiro)
> 2. ICMMI 2017 (Lukasz T. Stepien)
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> ----------------------------------------------------------------------
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> Message: 1
> Date: Fri, 10 Feb 2017 12:50:28 -0500
> From: Stewart Shapiro <shapiro.4 at osu.edu>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: [FOM] vagueness in mathematics?
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> gmail.com>
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>
> 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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> Message: 2
> Date: Fri, 10 Feb 2017 19:31:30 +0100
> From: "Lukasz T. Stepien" <sfstepie at cyf-kr.edu.pl>
> To: FOM at cs.nyu.edu
> Subject: [FOM] ICMMI 2017
> Message-ID: <346ee39f21ba4975d0454be2e2ba9bb0 at cyf-kr.edu.pl>
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> Dear FOM-ers
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> I send Announcement of THE 5th INTERNATIONAL CONFERENCE ON
> MAN-MACHINE INTERACTIONS ICMMI 2017,
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> which I have received from the Organizers of ICMMI 2017.
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>
> Best regards
>
>
> Lukasz T. Stepien
>
> ---
>
> Lukasz T. Stepien
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> ____________________________________________________________
> _______________________
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> ICMMI 2017 CALL FOR PAPERS
>
> THE 5th INTERNATIONAL CONFERENCE ON MAN-MACHINE INTERACTIONS ICMMI 2017
>
> October 3-6, 2017
>
> Cracow, Poland
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> http://icmmi.polsl.pl
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> Organised by the Institute of Informatics, the Silesian University of
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> ____________________________________________________________
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> Dear Colleague,
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> We would like to remind you that full paper submission deadline for
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> Conference: October 3-6, 2017
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> SOLICITED PAPERS
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