[FOM] vagueness in mathematics?
Robin Adams
robin.adams78 at gmail.com
Wed Feb 15 11:10:08 EST 2017
At the same time as this discussion, there is a discussion on the category
theory mailing list about the definition of "structure". Jean Benabou
pointed out that the word is used often, and in a way that suggests there
is a definition on which all mathematicians agree, but this is misleading.
There have been a few attempts (e.g. in Bourbaki), but none has been
generally accepted by the whole community of mathematicians.
A "forgetful functor" is a functor from a category whose objects have "more
structure", to a category whose objects have "less structure". We know
intuitively what we mean when we say that a group is monoid with some
"additional structure". We can write "Let U be the forgetful functor from
the category of groups to the category of monoids", and everyone knows what
we mean. But a formal definition is elusive.
Very like the case of "definite property" in Dedekind's set theory as
opposed to Zermelo's - everyone agrees on the individual cases, but there
is as yet no precise general definition.
So, in my opinion, "structure" and "forgetful functor" are vague concepts
in today's mathematics.
--
Robin
On 14 February 2017 at 18:06, <fom-request at cs.nyu.edu> wrote:
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> Today's Topics:
>
> 1. Reminder: Origins and Varieties of Logicism - II - Further
> Explorations - 20th-21st February, Vita-Salute San Raffaele
> University, Milan, Italy (Andrea Sereni)
> 2. CCA 2017 (Computability and Complexity in Analysis): 2nd call
> for papers (Daniel Gra?a)
> 3. Re: vagueness in mathematics? (tchow)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Mon, 13 Feb 2017 12:23:33 +0100
> From: Andrea Sereni <andrea.sereni at iusspavia.it>
> To: FOM at cs.nyu.edu
> Subject: [FOM] Reminder: Origins and Varieties of Logicism - II -
> Further Explorations - 20th-21st February, Vita-Salute San Raffaele
> University, Milan, Italy
> Message-ID: <64ce5577-68a8-d5c2-feab-7d5ce3502c6d at iusspavia.it>
> Content-Type: text/plain; charset="utf-8"; Format="flowed"
>
> *apologies for cross-posting*
>
> *
> *
>
> *Origins and Varieties of Logicism - II -** Further Explorations*
>
> *
> *
>
> 20th-21st February 2017
> Vita-Salute San Raffaele University
> Milan - Italy
>
>
> PhD programme in /Philosophy/ (Vita-Salute San Raffaele University)
> PhD programme in /Cognitive Neuroscience and Philosophy of Mind/ (School
> of Advanced Studies IUSS Pavia & Vita-Salute San Raffaele University)
> CRESA (Vita-Salute San Raffaele University)
> NeTS Centre @ IUSS
>
>
> /Under the auspices of//
> /Italian Network for the Philosophy of Mathematics, FilMat
>
>
> This workshop is a follow-up to a previous meeting on the /Originsand
> Varieties of
> Logicism/ (http://www.iusspavia.it/news.php?id=2262&menu=menu-news.html).
> Among traditional foundational programmes in the philosophy of
> mathematics, Logicism still attracts much attention, witnessed by a
> revival of studies on its originsand on logicist-oriented issues tightly
> connected with present research developments. By focussing on these
> issues, and extending the range of topics from the previous conference,
> the workshop aims at providing both further historical investigations on
> full-flegded Logicists and other authors standing in connection or
> opposition to such a view - such as Frege, Wittgestein, Ramsey, and
> Hilbert - and theoretical explorations of recent research - such as
> numerical cognition, ordinal abstraction, Neo-Logicism - in order to
> shed new light on the legacy of various kinds of Logicism for
> contemporary philosophy of mathematics.
>
> http://www.unisr.it/en/origins-and-varieties-of-logicism-ii-
> further-explorations/
> <http://www.unisr.it/en/origins-and-varieties-of-logicism-
> ii-further-explorations/>
> http://www.unisr.it/origins-and-varieties-of-logicism-ii-fur
> ther-explorations/
> <http://www.unisr.it/origins-and-varieties-of-logicism-ii-fu
> rther-explorations/>
>
>
> Programme
>
> _/Monday February 20/_
>
> 14.00-14.15 /Welcome greetings/: Massimo Reichlin (UniSR, PhD programme
> coordinator), Francesca Boccuni (UniSR), Andrea Sereni (IUSS, Pavia)
>
> 14.15-15.30*
> Bob Hale* (University of Sheffield)/
> Ordinals by Abstraction/
>
> 15.30-16.45*
> Salvatore Florio* (University of Birmingham)
> /Abstraction without Reference /
>
> 16.45-17.15 Coffee break
>
> 17.15-18.30 *
> Lieven Decock* (Vrije University Amsterdam)
> /Logicism and Numerical Cognition/
>
> _
> _
>
> _/Tuesday February 21/_
>
> 9.15-10.30*
> Michael Potter* (University of Cambridge)/
> Wittgenstein on Ramsey's Logicism/
>
> 10.30-11.00 Coffee break
>
> 11.00-12.15 *
> Fiona Doherty* (University of Cambridge)/
> Frege, Hilbert and Neo-Fregean Logicism: What's in a Name?/
>
> 12.15-13.30
> *Michael Hallett* (McGill University)
> /Frege, G?del and Creation/
>
>
> /
> /
>
> /Organized by/:
> Francesca Boccuni (Vita-Salute San Raffaele University)
> Andrea Sereni (School of Advanced Studies IUSS Pavia)
>
>
> /Conference Venue/:
> Vita-Salute San Raffaele University
> Via Olgettina, 58, 20132
> Milano, Italy
> */Room: Agnodice, Dibit 1/*
>
> /Directions//
> /http://www.unisr.it/en/luniversita/#how-to-reach-the-university
>
>
> Attendance is free. Everyone is invited!
>
> /Info/:
> boccuni.francesca at unisr.it <mailto:boccuni.francesca at unisr.it>
> andrea.sereni at iusspavia.it <mailto:andrea.sereni at iusspavia.it>//
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> ------------------------------
>
> Message: 2
> Date: Mon, 13 Feb 2017 18:36:44 +0000
> From: Daniel Gra?a <dgraca at ualg.pt>
> To: fom at cs.nyu.edu
> Subject: [FOM] CCA 2017 (Computability and Complexity in Analysis):
> 2nd call for papers
> Message-ID:
> <CAM4iN=KQ9-=syqEH6KeHUcU1wU=oDdYZUFJKbveQSYObY-wz_w at mail.gm
> ail.com>
> Content-Type: text/plain; charset=UTF-8
>
> ______________________________________________________________
>
> Second Call for Papers
> Fourteenth International Conference on
>
> Computability and Complexity in Analysis (CCA 2017)
> http://cca-net.de/cca2017/
>
> July 24-27, 2017, Daejeon, Republic of Korea (South Korea)
>
> Submission deadline: April 3, 2017
> ______________________________________________________________
>
>
> Topics
>
> * Computable analysis
> * Complexity on real numbers
> * Constructive analysis
> * Domain theory and analysis
> * Effective descriptive set theory
> * Theory of representations
> * Computable numbers, subsets and functions
> * Randomness and computable measure theory
> * Models of computability on real numbers
> * Realizability theory and analysis
> * Reverse analysis
> * Real number algorithms
> * Implementation of exact real number arithmetic
>
>
> Invited Speakers
>
> * Hee-Kap Ahn (Pohang, Republic of Korea)
> * Ver?nica Becher (Buenos Aires, Argentina)
> * Anders Hansen (Cambridge, UK)
> * Takayuki Kihara (Berkeley, USA)
> * Amaury Pouly (Max Planck Institute, Germany)
> * Linda Brown Westrick (Connecticut, USA)
>
>
> Scientific Programme Committee
>
> * Johanna Franklin (Hempstead, USA)
> * Daniel Gra?a, chair (Faro, Portugal)
> * Vassilios Gregoriades (Turin, Italy)
> * Peter Hertling (Munich, Germany)
> * Gyesik Lee (Anseong, Republic of Korea)
> * Alexander Melnikov (Auckland, New Zealand)
> * Takako Nemoto (Ishikawa, Japan)
> * Arno Pauly (Brussells, Belgium)
> * Crist?bal Rojas (Santiago, Chile)
> * Frank Stephan (Singapore)
> * Klaus Weihrauch (Hagen, Germany)
>
>
> Organising Committee
>
> * Sunyoung Kim (Yonsei University, Republic of Korea)
> * Sewon Park (KAIST, Republic of Korea)
> * Martin Ziegler, chair (KAIST, Republic of Korea)
>
>
> Submissions
>
> Authors are invited to submit 1-2 pages abstracts in PDF format,
> including references via the following web page:
>
> https://easychair.org/conferences/?conf=cca2017
>
> If full versions of papers are already available as technical report
> or arXiv version, then corresponding links should be added to the
> reference list. Final versions of abstracts might be distributed to
> participants in hardcopy and/or in electronic form.
>
>
> Dates
>
> * Submission deadline: April 3, 2017
> * Notification of authors: May 1, 2017
> * Final version: May 29, 2017
>
>
> Conference Web Page
>
> http://cca-net.de/cca2017/
>
> A co-located "Workshop on Real Verification" will take place on
> Friday, July 28, jointly organized by:
>
> * Gyesik Lee (Hankyong National University, Republic of Korea)
> * Martin Ziegler (KAIST, Republic of Korea)
>
> https://complexity.kaist.edu/CCA2017/workshop.html
>
>
> ------------------------------
>
> Message: 3
> Date: Mon, 13 Feb 2017 18:24:42 -0500
> From: tchow <tchow at alum.mit.edu>
> To: fom at cs.nyu.edu
> Subject: Re: [FOM] vagueness in mathematics?
> Message-ID: <3ddca6d40dc307220b90d3739b43aa89 at math.mit.edu>
> Content-Type: text/plain; charset=UTF-8; format=flowed
>
> Stewart Shapiro wrote:
>
> > 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?
>
> In my view, the question of whether mathematics successfully avoids
> vagueness boils down to the question of whether Kripkenstein is silly or
> not. The tacit consensus among mathematicians is that Kripkenstein is
> silly, and so we have access to a bedrock non-vague system of syntactic
> rules on which everything else can be solidly built.
>
> Below is my personal explanation of what Kripkenstein is, copied from a
> MathOverflow answer I wrote some time ago.
>
> Tim
>
> ---
>
> In Wittgenstein's Philosophical Investigations, he makes an argument
> about "private languages" that Saul Kripke later interpreted in a
> certain way. The basic point is that it is very difficult, if not
> impossible, to pin down what a "rule" is. Imagine that you are trying to
> teach a Martian the syntactic rule, "append a 1 to the end of a string."
> The Martian looks puzzled so you give some examples:
>
> 0 -> 01
> 101 -> 1011
> 0010 -> 00101
>
> and so forth. The Martian seems to get the idea, and does a few examples
> to confirm with you. The first few examples look good, but then all of a
> sudden the Martian comes up with
>
> 1111111 -> 111111110
>
> Say what? Somehow it seems that the Martian hasn't gotten the rule after
> all. Or maybe the Martian has extrapolated from your examples to a
> *different* rule? How do you make sure you communicate the rule you
> intend? If you have previously already agreed on some basic rules then
> you can build on those to define new rules, but how do you get started?
> It's hard to get more basic than "append a 1."
>
> Perhaps you could try building a physical device that optically scans
> its input and writes a 1 next to it. But any physical device will
> eventually fail to implement your intended "append a 1" rule when it
> reaches a certain physical limit, so the device doesn't unambiguously
> communicate your intended rule to the Martian either.
>
> No matter how you slice it, it seems that you can't guarantee that you
> have communicated your rule to the Martian, since any finite amount of
> interaction is consistent with infinitely many rules. Once we see this,
> we could take a more radical step and wonder, maybe *I'm* the Martian.
> Maybe all these years I've been assuming that I know what people mean
> when they specify syntactic rules, but actually I've just been lucky and
> haven't discovered the discrepancy between my understanding of what
> "append a 1" means and what everyone else means by it. (Here you can get
> a glimpse of where Wittgenstein's term "private language" comes into the
> discussion.) Even more radically, we could wonder whether the notion of
> a "rule" is incoherent. Perhaps there really is no such thing as a
> "rule" in the sense of some unambiguous finite description of something
> that applies to an infinite number of cases.
>
>
>
> ------------------------------
>
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>
> End of FOM Digest, Vol 170, Issue 15
> ************************************
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