[lit-ideas] Re: Philosophy Of Maths
- From: "John McCreery" <john.mccreery@xxxxxxxxx>
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Tue, 17 Jun 2008 12:47:58 +0900
On Tue, Jun 17, 2008 at 9:31 AM, John McCreery <john.mccreery@xxxxxxxxx>
wrote:
>
>
> On Tue, Jun 17, 2008 at 8:41 AM, Donal McEvoy <donalmcevoyuk@xxxxxxxxxxx>
> wrote:
>
>>
>>
>> We are tottering it seems on the large issue of realism/anti-realism in
>> the theory of knowledge and, in particular, of mathematical knowledge.
>
>
> Indeed. So perhaps it might be worthwhile to look a bit more carefully at
> what we are talking about. According to a Wiki article that looks pretty
> good
>
> In a private message, it was suggested to me that instead of relying on
Wiki, I Google "Benacerraf" I did, and while this is another reference to
Wiki (http://en.wikipedia.org/wiki/Paul_Benacerraf), it sounds pretty
interesting.
*Paul Benacerraf* is a philosopher of
mathematics<http://en.wikipedia.org/wiki/Philosophy_of_mathematics>
who
has been teaching at Princeton
University<http://en.wikipedia.org/wiki/Princeton_University> since
he joined the faculty in 1960. He was appointed Stuart Professor of
Philosophy in 1974, and recently retired as the *James S. McDonnell
Distinguished University Professor of Philosophy.* Born
inParis<http://en.wikipedia.org/wiki/Paris>,
his parents were Sephardic <http://en.wikipedia.org/wiki/Sephardic>
Jews<http://en.wikipedia.org/wiki/Jew>
from Morocco <http://en.wikipedia.org/wiki/Morocco>. His brother is the Nobel
Prize <http://en.wikipedia.org/wiki/Nobel_Prize>-winning immunologist Baruj
Benacerraf <http://en.wikipedia.org/wiki/Baruj_Benacerraf>.
Benacerraf is perhaps best known for his two papers *What Numbers Could Not
Be* (1965) and *Mathematical Truth* (1973), and for his highly successful
anthology on the philosophy of mathematics, co-edited with Hilary
Putnam<http://en.wikipedia.org/wiki/Hilary_Putnam>
.
In *What Numbers Could Not Be*, he argues against a
Platonist<http://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism>
view
of mathematics, and for
structuralism<http://en.wikipedia.org/wiki/Structuralism#Structuralism_in_the_philosophy_of_mathematics>,
on the ground that what is important about numbers is the abstract
structures they represent rather than the objects that number words
ostensibly refer to. In particular, this argument is based on the point
that Zermelo <http://en.wikipedia.org/wiki/Zermelo> and von
Neumann<http://en.wikipedia.org/wiki/Von_Neumann> give
distinct, and completely adequate, identifications of natural numbers with
sets.
In *Mathematical Truth*, he argues that no interpretation of mathematics
(available at that time) offers a satisfactory package of epistemology and
semantics; it is possible to explain mathematical truth in a way that is
consistent with our syntactico-semantical treatment of truth in
non-mathematical language, and it is possible to explain our knowledge of
mathematics in terms consistent with a causal account of epistemology, but
it is in general not possible to accomplish both of these objectives
simultaneously. He argues for this on the grounds that an adequate account
of truth in mathematics implies the existence of abstract mathematical
objects, but that such objects are epistemologically inaccessible because
they are causally inert and beyond the reach of sense perception. On the
other hand, an adequate epistemology of mathematics, say one that ties
truth-conditions to proof in some way, precludes understanding how and why
the truth-conditions have any bearing on truth.
John
--
John McCreery
The Word Works, Ltd., Yokohama, JAPAN
Tel. +81-45-314-9324
http://www.wordworks.jp/
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