[FOM] philosophically interesting mathematics (was Re: John Baez on David Corfield's book)
Alexander M Lemberg
sandylemberg at juno.com
Sun Oct 5 23:52:28 EDT 2003
My point is that a very deep and provocative set of questions arises from
this example. If you want this spelled out, what I mean is that the
coincidence of the uniqueness of this phenomenon to 4 dimensions and the
dimensionality of the phenomenal world suggests a deep connection between
physical and phenomenal actuality and mathematical possibility. I will
admit that the present stage of philosophical progress on what I have in
mind is more or less at the level of numerology. But that does not take
away from the clear philosophical interest of this discovery.
In my view, most of mathematics is of philosophical relevance and
interest, although often of a very different character from that of
foundational questions. Nevertheless, all philosophical questions are
ultimately related.
Also, progress on most non-foundational issues is less developed because
they are often more complicated and the questions less clearly defined,
and also because less work has been done on them. In many cases, the
philosophical import is in the body of mathematics itself.
What is interesting about Corfield to me is that he does take a broader
perspective than those who address strictly foundational issues.
I also don't get a sense, from the sample I have looked at, that Corfield
is a blind follower of "social constructivism" (a view which, for the
record, I regard as unacceptable). He makes numerous comments about
Lakatos which I haven't had a chance to digest, but they certainly do not
appear to be uncritical agreement.
Sandy
On Sat, 4 Oct 2003 19:32:57 -0400 Stephen G Simpson
<simpson at math.psu.edu> writes:
> Alexander M Lemberg Sat, 4 Oct 2003 16:07:01 -0600 writes:
>
> > exotic R^4, a differentiable manifold which is homeomorphic but
> not
> > diffeomorphic to the usual R^4. This phenomenon is unique to 4
> > dimensions.
>
> OK, good. Corfield and others asserted many times that there is
> lots
> of philosophically interesting mathematics other than f.o.m. I kept
> asking for examples, but none was forthcoming. I am glad to finally
> have an example on the table.
>
> However, I don't find this example completely convincing.
>
> What do you see as the philosophical interest of exotic R^4? Have
> philosophers used exotic R^4 in a serious way to address serious
> philosophical issues? Perhaps you are going to say something about
> our 4-dimensional world being possibly homeomorphic or diffeomorphic
> to R^4 or exotic R^4. I don't see that this is a philosophical
> question, but maybe you do, and I am willing to be persuaded.
>
> ---
>
> Stephen G. Simpson
> Professor of Mathematics
> Penn State University
> http://www.math.psu.edu/simpson/
>
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