Science And Math

Gonçalo Rodrigues op73418 at mail.telepac.pt
Mon Apr 22 15:51:00 EDT 2002


On Mon, 22 Apr 2002 13:49:00 -0400, Lulu of the Lotus-Eaters
<mertz at gnosis.cx> wrote:

>Gonçalo Rodrigues <op73418 at mail.telepac.pt> wrote previously:
>|mathematical field is roughly divided in the Platonist field, those who
>|believe that mathematical objects exist somewhere out there in space
>|...and the formalist field of those who believe mathematics is
>|essentialy a game we play on paper with symbols with no "real" meaning
>|attached to it besides the convenience in describing natural phenomena.
>
>There is at least one philosophical position missing from Rodrigues'
>enumeration.  The intuitionist approach to mathematics sees mathematical
>reasoning as exposing the "rules of thought."  These rules are not
>independently existing structures in the world (Platonism), but neither
>is mathematics merely symbolic (Formalism).  Instead, there is a
>substantial but procedural fact about the way mathematicians think about
>their subject matter.

Yes, you are right. But I wouldn't be surprised if they really are a
minority.

>
>It is obvious from my other little note in the thread that I feel
>sympathy with the Intuitionist idea about mathematics.  Of course, this
>sympathy is somewhat forclosed by my attachment to the Axiom of Choice.
>I'm not sure I'm able to quite go as far as Brouwer (even though I only
>guiltily fail to).
>
>Yours, Lulu...

If it were just the axiom of choice... The biggest problem for the
constructivist/intuitionist (and my apologies if my lumping them
together is just plainly wrong) is that no non-constructive proofs (for
example, existence proofs relying in double negation, e.g. object x does
not exist ==> a contradiction, therefore object x must exist) is
accepted.

These came about because these non-constructive proofs can yield some
really hair-raising examples (an orange can be split and reassembled in
such a way to yield two oranges, etc.)

But rest assured, there is *intuitinistic/constructivist* mathematics
being done (at least that is what my Topos theory friends tell me, if I
understand them right)

Best regards,
Gonçalo Rodrigues




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