FOM: NATURE of mathematics

[Vaughan Pratt]

Julio Cabillon's questions seem like fun, hope it's ok if I join in.

>1) Should mathematics on other planets (of other galaxies) be entirely
>   different from ours -- provided its existence? Should it be similar?
>   Isomorphic? What do you think? ...

I think no planet can claim to have advanced mathematics until it has
set theorists, category theorists, and large stadiums where people can
go to watch them wrestle, mentally or otherwise.  On planets with TV
networks or web browsers, people can watch from home.  More advanced
planets have think tanks where real mathematicians can go to get real
work done uninterrupted by sideshows.

>2) Do you endorse that "mathematical reality lies outside us, that our
>   function is to discover or observe it, and that the theorems which
>   we prove, and which we describe grandiloquently as our 'creations',
>   are simply our notes of our observations"? If so, I would appreciate
>   to know why?

"He's a real nowhere man, sitting in his nowhere land."  --crew of the
Yellow Submarine, taunting Jeremy Hillary Boob, Ph.D. (who took it in
very good humour, a sterling example for the rest of us).

>3) In the following incomplete sentence,
>
>          "Hamilton .......... the quaternions in 1843",
>   let us suppose we have to fill in the blank with a word.
>   [discovered / invented / constructed / found / conceived / ...]
>   What would be your choice?

Depends, is it Monday / Tuesday / Wednesday / Thursday / Friday /  ... ?

The nature of mathematics is too undetermined to support a fixed choice.

Personally I liked his symmetrization of the Newton-Lagrange laws of
motion even better.  He was much younger then, supporting the thesis
that the vigor of people's ideas matches that of their body.

>4) Do you think that the probability that there's another Mersenne prime
>   is greater than that that there's a mistake in the proof that the one
>   recently discovered is prime?

Yes, modulo the caveat that probability is not an absolute notion but
is always relative to a sample space.  Here we could quibble endlessly
over the sample space appropriate to making the question precise.

>5) In what sense can we say that mathematics is timeless true?

The sense is felt most strongly of the world's mathematics, and least
strongly of our own, when the proctor says "Pens down".

>6) In the year 2098 (say, in case our fragile planet still be there), do
>   mathematicians will believe that there *really* are infinitely
>   many natural numbers?... Yet, without doing futurology, do you
>   believe here and now that there *really* are infinitely many natural
>   numbers? If so, where are there?... Do you really believe that it is
>   conceivable that these infinitely many natural numbers did exist on
>   a Jurassic park despite the fact that there were no mathematicians
>   around to imagine them?

Jean-Yves Girard thinks numbers are resources, and Paul Ehrlich says
we're going to run out of natural resources before then.  If they're
right, all remaining numbers in 2098 will be unnatural.

Vaughan Pratt