FOM: NATURE of mathematics

[Vaughan Pratt]

>1) Should mathematics on other planets (of other galaxies) be entirely
>   different from ours -- provided its existence? Would you expect it
>   to be similar? ... What do you think? ... No kidding, please.

I just realized I misrepresented my answer to this when I 'fessed
up to my answers to Cabillon's questions 1, 5, and 6 having "no fom
content whatsoever."  The point of my answer to 1 was as serious as the
question itself.  I would expect other planets to have developed either
set theory or category theory at some point, and possibly both.

But on reflection I now think that my (decrypted) answer is unreasonable,
as now yet another reasonable possibility occurs to me.  Our hypothetical
alien mathematicians may be *much* smaller than us, so much so as to be
in an excellent position to observe the weirdness of quantum mechanics
even before noticing the regular passage of the stars in their sky.

In that case their earliest mathematics might have been Hilbert
spaces, or something even more suited to quantum mechanics, and all
their theorems would be about the structure of what can be embedded
in separable Hilbert space.  This would certainly include set theory,
category theory, etc. etc., but there is a lot of room in Hilbert space
and they may well find other structures of greater appeal than those.

In particular quantum weirdness (as we perceive it) might, for practical
reasons we find hard to imagine, make counting unnatural for them, and
both numbers and set theory might seem utterly alien to them, just as
quantum mechanics seems utterly alien to us giants.

One can imagine it being commonly said at some point in their mathematical
development that "only ten mathematicians in the world can explain
natural numbers."

Vaughan Pratt