[FOM] Resources on the empirical foundations of mathematics

Ben Crowell fomcrowell06 at lightandmatter.com
Wed Jan 11 11:48:24 EST 2006

>This seems to suggest that there is a finite limit N on the number
>of distinguishable points of space in the entire universe; thus a
>limit on the number of countable, observable, real objects.

I think this depends on both definitions and empirical facts.
As a matter of definition, it makes a difference whether "the
entire universe" means the part that's observable to us, or
whether it includes regions too far away for light to have had
time to reach us since the big bang. Also, the issue of whether
the universe has finite or infinite volume is something that can
be measured. The latest observations show that the universe is
very close to the dividing line between open and closed.
In cosmology without a cosmological constant, finite
volume also implies finite time before the universe collapses
and time ends, while infinite volume goes along with an infinite
amount of time in the future. (But this is all heavily modified
by the latest observations, which show the universe's expansion
is accelerating.) If the point is to put a limit on the size of
numbers that could be computed, then another issue is
thermodynamics; erasing a bit decreases the entropy of a
computer by a certain amount, so even if time and space turn
out to stretch infinitely far, there will be a time after which
computation can no longer exist.

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