On 01 Mar 2013, at 16:37, meekerdb wrote:
On 3/1/2013 7:05 AM, Bruno Marchal wrote:
I don't think that what you ask is possible, even if I am
pretty sure that x + 0 = x, x + s(y) = s(x + y), etc.
I'm not at all sure that there is successor for every x.
Then you adopt ultrafinitism, and indeed comp does not make sense
with such hypothesis, and UDA1-7 suggests that ultrafinitism
might save physicalism, but step 8 put a doubt on this.
The axiom that all natural numbers have a successor is used in
basically all scientific paper though.
It is assumed, but I'm not sure it is used in an essential way. I
recognize it difficult to do mathematics without it, but still it
may be only a convenience.
OK. I don't see the problem with this. Convenience is a fuzzy
notion. A brain too is convenient. Universes can be convenient. I
am not sure to see your point.
In physics we sometimes get big numbers, like 10^88 or 10^120, but
we never need 10^120 + 1.
But physics is no more assumed in the TOE derived from comp.
The number of chess games is about 10^120. The number of GO games is
far bigger.
And string theory points on 10^500 theories.
And my friends the Roses have never seen a gardener dying. Some rare
Roses have heard rumors that can happen, but all rational Roses knows
that belong to fiction.
Frankly, for a logician, 10^100 looks really like an infinitesimal :)
We make an axiom of succession and assume it applies to 10^120 like
other numbers,
But then working on the prime number distribution, some bound on some
function get *far* higher than that.
but maybe that is because it easier than thinking of axioms to
describe how we really calculate: 10^88 + 1 = 10^88.
Like it is easier to make the earth turning around the sun than the
contrary. If you are for complex theories a priori, ...
There's a book "Ad Infinitum" by Rotman that proposes something
along these lines, but he writes like a French philosopher so I
found it hard to tell whether his idea really works. But we do know
that real computers work, and their mathematics are finite.
That's correct on one level, and incorrect on another level. The
metamathematics of the finite mathematics is not finite. Some proof of
the correctness of some algorithm in numerical analysis can requires
big cardinals. Real computers might do finite things, but they do very
complex things which might involve very large numbers, or high
cardinal or ordinal, for human figuring out why they acts like they
can act.
With comp you can put infinities only in the mind, as I do, but we
have still to study that mind, and needs infinite mathematics for that.
With comp the ontology is finitist, but that ontology is seen by
inside, by finite creatures,which have to imagine very large
unboundable structure, needing stronger mathematics, just to get
notions of meaning, etc.
Also, some could argue that only the mathematical universal machine is
finite, and that a "real computer" (if that notion makes sense) is an
infinite analogical quantum field living in an infinite dimensional
Hilbert spaces.
I put my card on the table, and reason. I don't defend a truth about
some reality. To use the notion of universal machine to formulate the
mind-body problem in the frame of some hypothesis is a different task
than to program a computer to perform some "useful" task in real time.
Bruno
Brent
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