To: |
"[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx> |
---|---|

Cc: |
Avril Styrman <Avril.Styrman@xxxxxxxxxxx> |

From: |
Pat Hayes <phayes@xxxxxxx> |

Date: |
Thu, 7 Feb 2008 16:43:35 -0600 |

Message-id: |
<p06230902c3d1331024fd@[10.100.0.18]> |

At 10:06 PM +0200 2/7/08, Avril Styrman wrote:
PH: (actually CM quoted by PH) > You might try your luck on the (hopefully soon) also the general mathematical ideology. This is vanishingly unlikely. It is just easier to be a Platonist
when doing actual mathematics.
It is not just an ideology like some religion. It is evident that everything paradoxical should be pruned off from logic. Of course. But one can work paradox-free without adopting strict
finitism.
No, the error (repeated below) is to assume that an infinite set
must contain an infinite object. The natural numbers are an obvious
counterexample.
The whole Do you disagree with Wittgenstein? Yes. Almost everything he wrote, in fact.
I don't consider All *finite* sets of that form do, yes.
If the set has infinitely many members, there should be infinitely big No, that is simply a mistake. There are indeed arbitrarily large members, and in fact infinitely many of them, but there need not be
infinitely large members.Cantorists only try to escape the evident implications of ? What makes you feel that this is an implication? It seems
simply obvious to me that the set of natural numbers contains only
finite numbers but is itself infinite. I think this has been obvious
to me since I was a fairly young child. There is nothing paradoxical
about it. It has nothing to do with naming conventions.
What should be done is to see that the problem: if the series is generating, it has to have some speed of What kind of series are you talking about? If the speed is finite, intuitionism has turned into finitism. Nope. That simply does not follow. (Or wait: do you mean that
there are infinitely many of them? Yes, of course. Or do you mean,
some of them must be infinite? That is (to me, clearly) false.)
Um.. the point of the proof was to show that there cannot be such
an N. I postulate it in order to show that such a postulation leads to
a contradiction.
greatest natural number, because the border is vague. A vague border on a set of integers is a notion that cries
out for more detailed explication.
is Still a Finite Number. Of course it is: ALL natural numbers are finite. But what this
argument tells me is that the notion of 'the largest imaginable
number' is just as meaningless as 'the largest number', and indeed for
the same reason.
Logique et Analyse. Vol.42 No.165-166.
No, its not nonsense. I can imagine any number. I claim this is
true. Prove me wrong. You have to show that a number exists such that
I cannot ever imagine it. Your move.
* * * * * * * * * * > mathematics to some kind of dark political conspiracy. Mathematicians For what purpose? Humans use Mathematica thousands of times every
day to solve real problems in real engineering, and all of this would
have to be abandoned if we were to take strict finitism seriously.
Hilbert wasn't joking when he referred to 'Cantor's paradise'.
is the 'natural' part of the natural No, its not contradictory. You may dislike it, but it is
consistent.
, A subjective judgement. It provides a unique foundation for all
normal mathematics which has not been surpassed, and overcomes a host
of difficulties that late 19th-century mathematics was wrestling with
(convergences of infinite series, accounting for the irrationals,
etc.)
Nobody needs it, Mathematics needs it. and that is number can we understand? (1) What exactly do you mean by 'understand'? And (2) why does
this matter? We can certainly
refer to and reason about
and prove properties of very, very large numbers: much larger
than could possibly be physically represented as a numeral.Lloyd argues in [Seth Lloyd: Computational is the upper bound of the storage capacity of the known universe; Lloyd also So what? That is just the
numeral; but we can denote the
number in other, more compact, ways. In fact, you just did, using only
six characters.
It seems pretty easy to write down
A(A(A(5,5),5),5)
or for that matter
10|(10|(10|100))
both of which are way, way larger than the number of quarks in
the known universe. In fact I think that the latter is more than the
number of quarks that one could pack into the known universe if it
were packed solid.
Pat
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