To: |
Avril Styrman <Avril.Styrman@xxxxxxxxxxx>, "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx> |
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From: |
Pat Hayes <phayes@xxxxxxx> |

Date: |
Wed, 13 Feb 2008 13:23:02 -0600 |

Message-id: |
<p06230906c3d8d897e26f@[10.100.0.28]> |

At 10:15 AM +0200 2/13/08, Avril Styrman wrote:
John, Pat, Chris, 1. Its not simpler. It has to explain a host of new difficulties,
not the least of which is how it can account for the hundreds of
apparently obvious refutations of it (such as the elementary proof
that cannot be a largest integer.)
2. It is most certainly not uncontroversial. As this is an
elementary empirical fact, I will not provide arguments for it.
3. Continuous mathematics is certainly useful, I would claim
needed.
Pat and Chris are claiming that finitism is somehow No, because it seems obvious. In fact I have been interested in
strict finitism for many years and have read everything about it I can
find. I am not impressed by the arguments I have seen so far, all of
which seem very weak and/or obviously faulty.
(By the way, your claim, repeated below, that mathematicians
believe in infinity only because we were taught to as babies and are
too uncritical to think for ourselves, is rather insulting. You might
want to think about other ways to try to convince people.)
John That is not the only reason, although it was historically
important. But note, Cantor's demonstration of the uncountability of
the reals is all about *numbers*, not about points. It nowhere refers
to geometry: it uses only the idea of a decimal expansion. One could
reject the set-theoretic account of the continuum, and still Cantor's
diagonalization is crying out for a refutation from any finitist. I'd
be interested to see a careful refutation of it (or, for that matter,
of the ancient and elementary proof that there is no largest number,
which I gave in an earlier posting.)
...
I have. One way to look at the finitism-transfinitism debate Wait. This phrase is like 'the creationist/evolutionist debate'.
There is no "finitism-transfinitism debate". Nobody in the
philosophy of mathematics takes finitism seriously as a coherent
foundational philosophy.
> expansion would take more than the information Using what notation? Decimal numerals are only one way to refer
to numbers: mathematics provides all kinds of ways. If I can use
"Ack( ..)" then I can refer to much larger numbers than if I
am restricted to decimal numerals.
It will alway be finite. If I am allowed to write omega-n, then I can refer to infinite
ordinals using a finite vocabulary. Are you saying that infinity
cannot be referred to? That is obviously false.
Of course, there is no sense of typing Ack(Ack(5,5),5) because you cannot What can I not understand about its magnitude? I know it is
larger than Ack(5,5), for example. But in any case, what I can
understand about it and whether it is any use (and how do you know
what uses I might have for very large numbers?) are irrelevant to the
proposition that it EXISTS. It seems obvious to me that it exists, as
does its square; indeed, as does the result of raising it to its own
power (a
much bigger number than any that have been discussed
on this thread so far, but still one that has a successor.)PH: You just did talk about something infinite, by your casual use of
the three-dots notation, and saying 'the series'. That series is an
infinite object.
You must That does not establish that the limit is zero. To establish
that, you have to also show that if your accuracy is 10|3 digits,
then.. and if your accuracy is 10|4 digits, then, ... and so on. And
that 'and so on' has no finite stopping place.
The good part of finitism here is, that even if one serieses'. To turn your ad hominism on its head for a moment, I wonder if
you really understand the notion of a limit. Since it is easy to show
that the notion cannot be defined in a strict finitist framework (or,
to be more exact, that any definition of it reduces to triviality
there - just use induction to get to the finite end of the finite
series, and you will attain any approximable limit), this may not be
surprising.
Real analysis; differential geometry, for example.
Give one single example of something
Funny, those are the very reasons that most scientists and
virtually all mathematicians are NOT finitists.
> premises. It means taking the *axioms* of any branch of mathematics Fair enough. What about the Burali-Forti paradox. All Cantorist sets exist For the record, I utterly fail to follow this piece of your argument. BUt never mind... there should also exist the greatest ordinal greater and greater ordinals. This is a case of A and not A. No, its a case where one has to admit that naive intuitions break
down. That is why this is no longer classified as a paradox. There is
no greatest ordinal (in fact, I don't find this surprising at all: why
would one think there was??) The set of all sets is not itself a set
(or perhaps, does not exist); and many other examples like this are
all handled in the same way. So this is not a paradox in the sense
that Chris was asking for. Again, you might not
like this way
of thinking, but it is the way that has been used, and it makes sense
and appears to be internally consistent.In order to justify his cause, Cantor turned the paradox into a from an evident paradox. But one an say that about almost any new mathematical insight.
Such rhetoric isn't constructive or useful (and it certainly isnt
doing mathematics.)
I just wait to see in what exact way you disagree, because it seems as and you want to defend it. We think it is true because there are so many convincing
arguments for it, and it seems obviously, intuitively, true.
I believe that finitism is a better view, but where finitism is not enough. We already have.
> >Cantorists only try to escape the evident implications of
This is starting to get beyond merely insulting, and verging on
the insane. I knew that there were infinitely many numbers (though I
might not have expressed it that way) a long time before I had heard
of Cantor. Im not sure exactly, but I know it was before I was seven
years old, because I had an argument with a friend about it at that
age. His view was that there was a largest number but that only God
knew what it was. I said God could add one to it, and he said that
God
could, but wouldn't ever actually do it, so it would always
be the largest.
No, it isn't. If you want to refer to the ordered set, use
different brackets:
<1,2,3, ... , n>
This means that you don't even NO IT DOESN'T !!! Here, YOU explain to US why you believe this,
which seems so obviously false. Why can't see that the integers are an
obvious counter-example?
This interpretation of complete induction is totally as objective as that which you You have to explain what you mean by 'make sense' and
'unnatural'. And you have to do this in
mathematical
terms.
It isn't, and you havn't. , while finitism manages to do all that needs to be done, without paradoxes. Its rife with paradoxes. There's the largest-number paradox. All
finite sets of numbers have a largest member. If the set of all
numbers is finite, then there is a largest number, call it N.
N+1 is larger than N: paradox. Show me how to get past that one, and
I'll give you another.
> >Do you disagree with Wittgenstein? Yes.
> >If the set has infinitely many members, there should be infinitely big the set of all natural numbers. No. Nothing about random. I can rephrase the point without using
the a-word: for any given size, there is a number in the set larger
than that size. That is all I meant by 'arbitrary'.
> >Another way to explain this is that do you have any a vague border for the intelligible natural numbers. You can talk about it, but I'd like to know what you mean. I have
no idea what a vague border could be. Give us some of this
much-vaunted finitist mathematics. What is a vague boundary? What
properties does it have, what theorems are true of it?
> imagine it. Your move. I said nothing about application or utility. I said only that I
can imagine it (you havn't refuted this claim) and that it exists
(which I understand you to be denying?)
Your move. I understand them, yes. I don't claim that they are particularly
useful; only that they exist.
... the Cantorist conception is contradictory >
Yes, indeed I do.
> They most certainly need real analysis and differential geometry.
If you can show how to re-create these fields (and, say, topology,
catastrophe theory, etc.) within a strictly finitist philosophy of
mathematics, you will at the very least be able to publish many papers
in journals devoted to the philosophy of mathematics.
> the question is, just how big a number can we understand? be typed down? If there is no use, why postulate it? I DID write it down. And the 'use' is that its a lot easier to
allow it to exist than to deny its existence. It doesn't need to be
'postulated': its existence follows from the ordinary assumptions of
arithmetic.
Denying its existence is what would take a great
deal of explaining. See below.
Consider N!, where N is an integer. Clearly N! is greater than N,
for any N. Moreover, as N increases, N! increases much faster, so that
the amount by which N! is greater than N increases as N increases.
This means that if there is an upper limit to the set of numbers, even
if it is 'vague', that N! crosses that upper boundary a lot sooner
than N does. So there must be a lot of numbers between the largest N
such that N! vaguely exists and the largest N, all of which exist but
their factorials don't. How can that happen? Now re-run this argument
for any other rapidly-increasing function of N, say 2|N or N|N, or
even Ack(N,N). What happens to these functions past the point where
their value would be too large to be a 'real' integer, yet their
argument is still quite small? Do they somehow "stop
working"? But they are all defined by finitely describable, even
quite compact, algorithms which perform ordinary arithmetic
operations, so what is there to go wrong? Or do they actually produce
results, but these very, very big results are not counted as
"numbers"? Why not? But more to the point, whatever you call
them, if they exist then classical infinitary mathematics seems to
apply to them. Or... what?? You have to say SOMETHING about arguments
like this, other than that you don't like their conclusions.
is 1. That doesn't make finitist sense. Your arguments seem to
presuppose the denial of your own intended conclusions. By the way,
the mathematical theory you seem to be groping towards is measure
theory, which makes very good sense of how to talk of the
probabilities of selections from infinite sets. I think your point can
be stated as: the set of all 'accurate' numbers must have measure
zero. But that still doesn't account for how this (finite!) set can
have a 'vague' upper limit.
Pat
PS. There is a lot of fascinating work on rapidly-increasing
functions on integers. One can derive results which are close to
paradoxical-sounding, such as the existence of the busy beaver
function which, however, cannot be computed or even have a computable
upper bound. None of this requires infinitary mathematics, but it
poses quite a challenge for a strict finitist perspective. You can
follow this up in Wikipedia starting with 'large number'.
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