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At 10:06 PM +0200 2/7/08, Avril Styrman wrote:
PH: (actually CM quoted by
PH)
> You might try your luck on the
> >USENET groups sci.logic and sci.math, where actual
mathematicians
> >entertain themselves arguing with extremists, ideologues, and
assorted
> >crackpots who believe that there are deep logical or
conceptual flaws
> >in classical mathematics. Just for the record, I do
think the sort of
> >"strict" finitism (a.k.a.
"ultra-intuitionism") you appear to be
> >advocating is an interesting philosophical ideology,
I am not interested in the ideology in itself, but only how
the ideology should affect mathematics and all science for
that matter. It is currently a minor ideology, but will
be
(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.
PH:
> BTW, you make
> one logical error in an earlier post. An infinite
> series of finite things can grow indefinitely
> without any one of them actually becoming
> infinite. This is true even with a strict
> finitist understanding of "infinite".
What you call a 'logical error' is that I do not accept
Cantor's subjective interpretation of induction.
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
transfinitism is built on the Cantorist complete induction.
It is a pure invention:
I have always said you can?t speak of all numbers, because
there?s no such thing as ?all numbers?. But that?s only
the
_expression_ of a feeling. Strictly, one should say, . . .
?In arithmetic we never are talking about all numbers, and
if someone nevertheless does speak in that way, then he so
to speak invents something - nonsensical - to supplement
the arithmetical facts.? (Anything invented as a supplement
to logic must of course be nonsense).
-Ludwig Wittgenstein: Philosophical Remarks XII.129
Do you disagree with
Wittgenstein?
Yes. Almost everything he wrote, in fact.
I don't consider
all his stuff good, but his critique against transfinitism
is enjoyable.
Having a set {1,2,3, ..., n}, its order type is n and its cardinality
is n.
The same goes for the greatest member of the set, if we think in terms
of
the frontrunner. The set always has as big a member as is its
cardinality.
All *finite* sets of that form do, yes.
If the set has infinitely many members,
there should be infinitely big
memebers too.
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
complete induction. Sure, they don't want to have infitely big
naturals, so
they just deny them. This is the doctrine of the Cantorists: just deny
the
implications of your own theory.
? 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
complete induction leads into a paradox, and not to try to escape
into
naming conventions in order to make the paradox disappear. It does not
help
to appeal that omega-zero is only the order type of an inductive set:
this
is just appealing to a naming convention.
Also appealing to intuitonism does not help. The free generation has
a
problem: if the series is generating, it
has to have some speed of
generation.
What kind of series are you talking about?
If the speed is finite, intuitionism has
turned into finitism.
If the speed is transfinite, intuitionism has turned into
transfinitism. If
the question is left open, it helps nobody to introduce the free
generation.
PH:
> It is kind of fun to see if there is a coherent alternative,
though.
> How could one make sense of the thesis that there are only
finitely
> many integers? The counterproof seems very simple and is probably
one
> of the oldest mathematical proofs ever devised.
>
> 1 Suppose there were a largest integer.
> 2 Call it N.
> 3 Consider N+1.
> 4 N+1 is larger than N: contradiction.
> 5 Ergo, there is no largest integer.
It's no riddle. For me it makes a good good sense.
If there are integers that constantly grow on each step,
and there are infinitely many steps, then the integers
have to be infinite too.
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.)
Another way to explain this is that do you have any
idea of N? How big is N? If you have no idea about it,
and no use for it, why do you postulate N in the first
place?
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.
JS:
> Many 19th century mathematicians strongly objected to that
way
> of talking, and I sympathize with them. But those
mathematicians
> would *never* agree to a fixed upper bound on the integers,
such
> as 10**120, Ackermann(5 5), or any other finite integer.
The border does not have to be fixed, but vague: something
that can be understood. There is no clear border of the
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.
CM:
> The view isn't even coherent. If Ackerman(5 5) exists, why
not
> Ackerman(Ackerman(5 5) 5) -- a massively larger number? And
of course
> if *that* number exists, well, you get the idea.
Yes, I get the idea, and that is the classical rebuttal of
finitism, but it has flaws in it. I omitted the following
from Jean Paul Van Bendegem: Why the Largest Number
Imaginable
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.
Proceedings of the First Flemish-Polish Logico-Philosophical
Workshop, 1999.
* * * * * * * * * *
The procedure ?Give me any numeral n you can imagine, I will give
you the
next one? has to break down at a certain point. Ask any person to
imagine a
very large numeral, say, in decimal presentation. Usually what we do
is to
form a picture, say, we see a blackboard and it is covered with
ciphers all
over. But that won?t do. For once we have such a picture, it is
obvious that
it is communicable, hence that it is finitely expressible and hence
that
there is room to imagine the next numeral and to communicate it. Thus,
the
alternative must be that the numeral is so large that it cannot be
imagined,
thereby making it senseless to talk about the next one. I will return
to the
implicit paradoxical nature of what I just wrote. What is being asked
is to
imagine a numeral so huge that it cannot be imagined. Let me settle at
this
stage of the presentation an obvious counterargument: is it not a
silly
notion that I can imagine all numerals up to n, and then suddenly for
the
next one, my imagination fails me? What the argument shows is that
the
notion of the ?largest numeral imaginable? must be a vague notion.
This
observation is supported by the fact that paradoxes concerning
vague
predicates also apply to this situation. One of the most
straightforward
connections is with the Wang paradox, itself a variant of the Sorites
paradox.
Suppose that Imagines(x, n) is an abbreviation for ?person x is
capable of
imagining the numeral n?, then it is claimed that both (V =
forall):
(1) Imagines(x, 1)
(2) Vn(Imagines(x, n) -> Imagines(x, n + 1))
are extremely plausible. After all, (2) is nothing but a reformulation
of
the idea that the next numeral can always be imagined. But, given (1)
and
(2), the conclusion
(3) Vn(Imagines(x, n))
follows immediately by mathematical (complete) induction and that is
nonsense.
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.
* * * * * * * * * *
PH:
> Its not enough to just announce as an obvious doctrine
> that infinity is wrong; still less to seem to link
conventional
> mathematics to some kind of dark
political conspiracy. Mathematicians
> tend to be Platonists because they are driven to it by
following
> chains of thought which seem to be inevitable and conclusive. If
you
> want to announce an alternative, you have to tell us where the
less
> travelled paths branch off the mathematical highway.
Mathematicians want to say ''for all natural numbers''. But
all that a human needs
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
numbers. Therefore, the 'for all' in ''for all natural numbers''
should be interpreted as e.g. ''for all those naturals that
can be typed within the known universe''. And why is this
important and better? It is better than having the Cantorist
conception, because the Cantorist conception is
contradictory
No, its not contradictory. You may dislike it, but it is
consistent.
,
and leads to very obscure things.
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
why it is better to prune it off.
CM:
> Indeed -- which of course means that there are infinitely many
finite
> integers, and hence that there is a set that contains them, hence
a
> power set of that set, and off we go down the Cantorian bunny
trail! :-)
> You may not like where that leads, but it is very hard to argue
that
> there is a nonarbitrary point at which you can stop that line
of
> reasoning.
JS:
> The traditional view before Cantor was that infinity means
the
> absence of a fixed boundary, and that any reference to
infinity
> was a way of talking about a process that would exceed any
fixed
> boundary. Cantor changed the "language game" of
mathematics by
> talking about completed sets that had infinitely many
members.
Imagine set theory?s having been invented by a satirist as a
kind
of parody on mathematics. - Later a reasonable meaning was seen in
it and it was incorporated into mathematics. (For if one person
can see it as a paradise of mathematicians, why should not another
see it as a joke?)
-Ludwig Wittgenstein: Remarks on the Foundations of Mathematics.
IV.7.
Does anyone here you have any reason for not abandoning Cantorist
complete induction, and any reason for holding it?
JS:
> In fact, 10*120 is an absurdly small integer for today's
computers.
> It can be stored in less than 60 bytes, and you can download
> computational packages that do arithmetic on multi-word
integers
> that are much larger than that.
10*120= 10 in the power of 120. In physics that is some sort of a
limit. We
might understand the magnity of 10*83, but the question is, just how
big a
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
Capacity of the Universe. Physical Review Letters 88, 2002] that
10*120 bits
is the upper bound of the storage
capacity of the known universe; Lloyd also
argues that the ?calculation? of the known universe requires
10*122 bits,
even though there are 10*70 atoms, whatever he means by the term
?atom?.
Kornai argues [Andr¨ys Kornai: Explicit Finitism. International
Journal of
Thoretical Physics, 2003/2, p.301-307.] that the imaginable output of
the
so-called Ackermann functions, the fastest growing functions, with the
input
A(4, 4) are outside the arithmetic power of any civilization
restricted to
the material resources of the known universe. Friedman [Harvey
Friedmann:
Enormous Integers in the Real Life, 1999.] argues that A(5,5) is
the
benchmark for incomprehensibly large numbers. This sort of a number
would
have so many digits, that we simply do not have space in the known
universe
to store the numeral in any other form than as the Ackerman function
-form.
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.
So, A(5,5) can be considered as some sort of a limit, if 10*120
is not enough. Anyway, there is the limit of what can be
communicated or written down.
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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