PH:
> >Ah, I see, well, you are clearly more interested in ideology than
> >mathematics or knowledge engineering, and that is not something that
> >ought to be debated in this forum. (01)
And this was the reason why I originally wanted to initiate
the thread with the title ''Separate cs ontology from
philosophical ontology PO'' in SUO forum, but it got drifted
here somehow. The initial assumption was that PO should not
be discussed in forums like this, but discussions often
get driven into those questions anyway, like now. I don't
see this totally useless. (02)
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. "ultraintuitionism") you appear to be
> >advocating is an interesting philosophical ideology, (03)
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.
It is not just an ideology like some religion. It is evident
that everything paradoxical should be pruned off from logic. (04)
PH:
> Avril, you might enjoy "Ad Infinitum" by Brian
> Rotman, Stanford U. Press 1993. (05)
Thanks, but unfortunately it was not in the local library. Also these are
good general books about the topic:
Mary Tiles: The Philosophy of Set Theory: An Historical Introduction to
Cantor’s Paradise. Basil Blackwell, 1991.
A.W. Moore: The Infinite. Routledge, 2001. (second edition) (06)
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". (07)
What you call a 'logical error' is that I do not accept
Cantor's subjective interpretation of induction. The whole
transfinitism is built on the Cantorist complete induction.
It is a pure invention: (08)
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 (09)
Do you disagree with Wittgenstein? I don't consider
all his stuff good, but his critique against transfinitism
is enjoyable. (010)
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.
If the set has infinitely many members, there should be infinitely big
memebers too. 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 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 omegazero is only the order type of an inductive set: this
is just appealing to a naming convention. (011)
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. 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. (012)
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. (013)
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. (014)
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? (015)
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. (016)
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. (017)
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. (018)
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. Logique et Analyse. Vol.42 No.165166.
Proceedings of the First FlemishPolish LogicoPhilosophical
Workshop, 1999. (019)
* * * * * * * * * *
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. (020)
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): (021)
(1) Imagines(x, 1)
(2) Vn(Imagines(x, n) > Imagines(x, n + 1)) (022)
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 (023)
(3) Vn(Imagines(x, n)) (024)
follows immediately by mathematical (complete) induction and that is nonsense.
* * * * * * * * * * (025)
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. (026)
Mathematicians want to say ''for all natural numbers''. But
all that a human needs 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,
and leads to very obscure things. Nobody needs it, and that is
why it is better to prune it off. (027)
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. (028)
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. (029)
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. (030)
Does anyone here you have any reason for not abandoning Cantorist
complete induction, and any reason for holding it? (031)
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 multiword integers
> that are much larger than that. (032)
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? 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.301307.] that the imaginable output of the
socalled 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. (033)
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. (034)
I really appreciate all the comments, so please continue. (035)
Avril (036)
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