On Feb 7, 2008, at 2:06 PM, Avril Styrman wrote:
> 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)
Actually I wrote that. You should keep your attributions straight. (02)
> I am not interested in the ideology in itself, but only how the
> ideology should affect mathematics and all science for that matter. (03)
The "ideology" in question would makes natural science impossible.
Notably, strict finitism, even coherently argued, does not permit
enough real analysis (which involves not only the infinite, but the
uncountably infinite) to do physics. (04)
> It is currently a minor ideology, but will be (hopefully soon) also
> the general mathematical ideology. (05)
It will never be, largely because it is viewed almost universally
among actual mathematicians as, at worst, a delusion, and at best a
novelty, a curiousity, like developing an internally consistent
defense of a flat earth. (06)
> It is not just an ideology like some religion. It is evident that
> everything paradoxical should be pruned off from logic. (07)
Indeed. And if you could show one single paradox in contemporary
mathematics you might be on to something. Note this does not mean
producing an informal argument involving undefined terms with unstated
premises. It means taking the *axioms* of any branch of mathematics
-- let's say ZF set theory plus Choice, since pre-ZF set theory was a
fairly rich source of genuine paradox -- and *demonstrating* a
contradiction, that is, producing from those axioms a deduction of A
and not-A, for some proposition A. (Note if you can do this, you've
probably got a Fields Medal coming.) Until you can do that, any
claims to the effect that classical, infinitary mathematics is
paradoxical is on the same logical footing as, say, Scientology. (08)
> 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. The whole transfinitism is
> built on the Cantorist complete induction. It is a pure invention: (09)
Yeah, like a (more or less) spherical Earth. (010)
> 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? I don't consider all his stuff
> good, but his critique against transfinitism is enjoyable. (011)
Except W. never actually provides any critique in the sense of a
coherent, logically precise argument. Just a rambling series of
pictures, metaphors, and loose assertions like the one above. This is
the inevitable currency of the mathematical skeptic. (012)
> Having a set {1,2,3, ..., n}, its order type is n (013)
That is an unordered set. It has no order type. (014)
> 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 members too. (015)
Good grief. It's worse than I thought. We have here the locus
classicus of all anti-infinitary crackpottery. It is actually pretty
challenging to try to figure out exactly what mental quirk it is that
leads some people to deny there are infinite sets of natural numbers
on the grounds that, if there were, there would have to be an infinite
natural number. It is, truly, just the strangest thing. I mean, what
can one say in response? Imagine if someone argued that, because the
cries of small children in pain gave him pleasure, it was morally
permissible for him to poke small children with a sharp stick. What
could you say in response? Anyone proffering the argument would
obviously be confused (to say the least) about the concept of moral
permissibility. But if you were to try to point out, e.g., that
finding something pleasurable does not entail that it is morally
permissible, the reply would simply be that it obviously does. Trying
to clarify the relevant moral concepts to him would simply, in his
eyes, confirm his belief that it is you who is confused. I'm afraid
that those who find the inference from "S is infinite" to "S has an
infinite member" compelling are just as confused about relatively
simply mathematical concepts, and that both arguing with them and
trying to unconfuse them is just as fruitless. (016)
> I really appreciate all the comments... (017)
Well, up til now, I suspect. ;-) (018)
> ...so please continue. (019)
Or not. No more on this from me, for the reasons just noted. (020)
chris (021)
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