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fom
Posts:
1,099
Registered:
12/4/12
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Re: Cantor's absurdity, once again, why not?
Posted:
Mar 14, 2013 4:31 PM
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On 3/14/2013 6:03 AM, WM wrote:
> On 14 Mrz., 11:17, fom <fomJ...@nyms.net> wrote:
>
>> For example, Wittgenstein understood perfectly
>> well how to apply Cantor's argument and he
>> certainly is not thought of as believing in
>> a completed infinity.
>>
>> However, he also did not attack the mathematicians
>> who conducted investigations along those lines.
>
> He said:
>
> There is no path to infinity, not even an endless one. [§ 123]
>
> It isn't just impossible "for us men" to run through the natural
> numbers one by one; it's impossible, it means nothing. [?] you can?t
> talk about all numbers, because there's no such thing as all numbers.
> [§ 124]
>
> There's no such thing as "all numbers" simply because there are
> infinitely many. [§ 126]
>
> Generality in mathematics is a direction, an arrow pointing along the
> series generated by an operation. And you can even say that the
> arrow
> points to infinity; but does that mean that there is something ?
> infinity ? at which it points, as at a thing? Construed in that way,
> it must of course lead to endless nonsense. [§ 142]
>
> If I were to say "If we were acquainted with an infinite extension,
> then it would be all right to talk of an actual infinite", that would
> really be like saying, "If there were a sense of abracadabra then it
> would be all right to talk about abracadabraic sense perception". [§
> 144]
>
> Set theory is wrong because it apparently presupposes a symbolism
> which doesn't exist instead of one that does exist (is alone
> possible). It builds on a fictitious symbolism, therefore on
> nonsense. [§ 174]
>
> 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?)
>
>
> If it were said: "Consideration of the diagonal procedure shews you
> that the concept "real number" has much less analogy with the concept
> "cardinal number" than we, being misled by certain analogies,
> inclined
> to believe", that would have a good and honest sense. But just the
> opposite happens: one pretends to compare the "set" of real numbers in
> magnitude with that of cardinal numbers. The difference in kind
> between the two conceptions is represented, by a skew form of
> expression, as difference of extension. I believe, and I hope, that a
> future generation will laugh at this hocus pocus.
>
> What is mocking? It's relative.
Of course. And, I do not read much into those quotes.
Except for the one parenthetical remark referencing a historical
quote -- namely, Hilbert's description of "Cantor's paradise" --
there is no overt criticism of mathematicians in any personal
sense.
I prefer to recognize that most of Wittgenstein with which I
am familiar is not characterized by any overt criticisms of
that sort.
"Its object would be attained if it afforded
pleasure to one who read it with understanding"
TLP
"It means the book is written in good will, and
in so far as it is not so written, but out of
vanity, etc., the author would wish to see it
condemned."
Philosophical Remarks
"I should not like my writing to spare other
people the trouble of thinking. But, if possible,
to stimulate someone to thoughts of his own.
"I should like to have produced a good book."
Philosophical Investigations
And, at the risk of quoting Wittgenstein
as if I understood his philosophy clearly,
"Philosophy may in no way interfere with the actual
use of language; it can in the end only describe
it.
"For it cannot give it any foundation either.
"It leaves everything as it is.
"It also leaves mathematics as it is, and no
mathematical discovery can advance it. A 'leading
problem in mathematical logic' is for us a
problem of mathematics like any other.
"It is the business of philosophy, not to resolve
a contradiction by means of mathematical or
logico-mathematical discovery, but to make it
possible for us to get a clear view of the state
of mathematics that troubles us: the state of
affairs before the contradiction is resolved."
Philosophical Investigations
Like all men, Wittgenstein probably would have
liked his ideas to be better understood and
more widely appreciated. Undoubtedly, there
will be written passages and anecdotal stories
that may reflect badly with regard to retorts
toward others. For the most part, however,
he was a philosopher working as a philosopher.
There is not that much that appears to be more
than that.
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