[FOM] Wittgenstein
Alexander Zenkin
alexzen at com2com.ru
Wed Apr 30 04:11:46 EDT 2003
>Harvey:
>*DID LW WRITE ANYTHING THAT CAN AT LEAST BE REASONABLY INTERPRETED AS
BEING SIGNIFICANT FOR THE FOUNDATIONS OF MATHEMATICS? IF SO, EXACTLY
WHAT?*
I guess that the Cantor's diagonal argument is significant for
FOM, F.O.M., fom, f.o.m., etc. and therefore LW's attitude to the
argument is significant for all of these areas as well.
Unfortunately, at this moment I have no LW's books at hand, but
I think that one of the most professional defenders of Cantor's diagonal
argument, Prof. W.Hodges, gives a quite exact description of LW's
attitude to the FOM-problem. So, in his famous paper "An Editor Recalls
Some Hopeless Papers" (The Bulletin of Symbolic Logic, Volume 4, Number
1, pp. 1-17 1998) where he upholds marvelously a logical immaculacy of
Cantor's argument, W.Hodges writes, in particular:
"This may be the moment to mention a passage in Wittgenstein's
book `Remarks on the foundations of mathematics' [22], where he claims
(if I follow him right) that Cantor's argument has no deductive content
at all.
The theme of Wittgenstein's book is that mathematical statements
get any meaning they may have from rule-governed activities that involve
them. He singles out Cantor's argument because it would appear to have
no relation to any imaginable activity.
Except for one, namely the activity of writing out lists of
complete decimal expansions of real numbers. This is of course a daft
activity, doomed to failure. Ah, says Wittgenstein, that's what Cantor's
theorem must amount to ([22] p. 129):
Surely---if anyone tried day-in day-out 'to put all irrational
numbers into a series' we could say: "Leave it alone; it means nothing;
don't you see, if you established a series, I should come along with the
diagonal series!" This might get him to abandon his undertaking. Well,
that would be useful. And it strikes me as if this were the whole and
proper purpose of this method. It makes use of the vague notion of this
man who goes on, as it were idiotically, with his work, and it brings
him to a stop by means of a picture.
I think that the following short fragment of our paper "On One
Reconstruction of Wittgenstein's Objection Against Cantor's Diagonal
Proof". (VII scientific Conference "Modern Logic", Sankt-Petersburg,
2002. Proceedings, pp. 320-323.) will help to make clear a hidden
FOM-sense of the LW's statement that "Cantor's argument has no deductive
content at all."
"9. A 'funny' set-theoretical game for two honest tricksters.
As is known, Wittgenstein was very skeptical of Cantor's diagonal
argument.
Cantor's proof was suspected of being a senseless set-theoretical game
(see [23,24]): a man tries day-in day-out to put all irrational numbers
into a series; when all reals are put into a series and enumerated and
this "idiotical work" is finally ended, a trickster (the Cantor's
diagonal procedure) suddenly appears and says to the man: "Of course,
you have just enumerated all reals and you have utilized all natural
numbers, however, please, here is a new real, yet you lack even one
spare natural number in order to enumerate this single real.
Consequently, a number of reals is greater than a number of natural
numbers".
Consider now the following natural continuation of this,
according to Wittgenstein, "daft <set-theoretical> activity" [19]. So,
we already have one (Cantor's) trickster (hereafter: a Cheat-1) who is
able to do the following set-theoretical conjuring trick:
Always after (!) a given enumeration,
x1, x2, x3, . . . , (1)
has been presented, Cheat-1 shows (generates, defines, invents, takes
out his sleeve, etc.) a new (anti-)diagonal Cantor's real y1 which is
different from every real of the enumeration (1).
Now we introduce a second trickster (hereafter: a Cheat-2) who
is able to change the countable set of indexes {1,2,3, ...} in the given
enumeration (1) to another countable set of indexes, say {2,3,4,...}.
Naturally, without any modifying of the number and order of the reals
themselves in the initial sequence (1). Since the final result of
applying the Cantor's diagonal algorithm depends only upon the number
and order of the reals in (1) and does not depend upon any indexing of
these reals, the Cheat-1 (and all his admirers) simply has not
algorithmic tools to observe the artful algorithmic operation of the
Cheat-2.
However, I must especially emphasize here that the both of our
Cheats are absolutely honest in the sense that they do not violate any
law of any logic: simply Cheat-1 plays by the rule based upon the
property 'to be ACTUAl' of the ACTUALLY INFINITE sequence (1), whereas
Cheat-2 plays by the rule based upon the property 'to be INFINITE' of
the same ACTUALLY INFINITE sequence (1), or, more precisely, upon the
transitivity law of the equivalence relation (between all countable
sets).
Consider the following purely set-theoretical game and its
implications for Cantor. The starting state, in accordance with Cantor's
diagonal proof, is as follows.
"A given (arbitrary) enumeration (1) contains all real numbers
of X=[0,1]".
Step-1. Our Cheat-2 with stealth absconds with a single index, say '1',
and re-indexes the sequence (1) to form the sequence:
x2, x3, x4, . . . , (1*)
so that the sequence (1*) contains the same reals in the same order: x2
of (1*) is equal to x1 of (1), x3 of (1*) is equal to x2 of (1), and so
on.
Step-2. Cheat-1, using (1) or, what is the same, (1*), creates a new
real number, say y1, and makes the following claim: "So, now you lack
even one free natural number in order to index my new real number, y1.
Consequently, the number of all my real numbers is greater than the
number of all your natural numbers, i.e., the cardinality of X is
greater than the cardinality of N".
Step-3. Cheat-1, again openly, takes index '1' from his sleeve and
claims: "Your card is covered; here is a free, spare natural number to
index your non-indexed real number y1 ".
Step-4. A game referee indexes the new (Cantor's) real number y1 of
Cheat-1 using the number (actually the numeral) '1' of Cheat-2, puts it
in its natural first place within the given enumeration (1*) as follows:
y1, x2, x3, x4, . (1.1)
and claims: "Since Cheat-1 has no new reals at this moment, the
enumeration (1.1) now contains all the real numbers of X. Draw game:
0:0! ". - Consequently, the number of reals is not greater than the
number of natural numbers.
It is obvious that now our Cheats are free to return to Step-1
and repeat the steps of the game, generating a new Cantor's
(anti-)diagonal real y2 and a new sequence
y2, y1, x2, x3, x4, . (1.2)
Then again, generating new Cantor's real y3 and a new sequence
y3, y2, y1, x2, x3, x4, . (1.3)
And so on ad infinitum.
Consequently, and this is shown in [14-18,21,34], we have the
following POTENTIALLY INFINITE "reasoning" (here B="there is an
enumeration of all real numbers of X"):
B -- > not-B -- > B -- > not-B -- > B -- > not-B -- > B -- > . .
. (L)
Why the process (L) is just POTENTIALLY infinite? - Because
there is neither logical, nor mathematical reason to stop it. Such the
process (L) was described well (possibly for the first time) by Douglas
Hofstadter ("GODEL, ESCHER, BACH: an Eternal Golden Braid." CHAPTER
XIII, The Insidious Repeatability of the Diagonal Argument): "It is an
interesting exercise. But if you tackle it, you will see that no matter
how you twist and turn trying to avoid the Cantor "hook", you are still
caught on it. One might say that any self-proclaimed "table of all
reals" is hoist by its own petard." But one can add as well: the last is
true for every step 'not-B' in (L), however "you avoid the Cantor 'hook'
" at every next step 'B' of the process (L).
In [32,33] the necessary and sufficient conditions of
paradoxicality as a whole are formulated, and by means of the classical
model theory it has been proven rigorously that the true nature of the
"Liar" and similar paradoxes is described not by the traditional FINITE
form (here A="I am a liar")
[A -- > not-A]&[ not-A -- > A], (L1)
but by the following POTENTIALLY INFINITE "reasoning":
A -- > not-A -- > A -- > not-A -- > A -- > not-A -- > A -- > . . . (L2)
The formal coincidence of infinite paradoxical "reasonings" (L)
and (L2) is no mere accident. It shows that Cantor's diagonal proof
contains the infinite fragment (L) which is in fact a new "Liar"-type
set theoretic paradox (a paradox for both "naive" and "non-naive" set
theories). Until the potentially infinite "reasoning" of (L) is
completed, Cantor's conclusion "Consequently, the assumption that X is
countable is false" is from the standpoint of classical logic invalid,
since the "conclusion" contains a fatal, very insidious logic error
called "jump to a (very desired) conclusion". Since the potentially
infinite "reasoning" (L), according to Aristotle, can never be
completed, Cantor's diagonal proof is incapable of completion and,
consequently, the theorem concerning the uncountability of the continuum
is simply unprovable from the standpoint of classical logic.
The last confirms LW's statement that "Cantor's argument has no
deductive content at all."
[References]
Of course, LW was not a professional mathematician, and "didn't
publish serious mathematics" as H.Friedman accentuates, but, e.g., the
outstanding mathematician of the beginning of the XX c. H.Poincare who
published a lot of serious mathematics adhered to the same opinion
having a direct relation to Foundations of Mathematics:
"All Cantor's set theory is built on a sand [...]. Later
generations will regard Mengenlehre (set theory) as a disease from which
one has recovered. [...] Point set topology is a disease from which the
human race will soon recover."
And we have no reasons not to trust in the high mathematical
competence of H.Poincare.
Alexander Zenkin
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Prof. Alexander A. Zenkin,
Doctor of Physical and Mathematical Sciences,
Leading Research Scientist,
Department of Artificial Intelligence Problems,
Computing Center of the Russian Academy of Sciences,
Vavilov st. 40, 117967 Moscow GSP-1, Russia
e-mail: alexzen at com2com.ru
URL: http://www.com2com.ru/alexzen/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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