SW Letter

Here is (a slightly edited version of) the text of a letter I sent to Socialist Worker recently, which they chose not to publish:

Comrades,

Anindya Bhattacharyya rightly praises Alan Turing for his important contributions to computational science (Socialist Worker, 04/08/07, p. 11), but he should have told his readers that Kurt Gödel and Georg Cantor's controversial results should not be viewed in the same light.

Gödel's famous theorem, for example, depends on the validity of Cantor's 'diagonal argument' to show that real numbers are uncountable. This in turn only works if sets are interpreted as Platonic objects. Once that assumption is ditched (and which Marxist could object to that?), Cantor's construction of transfinite cardinals cannot proceed. And with that, the proof of Gödel's incompleteness theorem fails.  

Cantor was a self-confessed mystic, and Gödel an avowed Platonist. Normally that would not be enough to impugn a theorist's work, but in this case their conclusions cannot be derived otherwise. 

To be sure, the above claims are controversial -- but only for non-materialists. 

RL

----------oOo----------

The following material was not part of the original letter.

As Henri Poincaré (arguably the greatest mathematician of the 20^th Century) said:

Brouwer added:

[AST = Axiomatic Set Theory.]

More details here and here.

Cantor's mysticism is detailed in the following:

Aczel, A. (2000), The Mystery Of The Aleph. Mathematics, The Kabbalah, And The Search For Infinity (Simon & Schuster).

Versions of the diagonal argument can be found here:

Hunter, G. (1996), Metalogic (University of California Press, 2^nd ed.).

Moore, A. (2001), The Infinite (Routledge, 2^nd ed.).

Further reading:

Balaguer, M. (1998), Platonism And Anti-Platonism In Mathematics (Oxford University Press).

Crary, A., and Read, R. (2000) (eds.), The New Wittgenstein (Routledge).

--------, (2001), 'Prose Versus Proof: Wittgenstein On Gödel, Tarski And Truth', Philosophia Mathematica 3, 9, pp.901-928.

Frege, G. (1953), The Foundations Of Arithmetic (Blackwell).

Giaquinto, M. (2004), The Search For Certainty. A Philosophical Account Of Foundations Of Mathematics (Oxford University Press).

MacBride, F. (2003), 'Speaking With Shadows: A Study Of Neo-Logicism', British Journal for the Philosophy of Science 54, 1, pp.103-63.

Marion, M. (1993), 'Wittgenstein And The Dark Cellar Of Platonism', in Puhl (1993), pp.110-18.

--------, (1998), Wittgenstein, Finitism, And The Foundations Of Mathematics (Oxford University Press).

Puhl, K. (1993) (ed.), Wittgenstein's Philosophy Of Mathematics, Volume Two (Hölder-Pichler-Tempsky).

Rodych, V. (1997), 'Wittgenstein On Mathematical Meaningfulness, Decidability And Application', Notre Dame Journal of Formal Logic 38, 2, pp.195-224. [This links to a PDF. Other files versions can be accessed here.]

--------, (1999a), 'Wittgenstein On Irrationals And Algorithmic Decidability', Synthèse 118, pp.279-304.

--------, (1999b), 'Wittgenstein's Inversion Of Gödel's Theorem', Erkenntnis 51, pp.173-206.

--------, (2000), 'Wittgenstein's Critique Of Set Theory', Southern Journal of Philosophy 38, pp.281-319.

--------, (2002), 'Wittgenstein On Gödel: The Newly Published Remarks', Erkenntnis 56, 3, pp.379-97.

--------, (2003), 'Misunderstanding Gödel: New Arguments About Wittgenstein, And New Arguments By Wittgenstein', Dialectica 57, 3, pp.279-313.

--------, (2006), 'Who Is Wittgenstein's Worst Enemy?: Steiner On Wittgenstein On Gödel', Logique et Analyse 49, 193, pp.55-84.

--------, (2007), 'Wittgenstein's Philosophy Of Mathematics', The Stanford Encyclopedia of Philosophy (Spring 2007 Edition), Edward N. Zalta (ed.).

Sayward, C. (2001), 'On Some Much Maligned Remarks Of Wittgenstein On Gödel,' Philosophical Investigations 24:3: pp.262-70.

Shanker, S. (1987), Wittgenstein And The Turning-Point In The Philosophy Of Mathematics (State University of New York Press).

--------, (1988a), 'Wittgenstein's Remarks On The Significance Of Gödel's Theorem', in Shanker (1988b), pp.155-256.

--------, (1988b) (ed.), Gödel's Theorem In Focus (Croom Helm).

Zenkin, A. (2004) 'Logic Of Actual Infinity And G. Cantor's Diagonal Proof Of The Uncountability Of The Continuum', The Review of Modern Logic 9,  3&4, pp.27-82.

--------, (2005), 'Scientific Intuition Of Genii Against Mytho-"Logic" Of Cantor's Transfinite "Paradise"', Philosophia Scientia 9, 2, pp.145-63 (2005). An earlier version of this paper can be found here.

Papers available online:

Read, R. (No Date), 'Gödel’s Theorem Over-Interpreted: There Is No Such Thing As De Re Self-Reference'.

Zenkin, A. (2000). 'Fatal Mistake Of Georg Cantor'.

--------, (No Date), 'As To Logic Of Cantor's Diagonal Argument'.

Link:

I have just discovered this site which contains several original and illuminating discussions of issues connected with Cantor's Diagonal Method, along with a novel refutation. [Unfortunately, some of the links in this site do not seem to work. Unfortunately, too, the author appears to be a right-wing Blogger!]

Word count: 880

Latest Update: 28/05/08

© Rosa Lichtenstein 2008

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