**
Socialist Worker -- Gödel Letter**

**
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Microsoft's new browser,
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**~~~~~~oOo~~~~~~**

As is the case with all my
work, nothing here should be read as an attack
either on Historical Materialism [HM] -- *a scientific theory I fully accept --, *or,
indeed,
on revolutionary socialism. I remain as committed to the self-emancipation of the
working class and the dictatorship of the proletariat as I was when I first became a revolutionary
nearly thirty years ago.

**
The
difference between
Dialectical Materialism [DM] and HM, as I see it, is explained
here.**

**~~~~~~oOo~~~~~~**

**
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*In view of the recent controversy
over their handling of
allegations of rape -- advanced against a former leading member of the UK-SWP
--
I no longer associate myself with Socialist Worker, or with the UK-SWP. *

*
Summary Of My Main Objections To
Dialectical Materialism*

*
Abbreviations Used At This
Site*

**5)
Link**

**6)
Appendix: A Recent Paper On Infinitary Set
Theory**

Below is a
slightly edited version of a letter I sent to *Socialist Worker*
in the summer of 2007, which they chose not to publish (links added):

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 shouldn't be viewed in the same light.

Gödel's famous theorem, for example, depends on the validity of Cantor's 'diagonal argument' to show that the 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 wouldn't 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 to non-materialists.*

Rosa Lichtenstein

The following material wasn't part of the original letter.

Henri
Poincaré (arguably the greatest mathematician of the 20^{th}
Century) had this to say:

"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." [Quoted from here.]

And, here is Gauss (arguably the greatest mathematician of any century):

"I protest against the use of infinite magnitude as
something completed, which is never permissible in mathematics. Infinity is
merely a way of speaking, the true meaning being a limit which certain ratios
approach indefinitely close, while others are permitted to increase without
restriction." [Quoted from
here (this links to a PDF).]

"I don’t know what predominates in Cantor's theory -- philosophy or theology, but I am sure that there is no mathematics there." [Quoted from here (this links to a PDF).]

Weyl:

"...classical logic was abstracted from the mathematics of
finite sets and their subsets.... Forgetful of this limited origin, one
afterwards mistook that logic for something above and prior to all mathematics,
and finally applied it, without justification, to the mathematics of infinite
sets. This is the Fall and original sin of [Cantor's] set theory...." [Quoted
from
here (this links to a PDF).]

"Cantor's set theory as a whole is a pathological incident in the History of Mathematics and a moral crime against future generations (of meta-mathematicians and AST-people)". [Quoted from here.]

**
[AST =
Axiomatic Set Theory.]**

More details can be accessed here and here.

Cantor's mysticism is explained in the following work:

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

Versions of his '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.).

See also Berto (2009), pp.32-36.

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

Berto, F. (2009), *There's Something About Gödel. The Complete Guide To The
Incompleteness Theorem* (Wiley Blackwell). [The section on Wittgenstein is to
be found on pp.189-213.]

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

Dybjer, P., Lindström, S., Palmgren, E., and Sundholm, G.
(2012) (eds.),* Epistemology Versus Ontology: Essays On The Philosophy And
Foundations Of Mathematics In Honour Of Per Martin-Löf* (Springer Verlag).

Floyd, J. (2000), 'Wittgenstein, Mathematics And Philosophy', in Crary and Read (2000), pp.232-61.

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

--------, (2012), 'Wittgenstein's
Diagonal Argument: A Variation On Cantor And Turing', in Dybjer *et al*
(2012), pp.25-44. [This links to a PDF.]

--------, (forthcoming 1), *Wittgenstein On Gödel
And Turing*.

--------, (forthcoming 2), *The Uncaptive Eye: Wittgenstein, Mathematics And
Philosophy.*

[There are several more relevant papers of Juliet's accessible here.]

Floyd, J., and Putnam, H.
(2000), 'A Note On Wittgenstein's "Notorious Paragraph" About The Gödel
Theorem',* Journal of Philosophy ***97**, 11, pp.624-32.

--------, (2006), 'Bays, Steiner And Wittgenstein's
"Notorious" Paragraph About The Gödel Theorem', *Journal of Philosophy *
**103**, 2, pp.101-10.

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).

Potter, M. (2002), *Reason's Nearest Kin.
Philosophies Of Arithmetic From Kant To Carnap* (Oxford University Press).

--------, (2004), *Set Theory And Its
Philosophy* (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 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.

--------, (2011), 'Wittgenstein's
Philosophy Of Mathematics', *The Stanford Encyclopedia of Philosophy *
(Spring 2011 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).

Slater, H. (2002), *Logic Reformed*
(Peter Lang).

Smith, P. (2007), *An Introduction To
Gödel's Theorems* (Cambridge University Press).

Tiles, M. (2004), *The Philosophy Of Set
Theory. An Historical Introduction To Cantor's Paradise* (Dover Publications).

Wittgenstein, L. (1974), *Philosophical Grammar,
*edited by Rush Rhees, translated by Anthony Kenny (Blackwell).

--------, (1975), *Philosophical Remarks,
*edited by Rush Rhees, translated by Roger White and Raymond Hargreaves (Blackwell).

--------, (1976), *Wittgenstein's Lectures
On The Foundation Of Mathematics: Cambridge 1939*, edited by Cora Diamond* *(Harvester
Press).

--------, (1978),* Remarks On The
Foundations Of Mathematics, *edited Elizabeth Anscombe (Blackwell, 3^{rd} ed.).

--------, (2009), *Philosophical
Investigations, *translated by Elizabeth Anscombe, Peter Hacker and Joachim
Schulte
(Blackwell, 4^{th} ed.).

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.

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

Wildberger, N. (No Date), 'Set Theory: Should You Believe?' [This links to a PDF. The first few pages of this paper have been reproduced in the Appendix.]

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

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

A friend has just sent me a link to a site that is (partly) devoted to exposing the serious logical and mathematical flaws in Gödel's Incompleteness Theorem. The site also contains links to on-line papers that argue to the same end.

**Link**:

Then there is this site, which contains several original and illuminating discussions of issues connected with Cantor's Diagonal Method, along with a novel refutation. [However, some of the links at this site do not seem to work. Unfortunately, the author appears to be a right-wing Blogger!]

Here is an extract from a recent
paper on *Infinitary Set Theory* (links added):

**Set Theory: Should You Believe?
**

Professor N J Wildberger

School of Mathematics, University of New South Wales

"I protest against the use of infinite magnitude as something completed, which
is never permissible in mathematics. Infinity is merely a way of speaking, the
true meaning being a limit which certain ratios approach indefinitely close,
while others are permitted to increase without restriction." (Gauss)

"I don’t know what predominates in Cantor’s theory -- philosophy or theology,
but I am sure that there is no mathematics there." (Kronecker)

"...classical logic was abstracted from the mathematics of finite sets and their
subsets.... Forgetful of this limited origin, one afterwards mistook that logic
for something above and prior to all mathematics, and finally applied it,
without justification, to the mathematics of infinite sets. This is the Fall and
original sin of [Cantor's] set theory..." (Weyl)

**Modern Mathematics As Religion
**

Modern mathematics

If mathematics made complete sense it would be a lot easier to teach, and a lot easier to learn. Using flawed and ambiguous concepts, hiding confusions and circular reasoning, pulling theorems out of thin air to be justified 'later' (i.e. never) and relying on appeals to authority don't help young people, they make things more difficult for them.

If mathematics made complete sense there would be higher standards of rigour, with fewer but better books and papers published. That might make it easier for ordinary researchers to be confident of a small but meaningful contribution. If mathematics made complete sense then the physicists wouldn't have to thrash around quite so wildly for the right mathematical theories for quantum field theory and string theory. Mathematics that makes complete sense tends to parallel the real world and be highly relevant to it, while mathematics that doesn't make complete sense rarely ever hits the nail right on the head, although it can still be very useful.

So where exactly are the logical problems? The troubles stem from the

Most of the problems with the foundational aspects arise from mathematicians' erroneous belief that they properly understand the content of public school and high school mathematics, and that further clarification and codification is largely unnecessary. Most (but not all) of the difficulties of Set Theory arise from the insistence that there exist 'infinite sets', and that it is the job of mathematics to study them and use them.

In perpetuating these notions, modern mathematics takes on many of the aspects of a

Training of the young is like that in secret societies -- immersion in the cult involves intensive undergraduate memorization of the standard thoughts before they are properly understood, so that comprehension often follows belief instead of the other (more healthy) way around. A long and often painful graduate school apprenticeship keeps the cadet busy jumping through the many required hoops, discourages critical thought about the foundations of the subject, but then gradually yields to the gentle acceptance and support of the brotherhood. The ever-present demons of inadequacy, failure and banishment are however never far from view, ensuring that most stay on the well-trodden path.

The large international conferences let the fellowship gather together and congratulate themselves on the uniformity and sanity of their world view, though to the rare outsider that sneaks into such events the proceedings no doubt seem characterized by jargon, mutual incomprehensibility and irrelevance to the outside world. The official doctrine is that all views and opinions are valued if they contain truth, and that ultimately only elegance and utility decide what gets studied. The reality is less ennobling -- the usual hierarchical structures reward allegiance, conformity and technical mastery of the doctrines, elevate the interests of the powerful, and discourage dissent.

There is no evil intent or ugly conspiracy here -- the practice is held in place by a mixture of well-meaning effort, inertia and self-interest. We humans have a fondness for believing what those around us do, and a willingness to mold our intellectual constructs to support those hypotheses which justify our habits and make us feel good.

The reason that mathematics doesn't make complete sense is quite easy to explain when we look at it from the educational side. Mathematicians, like everyone else, begin learning mathematics before kindergarten, with counting and basic shapes. Throughout the public and high school years (K-12) they are exposed to a mishmash of subjects and approaches, which in the better schools or with the better teachers involves learning about numbers, fractions, arithmetic, points, lines, triangles, circles, decimals, percentages, congruences, sets, functions, algebra, polynomials, parabolas, ellipses, hyperbolas, trigonometry, rates of change, probabilities, logarithms, exponentials, quadrilaterals, areas, volumes, vectors and perhaps some calculus. The treatment is non-rigorous, inconsistent and even sloppy. The aim is to get the average student through the material with a few procedures under their belts, not to provide a proper logical framework for those who might have an interest in a scientific or mathematical career.

In the first year of university the student encounters calculus more seriously and some linear algebra, perhaps with some discrete mathematics thrown in. Sometime in their second or third year, a dramatic change happens in the training of aspiring pure mathematicians. They start being introduced to the idea of

Do you suppose the curriculum at this point has time or inclination to return to the material they learnt in public school and high school, and finally organize it properly? When we start to get really picky about logical correctness, doesn't it make sense to go back and ensure that all those subjects that up to now have only been taught in a loose and cavalier fashion get a proper rigorous treatment? Isn't this the appropriate time to finally learn what a

The first reason is that even the professors mostly don't know! They too have gone through a similar indoctrination, and never had to

The modern mathematician walks around with her head full of the tight logical relationships of the specialized theories she researches, with only a rudimentary understanding of the logical foundations underpinning the entire subject. But the worst part is, she is largely unaware of this inadequacy in her training. She and her colleagues

Mathematicians like to reassure themselves that foundational questions are resolved by some mumbo-jumbo about 'Axioms' (more on that later) but in reality successful mathematics requires familiarity with a large collection of 'elementary' concepts and underlying linguistic and notational conventions. These are often unwritten, but are part of the training of young people in the subject. For example, an entire essay could be written on the use, implicit and explicit, of

The second reason is that any attempt to lay out elementary mathematics properly would be resisted by both students and educators as not going forward, but backwards. Who wants to spend time worrying about the correct approach to polynomials when

But there are two foundational topics that are introduced in the early undergraduate years:

Or some such nonsense. Set theory as presented to young people

I think we can agree that (finite) set theory is understandable. There are many examples of (finite) sets, we know how to manipulate them effectively, and the theory is useful and powerful (although not as useful and powerful as it should be, but that’s a different story).

So what about an 'infinite set'? Well, to begin with, you should say precisely what the term means. Okay, if

The dubious nature of Cantor's definition was spectacularly demonstrated by the contradictions in 'infinite set theory' discovered by Russell and others around the turn of the twentieth century. Allowing any old 'infinite set' à la Cantor allows you to consider the 'infinite set' of 'all infinite sets', and this leads to a self-referential contradiction. How about the 'infinite sets' of 'all finite sets', or 'all finite groups', or perhaps 'all topological spaces which are homeomorphic to the sphere'? The paradoxes showed that unless you are very particular about the exact meaning of the concept of 'infinite set', the theory collapses. Russell and Whitehead spent decades trying to formulate a clear and sufficiently comprehensive framework for the subject.

Let me remind you that mathematical theories are not in the habit of collapsing. We do not routinely say, "Did you hear that Pseudo-convex cohomology theory collapsed last week? What a shame! Such nice people too."

So did analysts retreat from Cantor's theory in embarrassment? Only for a few years, till Hilbert rallied the troops with his battle-cry "No one shall expel us from the paradise Cantor has created for us!" To which Wittgenstein responded "If one person can see it as a paradise for mathematicians, why should not another see it as a joke?"

Do modern texts on set theory bend over backwards to say precisely

The bulwark against such criticisms, we are told, is having the appropriate collection of 'Axioms'! It turns out, completely against the insights and deepest intuitions of the greatest mathematicians over thousands of years, that it all comes down to

The rest of the above paper can be accessed here. [This links to a PDF.]

**Word count: 4,170**

**Latest Update: 27/04/16**

**© Rosa Lichtenstein 2016**

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