Wittgenstein's views of numbers

[adammclean]

I am a bit perplexed by Wittgenstein' remarks in his book on the Philosophy of Mathematics.
I read this many years ago and found it almost impenetrable. Now I am trying again to make some assessment of Wittgenstein's view of numbers.

It seems he tried to remove many of the difficult problems of mathematics by merely saying that such ideas had no meaning. Thus he dismissed set theory as nonsense, and would not accept the continuum of real numbers. Indeed, in his attempt to finitise mathematics he would not accept that the decimal expansions of irrational numbers had any meaning, beyond whatever point one had actually calaculated it. Thus, for Wittgenstein, the question - is there any appearance of seven sevens (...7777777...) in the decimal expansion of PI ?, had no meaning - and one could only place a question as, is there any appearance of seven sevens (...7777777...) in the decimal expansion of PI up to the 10,000 digit (or whatever)?.

The mathematics which Wittgenstein could envigage, would appear to have to be considerably pruned - out goes set theory, topology, analysis of continuous functions, infinity, and so on.

Does anyone have any views on Wittgenstein's philosophy of mathematics ? Are there any ideas in his philosophy relevant to late 20th/ early 21st century mathematics, or should we merely see his work as a flawed critique of the mathematics of his era.

 

[SW VandeCarr]

I have only a superficial knowledge of Wittgenstein, but it sounds like he believed in a strictly constructivist view of mathematics. If so, he probably did not believe in inductive proofs and possibly analytic continuation. Such an approach can be internally consistent, but very limited. It certainly isn't mainstream mathematics.

 

[JSuarez]

I am a bit perplexed by Wittgenstein' remarks in his book on the Philosophy of Mathematics.
I read this many years ago and found it almost impenetrable.

Join the club. It's a pretty large one.

Thus, for Wittgenstein, the question - is there any appearance of seven sevens (...7777777...) in the decimal expansion of PI ?, had no meaning - and one could only place a question as, is there any appearance of seven sevens (...7777777...) in the decimal expansion of PI up to the 10,000 digit (or whatever)?.

These were not Wittgenstein's ideas, but Brouwer's, from whom he was profoundly infuenced (they were both prone to mysticism and subjectivism), but that seems to be his view as well.

The mathematics which Wittgenstein could envigage, would appear to have to be considerably pruned - out goes set theory, topology, analysis of continuous functions, infinity, and so on.

This is not exactly true; Brouwer's intuitionism indeed rejected many things, but it had substitutes: "spreads" for sets, a different model for the real line, that implied an alternative version of real analysis (where all the functions were continuous and every continuous function was absolutely continuous), etc. But it was restrictive.

should we merely see his work as a flawed critique of the mathematics of his era.

Well, that's my opinion, but I'm no fan.

But there's a recent review article on Wittgenstein's philosophy of mathematics here:

http://plato.stanford.edu/entries/wittgenstein-mathematics/

That is probably more charitable than my opinion.

 

[adammclean]

But there's a recent review article on Wittgenstein's philosophy of mathematics here:
http://plato.stanford.edu/entries/wittgenstein-mathematics/
That it's probably more charitable my opinion.

Yes this article strives to document Wittgenstein's approach, without the author taking a rhetorical position. I found it very informative. The mathematics that would result from pruning out the law of the excluded middle and the induction axiom in Peano arithmetic, amongst others, would be almost unrecognisable to us today. Perhaps this disconnect indicates that Wittgenstein's approach was a device to try and outlaw certain aspects of mathematics which he found pathological, rather than a foundation on which mathematics could realistically be built.
Infinities in mathematics are challenging to our intuition, but perhaps are something we have to take delight in, rather than running away from.
Wittgenstein had a profound influence on mid and late 20th century philosophy, so one does want to look favourably on his views on the foundation of mathematics. But I, for one, though open to trying to appreciate his ideas on mathematics, am struggling a bit to see it clearly. I cannot just throw away the evolution of mathematics since the 1930's, as surely this is one of the great eras of mathematical exploration and creativity and yet I feel a mind as influential as Wittgenstein's should not be easily dismissed.

 

[adammclean]

This is not exactly true; Brouwer's intuitionism indeed rejected many things, but it had substitutes: "spreads" for sets, a different model for the real line, that implied an alternative version of real analysis (where all the functions were continuous and every continuous function was absolutely continuous), etc. But it was restrictive.

Thanks for this. I will try and take a look at Brouwer's mathematical ideas.

 

[adammclean]

I just found this interesting article.

Wittgenstein's Analysis of the Paradoxes in His Lectures on the Foundations
of Mathematics
Charles S. Chihara
The Philosophical Review, Vol. 86, No. 3 (Jul., 1977), 365-381.

It is an account of Wittgenstein's 1939 lectures on the foundations of mathematics. Alan Turing was in the audience and made some telling points and challenged Wittgenstein. It is interesting that Wittgenstein seemed to hold the view that contradictions and inconsistences in a formal system should just be ignored. To him these often seemed to be merely word games with no consequences for the formal system if we ignored them. Not something that Turing could go along with.

 

[rhh1]

Ray Monk offers an interesting and discursive discussion of Wittgenstein's views on mathematics at:http://www.scribd.com/doc/12957037/wittgenstein-and-his-interpreters

Monk R. (2007) 'Bourgeois, Bolshevik or Anarchist? The Reception of Wittgenstein's Philosophy of Mathematics' in Kahane, G., Kanterian, E .and Kuusels, O. Wittgenstein and His Interpreters: Essays in Memory of Gordon Baker Oxford: Blackwell

 

[disregardthat]

I have only a superficial knowledge of Wittgenstein, but it sounds like he believed in a strictly constructivist view of mathematics. If so, he probably did not believe in inductive proofs and possibly analytic continuation. Such an approach can be internally consistent, but very limited. It certainly isn't mainstream mathematics.

He did believe in inductive proofs. However he interpreted them slightly differently than one commonly would. We would say that the conclusion of the inductive proof is that the relevant statement is true for all positive integers. Wittgenstein would say that the conclusion is not a statement about all positive integers, but rather a way of proving it for any integer. So given m, we can by the inductive argument construct a sequence of implications such that the statement for m is proved. He said that "the statement is true for all integers/any integer" is a proxy statement, not a mathematical one. He stresses the importance of being careful about what are genuine mathematical propositions and what are not.

This perspective is understandable as he does not favor the notion of an actual infinite set. There is no such thing as a "list" of the natural numbers. He argues that our notion of the really large is mistakenly applied to the infinite. He considered the infinity of the integers not as a completed infinite list as such, but rather as an unrestricted potential of creating new numbers.

“the mistake in the set-theoretical approach consists time and again in treating laws and enumerations (lists) as essentially the same kind of thing and arranging them in parallel series so that one fills in gaps left by the other.”

This was basically his stance towards all forms of (countable) infinity, the unrestricted potential to continue the series/sequence/list/process. Note that to him (not to Brouwer!) it is essential that the potential to continue is given, that you actually have an algorithm. So free-choice sequences was strictly disallowed (Brouwer approved of them I believe), we would e.g. need a recurrence relation in order to say that it defines an "infinite" sequence. Naturally he disapproved of the axiom of choice as well. His view was also strictly formalist while Brouwers view was not. I don't think you can draw the parallels between them too far.

His opinion of set theory is seen in this quote:

"Set theory attempts to grasp the infinite at a more general level than the investigation of the laws of the real numbers. It says that you can't grasp the actual infinite by means of mathematical symbolism at all and therefore it can only be described and not represented. … One might say of this theory that it buys a pig in a poke. Let the infinite accommodate itself in this box as best it can."

So basically he thought set-theorists axiomatized their flawed intuition of the infinite.

 

[disregardthat]

These were not Wittgenstein's ideas, but Brouwer's, from whom he was profoundly infuenced (they were both prone to mysticism and subjectivism), but that seems to be his view as well.

If I'm not mistaken I believe Brouwer didn't actually claim that such propositions were meaningless, only that the law of excluded middle didn't apply to such propositions, but he accepted them as meaningful. Wittgenstein rejected them as genuine mathematical propositions because of this, as in his view every genuine mathematical proposition must be verifiable by means of calculation. This is one important distinction between Brouwer and Wittgenstein.

The mathematics that would result from pruning out the law of the excluded middle and the induction axiom in Peano arithmetic, amongst others, would be almost unrecognisable to us today. Perhaps this disconnect indicates that Wittgenstein's approach was a device to try and outlaw certain aspects of mathematics which he found pathological, rather than a foundation on which mathematics could realistically be built.

Wittgenstein did not reject the law of excluded middle, he thought it essential to mathematics.