An Incident in the History of the Philosophy of Mathematics or
What did Wittgenstein mean by Cantor's theories being a "cancerous growth" on mathematics?
A question asked;
When commenting on Cantor's ideas of uncountable sets and different levels of infinity, Wittgenstein called it a "cancerous growth on the body of mathematics". Cantor's (and others such as Dedekind) ideas have since provided the basis for much of the development of mathematics thereafter. What could have led Wittgenstein to make such a remark? What did he mean by it?
The hard part in answering this question is trying to explain the pure mathematics in everyday language so that a common reader will know what was at issue between Wittgenstein and those, such as Bertrand Russell, who thought that Cantor, Weirstrass and Dedekind provided a solution to metaphysical problems of the foundations of mathematics. If I get the basic statement of the background wrong please correct me. Still I think it is necessary to state the problem in everyday language because one must have a clear view of how much Cantor's discovery went against common sense. If the reader can understand this she will also be able to understand why so many philosophers and mathematicians thought that Cantor's theories of the infinite did not say anything that made sense. But more important for this note the reader will be able to see how Wittgenstein's view differed from the other condemnations of Cantor's line of thinking.
Cantor considered the problems of infinite sets. The common logic since Aristotle had been that the infinite was not actual but only potential.. But against common logic Cantor showed that there are sets larger than the infinite sets of natural numbers. He showed specifically that no infinite set could have as many elements as all possible subsets of that infinite set. This led to a revolution in how we conceived of set theory and of the infinite. The infinite could no longer be considered an anomaly. In other words their were different "kinds" of infinite sets. (Oh mathematicians forgive my simplicity!) What Cantor was able to show was that infinity was "actual" not just an unimaginably large number, not just "potential". He showed there are infintie sets that are larger than other sets that are also infinite. The best example is the set of all natural numbers versus the set of all irrational numbers. Both sets are infinite sets. But the set of all irrational numbers is "larger," or contains more members, than the set of all natural numbers. (Forgive me. I have merely stated the same notion in a number of ways while avoiding technical language. I did this in the hopes that non-mathematical readers will get my drift. Possibly I'm just furthering your confusion. Also for those of you who may belong to the school of mathematical realists forgive me for stating all of this as if it were just another kind of reality.)
When a mathematician comes to such conclusions philosophers sneeze. Why? Because to decide that the infinite set of irrational numbers is larger than the infinite set of natural numbers is to indirectly decide questions posed at the origins of Aristotle's metaphysics, i.e. the metaphysical status of the infinite. Philosophers of Mathematics recognized this if no one else did. Russell accepted the mathematics but spent much time trying to ground the insight into his own formal logic.
Wittgenstein rejected Cantor but he was not the only one.. Ponicare said, "There is no actual infinity; Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre as a disease from which one has recovered " Brouwer said that: Cantor's theory was "a pathological incident in the history of mathematics from which future generations will be horrified." Another quote in my notebook is from Wittgenstein. "Cantor's argument has no deductive content at all.' Yet I would distinguish this reaction from Ponicare and Brouwer. I take Wittgenstein to mean that he would not argue with the mathematics but would just proclaim it all irrelevant to any philosophical or logical view.
I think most of these reactions were simply a matter of an inability to reconceive ancient notions. But many mathematicians seized on Cantor's theory. Some philosophers were horrified. It didn't seem gentlemanly that these theories were being used as solutions to ancient problems philosophy. Also, the mathematicians who ceased on Cantor's theories treated them as if they were the second coming of the Pythagorean theorem or a new discovery of Pi. Cantor's theories made much of what was said previously in the philosophy of mathematics hard to justify. There were philosophers who were simply exasperated. Why don't mathematicians stop this nonsense, leave us alone, and get back to their equations? What I wonder is, if there were many mathematicians with a philosophical bent who were discouraged by the narrowness of the philosophers. This is an historical determination that is hard to make. No one can ever know what was lost by way of dogmatism.
Wittgenstein was one of those who looked at all of this as an attempt to establish a New Pythagorean Cult around pure mathematics and formal logic. But even though I reject his view I think it should be fully understood. At base Wittgenstein had interesting reasons, that I think can't be easily countered, unless one is a thorrough going rationalist or believes in a pragmatic realism that states in the long run we just work and see what works. (I am somewhere within those choices.)
Wittgenstein's view of mathematics was unique and I doubt one could find more than two people who would have agreed with him in 1932. But I don't think he cared much about who agreed with him, except for Turing. When he was giving a course on these subjects it seems that the only person he cared to 'make see' his point of view was Turing, who would argue with W all the way. Wittgenstein thought that "belief" in mathematics was a kind of religion among intellectuals. He would throw out what must have seemed like Delphic statements at the time such as:
"There is no religious denomination in which the misuse of metaphysical expressions has been responsible for so much sin as it has in mathematics."
"I shall try again and again to show that what is called a mathematical discovery had much better be called a mathematical invention."
The quote about "cancerous growth" is not referring directly to Cantor but rather to Russell's discussion of Cantor, Weirstrass and Dedekind.. Russell believed that pure mathematics had laid the foundations which could ground mathematics in formal logic. For Wittgenstein, these mathematicians' solutions to problems of the infinitesimal, the infinite and continuity and Russell's acceptance of these solutions as great achievements of mathematical logic had "deformed the thinking of mathematicians and philosophers." But Wittgenstein's position was not the same as other philosophers and mathematicians who criticized Cantor, et. al. He did not question the mathematics of the solutions or criticize their premises, he questioned whether these solutions were solutions to mathematical problems at all. More precisely he re-categorized the solutions to another context outside of mathematics and tried to demonstrate that the new context where these solutions must be discussed could be either accepted or rejected without effecting mathematics or logic at all.
Wittgenstein's reference to the 'cancerous growth' on mathematics encapsulates two related notions: In his view mathematicians had grafted onto mathematics the following: (1) the idea that mathematics somehow gave answers to what Wittgenstein believed were metaphysical questions and (2) the idea that when doing certain kinds of 'pure mathematics' what you were doing had some connection to that other kind of game called 'formal logic.'
It was these metaphysical 'answers' and the development of a formal logic that were the 'cancerous growth'. Cantor (and the way others developed Cantor) was just an example of this 'cancerous growth.' To the extent that I understand the issues here I think that Wittgenstein was being dogmatic. To the extent that I understand W'ittgenstein's concern I think he was trying to get the best mathematicians (mainly Turing, who he much admired) to see how both mathematics and formal logic had no real 'foundation' but could be restated in ways that were not 'elegant'. These 'non-elegant' restatements would be equally 'true' in that they would come to the same conclusion without flaws but would seem absurd. I think Wittgenstein was saying that sometimes the elegance of the solution tricks us into accepting it as fundamental or correct.
If I remember, correctly some of what Wittgenstein wrote in his notebooks on these subjects was recently published (4 years ago?). It seems to me that much of Wittgenstein's rhetoric seems to come from the fact that he simply could not get Turing to see that his (Wittgenstein's) picture of mathematics was one possible view of the cathedral. He just thought that all mathematicians were misled on the "reality" of Cantor's proofs and then compounded it all by developing false notions about proclaiming that here - at last -- was the foundation of mathematics.
Of course I may be too hard on Wittgenstein here. There was something in his whole notion about how the "game" of mathematics should be played in order to make sense in the world that also led him to reject Godel's theorem. Who knows maybe in the end we will find that the way Wittgenstein viewed the "game" of mathematics was a sort of anti-foundational foundationalism. I trust I am being appropriately obscure!
Again these are very complicated questions and unfortunately unlike during the 80 years between 1860 and 1940 we don't seem to have great mathematicians who are interesting philosophers and great philosophers who are good mathematicians. The other possibility is that I don't know what I am talking about. It has been a long time since I studied these topics, a long time since those courses where very smart and inarticulate professors tried to explain to me (a very dumb but articulate student) the elegance of pure math. At the time I agreed with Wittgenstein on at least one point. The elegance seemed purely imaginary.
New York City
9 March 2006 (originally written - 5 Feb 2005)
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