Again we have the usual mixed bag (i.e. something for everyone). Most to my taste was an entry from Jerry Monaco at Shandean Postscripts (a new one for me). I don't know much about Wittgenstein's philosophy of mathematics but it comes as no suprise to hear that (as elsewhere) he was no fan of platonism. Unfortunately, in rejecting platonism in this context it's really easy to say things like this:
"There is no religious denomination in which the misuse of metaphysical expressions has been responsible for so much sin as it has in mathematics."and
"I shall try again and again to show that what is called a mathematical discovery had much better be called a mathematical invention."(Jerry doesn't give the references for these quotes but I suppose RFM would be a good place to look.) Like a lot of what Richard Rorty says (i.e. about everything), this makes it sound like we are straightforwardly rejecting realism in favor of antirealism: (the objects in question are) not found but made. But of course when we wrack our brains for a valid proof of this or that mathematical claim we don't want to think of ourselves as simply making stuff up. This sends us back to anti-anti-realism; but in math, as I say (i.e. more so even than in the case of, say, physical objects, where it's hard enough), there doesn't seem to be anywhere to go but back to platonism. I say seem, because as far as I can see, Wittgenstein's view was that neither realism nor anti-realism was satisfactory. That said, it does seem that platonism (or realism, in this sense – although of course there is another sense in which, as in the title of Diamond's book, it was a "realistic spirit" which Wittgenstein aimed to capture) is mainly what set Wittgenstein off.
So what exactly is it that set him off here? Jerry's post relates how Cantor's, well, "discovery" of the various levels of infinity caused great excitement among mathematicians, some of whom apparently thought that this solved whatever philosophical difficulties remained concerning not only the status of the infinite but the very foundations of mathematics. As Jerry explains it,
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.If I understand this (an even proposition), that the infinite was "not actual but only potential" was Aristotle's solution to Zeno's paradoxes of motion: a metaphysical matter. As a result,
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.While some people simply rejected the mathematics (i.e. Cantor was just wrong about infinity), Jerry says:
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.This is certainly the view that we would like to (be able to) take, as it would explain the remark that nondenumerable infinities were "a cancerous growth on the body of mathematics" (which Jerry takes to refer to Russell's discussion rather than the math itself) while still allowing their utility in purely mathematical terms. But we still have those remarks about mathematics being an "invention," which is hard to take. This may be why Jerry says that ultimately he rejects Wittgenstein's view of mathematics.
So again, what Jerry has Wittgenstein reject (in addition to certain ideas about the connection of math to formal logic, e.g. Russell's project in Principia Mathematica, which I will not discuss) is "the idea that mathematics somehow gave answers to what Wittgenstein believed were metaphysical questions." Okay, but I think we can say a bit more.
As I see it, the issue here is not simply one of mathematicians failing to mark the boundary between mathematics and philosophy (similar to the way in which some misguided souls think that all by itself, without what it sees as "metaphysics" (by which it really means philosophy) science can solve the free will problem, or determine when life begins, or whatever). This is indeed part of it; but Wittgenstein has the same attitude toward philosophical "discoveries." This is the notoriously poorly understood sense of the famous remark at PI §133: "The real discovery is the one [...] that gives philosophy peace." In other words: If you really haven't understood what I've said up to now (in §§1-132, or maybe §§89-132), and insist on making a philosophical "discovery," then what you need to "discover" is whatever, upon "learning" which, we will be able to shake our obsession with ... making discoveries (i.e. in philosophy).
Our problem in philosophy is that we insist on looking outside ourselves for the answer to our problems (including this one), when instead Wittgenstein sees philosophical problems as concerning ourselves and our (bad) habits, and philosophical work as work on oneself. The point is not simply to switch from outer to inner (as some phenomenologists and/or existentialists would have it, responding to much the same Cartesian conception of philosophy, if then thereby reinstating it). Instead, it concerns the relation between "outer" and "inner" – not as objects (which makes the issue again one of knowledge, and we're back where we started, caught in the Cartesian snare), but (for lack of a better term – and this one at least makes a connection to part II of PI) as aspects of our experience. When we "learn to see the world aright," what we have managed to do is not finally to load the truth into our "belief box," but instead to find our way around – to see not, or not simply, how things are, but (metaphorically speaking) where we are.
This is not to say that knowledge (or "discoveries") cannot be part of our road to understanding, just that they cannot substitute for it – or worse, constitute it. It's this latter absurdity which I think is causing Wittgenstein's understandably frustrated dogmatism on this point, as well as the related one Jerry discusses.
But that tells us more about Wittgenstein than about the mathematics he seems to have spurned. For an excellent look at the mathematics (and the philosophy of same), I recommend Shaughan Lavine's Understanding the Infinite.
Now, go check out the Carnival yourself!