Comment by bifftastic
5 years ago
Can you say more of either of these? Especially the diagonalization argument which seems difficult to, er, argue with.
5 years ago
Can you say more of either of these? Especially the diagonalization argument which seems difficult to, er, argue with.
I'm not an expert on Wittgenstein, but I think the critique of Cantor's diagonal argument is more to do with its implication that an infinite set can be "bigger" than another. To say so is more of a semantics argument than a mathematical one as it takes for granted the meaning of "infinite". That is, if you define an infinite set as being inherently without size, it makes no sense to then assign it comparitive sizes via cardinality.
I think it has to do with Wittgenstein's constructivism. Since humans can never actually construct an infinite set, one cannot say that such a thing exists as a constructivist. As such, infinity would be a fuzzy without-size concept that Witty was suspicious of. But if one does think mathematical objects are real, then it's not a problem.
I'm guessing constructivists have found better ways to approach infinity than Witty, without conceding that infinity is real.
Wittgenstein was not a constructivist, though. At least in his later years he explicitly argued against any -ism as a position and tried to avoid philosophical theses as a whole. (Whether he succeeded with this undogmatic approach is another question of course, but to say that he was a constructivist would commit him to a position that he did not advocate for, even if many of his remarks can certainly be read in such a way if they are read in isolation.)
I don't get the sense that Wittenstein was suspicious of infinity as an idea / concept in itself, only of its treatment in mathematical logic. Otherwise much of his writings on it could've been far more terse.
You can find what he had to say on Cantor's argument here: http://www.logicmuseum.com/cantor/wittgensteinquotes.htm
These are excerpts, some paragraphs are missing, but the gist is more or less there and you can find the full discussion in 'Remarks on the Foundations of Mathematics'.
Even the Remarks on the Foundations of Mathematics are only a selection of some of Wittgenstein's notebooks made by the editors of his posthumous publications. It is not a work by Wittgenstein in the same sense as the Tractatus or the Philosophical Investigations.
The Part II of the Remarks on the Foundations of Mathematics, which is the part on Cantor, is especially questionable. It is compiled from two documents of Wittgenstein's Nachlass, Ms-117 and Ms-121, but only about a third of the remarks in Ms-121 were included in the published book. There are also other notebooks were Wittgenstein discussed Cantor (most notably Ms-162a and Ms-126b) and which were not published at all. The editors made the decision to exclude some of these remarks because Wittgenstein's notebooks often contain merely drafts of remarks that were never fully developed, so it would be wrong to say that the published book is a deliberate misrepresentation of Wittgenstein's thoughts, but the situation is definitely quite nuanced and complex.
I personally think that the Remarks on the Foundations of Mathematics are very interesting, but that it's a disservice to Wittgenstein and any reader of his works to read his remarks on the philosophy of mathematics as a disconnected collection of aphorisms. The context really matters and his more mathematical remarks only make sense against the backdrop of the Philosophical Investigations and perhaps even his later writings such as On Certainty.