(November 8, 2021 at 1:36 pm)vulcanlogician Wrote:Quote:Meno has a very interesting argument, from mathematics, about the nature of knowledge and memory. it is interesting to see how the slave arrives at the correct answers, but only through leading questions.
There is also a dialog, I don't recall which one, in which Theatetus mentions that he has a proof that `square roots' are irrational unless the integer is a perfect square. This is interesting as part of the history of mathematical ideas. The discovery of irrational ratios was a blow to the Pythagorean philosophy and the investigation of the concept of irrational ratios by Theatetus was quite important. There is even a quote in Aristotle using Theatetus' definition as opposed to the more well-known definition by Eudoxus. In any case, that Theatetus claims to have a proof is interesting because the only natural proof at that stage would have been one using prime numbers and unique factorization.
There were plenty of blows dealt to mathematical realism in the 19th and 20th centuries: Bertrand Russell's criticism of Frege, and a few others that escape me right now. The thing about that is that while mathematical realism became much more dissatisfying of a theory after those revelations, mathematical fictionalism and other competing theories still remain as dissatisfying as, if not more dissatisfying, than mathematical realism.
I think the main point of the slave being taught to understand a mathematical principle is that there is a real principle that is kind of "there" to be discovered. It's kind of hard to dispute that. But if you want to claim that it is true that the principle is "there" waiting to be discovered, you are presented with all kinds of metaphysical puzzles. Some of these questions are things only philosophers care about. There are distinctions there that you can kind of gloss over or easily explain away. But you can't [i]completely explain the issue away, and (for better or worse) philosophers have been chewing on that riddle for centuries.
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Russell, Skolem, Godel (even though Godel was a realist!), Cohen, etc. Personally, I am a mathematical formalist with a thread that we choose axioms to conform to intuition and utility.
The interesting thing about Meno is that Socrates wasn't claiming realism in math (although that is there) as much as he was claiming that we all already know the truth, but just have to be reminded of it. This was at least as much an argument for life before we are born as it was an argument about mathematical realism.
As a formalist, I tend to see math as part of an intricate game, based on assumptions made about sets, numbers etc. So, for me, the question of the 'existence' of mathematical objects )or even mathematical arguments) becomes an analysis of games.
For example, it is pretty clear, I think, that the game of chess was invented. It isn't a universal form that is part of the nature of the universe. So, when I show you a board position and claim there is mate in 3 moves, in what sense does that sequence of moves 'exist'? As I see it, we invent the rule and then discover the consequences of those rules. How that works philosophically, I do not know.
I see math as being similar: we invent the axioms and rules of deduction, but we then discover the consequences of those rules. We choose certain rules because of their appeal to intuition or their utility (for example, the inductive principle in Peano Arithmetic), but then discover the consequences of those rules.
One idea I like to play with is what would happen if we discovered an inconsistency in mathematics (since Godel, we know this is a possibility). How would our system change? Which rules would we keep and which would we allow ourselves to discard? The last time this happened (Russel and his paradox), there was a massive overturning of how mathematics was done. What would happen the next time?
