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Thanks for the detailed response. This critique of maths just doesn't resonate with me at all, though. >that it is logically possible for this to happen means

Thanks for the detailed response. This critique of maths just doesn't resonate with me at all, though. >that it is logically possible for this to happen means our foundations/definitions/models are all screwed up Why? Simply because it feels unintuitive to you? Where is the useful alternative proposed by this theory? Is there any sound definition of what is 'metaphysically real' and why there is any reason to dismiss things which are not? For example consider FLT - the proof surely uses fields and concepts with countless 'absurdities' inside them. And yet it proved something quite real and tangible. Consider that there may be no alternative proof of FLT. In that case, doesn't that validate those absurd concepts and give them a reality of their own that doesn't require any 'metaphysical' form? They have revealed a truth about the physical natural numbers we can all understand. To me that gives them a very compelling 'realness'. Of course it could be that you can throw out all the absurd definitions and still build a useful enough system to prove such things - I am sure people have tried. Maybe the point is that not enough people have tried?

georgewsinger 6y ago
Not being facetious: I suggest you read the book. It discusses precisely these objections. I should say also the only reason I read this book was because I liked the author's other books which have nothing to do with mathematics. Before reading it I thought "man who gives a shit about philosophy of mathematics? can we just do some real math please?", but this book changed my mind about that. Also: I agree that traditional mathematics is useful. I trust that FLT true (not to mention it's an awesome achievement). And I even trust that proofs which make heavy use of concepts I find philosophically dubious will almost always turn out to still hold when placed under the "right" sort of foundations. There are some exceptions to this though. For example, I absolutely don't trust stuff like this: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox By "don't trust", I mean I seriously doubt there is anything at all out there in the universe (or in the metaphysically possible universe) that behaves like this. Instead, it's just a logical result which leans heavily on some problematic mathematical assumptions which were never actually justified. It should be taken as a reductio ad absurdem that something went wrong with the inputs (or at least evidence as such). 100+ years ago, people spent time wrestling with the foundations of math, but now it's looked down upon as a waste of time/not "serious" math, etc.

georgewsinger 6y ago

Not being facetious: I suggest you read the book. It discusses precisely these objections. I should say also the only reason I read this book was because I liked the author's other books which have nothing to do with mathematics. Before reading it I thought "man who gives a shit about philosophy of mathematics? can we just do some real math please?", but this book changed my mind about that. Also: I agree that traditional mathematics is useful. I trust that FLT true (not to mention it's an awesome achievement). And I even trust that proofs which make heavy use of concepts I find philosophically dubious will almost always turn out to still hold when placed under the "right" sort of foundations. There are some exceptions to this though. For example, I absolutely don't trust stuff like this: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox By "don't trust", I mean I seriously doubt there is anything at all out there in the universe (or in the metaphysically possible universe) that behaves like this. Instead, it's just a logical result which leans heavily on some problematic mathematical assumptions which were never actually justified. It should be taken as a reductio ad absurdem that something went wrong with the inputs (or at least evidence as such). 100+ years ago, people spent time wrestling with the foundations of math, but now it's looked down upon as a waste of time/not "serious" math, etc.