In the actual physical world we live in, statements like “there is a square of area two” might not be exactly true. There certainly is no evidence that they are exactly true. (Whereas, in contrast, two plus two really gives you exactly four bananas.)
It’s certainly true that much human-developed math was developed to serve practical purposes, and therefore does accurately model aspects of the real world. But the math isn’t made less true because the physical world deviates slightly from it; likewise the math that’s less tied to the physical world isn’t less true. There are lots and lots of theorems that are interesting, and even useful, but that don’t seem to have much to do with anything physical. (E.g., number theory, or abstract algebra.)
We should maybe taboo the word “true”, since for a mathematical theorem to be true is not exactly the same as for an interpreted sentence about the physical world. How would you then formulate the sentence “the math that’s less tied to the physical world isn’t less true”?
In this case, I mean something like “if you start off with consistent and true beliefs, adding more true beliefs won’t lead to self contradiction.” I can define self-contradiction formally, as asserting both a statement and its formal negation.
This may seem slightly circular, but I think it’s still a useful definition that captures what I want. I also think some circularity is useful to capture what we mean by an axiomatic system.
In the actual physical world we live in, statements like “there is a square of area two” might not be exactly true. There certainly is no evidence that they are exactly true. (Whereas, in contrast, two plus two really gives you exactly four bananas.)
It’s certainly true that much human-developed math was developed to serve practical purposes, and therefore does accurately model aspects of the real world. But the math isn’t made less true because the physical world deviates slightly from it; likewise the math that’s less tied to the physical world isn’t less true. There are lots and lots of theorems that are interesting, and even useful, but that don’t seem to have much to do with anything physical. (E.g., number theory, or abstract algebra.)
We should maybe taboo the word “true”, since for a mathematical theorem to be true is not exactly the same as for an interpreted sentence about the physical world. How would you then formulate the sentence “the math that’s less tied to the physical world isn’t less true”?
In this case, I mean something like “if you start off with consistent and true beliefs, adding more true beliefs won’t lead to self contradiction.” I can define self-contradiction formally, as asserting both a statement and its formal negation.
This may seem slightly circular, but I think it’s still a useful definition that captures what I want. I also think some circularity is useful to capture what we mean by an axiomatic system.