they would also get that if they learned more math. eventually they tell you that a lot of concrete rules are more like conventions and assumptions that we have collectively decided are “reasonable”. don’t get me wrong, those conventions are still extremely useful. calculus, for example, has made a lot of problems way easier to solve. but it’s not like moses came down from the mountain with the fundamental theorem of calculus etched in stone. you still need to assume things in order to be able to do calculus, and the ways in which calculus is taught and understood has changed a decent amount over the years (infinitesimals to limits, riemann sums to measures, the introduction of differential forms, etc)
For real. A few times, I’ve been like “What makes you think matter is more fundamentally real than consciousness?” and received an argument that you can measure matter and make mathematical proofs about it.
And I’m just… dumbfounded by the lack of awareness that they’re essentially using a mere mention of math itself to dismiss the significance of axioms.
my experience studying math has been that if someone uses the word “mathematical” when they’re trying to argue something, then there is a decent chance they don’t really know what they’re talking about. if they did, they would probably use a more specific term or cite a theorem or proof. math is not a monolith.
your anecdote is a pretty spectacular example of that. how nice it would be if we could “mathematically prove” that ZFC is objectively true. and also how nice it would be if we could “mathematically define” what it even means for something to be “true” or “objectively true”.
Side note, in my experience people often misuse the word “math” to mean "arithmetic, as in “I did the math” or “Your math is wrong” when they’re just adding up some numbers lol.
yeah that one can be pretty rough too. i think i’ve become a bit desensitized to it over the years, and paul lockhart’s lament has helped me cope a bit, but the pain is still there.
the incompleteness theorems are a part of this broader point as well, since they basically say we can’t choose a perfect system that has everything we want. but still, the incompleteness theorems themselves require making assumptions. you still need to assume some axioms for them to apply, in addition to picking a set of logical rules to follow. and those logical rules aren’t set in stone either. some mathematicians don’t subscribe to the law or the excluded middle, and it makes for some interesting mathematics. for example, it lets you define an infinitesimal as something that’s basically “not not zero”, while still being different from zero.
they would also get that if they learned more math. eventually they tell you that a lot of concrete rules are more like conventions and assumptions that we have collectively decided are “reasonable”. don’t get me wrong, those conventions are still extremely useful. calculus, for example, has made a lot of problems way easier to solve. but it’s not like moses came down from the mountain with the fundamental theorem of calculus etched in stone. you still need to assume things in order to be able to do calculus, and the ways in which calculus is taught and understood has changed a decent amount over the years (infinitesimals to limits, riemann sums to measures, the introduction of differential forms, etc)
For real. A few times, I’ve been like “What makes you think matter is more fundamentally real than consciousness?” and received an argument that you can measure matter and make mathematical proofs about it.
And I’m just… dumbfounded by the lack of awareness that they’re essentially using a mere mention of math itself to dismiss the significance of axioms.
my experience studying math has been that if someone uses the word “mathematical” when they’re trying to argue something, then there is a decent chance they don’t really know what they’re talking about. if they did, they would probably use a more specific term or cite a theorem or proof. math is not a monolith.
your anecdote is a pretty spectacular example of that. how nice it would be if we could “mathematically prove” that ZFC is objectively true. and also how nice it would be if we could “mathematically define” what it even means for something to be “true” or “objectively true”.
Side note, in my experience people often misuse the word “math” to mean "arithmetic, as in “I did the math” or “Your math is wrong” when they’re just adding up some numbers lol.
yeah that one can be pretty rough too. i think i’ve become a bit desensitized to it over the years, and paul lockhart’s lament has helped me cope a bit, but the pain is still there.
https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems
the incompleteness theorems are a part of this broader point as well, since they basically say we can’t choose a perfect system that has everything we want. but still, the incompleteness theorems themselves require making assumptions. you still need to assume some axioms for them to apply, in addition to picking a set of logical rules to follow. and those logical rules aren’t set in stone either. some mathematicians don’t subscribe to the law or the excluded middle, and it makes for some interesting mathematics. for example, it lets you define an infinitesimal as something that’s basically “not not zero”, while still being different from zero.