Mathematicians can do mathematics without a knowledge of foundations because the human brain and organism are highly organized, but in a different manner than a formal conceptual system. That human thinking and understanding is, on the whole, not a matter of logically manipulating formalized ideas is generally accepted in philosophy today, as in the later Wittgenstein.
Mathematicians don't care about foundations (2022)
31–36 of 36 posts
Re: Mathematicians don't care about foundations (2022)
#32For most mathematicians, most of the time, the mathematics they personally do is, in relation to different mathematical foundations, like architecture is to atoms.
Architects know and care that they are building things made of atoms. And then ... pretty much don't think about atoms because the objects and relationships they are working on are abstractions well above the fine details of atoms.
And having designed many structures with architectural methods, and seeing those buildings built and stand, it doesn't worry them to hear physicists arguing that maybe atoms are different than they thought. They figure, their experience with architecture and its artifacts has proven to be reliable, so there is no realistic threat of some new quantum theory undermining their work.
On infrequent situations, where their work needs to deal with some special property of some material, they don't have any issue dipping down any number of levels of abstraction. But as a practical matter, that is infrequent for most math.
Re: Mathematicians don't care about foundations (2022)
#33IMO it's not far off how most python or javascript devs don't care about registers or cache misses. Someone's thought deeply about those things so you don't have to. Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem use…
> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed. In a way mathematicians can afford t…
> The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack
This is a very naive take - the very direct translation of what you're saying doesn't happen does in happen in analysis all the time: there are many inequalities which can be "stacked" to prove a bound on something but their factors are too large so you cannot just stack them if you need a fixed bound for your proof to go through. Unsurprisingly this is exactly how actual runtime analysis also works (it's unsurprising because they're both literally math).
Re: Mathematicians don't care about foundations (2022)
#34For most mathematicians, most of the time, the mathematics they personally do is, in relation to different mathematical foundations, like architecture is to atoms. Architects know and care that they are building things made of atoms. And then ... pretty much don't think about atoms because the objects and relationships they are working on are abstractions well above the fine details of atoms. And having designed many…
Thanks, this is a very good analogy.
Re: Mathematicians don't care about foundations (2022)
#35Earlier quoted context omitted.
> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed. In a way mathematicians can afford t…
> The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack This is a very naive take - the very direct translation of what you're saying doesn't happen does in happen in analysis all the time: there are many inequalities which can be "stacked" to prove a bound on something but their factors are too large so you cannot just stack them if you need a fixed bound for your proo…
I think you are taking it a bit out of context here. Obviously, I assumed in that phrase, that mathematicians are stacking suitable methods and proofs. If some factors are too large to not fit in some bound, then obviously that's not something you would stack. But once you have suitable proofs and proven correct and suitable methods, you can stack, and correctness does not go out of the window. Correctness remaining, you will be able to get to a proven correct result.
Of course proving things in mathematics is also a lot harder, usually, than computer programming, and it is probably still easy to make mistakes.
Re: Mathematicians don't care about foundations (2022)
#36IMO it's not far off how most python or javascript devs don't care about registers or cache misses. Someone's thought deeply about those things so you don't have to. Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem use…
> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). I'd be careful about generalizing that to all or most 'mathematicians'. E.g., people working in a lot of fields won't bat an eye at invoking the real numbers when the rational or algebraic numbers would do.
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