My current rule of thumb, which largely seems to work, is to ask “what are the real-world consequences of propositions in this discipline being right or wrong?”
Interesting heuristic. I’d be intrigued to hear what it says about pure mathematics?
One of my strongest mathematical interests is graph theory, in part because networks are incredibly pure abstract mathematical objects which you can draw lots of conclusions about on a purely logical basis, and in part because they can be used to model so many real-world phenomena. As a result, even modest propositions in that particular area have lots of real-world consequences.
History also strongly suggests that even the most historically useless pure maths can have tremendously important applications and consequences further down the line, some choice examples being radon transformations, modular arithmetic and eigenspace. It would be an incredibly bold statement to say a particular area of pure maths is completely without real-world consequence. There’s an awful lot of remaining time for even the most esoteric theorem to be put to use.
Apologies in advance for nitpicking, but the heuristic is to ask what are the real-world consequences of propositions in this discipline being right or wrong, not whether the discipline has real-world consequences. So what are the propositions of mathematics that can be right or wrong? Clearly a published theorem can be right or wrong, but most are correct. What can be right or wrong is what areas of pure mathematics people consider to be interesting. I would say that these propositions can be right or wrong, and do have “real world” consequences. People used to think graph theory was not interesting—they were wrong.
I’m not sure interestingness is really the focus of the issue. I’m sure feminist post-structuralist discourse analyses of the recent banking crisis are very interesting to people interested in the subject, but I still don’t think it has any power to deduce true facts about the universe.
I do have another heuristic which is a little less straightforward to apply, but a little more selective: would a society of humans kept isolated from our own for thousands of years develop a similar discipline with the same essential elements as our version? Pure maths definitely passes that one, while something like Jungian analysis probably wouldn’t.
There’s a lot of places where we’d have catastrophic consequences if we had wrong beliefs about pure mathematics. For example, public-key crypto would fall apart if mathematicians were severely mistaken about finite groups and their relationship to prime numbers. And we couldn’t be very wrong about real analysis before we’d notice something the matter with calculus.
I would have said that math is a degenerate case for such a heuristic because we so seldom are wrong about it.
Interesting heuristic. I’d be intrigued to hear what it says about pure mathematics?
Depends on which area of pure mathematics.
One of my strongest mathematical interests is graph theory, in part because networks are incredibly pure abstract mathematical objects which you can draw lots of conclusions about on a purely logical basis, and in part because they can be used to model so many real-world phenomena. As a result, even modest propositions in that particular area have lots of real-world consequences.
History also strongly suggests that even the most historically useless pure maths can have tremendously important applications and consequences further down the line, some choice examples being radon transformations, modular arithmetic and eigenspace. It would be an incredibly bold statement to say a particular area of pure maths is completely without real-world consequence. There’s an awful lot of remaining time for even the most esoteric theorem to be put to use.
Apologies in advance for nitpicking, but the heuristic is to ask what are the real-world consequences of propositions in this discipline being right or wrong, not whether the discipline has real-world consequences. So what are the propositions of mathematics that can be right or wrong? Clearly a published theorem can be right or wrong, but most are correct. What can be right or wrong is what areas of pure mathematics people consider to be interesting. I would say that these propositions can be right or wrong, and do have “real world” consequences. People used to think graph theory was not interesting—they were wrong.
I’m not sure interestingness is really the focus of the issue. I’m sure feminist post-structuralist discourse analyses of the recent banking crisis are very interesting to people interested in the subject, but I still don’t think it has any power to deduce true facts about the universe.
I do have another heuristic which is a little less straightforward to apply, but a little more selective: would a society of humans kept isolated from our own for thousands of years develop a similar discipline with the same essential elements as our version? Pure maths definitely passes that one, while something like Jungian analysis probably wouldn’t.
There’s a lot of places where we’d have catastrophic consequences if we had wrong beliefs about pure mathematics. For example, public-key crypto would fall apart if mathematicians were severely mistaken about finite groups and their relationship to prime numbers. And we couldn’t be very wrong about real analysis before we’d notice something the matter with calculus.
I would have said that math is a degenerate case for such a heuristic because we so seldom are wrong about it.