I spent just a little more time learning history and disagree even more strongly.
Calculus and differential equations were developed (twice) with the explicit purpose of describing the behavior of the world around us.
The theory of determinants and later of linear algebra were developed with the explicit purpose of solving systems of linear equations which arise in the problem of predicting the world around us.
The calculus of variations and later functional analysis were developed with the explicit purpose of understanding the particular differential equations which arise in the problem of predicting the world around us (laws of motion, heat equations, etc.)
Probability was developed to allow people to understand and calculate probabilities, whose usefulness was already understood (insurance predates the study of probability).
Statistics was developed to understand large quantities of demographic data, whose existence predated the study of statistics.
Group theory and representation theory were developed to understand a problem unconnected to the world around us. The actual importance of finite group theory apart from representation theory appears to be extremely close to zero. The importance of representation theory in physical developments over the last century also appears to be extremely small (although the formalism is used extensively in theoretical treatments) but I don’t know enough to say with confidence. I would unquestionably have argued against the development of group theory, but I am not convinced that this would have been a bad thing.
Number theory was developed to understand a problem unconnected from reality. Number theory has apparently contributed almost nothing to society since its creation. You could argue that the development of cryptography was dependent on at least a rudimentary understanding of number theory, but given the existence of lattice cryptography and the early emergence of its predecessors (more or less concurrent with RSA), you would almost certainly lose this argument.
Non-Euclidean geometry was developed without connection to reality. It became applicable with the observation that the universe was best described by non-Euclidean geometry. I would unquestionably have argued against working on non-Euclidean geometry before the development of general relativity; the main question was whether the existence of non-Euclidean geometry facilitated the discovery of general relativity. It seems that Einstein explicitly suggested that spacetime may non-Euclidean before learning that non-Euclidean geometry had been extensively studied. This leads me to suspect that work on non-Euclidean geometry was not essential in the development of general relativity, and that it could just as well have been done after it became relevant.
Of course I can provide a long list of fields unconnected to reality. I cannot think of any significant contributions from any of them. If you can think of a good counterexample here, feel free to suggest it (I think group theory is far and away the best).
Non-Euclidean geometry was developed without connection to reality.
Axiomatic hyperbolic geometry was a game about arbitrary axioms.
Perhaps that’s why Gauss didn’t publish on it. What he did publish
was his work on extrinsic differential geometry inspired by his
work as a surveyor. The Gauss-Bonnet theorem for triangles answers
a question a surveyor would ask.
Riemann said that his intrinsic differential geometry was an attempt to understand space.
Number theory was developed to understand a problem unconnected from reality.
Yes, I think that’s what Gauss meant when he called number theory
the queen of mathematics. But I think number theory was a lot
narrower back then. A lot of things that would now be called
number theory were instead grouped under “solving equations.” I
think that when Abel showed that one couldn’t solve the quintic by
radicals and when he showed that one could solve it by
hypergeometric functions, he thought he was studying the same
field.
2. Applications
Group theory and number theory are endemic in CS. It’s not just
cryptography. Consider coding theory.
For an application of 20th C math,
Margulis’s expanders were for decades the only explicit ones
(and one does want non-random expanders for randomness extraction).
The actual importance of finite group theory apart from
representation theory appears to be extremely close to zero.
I’m not sure what you mean. Perhaps that when finite group theory turned inwards and tried to classify finite simple groups, it stopped having applications? Maybe I’d buy that. It seems like an argument for the “interconnected” position against the “interesting” position, but fairly neutral for the “applied” position.
Anyhow, this seems to discount decades of 19th C struggle to clarify the meaning of an abstract group, which is clearly important if groups are important. It’s hard to see in retrospect what was so difficult. One might credit this advance to set theory, the idea that one should talk about abstract sets (like the set of cosets).
So I think set theory is a quite inward-looking subject that turned out to have great clarifying impact on mathematics. But maybe it’s not necessary—do physicists think about groups without it? Similarly, category theory clarified a lot of math, perhaps not in ways that have yet reached the physicists (the way set theory could), but it was been picked up for its own sake in CS, both in type theory and in parallel computation.
I spent just a little more time learning history and disagree even more strongly.
Calculus and differential equations were developed (twice) with the explicit purpose of describing the behavior of the world around us.
The theory of determinants and later of linear algebra were developed with the explicit purpose of solving systems of linear equations which arise in the problem of predicting the world around us.
The calculus of variations and later functional analysis were developed with the explicit purpose of understanding the particular differential equations which arise in the problem of predicting the world around us (laws of motion, heat equations, etc.)
Probability was developed to allow people to understand and calculate probabilities, whose usefulness was already understood (insurance predates the study of probability).
Statistics was developed to understand large quantities of demographic data, whose existence predated the study of statistics.
Group theory and representation theory were developed to understand a problem unconnected to the world around us. The actual importance of finite group theory apart from representation theory appears to be extremely close to zero. The importance of representation theory in physical developments over the last century also appears to be extremely small (although the formalism is used extensively in theoretical treatments) but I don’t know enough to say with confidence. I would unquestionably have argued against the development of group theory, but I am not convinced that this would have been a bad thing.
Number theory was developed to understand a problem unconnected from reality. Number theory has apparently contributed almost nothing to society since its creation. You could argue that the development of cryptography was dependent on at least a rudimentary understanding of number theory, but given the existence of lattice cryptography and the early emergence of its predecessors (more or less concurrent with RSA), you would almost certainly lose this argument.
Non-Euclidean geometry was developed without connection to reality. It became applicable with the observation that the universe was best described by non-Euclidean geometry. I would unquestionably have argued against working on non-Euclidean geometry before the development of general relativity; the main question was whether the existence of non-Euclidean geometry facilitated the discovery of general relativity. It seems that Einstein explicitly suggested that spacetime may non-Euclidean before learning that non-Euclidean geometry had been extensively studied. This leads me to suspect that work on non-Euclidean geometry was not essential in the development of general relativity, and that it could just as well have been done after it became relevant.
Of course I can provide a long list of fields unconnected to reality. I cannot think of any significant contributions from any of them. If you can think of a good counterexample here, feel free to suggest it (I think group theory is far and away the best).
Origins
Axiomatic hyperbolic geometry was a game about arbitrary axioms. Perhaps that’s why Gauss didn’t publish on it. What he did publish was his work on extrinsic differential geometry inspired by his work as a surveyor. The Gauss-Bonnet theorem for triangles answers a question a surveyor would ask. Riemann said that his intrinsic differential geometry was an attempt to understand space.
Yes, I think that’s what Gauss meant when he called number theory the queen of mathematics. But I think number theory was a lot narrower back then. A lot of things that would now be called number theory were instead grouped under “solving equations.” I think that when Abel showed that one couldn’t solve the quintic by radicals and when he showed that one could solve it by hypergeometric functions, he thought he was studying the same field.
2. Applications
Group theory and number theory are endemic in CS. It’s not just cryptography. Consider coding theory. For an application of 20th C math, Margulis’s expanders were for decades the only explicit ones (and one does want non-random expanders for randomness extraction).
I’m not sure what you mean. Perhaps that when finite group theory turned inwards and tried to classify finite simple groups, it stopped having applications? Maybe I’d buy that. It seems like an argument for the “interconnected” position against the “interesting” position, but fairly neutral for the “applied” position.
Anyhow, this seems to discount decades of 19th C struggle to clarify the meaning of an abstract group, which is clearly important if groups are important. It’s hard to see in retrospect what was so difficult. One might credit this advance to set theory, the idea that one should talk about abstract sets (like the set of cosets).
So I think set theory is a quite inward-looking subject that turned out to have great clarifying impact on mathematics. But maybe it’s not necessary—do physicists think about groups without it? Similarly, category theory clarified a lot of math, perhaps not in ways that have yet reached the physicists (the way set theory could), but it was been picked up for its own sake in CS, both in type theory and in parallel computation.