In the case of calculus, differential equations, statistics, functional analysis, linear algebra, group theory, and numerical methods, the important results for modern work were in fact developed after their usefulness could be appreciated by an intelligent observer.
This is simply untrue, unless you’ve rigged the definition of “intelligent observer” and added a dose of hindsight bias. It is unlikely that the true extent of the “practical” importance of calculus today could have been predicted by even the most imaginative of Newton’s fellow Cambridge dons in the late 17th century. In that era, thinking about things like the orbits of celestial bodies was “idle speculation” par excellence. It’s hard to appreciate this, because it seems so obvious in retrospect that we would have space rockets, doesn’t it? Not to mention the use of differential equations in fields like economics, whose existence was a century away but were just so clearly on humanity’s horizon, right?
Functional analysis is a particularly interesting choice of example. The fact that its application to quantum mechanics (which is what I presume you were thinking of) arose concurrently with the development of the subject itself was largely a fortuitous (if serendipitous) coincidence. The actual “physical” roots of the subject were more indirect, via differential/integral equations and the calculus of variations (18th-century physics, in other words), and it was basically the result of mathematicians’ attempt to turn these somewhat ad-hoc disciplines into nice-looking abstract theories.
As for group theory, its origins lie in the attempt to solve the quintic by radicals—about as “useless” an undertaking as could be imagined. (The cubic and quartic formulas already being much too complicated for practical use.)
Realistically, an argument like yours, made back in the day, would have shown that Newton should have devoted his life to inventing better agricultural tools. And it might have been a good argument—applied to someone other than Newton. (They could really have used better agricultural tools, no doubt.)
If you don’t feel satisfied doing math, or think you could make a better contribution doing something else, you shouldn’t be doing it. But don’t make the mistake of pretending that your argument generalizes.
And see here regarding the nature of mathematics’ usefulness, which doesn’t reside in specific “applications”.
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 don’t know much history, but am inclined to disagree with most of your claims (your statement about group theory is completely correct. I might be able to salvage my claim by restricting to the subset of group theory I care about, which is really more linear algebra and representation theory, but I don’t know if the history would support me even then. Apologies for my error)
It is unlikely that the true extent of the “practical” importance of calculus today could have been predicted by even the most imaginative of Newton’s fellow Cambridge dons in the late 17th century.
I don’t understand this. Newton made the observation that calculus described not only the orbits of the bodies but also the behavior of the everyday objects humans interact with (and in fact described motion in general) before formally developing calculus—at least, thats how the normal version of the history goes (I have no idea how accurate it is). Are you claiming that an intelligent observer would doubt the importance of describing the motion of objects around them, or what?
The actual “physical” roots of the subject were more indirect, via differential/integral equations and the calculus of variations (18th-century physics, in other words), and it was basically the result of mathematicians’ attempt to turn these somewhat ad-hoc disciplines into nice-looking abstract theories.
I was talking about the applications of functional analysis to understanding differential equations, which are (as I understand it) the actual point of functional analysis. Not coincidentally, functional analysis was developed in response to the obviously important problem of understand differential equations. Its not like someone sat down and developed functional analysis, and then it happened to later be discovered that it was a powerful technique for understanding the world.
Happening to provide a formalization for quantum mechanics is really not important in my view. If you think that no formalization of quantum mechanics would exist if mathematicians hadn’t thought of functional calculus, I think you are very confused.
If you don’t feel satisfied doing math, or think you could make a better contribution doing something else, you shouldn’t be doing it. But don’t make the mistake of pretending that your argument generalizes.
I am actually curious knowing whether I should be doing math. If I should, my life is much easier. I would like to have an honest discussion about the utility of math apart from specific applications. I tend to agree that the use of math is not in immediate applications. I also believe that you can foresee that calculus is useful, or differential equations, or any of the other things I mentioned, or even negative or imaginary numbers, and that this is not just hindsight bias but actual discrimination. This seems like a factual question which we have some hope of resolving (though not too much).
The actual “physical” roots of the subject were more indirect, via differential/integral equations and the calculus of variations (18th-century physics, in other words), and it was basically the result of mathematicians’ attempt to turn these somewhat ad-hoc disciplines into nice-looking abstract theories.
I was talking about the applications of functional analysis to understanding differential equations, which are (as I understand it) the actual point of functional analysis.
This sounds like violent agreement to me. Your disagreement is the utility of the application of functional analysis to differential equations. Is it a practical problem to know when Dirichlet’s principle applies? Or, if you insist that functional analysis dates from Leray, I am told that physicists do not care about the mathematical problem of whether the Navier-Stokes equation has smooth solutions—water flows, and that is good enough for them.
Realistically, an argument like yours, made back in the day, would have shown that Newton should have devoted his life to inventing better agricultural tools. And it might have been a good argument—applied to someone other than Newton. (They could really have used better agricultural tools, no doubt.)
1) Are you agreeing with paulfchristiano that he should abandon pure math today and choose some more productive occupation, unless he’s as exceptional as Newton was in his time?
2) Why do you think the world would be worse off now if Newton had chosen to invent agricultural tools, or otherwise maximize instrumental good in his own time? How about if everyone else used the same rule too? I think we’d have a pretty awesome world today! Isn’t this the proper test of whether an argument “generalizes”?
This is simply untrue, unless you’ve rigged the definition of “intelligent observer” and added a dose of hindsight bias. It is unlikely that the true extent of the “practical” importance of calculus today could have been predicted by even the most imaginative of Newton’s fellow Cambridge dons in the late 17th century. In that era, thinking about things like the orbits of celestial bodies was “idle speculation” par excellence. It’s hard to appreciate this, because it seems so obvious in retrospect that we would have space rockets, doesn’t it? Not to mention the use of differential equations in fields like economics, whose existence was a century away but were just so clearly on humanity’s horizon, right?
Functional analysis is a particularly interesting choice of example. The fact that its application to quantum mechanics (which is what I presume you were thinking of) arose concurrently with the development of the subject itself was largely a fortuitous (if serendipitous) coincidence. The actual “physical” roots of the subject were more indirect, via differential/integral equations and the calculus of variations (18th-century physics, in other words), and it was basically the result of mathematicians’ attempt to turn these somewhat ad-hoc disciplines into nice-looking abstract theories.
As for group theory, its origins lie in the attempt to solve the quintic by radicals—about as “useless” an undertaking as could be imagined. (The cubic and quartic formulas already being much too complicated for practical use.)
Realistically, an argument like yours, made back in the day, would have shown that Newton should have devoted his life to inventing better agricultural tools. And it might have been a good argument—applied to someone other than Newton. (They could really have used better agricultural tools, no doubt.)
If you don’t feel satisfied doing math, or think you could make a better contribution doing something else, you shouldn’t be doing it. But don’t make the mistake of pretending that your argument generalizes.
And see here regarding the nature of mathematics’ usefulness, which doesn’t reside in specific “applications”.
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.
I don’t know much history, but am inclined to disagree with most of your claims (your statement about group theory is completely correct. I might be able to salvage my claim by restricting to the subset of group theory I care about, which is really more linear algebra and representation theory, but I don’t know if the history would support me even then. Apologies for my error)
I don’t understand this. Newton made the observation that calculus described not only the orbits of the bodies but also the behavior of the everyday objects humans interact with (and in fact described motion in general) before formally developing calculus—at least, thats how the normal version of the history goes (I have no idea how accurate it is). Are you claiming that an intelligent observer would doubt the importance of describing the motion of objects around them, or what?
I was talking about the applications of functional analysis to understanding differential equations, which are (as I understand it) the actual point of functional analysis. Not coincidentally, functional analysis was developed in response to the obviously important problem of understand differential equations. Its not like someone sat down and developed functional analysis, and then it happened to later be discovered that it was a powerful technique for understanding the world.
Happening to provide a formalization for quantum mechanics is really not important in my view. If you think that no formalization of quantum mechanics would exist if mathematicians hadn’t thought of functional calculus, I think you are very confused.
I am actually curious knowing whether I should be doing math. If I should, my life is much easier. I would like to have an honest discussion about the utility of math apart from specific applications. I tend to agree that the use of math is not in immediate applications. I also believe that you can foresee that calculus is useful, or differential equations, or any of the other things I mentioned, or even negative or imaginary numbers, and that this is not just hindsight bias but actual discrimination. This seems like a factual question which we have some hope of resolving (though not too much).
This sounds like violent agreement to me.
Your disagreement is the utility of the application of functional analysis to differential equations. Is it a practical problem to know when Dirichlet’s principle applies? Or, if you insist that functional analysis dates from Leray, I am told that physicists do not care about the mathematical problem of whether the Navier-Stokes equation has smooth solutions—water flows, and that is good enough for them.
1) Are you agreeing with paulfchristiano that he should abandon pure math today and choose some more productive occupation, unless he’s as exceptional as Newton was in his time?
2) Why do you think the world would be worse off now if Newton had chosen to invent agricultural tools, or otherwise maximize instrumental good in his own time? How about if everyone else used the same rule too? I think we’d have a pretty awesome world today! Isn’t this the proper test of whether an argument “generalizes”?