I’m getting a PhD in math and I have had similar thoughts. A few quick remarks about your specific arguments followed by my personal take:
In rebuttal 1, you mention number theory as an example where the application to crypto took a long time to be apparent. I’m a number theorist and I don’t find this argument compelling. Much of the number theory that is used in crypto is not deep. There’s nothing about Diffie-Hellman or RSA that requires the hundreds of years of number theory research that has gone into it. One could explain the algorithms for such procedures to a mathematician in the early 1800s with little effort (although the idea of having very efficient methods of arithmetic might strike them as very odd). Moreover, while other parts of number theory have turned out to be relevant it is still a very tiny fraction of all number theory.
A possibly better example would be non-Euclidean geometries which really were studied in detail in the 19th century before they were found to have practical applications.
As for myself, I enjoy math a lot, and I suspect that I will be more productive at areas I enjoy than areas I enjoy less. This might be a rationalization but it connects to another aspect: I’m not a good utilitarian. Given the choice between me having a happy life and being somewhat productive for humanity and given the choice between being less happy and more productive for humanity I’ll choose being more happy. (When I phrase it that way it triggers far more reactions that lead me to not want to do that. But what decisions I make on a day to day basis about what to think about and what to do with my time are very much near mode).
In rebuttal 1, you mention number theory as an example where the application to crypto took a long time to be apparent. I’m a number theorist and I don’t find this argument compelling.
I also find this argument completely uncompelling. It gets brought up a lot though (and 4 years ago I gave it as justification for doing mathematics in a serious conversation, instead of engaging in an honest conversation about whether I should be doing math). It is slightly better than you make it sound, because without many years of number theory we would have basically no confidence about the algorithmic difficulty of number-theoretic problems.
A possibly better example would be non-Euclidean geometries which really were studied in detail in the 19th century before they were found to have practical applications.
I think this and most other positive examples suffer from a common objection; although you can do the math and later find an application, you could just as well wait until the application appears and then do the math. I think this objection is particularly strong here, because the need for the math was recognized by people who didn’t know the math existed (I think?)
although you can do the math and later find an application, you could just as well wait until the application appears and then do the math.
...combined with the fact that a lot of pure math does not (or havent yet anyways) lead to applications. It pays to put effort only into math that is immediately practically useful.
We need proper counterfactuals here, cases where a practical use of math counterfactually would not have been possible without previous development as pure math. And also, what-if the pure mathematicians have been directly working on practical math instead?
We need proper counterfactuals here, cases where a practical use of math counterfactually would not have been possible without previous development as pure math.
I think a decent argument could be made that Einstein would have been unlikely to have been able on his own to work out the necessary math he used in special and general relativity. On the other hand, this is much more of a severe issue for gen relativity, and it isn’t implausible that once he had constructed the basic theory others would have listened to him enough to work out the underlying math.
It is slightly better than you make it sound, because without many years of number theory we would have basically no confidence about the algorithmic difficulty of number-theoretic problems.
I’m not sure about this. RSA is published in 1978 and Diffie-Hellman was published in 1976. There was some amount of work on trying to efficiently factor integers before that and most of that work was focused on factoring numbers of special forms, but not nearly as much as their was in the next decade. And before DH, there was very little work on the discrete log question. In the case of factoring the difference in the work level can be seen in the drastic improvements in factoring in the few years after (especially the number field sieve and elliptic curve sieve.) Similarly, determining if a number was prime dropped from being almost as difficult as factoring in the mid 1970s to being provably in P 30 years later, and I don’t think almost anyone in the late 1970s saw that coming (although Miller-Rabin did sort of point in that direction).
I feel the same way you do, including the last paragraph, with a possible exception: I do not endorse the fact that I am a bad utilitarian.
Because of this I’m currently attempting to invoke a crisis moment in which I can effectively change fields, although I suspect that I will stay in graduate school to get my Ph.D. and use the opportunity to get more of an education.
I’m getting a PhD in math and I have had similar thoughts. A few quick remarks about your specific arguments followed by my personal take:
In rebuttal 1, you mention number theory as an example where the application to crypto took a long time to be apparent. I’m a number theorist and I don’t find this argument compelling. Much of the number theory that is used in crypto is not deep. There’s nothing about Diffie-Hellman or RSA that requires the hundreds of years of number theory research that has gone into it. One could explain the algorithms for such procedures to a mathematician in the early 1800s with little effort (although the idea of having very efficient methods of arithmetic might strike them as very odd). Moreover, while other parts of number theory have turned out to be relevant it is still a very tiny fraction of all number theory.
A possibly better example would be non-Euclidean geometries which really were studied in detail in the 19th century before they were found to have practical applications.
As for myself, I enjoy math a lot, and I suspect that I will be more productive at areas I enjoy than areas I enjoy less. This might be a rationalization but it connects to another aspect: I’m not a good utilitarian. Given the choice between me having a happy life and being somewhat productive for humanity and given the choice between being less happy and more productive for humanity I’ll choose being more happy. (When I phrase it that way it triggers far more reactions that lead me to not want to do that. But what decisions I make on a day to day basis about what to think about and what to do with my time are very much near mode).
I also find this argument completely uncompelling. It gets brought up a lot though (and 4 years ago I gave it as justification for doing mathematics in a serious conversation, instead of engaging in an honest conversation about whether I should be doing math). It is slightly better than you make it sound, because without many years of number theory we would have basically no confidence about the algorithmic difficulty of number-theoretic problems.
I think this and most other positive examples suffer from a common objection; although you can do the math and later find an application, you could just as well wait until the application appears and then do the math. I think this objection is particularly strong here, because the need for the math was recognized by people who didn’t know the math existed (I think?)
voted up for this:
...combined with the fact that a lot of pure math does not (or havent yet anyways) lead to applications. It pays to put effort only into math that is immediately practically useful.
We need proper counterfactuals here, cases where a practical use of math counterfactually would not have been possible without previous development as pure math. And also, what-if the pure mathematicians have been directly working on practical math instead?
I think a decent argument could be made that Einstein would have been unlikely to have been able on his own to work out the necessary math he used in special and general relativity. On the other hand, this is much more of a severe issue for gen relativity, and it isn’t implausible that once he had constructed the basic theory others would have listened to him enough to work out the underlying math.
I’m not sure about this. RSA is published in 1978 and Diffie-Hellman was published in 1976. There was some amount of work on trying to efficiently factor integers before that and most of that work was focused on factoring numbers of special forms, but not nearly as much as their was in the next decade. And before DH, there was very little work on the discrete log question. In the case of factoring the difference in the work level can be seen in the drastic improvements in factoring in the few years after (especially the number field sieve and elliptic curve sieve.) Similarly, determining if a number was prime dropped from being almost as difficult as factoring in the mid 1970s to being provably in P 30 years later, and I don’t think almost anyone in the late 1970s saw that coming (although Miller-Rabin did sort of point in that direction).
I feel the same way you do, including the last paragraph, with a possible exception: I do not endorse the fact that I am a bad utilitarian.
Because of this I’m currently attempting to invoke a crisis moment in which I can effectively change fields, although I suspect that I will stay in graduate school to get my Ph.D. and use the opportunity to get more of an education.