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 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).