5 ms·

If the author is right, what subfields of mathematics will likely be the most salient? Linear Algebra? Stats? Category Theory? Isn't it possible that the specie

If the author is right, what subfields of mathematics will likely be the most salient? Linear Algebra? Stats? Category Theory? Isn't it possible that the species of math you invest in will turn out to not be so valuable in an AI-driven future? Or is the hope that a baseline mathematical fluency will help engineers pivot no matter what?

mamon 7y ago
You cannot really predict that. Number theory was once considered "pure mathematics" in a sense that one could not even imagine practical applications for checking if a number is a prime or not. And then someone invented cryptography... EDIT: For all those nitpicking downvoters: yes, I meant public key cryptography.
- zeroonetwothree 7y ago
  Cryptography predates number theory. Probably you mean public key cryptography.
- lou1306 7y ago
  To be fair, cryptography and number theory have coexisted without much overlap from centuries (millennia?). Then, the rise of mechanized cryptanalysis forced us to look for hard-to-break ways to encrypt stuff, and prime factorization was a very good candidate.
- ska 7y ago
  That's not really a useful definition of "pure" mathematics. Also cryptography is one of the oldest applications, and has interacted with a variety of sub-disciplines in mathematics for basically ever.
moralestapia 7y ago
Category theory is fine but won't give you "anything new". I wasted a lot of time and money as plenty of big names were praising it (e.g. John Baez) so I followed them on a few conferences/talks. In the end, it's a nice theoretical framework to think about how to structure your code but that's it. If you do want to get involved (and, like me, are coming from a computational background) the book from Bartosz (i.e. the guy from the post) tells you enough to get a good overview of the thing; it's also available for free online. A second recommendation, a bit odd maybe, do not waste your time trying to engage with that community as most people there are incredibly smug and toxic (but if you don't believe me feel free to try it, ymmv). They are now pushing this idea that they're 'opening' the field to everybody and would like as much people to jump on their bandwagon, but they are just trying to get some traction to keep leeching off grants, etc... they couldn't care less about you, I experienced this first-hand. You can learn most of these things on your own anyway, as it is usual with most mathy things. I don't disagree with the premise of the post, when the time comes math will save your ass, for sure. Now, what works? As you mentioned linear algebra and some intermediate statistics would greatly improve your chances. I would drop some geometry as well, geometry has an inherent 'visual/spatial' feeling to it but try to learn it in such way that you become familiar of dealing with it in a completely abstract way. Regarding this, I have noticed a small surge in interest for clifford algebras, geometric algebras, etc... if you have the time pay attention to it, there are some greats talks about it on youtube and this one has an amazing community that I would definitely recommend to engage with. Also interesting to read about, some other algebras that are defined mathematically and are at the foundation of CS, things like lambda calculus come to mind. You don't have to become an expert on these fields to start seeing the benefits. Just build of the habit of reading about one or two topics per week, what they do, how they do it, and before you know a year has gone by and you will feel much more confident in your skills. Anything that requires more attention will naturally drag you to it. Finally, this guy Robert Ghrist is amazing (really!) https://www.math.upenn.edu/~ghrist/ https://www.math.upenn.edu/~ghrist/, check him out as well, his talks and other content. He also wrote a book on applied topology (free online as well) which is amazing. Even if you don't understand anything about it, give it a quick skim to have an idea of all the incredible things math can you do for you that you could not have imagined existed!
- tobmlt 7y ago
  Seconded. Robert Ghrist’s Elementary Applied Topology is one of my favorite books. A real eye opener if you haven’t seen some of it before.
- bhewer 7y ago
  'Category theory is fine but won't give you "anything new"'. This is far from the truth and disregards a significant body of work in programming language theory (not to mention algebraic geometry and topology). Perhaps most famously, monads, which have been widely adopted in the Haskell and Scala communities, originated in the category theory literature. Eugenio Moggi famously described how monads could be used to structure the denotational semantics of a programming language. Monads are just the most obvious example of 'new' concepts which originate in the category theory literature and are practically realised by the study of categorical semantics of programming languages. A brief investigation of the literature will reveal a vast array of concepts such as initial algebras/final coalgebras (used to implement safe recursion schemes), adjunctions (giving rise to free constructions such as the infamous free monad), profunctor optics (bidirectional data accessors). This is just to give a few examples in PLT and does not even touch upon lesser-known constructions in type theory. ‘In the end, it's a nice theoretical framework to think about how to structure your code but that's it.’ This comes across as overly dismissive of the problem of ‘how to structure code’, which is in no way trivial. Formalising desirable properties of programs, and designing languages/constructs which restrict to programs exhibiting these desirable properties, is a general focus of programming language theory. ’A nice theoretical framework’ in which such problems can be formalised is very significant, and the comments that programming language theorists working on categorical semantics are “incredibly smug and toxic” and “are just trying to get some traction to keep leeching off grants” is especially insulting to many wonderful people working in this domain.
photon_lines 7y ago
The author isn't right. He's right in some aspects (I'd say in 5% on what he conjectures), but for the most part, programmers and programming will be in quite a high demand for quite a foreseeable time in the future. Yes, C / C++ programming might become less important (see: memory wall) and I'd say highly parallel computation will take precedence, but this won't replace computer science nor programming...it'll just create the need to programmers to transition from one skill set into another. Also, as a note: the field of computer science is a branch of mathematics. Whenever you deal with data structures or algorithms, you're dealing with how to procedurally design information structures and procedures to solve problems and model the real world. Mathematics? Well, depending on what branch you're into, it's using symbols and information ...to, well, do the same thing! Out of all of the professions, it seems rather silly to me to think that our own profession is in danger of being replaced. I would think that it's the LEAST likely one, since programmers, to me at least, represent workers who are paid to communicate and find ways to solve problems in the real world using a tool which will continue to improve and enable our species to excel and automate jobs filled with repetitive drudgery. Out of all of the professions, I’m not sure why the author would think that ours is likely to go out of demand soon. I would conjecture that us playing a prominent role in automating other jobs ensures that we stay in demand for quite a long time! As far as the most useful mathematical branches are concerned – if you’re interested, the branches I find have the highest ability to help in solving real world problems are: calculus, linear algebra, probability and statistics, computer science (as already mentioned), convex and computational optimization, quantum mechanics, complex analysis, ordinary and partial differential equations, data mining and analysis, information theory … well, I could keep going, but I think you’d greatly benefit from checking out a great book which gives a pretty good overview on the key areas in applied mathematics: The Princeton Companion to Applied Mathematics. The above are just my own guesses, and it’s highly dependent on which problem you’re looking to solve.
analog31 7y ago
At some level, just being able to do math. Most people hate math, and there will always be some security in being willing to do things that most people hate. Engineers are assured, through the grapevine, that they will never use their math after college. They're right. Most of them will get so busy with organizing and arranging building blocks, that they'll forget their math. It makes sense to delegate the math work to a small fraction of people who enjoy it. Those people will never be great in number, and might not get paid any more, but will be valuable enough to have stable careers. The math doesn't even need to be particularly hard. At least, so I hope. ;-)
csomar 7y ago
The author is off his rocker. Just because he used cute phrases like "a monad is defined as a monoid in the category of endofunctors" doesn't mean he knows what he is talking about. Rust is the best shot to replace C, and it has yet to figure out how to manage memory without a complicated ownership/borrowing/lifetimes model and not impact performance. That and Rust is probably 5-6 years ahead of its time. The future where Math, Cryptocurrencies, AI have eaten the world is still very far away. So in my opinion, your best bet/insurance for the future is to learn Rust (which I'm doing); that being said I don't know what I'm talking about.

mamon 7y ago

You cannot really predict that. Number theory was once considered "pure mathematics" in a sense that one could not even imagine practical applications for checking if a number is a prime or not. And then someone invented cryptography... EDIT: For all those nitpicking downvoters: yes, I meant public key cryptography.

zeroonetwothree 7y ago
Cryptography predates number theory. Probably you mean public key cryptography.
lou1306 7y ago
To be fair, cryptography and number theory have coexisted without much overlap from centuries (millennia?). Then, the rise of mechanized cryptanalysis forced us to look for hard-to-break ways to encrypt stuff, and prime factorization was a very good candidate.
ska 7y ago
That's not really a useful definition of "pure" mathematics. Also cryptography is one of the oldest applications, and has interacted with a variety of sub-disciplines in mathematics for basically ever.

moralestapia 7y ago

Category theory is fine but won't give you "anything new". I wasted a lot of time and money as plenty of big names were praising it (e.g. John Baez) so I followed them on a few conferences/talks. In the end, it's a nice theoretical framework to think about how to structure your code but that's it. If you do want to get involved (and, like me, are coming from a computational background) the book from Bartosz (i.e. the guy from the post) tells you enough to get a good overview of the thing; it's also available for free online. A second recommendation, a bit odd maybe, do not waste your time trying to engage with that community as most people there are incredibly smug and toxic (but if you don't believe me feel free to try it, ymmv). They are now pushing this idea that they're 'opening' the field to everybody and would like as much people to jump on their bandwagon, but they are just trying to get some traction to keep leeching off grants, etc... they couldn't care less about you, I experienced this first-hand. You can learn most of these things on your own anyway, as it is usual with most mathy things. I don't disagree with the premise of the post, when the time comes math will save your ass, for sure. Now, what works? As you mentioned linear algebra and some intermediate statistics would greatly improve your chances. I would drop some geometry as well, geometry has an inherent 'visual/spatial' feeling to it but try to learn it in such way that you become familiar of dealing with it in a completely abstract way. Regarding this, I have noticed a small surge in interest for clifford algebras, geometric algebras, etc... if you have the time pay attention to it, there are some greats talks about it on youtube and this one has an amazing community that I would definitely recommend to engage with. Also interesting to read about, some other algebras that are defined mathematically and are at the foundation of CS, things like lambda calculus come to mind. You don't have to become an expert on these fields to start seeing the benefits. Just build of the habit of reading about one or two topics per week, what they do, how they do it, and before you know a year has gone by and you will feel much more confident in your skills. Anything that requires more attention will naturally drag you to it. Finally, this guy Robert Ghrist is amazing (really!) https://www.math.upenn.edu/~ghrist/ https://www.math.upenn.edu/~ghrist/, check him out as well, his talks and other content. He also wrote a book on applied topology (free online as well) which is amazing. Even if you don't understand anything about it, give it a quick skim to have an idea of all the incredible things math can you do for you that you could not have imagined existed!

tobmlt 7y ago
Seconded. Robert Ghrist’s Elementary Applied Topology is one of my favorite books. A real eye opener if you haven’t seen some of it before.
bhewer 7y ago
'Category theory is fine but won't give you "anything new"'. This is far from the truth and disregards a significant body of work in programming language theory (not to mention algebraic geometry and topology). Perhaps most famously, monads, which have been widely adopted in the Haskell and Scala communities, originated in the category theory literature. Eugenio Moggi famously described how monads could be used to structure the denotational semantics of a programming language. Monads are just the most obvious example of 'new' concepts which originate in the category theory literature and are practically realised by the study of categorical semantics of programming languages. A brief investigation of the literature will reveal a vast array of concepts such as initial algebras/final coalgebras (used to implement safe recursion schemes), adjunctions (giving rise to free constructions such as the infamous free monad), profunctor optics (bidirectional data accessors). This is just to give a few examples in PLT and does not even touch upon lesser-known constructions in type theory. ‘In the end, it's a nice theoretical framework to think about how to structure your code but that's it.’ This comes across as overly dismissive of the problem of ‘how to structure code’, which is in no way trivial. Formalising desirable properties of programs, and designing languages/constructs which restrict to programs exhibiting these desirable properties, is a general focus of programming language theory. ’A nice theoretical framework’ in which such problems can be formalised is very significant, and the comments that programming language theorists working on categorical semantics are “incredibly smug and toxic” and “are just trying to get some traction to keep leeching off grants” is especially insulting to many wonderful people working in this domain.

photon_lines 7y ago

The author isn't right. He's right in some aspects (I'd say in 5% on what he conjectures), but for the most part, programmers and programming will be in quite a high demand for quite a foreseeable time in the future. Yes, C / C++ programming might become less important (see: memory wall) and I'd say highly parallel computation will take precedence, but this won't replace computer science nor programming...it'll just create the need to programmers to transition from one skill set into another. Also, as a note: the field of computer science is a branch of mathematics. Whenever you deal with data structures or algorithms, you're dealing with how to procedurally design information structures and procedures to solve problems and model the real world. Mathematics? Well, depending on what branch you're into, it's using symbols and information ...to, well, do the same thing! Out of all of the professions, it seems rather silly to me to think that our own profession is in danger of being replaced. I would think that it's the LEAST likely one, since programmers, to me at least, represent workers who are paid to communicate and find ways to solve problems in the real world using a tool which will continue to improve and enable our species to excel and automate jobs filled with repetitive drudgery. Out of all of the professions, I’m not sure why the author would think that ours is likely to go out of demand soon. I would conjecture that us playing a prominent role in automating other jobs ensures that we stay in demand for quite a long time! As far as the most useful mathematical branches are concerned – if you’re interested, the branches I find have the highest ability to help in solving real world problems are: calculus, linear algebra, probability and statistics, computer science (as already mentioned), convex and computational optimization, quantum mechanics, complex analysis, ordinary and partial differential equations, data mining and analysis, information theory … well, I could keep going, but I think you’d greatly benefit from checking out a great book which gives a pretty good overview on the key areas in applied mathematics: The Princeton Companion to Applied Mathematics. The above are just my own guesses, and it’s highly dependent on which problem you’re looking to solve.

analog31 7y ago

At some level, just being able to do math. Most people hate math, and there will always be some security in being willing to do things that most people hate. Engineers are assured, through the grapevine, that they will never use their math after college. They're right. Most of them will get so busy with organizing and arranging building blocks, that they'll forget their math. It makes sense to delegate the math work to a small fraction of people who enjoy it. Those people will never be great in number, and might not get paid any more, but will be valuable enough to have stable careers. The math doesn't even need to be particularly hard. At least, so I hope. ;-)

csomar 7y ago

The author is off his rocker. Just because he used cute phrases like "a monad is defined as a monoid in the category of endofunctors" doesn't mean he knows what he is talking about. Rust is the best shot to replace C, and it has yet to figure out how to manage memory without a complicated ownership/borrowing/lifetimes model and not impact performance. That and Rust is probably 5-6 years ahead of its time. The future where Math, Cryptocurrencies, AI have eaten the world is still very far away. So in my opinion, your best bet/insurance for the future is to learn Rust (which I'm doing); that being said I don't know what I'm talking about.