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> There's a saying in mathematics that when any field matures it turns into algebra. Life crossed a threshold from chemistry to biology on Earth, and developed

> There's a saying in mathematics that when any field matures it turns into algebra. Life crossed a threshold from chemistry to biology on Earth, and developed exponentially from there. Order in mathematics crosses a similar threshold, as it becomes sufficiently structured to support algebraic reasoning. The subjective experience is like ice melting into a churning liquid, or a land-locked creature learning to fly. Once one has this experience, one cannot imagine thinking any other way. Hi, this seems wrong to me. First, as far as I can tell, such a saying does not exist. I've never heard it, despite having a fair amount of experience with mathematics (though not as much as you), and I can't find it through Googling. Second, and more substantively, it is not the case that mathematical fields inevitably turn into algebra. One only has to look at PDE, probability, and analytic number theory to see this is the case (to give just a few examples). All are highly mature fields and essentially non-algebraic. This is not to say that ideas from algebra are not occasionally useful, just that it should be obvious to anyone who opens an introductory graduate text in PDE that the subject has not "turn[ed] into algebra." > The killer app for Haskell is parallelism. To this day it has the best implementation of parallelism; one can achieve a 7x speedup on 8 cores by adding a handful of lines to a program. This ease is a consequence of the functional model, and of considerable effort by Haskell's developers. I don't think this statement is entirely wrong, but I don't think it's entirely right either. There are plenty of languages where one can get parallelism with just a few extra lines of code; Julia comes to mind immediately. Easy parallelism is hardly a Haskell-specific, or even FP-specific, feature. I would of course agree that designing things without mutable state makes writing parallel code easier, but this can be done in many high-level languages these days. But that is a nitpick. A more substantive objection is that the sole purpose of parallelism is performance, and idiomatic Haskell is generally speaking slower, sometimes a lot slower, than the same algorithm written in C++ or similar languages. (The adjective idiomatic is important here – I'm aware that one can with enough contortions write C-like code in Haskell, but this response undercuts of the point of highlighting Haskell-specific features like immutability, laziness, etc.) In particular, a few years ago it seemed like the people implementing parallel Haskell features didn't understand or care about fundamental things like cache locality. Maybe this has changed? There is also an amusing story about the "ease" of writing parallel quicksort in Haskell [1]. > The real point of lazy evaluation is that it best supports this algebraic reasoning, as carbon best supports life. It's worth noting that Simon Peyton Jones is on record as saying that any "Haskell 2" should not be lazy [2]. [1] https://sudonull.com/post/75314-Parallel-quick-sort-on-Haskell-and-how-hard-it-was-to-write https://sudonull.com/post/75314-Parallel-quick-sort-on-Haske... [2] https://news.ycombinator.com/item?id=1924061 https://news.ycombinator.com/item?id=1924061

marcosdumay 6y ago
Probability has surely morphed into an algebra, and what you study in modern introductory books is the conversion of the algebra into "everyday language" so that students don't get scared. It has very little similarity to what probability looked like when it was being developed. Also, you seem to be overstating the maturity of PDE (people are pretty much still trying to turn it into algebra, but it may not be possible) and number theory (that's an active research area). I have never seen that saying either, but it seems quite correct, with the exception of the areas where an algebra is impossible, so the field matures without becoming one.
- spekcular 6y ago
  On what basis do you say that probability has "morphed into an algebra"? Consider all of the popular research topics today: the KPZ equation, Schramm-Loewner evolution, spin glasses, percolation and critical phenomena, rigorous statistical mechanics in general [1], random matrix theory, large deviations, stochastic analysis and (P)SDEs. Obviously, I'm forgetting some, but none of these are primarily algebraic. Some algebraic methods are used (orthogonal polynomials in random matrix theory, and determinants and a whole host of other things to study integrable models in the KPZ universality class, etc.), but clearly groups/rings/fields are not playing a major role. For a more systematic approach you could look at recent issues of Annals of Probability, Annals of Applied Probability, and similar journals. There's not going to be a lot of "modern algebra" (of the flavor you see in algebraic geometry) there. The same comments apply to PDE and analytic number theory. Both are obviously mature fields (worked on for a long time by many people, with a lot of great discoveries), but again algebra does not play a central role in either. In particular I am not aware of any PDE specialists whose research agenda consists of "trying to turn it into algebra." [1]: https://www.unige.ch/math/folks/velenik/smbook/ https://www.unige.ch/math/folks/velenik/smbook/
  - bitdizzy 6y ago
    I know vaguely from a few friends that there is some cross pollination between algebraic geometry and PDE's but both fields are so huge that this doesn't characterize either field. I think the gist of it is that the moduli space of PDE solutions are frequently interesting as varieties. There's this rare unclosed survey-like mathoverflow question that highlights some of these points: https://mathoverflow.net/questions/66046/which-nonlinear-pdes-are-of-interest-to-algebraic-geometers-and-why https://mathoverflow.net/questions/66046/which-nonlinear-pde...

marcosdumay 6y ago

Probability has surely morphed into an algebra, and what you study in modern introductory books is the conversion of the algebra into "everyday language" so that students don't get scared. It has very little similarity to what probability looked like when it was being developed. Also, you seem to be overstating the maturity of PDE (people are pretty much still trying to turn it into algebra, but it may not be possible) and number theory (that's an active research area). I have never seen that saying either, but it seems quite correct, with the exception of the areas where an algebra is impossible, so the field matures without becoming one.

spekcular 6y ago
On what basis do you say that probability has "morphed into an algebra"? Consider all of the popular research topics today: the KPZ equation, Schramm-Loewner evolution, spin glasses, percolation and critical phenomena, rigorous statistical mechanics in general [1], random matrix theory, large deviations, stochastic analysis and (P)SDEs. Obviously, I'm forgetting some, but none of these are primarily algebraic. Some algebraic methods are used (orthogonal polynomials in random matrix theory, and determinants and a whole host of other things to study integrable models in the KPZ universality class, etc.), but clearly groups/rings/fields are not playing a major role. For a more systematic approach you could look at recent issues of Annals of Probability, Annals of Applied Probability, and similar journals. There's not going to be a lot of "modern algebra" (of the flavor you see in algebraic geometry) there. The same comments apply to PDE and analytic number theory. Both are obviously mature fields (worked on for a long time by many people, with a lot of great discoveries), but again algebra does not play a central role in either. In particular I am not aware of any PDE specialists whose research agenda consists of "trying to turn it into algebra." [1]: https://www.unige.ch/math/folks/velenik/smbook/ https://www.unige.ch/math/folks/velenik/smbook/
- bitdizzy 6y ago
  I know vaguely from a few friends that there is some cross pollination between algebraic geometry and PDE's but both fields are so huge that this doesn't characterize either field. I think the gist of it is that the moduli space of PDE solutions are frequently interesting as varieties. There's this rare unclosed survey-like mathoverflow question that highlights some of these points: https://mathoverflow.net/questions/66046/which-nonlinear-pdes-are-of-interest-to-algebraic-geometers-and-why https://mathoverflow.net/questions/66046/which-nonlinear-pde...