Great! These are some of my favorite essays on mathematics and I’m excited to see what other rationalists think of them.
I think many mathematicians would object to lumping algebra and analysis together; I know a lot of people who are great at algebra and terrible at analysis. My clusters here are not terribly well-formed but my rough sense is that algebra, analysis, and geometry are three fairly different things.
Somewhat weirdly, I have seen analysis usually described as belonging squarely into the intuition cluster. And I actually partially discovered my love of analysis after I read the corn-eating post on analysis vs. algebra, realized that I eat corn like an analyst but thought of myself as an algebraist, and then realized that all the algebra perspectives I like most are coming from the intuition/geometry perspective (Linear Algebra Done Right and 3Blue1Brown’s videos being two of my top 3 educational resources, and both being heavily intuition-based as opposed to algebra-based).
(I do not know whether the corn-eating thing is real in any meaningful sense, but it did get me to reconsider my perspective on mathematics)
So, I was an undergrad mathematician, and planned to become an academic, but bailed out of my PhD and became a programmer instead. I made notes as I was reading the various articles.
My strong suit in maths was analysis. I just never ‘got’ algebra at all and didn’t touch it after the first year.
Weirdly I was very good at linear algebra/matrices/spectral theory/fourier analysis. But all that seemed like a geometrical, intuitive theory about high-dimensional spaces to me. I had very strong reliable intuition there, but I never had any intuition, or idea about how one might go about acquiring one, for rings/fields/groups or mathematical logic.
I never liked any sort of symbol-manipulation. I felt I understood things if and only if I could make mental pictures of what was going on that would imply the answers ‘as if by magic’.
M. Meray’s endeavours seem unappealing. I appreciate them in the abstract but cannot imagine getting interested. Prof. Klein’s conducting sphere seems a fascinating masterstroke.
Feynman/Einstein are definitely ‘what I’d be if I was twenty times better’. I recognise their ways of thinking, at least as they explained them.
I agree wholeheartedly with Arnold’s rant.
I’m ambivalent on the problem-solver/theorizer distinction. I think I’m more of a theorizer, but problem-solving is important and they both matter. I’d have been proud to have contributed in either way.
Maths is very visual for me. The symbols mean nothing without the pictures.
As a programmer, I:
loathe OO
love lisp, and found it mind blowing when I first found it. By default I use a lisp variant called Clojure both personally and professionally, although I’ve tried almost everything. I avoid java and c++ if I can.
have occasionally tried Haskell, and feel that I ought to understand it, but it feels like programming with one hand tied behind my back. An awful lot of extra effort for no gain.
am quite fond of python, although I use it as a watered-down lisp and avoid all its OO facilities.
adored “Why Arc isn’t especially Object Oriented”
have never tried template metaprogramming, C++ is just too dirty for me, although I love C itself.
like both vi and emacs, and was originally a vi user, but these days I use emacs almost exclusively, and have done ever since I discovered what a joy it is as a lisp editor.
I think that all, with the exception of emacs, puts me strongly on the analysis/intuition side of things and weakly confirms the suggested dichotomy and its relationship to programming styles.
But it’s been a long time since I ate corn-on-the-cob. When I try to visualise it I see myself eating it in rows rather than spirals. But I don’t want to go out and find some, because then I’d bias the result. Somehow I have to catch myself in the act of eating it unconsciously. Any suggestions?
Am a computer scientist, working on AI alignment theory.
I’m probably one of the people where I work who is more sympathetic to MIRI-style ways of thinking about alignment.
Leaned towards a type of thinking that I labelled “algebraic” as a math undergrad.
My best course in undergrad was intro to analysis, but it was taught by a PDEs guy. Our department only had one real analyst, and was predominantly composed of algebra people.
My favourite take on linear algebra involves a ‘geometric’ approach, e.g. thinking of linear operators, not matrices, and taking this sort of view of the singular value decomposition.
I wish that everybody would always denote vectors with bra-ket notation.
My primary academic contribution to CS was to take a bunch of proofs about one family of probability distributions, and see if they worked on a different family of probability distributions (if this doesn’t sound CS-y.… uh, I’m a fake CS boy).
I really like Haskell, and deviations from it really bother me. In particular, the bits I like are the fact that it’s functional and strictly typed.
I mostly use Python because it’s easier.
Object-oriented programming seems weird and creepy to me.
I use emacs, and have briefly tried vi-type things but they never stuck.
When evaluating arguments, I tend to ask questions like “is this argument symmetric in the appropriate variables”, “if you take this variable to 0 or infinity, does the argument still work”, or “does this type check”. I could translate this into terms that make more sense for verbal/non-mathematical arguments, but honestly this is how I think of it.
When eating corn on the cob, I think I do it in spirals.
I only eat corn on the cob at my family home where I grew up, which is a different part of my life than the part that contains everything else on this list.
Looking over the post, I guess that I’m basically an algebraist except for the way I eat corn?
I eat corn like an analyst, vastly prefer Lisp to Haskell, use Vim, identify much more strongly with the personality description of the analyst, and while I haven’t done much higher math, have a deep and abiding love for the delta-epsilon definition of a limit.
Very curious to hear other results, either successful or not.
[+] did my PhD in a fairly analysis-y field (but also fairly geometrical, contrary to the analysis+algebra/geometry split kinda-implied by the OP here)
[+] prefer Lisp to Haskell but [-] feel vaguely guilty about that from time to time and feel I really “ought” to learn Haskell properly
[+] use Vim but [-] only because Emacs was bad for my wrists
[-] don’t much care for fancy C++ template metaprogramming but [+] also don’t much care for hardcore OO programming, though [-] I don’t by any means object to OO, “design patterns”, etc.
Eating corn on the cob is messy and gets stuff stuck in my teeth. It’s also slow. I always find a knife (even just a plastic butter knife), cut the corn off, and eat it with a fork or spoon. What category does that fit in? Until I started doing this, I think I kept experimenting with eating in different patterns. I have no idea what it’s like to eat corn without trying to optimize the process.
My feeling is that that’s probably analysis-style rather than algebra-style. (Even though the actual order of corn-kernel removal is more like that of algebraists.) Are any of the other distinctions that allegedly correlate with it ones that you can match up with your life? Of course they won’t be if you’re not a mathematics/software type.
(It would be very interesting to know whether the algebra/analysis divide among mathematicians is a special case of something that applies to a much broader range of people, and corn-eating might be a way to explore that. But I don’t think cornology is far enough advanced yet to make confident conjectures about what personality features might correlate with different modes of corn-eating.)
I’m a software engineer and my degree in college required a good chunk of advanced math. I am currently in the process of trying to relearn the math I’ve forgotten, plus some, so I’m thinking that if this analysis/algebra dichotomy points at a real preference difference, knowing which I am might help me choose more effective learning sources.
But I find it hard to point to one category or another for most aspects. Even the corn test is inconclusive! (I agree that it sounds more like an analysis thing to do.)
I love the step-by-step bits of algebra and logic, but I also love geometry.
I think I do tend to form an “idiosyncratic mental model of specific problems.” As I come to understand problems more, I feel like they have a quality or character that makes them recognizable to me. I did best in school when teaching myself from outside sources and then using the teacher’s methods to spot check and fill in gaps in my models.
I think object oriented programming is very useful, and functional programming is very appealing.
I use(d) vi/vim because that’s what I know well enough to function in. I barely touched emacs a couple times, was like, “dafuq is this?” and went back to vim. I never gave emacs a fair chance.
I think I lean towards ‘building up’ my understanding of things in chunks, filling in a bigger picture. But the skill of ‘breaking down’ massive concepts into bite-sized chunks seems like an important way to do this!
My tentative self diagnoses is that I have a weak preference for analysis. Reading more of the links in the OP might help me confirm this.
I just start gnawing on the corn cob somewhere at random, like the horrible physicist I am :) But the ‘analysis’ style makes more sense to me of the two, it had never even occurred to me that you could eat corn in the ‘algebra’ style.
I also think about linear algebra in a very visual way. I’m missing that for a lot of group theory, which was presented to us in a very ‘memorise this random pile of definitions’ way. Some time I want to go back and fix this… when I can get it to the top of the very large pile of things I want to learn.
That’s one of the more useful posts I’ve read in a while since it gives me a way to consolidate a bunch of other loose thoughts that have been kicking around. Thanks.
I’m on the boring side of all dichotomies in the OP, and the one with corn too. Funnily, my visual imagination is pretty good (mental rotation etc.) I just never seem to use it for math or programming, it’s step-by-step all the way.
I hate eating corn on the cob, I don’t remember the last time I did it, and I can’t even really inner sim doing it. Mathematically I spend a lot of time talking about algebra but am also, I think, better at analysis than other mathematicians would predict based on my reputation.
Ah, that probably needs clarifying… I was using ‘analysis’ in the sense of ‘opposed to synthesis’ as one of the dichotomies, rather than the mathematical sense of ‘analysis’. I.e. ‘breaking into parts’ as opposed to ‘building up’. That’s pretty confusing when one of the other dichotomies is algebra/geometry!
I agree that algebra and (mathematical) analysis are pretty different and I wouldn’t particularly lump them together. I’d personally probably lump it with geometry over algebra if I had to pick, but that’s likely to be a feature of how I learn and really it’s pretty different to either.
Great! These are some of my favorite essays on mathematics and I’m excited to see what other rationalists think of them.
I think many mathematicians would object to lumping algebra and analysis together; I know a lot of people who are great at algebra and terrible at analysis. My clusters here are not terribly well-formed but my rough sense is that algebra, analysis, and geometry are three fairly different things.
Somewhat weirdly, I have seen analysis usually described as belonging squarely into the intuition cluster. And I actually partially discovered my love of analysis after I read the corn-eating post on analysis vs. algebra, realized that I eat corn like an analyst but thought of myself as an algebraist, and then realized that all the algebra perspectives I like most are coming from the intuition/geometry perspective (Linear Algebra Done Right and 3Blue1Brown’s videos being two of my top 3 educational resources, and both being heavily intuition-based as opposed to algebra-based).
(I do not know whether the corn-eating thing is real in any meaningful sense, but it did get me to reconsider my perspective on mathematics)
So, I was an undergrad mathematician, and planned to become an academic, but bailed out of my PhD and became a programmer instead. I made notes as I was reading the various articles.
My strong suit in maths was analysis. I just never ‘got’ algebra at all and didn’t touch it after the first year.
Weirdly I was very good at linear algebra/matrices/spectral theory/fourier analysis. But all that seemed like a geometrical, intuitive theory about high-dimensional spaces to me. I had very strong reliable intuition there, but I never had any intuition, or idea about how one might go about acquiring one, for rings/fields/groups or mathematical logic.
I never liked any sort of symbol-manipulation. I felt I understood things if and only if I could make mental pictures of what was going on that would imply the answers ‘as if by magic’.
M. Meray’s endeavours seem unappealing. I appreciate them in the abstract but cannot imagine getting interested. Prof. Klein’s conducting sphere seems a fascinating masterstroke.
Feynman/Einstein are definitely ‘what I’d be if I was twenty times better’. I recognise their ways of thinking, at least as they explained them.
I agree wholeheartedly with Arnold’s rant.
I’m ambivalent on the problem-solver/theorizer distinction. I think I’m more of a theorizer, but problem-solving is important and they both matter. I’d have been proud to have contributed in either way.
Maths is very visual for me. The symbols mean nothing without the pictures.
As a programmer, I:
loathe OO
love lisp, and found it mind blowing when I first found it. By default I use a lisp variant called Clojure both personally and professionally, although I’ve tried almost everything. I avoid java and c++ if I can.
have occasionally tried Haskell, and feel that I ought to understand it, but it feels like programming with one hand tied behind my back. An awful lot of extra effort for no gain.
am quite fond of python, although I use it as a watered-down lisp and avoid all its OO facilities.
adored “Why Arc isn’t especially Object Oriented”
have never tried template metaprogramming, C++ is just too dirty for me, although I love C itself.
like both vi and emacs, and was originally a vi user, but these days I use emacs almost exclusively, and have done ever since I discovered what a joy it is as a lisp editor.
I think that all, with the exception of emacs, puts me strongly on the analysis/intuition side of things and weakly confirms the suggested dichotomy and its relationship to programming styles.
But it’s been a long time since I ate corn-on-the-cob. When I try to visualise it I see myself eating it in rows rather than spirals. But I don’t want to go out and find some, because then I’d bias the result. Somehow I have to catch myself in the act of eating it unconsciously. Any suggestions?
Am a computer scientist, working on AI alignment theory.
I’m probably one of the people where I work who is more sympathetic to MIRI-style ways of thinking about alignment.
Leaned towards a type of thinking that I labelled “algebraic” as a math undergrad.
My best course in undergrad was intro to analysis, but it was taught by a PDEs guy. Our department only had one real analyst, and was predominantly composed of algebra people.
My favourite take on linear algebra involves a ‘geometric’ approach, e.g. thinking of linear operators, not matrices, and taking this sort of view of the singular value decomposition.
I wish that everybody would always denote vectors with bra-ket notation.
My primary academic contribution to CS was to take a bunch of proofs about one family of probability distributions, and see if they worked on a different family of probability distributions (if this doesn’t sound CS-y.… uh, I’m a fake CS boy).
I really like Haskell, and deviations from it really bother me. In particular, the bits I like are the fact that it’s functional and strictly typed.
I mostly use Python because it’s easier.
Object-oriented programming seems weird and creepy to me.
I use emacs, and have briefly tried vi-type things but they never stuck.
When evaluating arguments, I tend to ask questions like “is this argument symmetric in the appropriate variables”, “if you take this variable to 0 or infinity, does the argument still work”, or “does this type check”. I could translate this into terms that make more sense for verbal/non-mathematical arguments, but honestly this is how I think of it.
When eating corn on the cob, I think I do it in spirals.
I only eat corn on the cob at my family home where I grew up, which is a different part of my life than the part that contains everything else on this list.
Looking over the post, I guess that I’m basically an algebraist except for the way I eat corn?
That post is hilarious, and fascinating.
I eat corn like an analyst, vastly prefer Lisp to Haskell, use Vim, identify much more strongly with the personality description of the analyst, and while I haven’t done much higher math, have a deep and abiding love for the delta-epsilon definition of a limit.
Very curious to hear other results, either successful or not.
I eat corn like an analyst, and
[+] did my PhD in a fairly analysis-y field (but also fairly geometrical, contrary to the analysis+algebra/geometry split kinda-implied by the OP here)
[+] prefer Lisp to Haskell but [-] feel vaguely guilty about that from time to time and feel I really “ought” to learn Haskell properly
[+] use Vim but [-] only because Emacs was bad for my wrists
[-] don’t much care for fancy C++ template metaprogramming but [+] also don’t much care for hardcore OO programming, though [-] I don’t by any means object to OO, “design patterns”, etc.
That is an amazing post.
Eating corn on the cob is messy and gets stuff stuck in my teeth. It’s also slow. I always find a knife (even just a plastic butter knife), cut the corn off, and eat it with a fork or spoon. What category does that fit in? Until I started doing this, I think I kept experimenting with eating in different patterns. I have no idea what it’s like to eat corn without trying to optimize the process.
My feeling is that that’s probably analysis-style rather than algebra-style. (Even though the actual order of corn-kernel removal is more like that of algebraists.) Are any of the other distinctions that allegedly correlate with it ones that you can match up with your life? Of course they won’t be if you’re not a mathematics/software type.
(It would be very interesting to know whether the algebra/analysis divide among mathematicians is a special case of something that applies to a much broader range of people, and corn-eating might be a way to explore that. But I don’t think cornology is far enough advanced yet to make confident conjectures about what personality features might correlate with different modes of corn-eating.)
I’m a software engineer and my degree in college required a good chunk of advanced math. I am currently in the process of trying to relearn the math I’ve forgotten, plus some, so I’m thinking that if this analysis/algebra dichotomy points at a real preference difference, knowing which I am might help me choose more effective learning sources.
But I find it hard to point to one category or another for most aspects. Even the corn test is inconclusive! (I agree that it sounds more like an analysis thing to do.)
I love the step-by-step bits of algebra and logic, but I also love geometry.
I think I do tend to form an “idiosyncratic mental model of specific problems.” As I come to understand problems more, I feel like they have a quality or character that makes them recognizable to me. I did best in school when teaching myself from outside sources and then using the teacher’s methods to spot check and fill in gaps in my models.
I think object oriented programming is very useful, and functional programming is very appealing.
I use(d) vi/vim because that’s what I know well enough to function in. I barely touched emacs a couple times, was like, “dafuq is this?” and went back to vim. I never gave emacs a fair chance.
I think I lean towards ‘building up’ my understanding of things in chunks, filling in a bigger picture. But the skill of ‘breaking down’ massive concepts into bite-sized chunks seems like an important way to do this!
My tentative self diagnoses is that I have a weak preference for analysis. Reading more of the links in the OP might help me confirm this.
I just start gnawing on the corn cob somewhere at random, like the horrible physicist I am :) But the ‘analysis’ style makes more sense to me of the two, it had never even occurred to me that you could eat corn in the ‘algebra’ style.
I also think about linear algebra in a very visual way. I’m missing that for a lot of group theory, which was presented to us in a very ‘memorise this random pile of definitions’ way. Some time I want to go back and fix this… when I can get it to the top of the very large pile of things I want to learn.
That’s one of the more useful posts I’ve read in a while since it gives me a way to consolidate a bunch of other loose thoughts that have been kicking around. Thanks.
I’m on the boring side of all dichotomies in the OP, and the one with corn too. Funnily, my visual imagination is pretty good (mental rotation etc.) I just never seem to use it for math or programming, it’s step-by-step all the way.
I hate eating corn on the cob, I don’t remember the last time I did it, and I can’t even really inner sim doing it. Mathematically I spend a lot of time talking about algebra but am also, I think, better at analysis than other mathematicians would predict based on my reputation.
Ah, that probably needs clarifying… I was using ‘analysis’ in the sense of ‘opposed to synthesis’ as one of the dichotomies, rather than the mathematical sense of ‘analysis’. I.e. ‘breaking into parts’ as opposed to ‘building up’. That’s pretty confusing when one of the other dichotomies is algebra/geometry!
I agree that algebra and (mathematical) analysis are pretty different and I wouldn’t particularly lump them together. I’d personally probably lump it with geometry over algebra if I had to pick, but that’s likely to be a feature of how I learn and really it’s pretty different to either.