1.) part of why I could never do anything but analysis, and the super-concrete parts of it at that (my thesis involved a lot of spherical harmonics and Heisenberg group stuff, all calculations on special cases, no big general theorems) is that I need to visualize shapes and sizes! If it’s basically in R^n (or Hilbert spaces) then we’re groovy; if it’s too much more general, how do you do anything except clumsily trying to patch together definitions and theorems’ inputs and outputs into a sequence that leads somewhere? I had this really sweet professor try to get across the “yoga of commutative algebra” and I was like “maybe if there were about 10x more examples and pictures I would get any of this but right now my brain will Not.” I still don’t know what algebra/topology people are doing with their heads. (though I suspect Thurston was actually visualizing, in extremely trippy ways).
2.) yes, of course, I have had mathematical Experiences. ranging from “literal fever dreams about sliding down a rainbow-hued complex exponential” to “acquiring synaesthesia colors while studying” to more normal stuff like “the Right way to learn involves doing little hand movements and singing a little song and laughing out loud when you Get It”.
@Lucy Keer has several (older) posts about "the two kinds of mathematicians," basically algebraists (like me) and analysts (like you). Many people have observed this (and Lucy has collected some discussions from the literature).
I suspect, from my own experience, that we specialize into one or the other way of doing things (perhaps based on innate abilities, or due to circumstances). Then it is possible to develop the other one to some extent. I did, although I was never as good at shape rotation as symbolcelling.
I'm an algebra-topology guy with essentially zero grasp of analysis, and I'm terrible at symbol manipulation! Algebra is like playing with legos -- clicking shapes together -- to me
As part of mathematical education, it seems important to explain that different people approach the stuff in different ways, which is fine; and different thinking styles work better in different parts of math; and one can develop the styles one is bad at, at least to some degree and with an expectable level of pain and frustration.
Something along the lines of “math is a mind-body thing, except you don’t move your body” has sometimes helped me explain it to athletes.
Most sports (especially things like golf) also recruit the same spatial/kinesthetic/visual/proprioception faculties as you are talking about, so they do have some intuitive sense of what that means.
David, Great post about envisioning, and I'm glad to see you diving into 'Mathematica'. Knowing that you already have too much to read and write, I will nonetheless highly recommend also diving into the quantitative psychological research of Todd Thrash, especially his paper, 'The creation and curation of all things worthy: Inspiration as vital force in persons and cultures'. (Happy to send you a pdf if you can't find it online.)
It has been a revelation for me. His articulation of the three basic aspects of the psychological process of inspiration (evocation, transcendence, and transmission) is a perspective that spontaneously inspired a new way for me to fit together myriad threads in my own developing philosophy. I think Thrash's perspective will help illuminate and clarify your nascent conceptualization of envisioning. Here is the abstract of his paper:
'Inspiration is an epistemic-motivational episode in which an individual is awoken to what is valuable and imbued with the spirit to express this understanding. In this article, I tell the story of my research program on inspiration. I begin by describing the origin of my research program as a corrective to the emphasis on personal agency in the motivation literature. I then review work on foundational issues of conceptualization, typology, methodology, and construct validation. Next I review research on the role of inspiration in the creative process. Inspiration is a response to, not a source of, creative ideas. Its function is to motivate transmission (e.g., actualization) of an idea in the form of a tangible product. Inventors prone to inspiration receive more patents, and inspired writers are more creative, productive, and efficient. Next I situate inspiration within a broader cultural-evolutionary context. Inspiration is infectious, such that inspired writers inspire their readers. Processes of transmission and contagion suggest that the core significance of inspiration lies in its procultural function. Inspiration, I argue, is a genetically evolved adaptation that catalyzes and guides cultural evolution by compelling selective retention of cultural wisdom in the form of cultural artifacts (e.g., scripture, poetry, inventions, actualized selves), thereby making the species more adaptable to local and changing environments. Finally, I discuss benefits to well-being. Inspiration marks moments of peak functioning that integrate the hedonic well-being of the primitive creature, the eudaimonic well-being of the social creature, and the self-transcendent well-being of the cultural creature.'
Thrash specifically has some very insightful things to say about the intertwined relationship between inspiration and creativity:
'The actualization-of-creativity model has important implications. First, it reveals that a simple explanatory model of the form “inspiration causes creativity” or “creativity causes inspiration” is not adequate. Inspiration is a cause of creativity in a product but an effect of creativity in an idea. This conclusion qualifies the above preliminary conclusion of bidirectional effects between inspiration and creativity.'
So if you're looking at ways to describe moments of inspirational envisioning with no woo woo baggage, you should definitely check out the paper, and Thrash's research generally.
Some of this fits (the kinesthetic-proprioceptive aspect particularly), although I absolutely don't understand the reluctance to talk about the lived-experience -- that's the only way I can talk about this stuff. Then again, it's easier for me to write formal mathematics than read it.
Another thing I never see anyone mention is that doing math for me was generally quite painful.
"Math is painful, let's go shopping" is probably everyone's experience!
Maybe what is left unsaid is that it's painful even for people who are very good at it. It takes an extraordinarily high tolerance for frustration and feeling stupid and gritting your way through anyway.
Presumably, those who persist are those who find it rewarding as well. Generally not at the same time, if my experience is anything to go by!
Yeah, I guess I meant researchers don't talk about it. Like admitting you're afraid of going into battle! Obviously most people find it painful and will tell you at great length, though.
>And the reality is, if you look for foundations for mathematics… People go into mathematics thinking they’re going to find absolute truth, and if you dig deep enough under those supposed absolute truths, you find it is clouds. There is no foundation other than clouds. It’s clouds all the way down. And I think a lot of mathematicians have read about this, and they realize there’s something scary there; and this is another part, probably, of why there’s a taboo about real talk about what math is, because it’s on sand, or clouds.
The story centers on a mathematician who uncovers a paradox that undermines her long-held belief in the absolute consistency of arithmetic. This discovery triggers a profound personal crisis, which reverberates in her relationship with her husband.
Major themes include the destabilizing power of revelation—how realizing a trusted system may be flawed can unravel one’s sense of self—and the limits of empathy when confronting someone else’s existential crisis.
I think it's a great fictional exploration of what that "something scary" might look like for people who are looking to mathematics for absolute truth. It certainly did for me when I first read it as an undergraduate in math.
>I was a math undergraduate. I saw people struggling with this. And for a lot of them, the problem was, and I found this really difficult myself. I got better at it.
I also was a math undergraduate, and I never got (much) better at that. I was decent enough at symbol manipulation, but that also never really felt natural or enjoyable to me, so I eventually declared myself talentless and moved on. But I'm still curious about the talent question. You read about all those geniuses who proved famous theorems at the age of 12 or whatever, and surely it came easy to them? Or is that just cope and most mathematicians have to work hard at it before they develop both their intuitions and their skills to the point where it feels "natural"?
>And when people realized at the beginning, early 20th century, that this doesn’t actually work, there was a major crisis, and it kind of looked like mathematics might just completely fall apart.
Yep, in my first year at the uni I kind of randomly stumbled upon the book The Loss of Certainty by Morris Kline about the foundation crisis, which was an entirely new perspective for me then, and probably the most exciting experience from that time. It was like I stumbled upon an actually taboo territory, a shameful secret that the math, and the wider rationality world swept under the rug while still trying to maintain the "absolute truth" facade against all odds. But of course I later learned about the anti-rationality pomo crusade, ascendant in other quarters, which is probably even worse in the grand scheme of things. An "enlightened" "meta-rational" synthesis of this mess is still far off from any respectable mainstream.
> You read about all those geniuses who proved famous theorems at the age of 12 or whatever, and surely it came easy to them? Or is that just cope and most mathematicians have to work hard at it before they develop both their intuitions and their skills to the point where it feels "natural"?
These seem like important questions! Many people have strong opinions about them, because they think these have fundamental moral and political consequences. As far as I can tell, there is very little empirical research, so the opinions probably aren't meaningful. I don't have one, except that science should be done on this!
> probably the most exciting experience from that time. It was like I stumbled upon an actually taboo territory, a shameful secret
But what was almost equally surprising to me was that basically nobody else cared about any of that! People were perfectly happy to take part in the respectable facade, entirely dismissing any such taboos. I also never really understood that. Clearly they were smart enough to understand it in principle, but probably you also need a contrarian personality to be fascinated by these topics.
My impression is that a lot of people just really like doing math, and don’t like thinking about what it means. Maybe partly because that’s a quite different activity, and maybe partly because it’s anxiety-provoking if you look into it.
I don’t feel like I fully understand what “liking doing math” is. It seems to be similar to liking doing puzzles like Rubik’s Cube or Sudoku, only much more difficult. I don’t like puzzles at all.
I do like learning math when I feel I’ve got some insight from it. I’m not interested in solving mathematical problems for their own sake.
But isn't it presumptuous of us non-naturals at math to pontificate about what it means to do math for those who actually enjoy doing it? You need the first-person experience to have the full picture...
Thank you for the wonderful discussion, and the great anecdotes. I love the all-seeing Cthulhu-Turing machine! Also looking forward to the metarationality chapters you gestured at!
Your thoughts reminded me strongly of the talk "Mathematics, Morally" by Eugenia Cheng. Here's a relevant excerpt:
"The key characteristic about proof is not its infallibility, not its ability to convince but its transferrability. Proof is the best medium for communicating my argument to X in a way which will not be in danger of ambiguity, misunderstanding, or defeat. Proof is the pivot for getting from one person to another, but some translation is needed on both sides.
So when I read an article, I always hope that the author will have included a reason and not just a proof, in case I can convince myself of the result without having to go to all the trouble of reading the fiddly proof. When this does happen, the benefits are very great. But is it always possible?"
Thank you! That paper was fascinating. I'm not sure what to think about it... I'll probably be turning it over in my mind for years to come.
I took a graduate-level seminar in category theory with G-C Rota. Pretty much the only thing I remember from it was his saying that there are two kinds of proofs. There's the ones where you go through it and check that every step is correct and at the end you don't know anything (other than that something is true, maybe, but who cares). And there's the ones where you get a picture in your head and you say "oh, I see, so that's why!"
Which seems to be pretty much the point Cheng makes here? Or, at least one of them.
You could take Rota's observation without the context of category theory, and many people might agree with it, without being able to do anything useful with it. To me, the point of Cheng's talk is first that
(1) "mathematical morality" is a meaningful and useful notion,
and then that
(2) it is closely related with category theory.
(1) seems to be more or less Rota's point, and is also what Cheng spends most of her talk building a case for. She then basically asserts (2), accompanied by some comments about the importance of analogies.
A solid argument for (2) would involve plenty of examples and greatly lengthen the talk, but it's essentially the same argument as
(3) category theory is useful for doing maths.
It's just that (1) is of clear and direct relevance to lots of people, and then leads to (2) when examined more closely; while (3) is less obviously motivated and its framing verges on "religious wars" territory, so many people will never bother to look into it seriously.
I am a fan of yours who's mostly content the just being a passive reader. But since I am an academic mathematicians and have thought about these questions a lot, I feel somewhat obligated to comment. Sorry this will be a long response.
On the role of intuition and meta-level thinking in mathematics
It is definitely true that intuition and meta-level thinking is central to how a mathematicians work, and I think teachers should definitely discuss it with their students, and encourage them to think about it. I am, however, quite cautious about how much it can be "taught". I have always avoided learning about other people's intuition because I believe there is probably very few practical methods that applies universally, very much like meta or philosophical thinking does not make one automatically better at concrete tasks. For things I want to do myself, I sometimes deliberately not read about it to avoid "spoilers". To me, the intuition is gained when working on concrete problems, and what is learned is often not generalizable. Often, what I got is a vague sense of confidence or unease depending on the problem. Another reason for my caution is that I have seen people obsessed with building intuition or meta-understanding become mediocre mathematicians, because they think the formalization part as unimportant grunt work.
On the role of foundations in mathematics
There is no foundation of mathematics that is evidently the true one. This is because the methods of formal logic cannot be a universal truth, as the foundation crisis demonstrated. A formal system that includes all the established mathematics, and also does not have (somewhat) weird consequences does not exist, hence we have accepted that mathematics does not represent the universal truth. However, this concern does not make the real number less real for me. There is a set of axioms that describe the number system leading to theorems that seems right, which serve as a good foundation for topics like differential geometry and partial differential equations. To me, the real number axioms feels "right", and it has a solid foundation in set theory. Asking the foundation to not have any weird consequences (and indeed, many people don't think those consequences as weird, just as a matter of fact) is too much to ask.
I do think your professor should have given a better answer on "what is real number". I was lucky to have been taught that real numbers can be defined by a set of axioms, and among which the completeness axiom has a number of different equivalent formulations. They represent different views into how the real number is "complete" or "continuous", and it gave me almost a platonic idea of what real number is.
On epiphanies
I did have a few epiphanies during my career, and it was extremely rewarding. One of my professors describe a magic moment when "something clicks and the student is transformed from a plain apprentice into a master of his/her own subject". Often, this is the moment when you prove your first truly original result.
On teaching how to do mathematics
I am not embarrassed at talking about how I do mathematics, but it's hard to find any particularly interesting thing to say. I have a few practices, but none of them feels that special or mystical. The way I learn things is try to internalize it, trying to come up with different proofs and connecting with different ideas. I found this very hard to teach to students because what "internalize" mean is very personal.
Let me also say that I represent your average non-genius mathematician, so I don't have any insight on how to prove breakthrough theorems. With that said, one really nice thing about mathematics is that we generally understand each other. For example, I don't find Terrence Tao to be an incomprehensible genius (based on his blog, I have not met him in person). His approach seems very reasonable and doable for a non-genius person, just without his level of consistency and power.
Thank you for this thoughtful and extensive comment!
Your practice of not learning about other people's intuitions is very interesting. It would not have occurred to me, but I can completely understand why this could be a good idea. There's a lot of things I do want to do for myself, even though I know there's a well-known solution approach, because ignoring that and doing it my way may lead to better understanding (and occasionally a better solution than the standard one).
And, the emergence of "a vague sense of confidence or unease" is familiar (in other disciplines besides mathematics). It seems important. I wonder if one can increase one's sensitivity to that, so it is more accurate and salient as a guide to how to proceed.
> I don't find Terrence Tao to be an incomprehensible genius
Yes... I wonder how much of that is because he's an outstanding communicator, who makes actually extremely difficult things seems relatively easy.
Somewhat relatedly, I think I remember reading someone saying that part of his special sauce is a willingness to grind through pages and pages of tiresome algebraic simplifications which don't seem to be going anywhere, long after most people would have given up, on the off chance that suddenly something understandable will pop out. That is almost the opposite of intuition! And sometimes it works.
It's sometimes said that people are generally good at, and specialize in, one or the other of those modes. Perhaps part of what makes Tao successful is being good at both?
My next post (scheduled for tomorrow morning) has more reflections on the incident in which I didn't get an answer to my question about the reals in class. I got out my copy of Rudin, and made an interesting discovery!
I'd be interested in your comments on that, if it prompts any interesting thoughts for you.
Thanks for your reply! To add to the discussion about intuition, there is a sense that mathematics can only be learned "by doing mathematics", which roughly means the process of problem solving, proving theorems, developing theories, formulating conjectures etc. I think some intuitions can only be gained by "doing mathematics", and I value those intuitions so much that I don't want it spoiled by preconceptions.
Terrence Tao's blog was very instrumental for my growth during my PhD years. I saw one of the highest regarded mathematicians use an approach that feels replicable and do not require being a genius. My understanding of his approach is after first understanding things on a formal level, then try to further build informal principles and intuition, but then as much as one can, also formalise those intuitions (for example, as alternative formulation of theorems). This reminds me of the higher levels of rationality you mentioned in the QnA (I don't remember the exact wording).
I am by no means implying I can do what Tao does. By being able to understand formal mathematics on an alarmingly fast rate (I heard many stories about this) and then producing synthesis and intuition at a fast rate, he can go to much further heights than most of us can.
There are of course mathematicians who are closer to the "incomprehensible genius", Ramanujan comes to mind. But modern mathematicians tend to comprehensible, maybe it's linked to the increasing teaching duty of mathematicians making them better communicators.
wow, that comic is gorgeous, thanks for posting it!
"what is a real number, really?" is a great question and a great example of the thing i was trying to talk about on twitter awhile back about the surprisingly specific philosophical commitments you have to make to be okay with doing modern mathematics, which are almost never made explicit and so which students either have to make intuitively by osmosis or not at all. i could go on a long rant about this. for starters, it's quite strange that provably almost no real numbers can be written down explicitly in any way whatsoever, and i think a student would be perfectly in their right to say that this is a bizarre state of affairs foundationally and that it indicates something strange is going on. people really do not want to talk about this!
These descriptions the experience of mathematical insight seem very unlike my personal experience.
A possible example of “what is is like”.
Imaginethat you are trying to demodulate a (digital) radio signal. One of the early steps in this is you want to determine when the signal actually starts. If the signal is binary, you can compute a correlation with a sync word at the start:
Sum_0^n a_i . x_{i+t}
… and declare you have found the start when you see a high correlation,
But … what if the signal uses 4 level modulation, and is sending two bits at a time?
So, I try to figure out what the n-ray generalisation of correlation is.
Then, while walking in the park … no, that is just dumb.
The thing that generalises to the n-ary case is the Euclidean distance, not the. correlation:
Mathematical experience is diverse. Which makes it even more important to share many personal stories! If there's only a few, from Great Mathematicians, and one's own experience is different, one might wrongly conclude that one is Not Capable. Whereas, in fact, there are many different ways of being capable (it seems!).
@Sarah Constantin's comment here, and my reply to it, are about that.
couple data points:
1.) part of why I could never do anything but analysis, and the super-concrete parts of it at that (my thesis involved a lot of spherical harmonics and Heisenberg group stuff, all calculations on special cases, no big general theorems) is that I need to visualize shapes and sizes! If it’s basically in R^n (or Hilbert spaces) then we’re groovy; if it’s too much more general, how do you do anything except clumsily trying to patch together definitions and theorems’ inputs and outputs into a sequence that leads somewhere? I had this really sweet professor try to get across the “yoga of commutative algebra” and I was like “maybe if there were about 10x more examples and pictures I would get any of this but right now my brain will Not.” I still don’t know what algebra/topology people are doing with their heads. (though I suspect Thurston was actually visualizing, in extremely trippy ways).
2.) yes, of course, I have had mathematical Experiences. ranging from “literal fever dreams about sliding down a rainbow-hued complex exponential” to “acquiring synaesthesia colors while studying” to more normal stuff like “the Right way to learn involves doing little hand movements and singing a little song and laughing out loud when you Get It”.
Fascinating, thanks!
@Lucy Keer has several (older) posts about "the two kinds of mathematicians," basically algebraists (like me) and analysts (like you). Many people have observed this (and Lucy has collected some discussions from the literature).
I suspect, from my own experience, that we specialize into one or the other way of doing things (perhaps based on innate abilities, or due to circumstances). Then it is possible to develop the other one to some extent. I did, although I was never as good at shape rotation as symbolcelling.
I'm an algebra-topology guy with essentially zero grasp of analysis, and I'm terrible at symbol manipulation! Algebra is like playing with legos -- clicking shapes together -- to me
*Three* kinds of mathematicians...
As part of mathematical education, it seems important to explain that different people approach the stuff in different ways, which is fine; and different thinking styles work better in different parts of math; and one can develop the styles one is bad at, at least to some degree and with an expectable level of pain and frustration.
Something along the lines of “math is a mind-body thing, except you don’t move your body” has sometimes helped me explain it to athletes.
Most sports (especially things like golf) also recruit the same spatial/kinesthetic/visual/proprioception faculties as you are talking about, so they do have some intuitive sense of what that means.
Nice, thank you!
David, Great post about envisioning, and I'm glad to see you diving into 'Mathematica'. Knowing that you already have too much to read and write, I will nonetheless highly recommend also diving into the quantitative psychological research of Todd Thrash, especially his paper, 'The creation and curation of all things worthy: Inspiration as vital force in persons and cultures'. (Happy to send you a pdf if you can't find it online.)
It has been a revelation for me. His articulation of the three basic aspects of the psychological process of inspiration (evocation, transcendence, and transmission) is a perspective that spontaneously inspired a new way for me to fit together myriad threads in my own developing philosophy. I think Thrash's perspective will help illuminate and clarify your nascent conceptualization of envisioning. Here is the abstract of his paper:
'Inspiration is an epistemic-motivational episode in which an individual is awoken to what is valuable and imbued with the spirit to express this understanding. In this article, I tell the story of my research program on inspiration. I begin by describing the origin of my research program as a corrective to the emphasis on personal agency in the motivation literature. I then review work on foundational issues of conceptualization, typology, methodology, and construct validation. Next I review research on the role of inspiration in the creative process. Inspiration is a response to, not a source of, creative ideas. Its function is to motivate transmission (e.g., actualization) of an idea in the form of a tangible product. Inventors prone to inspiration receive more patents, and inspired writers are more creative, productive, and efficient. Next I situate inspiration within a broader cultural-evolutionary context. Inspiration is infectious, such that inspired writers inspire their readers. Processes of transmission and contagion suggest that the core significance of inspiration lies in its procultural function. Inspiration, I argue, is a genetically evolved adaptation that catalyzes and guides cultural evolution by compelling selective retention of cultural wisdom in the form of cultural artifacts (e.g., scripture, poetry, inventions, actualized selves), thereby making the species more adaptable to local and changing environments. Finally, I discuss benefits to well-being. Inspiration marks moments of peak functioning that integrate the hedonic well-being of the primitive creature, the eudaimonic well-being of the social creature, and the self-transcendent well-being of the cultural creature.'
Thrash specifically has some very insightful things to say about the intertwined relationship between inspiration and creativity:
'The actualization-of-creativity model has important implications. First, it reveals that a simple explanatory model of the form “inspiration causes creativity” or “creativity causes inspiration” is not adequate. Inspiration is a cause of creativity in a product but an effect of creativity in an idea. This conclusion qualifies the above preliminary conclusion of bidirectional effects between inspiration and creativity.'
So if you're looking at ways to describe moments of inspirational envisioning with no woo woo baggage, you should definitely check out the paper, and Thrash's research generally.
Some of this fits (the kinesthetic-proprioceptive aspect particularly), although I absolutely don't understand the reluctance to talk about the lived-experience -- that's the only way I can talk about this stuff. Then again, it's easier for me to write formal mathematics than read it.
Another thing I never see anyone mention is that doing math for me was generally quite painful.
"Math is painful, let's go shopping" is probably everyone's experience!
Maybe what is left unsaid is that it's painful even for people who are very good at it. It takes an extraordinarily high tolerance for frustration and feeling stupid and gritting your way through anyway.
Presumably, those who persist are those who find it rewarding as well. Generally not at the same time, if my experience is anything to go by!
Yeah, I guess I meant researchers don't talk about it. Like admitting you're afraid of going into battle! Obviously most people find it painful and will tell you at great length, though.
>And the reality is, if you look for foundations for mathematics… People go into mathematics thinking they’re going to find absolute truth, and if you dig deep enough under those supposed absolute truths, you find it is clouds. There is no foundation other than clouds. It’s clouds all the way down. And I think a lot of mathematicians have read about this, and they realize there’s something scary there; and this is another part, probably, of why there’s a taboo about real talk about what math is, because it’s on sand, or clouds.
I'm reminded of an excellent short story by Ted Chiang called "Division by Zero": https://fantasticmetropolis.com/i/division
The story centers on a mathematician who uncovers a paradox that undermines her long-held belief in the absolute consistency of arithmetic. This discovery triggers a profound personal crisis, which reverberates in her relationship with her husband.
Major themes include the destabilizing power of revelation—how realizing a trusted system may be flawed can unravel one’s sense of self—and the limits of empathy when confronting someone else’s existential crisis.
I think it's a great fictional exploration of what that "something scary" might look like for people who are looking to mathematics for absolute truth. It certainly did for me when I first read it as an undergraduate in math.
Splendid, thank you!
>I was a math undergraduate. I saw people struggling with this. And for a lot of them, the problem was, and I found this really difficult myself. I got better at it.
I also was a math undergraduate, and I never got (much) better at that. I was decent enough at symbol manipulation, but that also never really felt natural or enjoyable to me, so I eventually declared myself talentless and moved on. But I'm still curious about the talent question. You read about all those geniuses who proved famous theorems at the age of 12 or whatever, and surely it came easy to them? Or is that just cope and most mathematicians have to work hard at it before they develop both their intuitions and their skills to the point where it feels "natural"?
>And when people realized at the beginning, early 20th century, that this doesn’t actually work, there was a major crisis, and it kind of looked like mathematics might just completely fall apart.
Yep, in my first year at the uni I kind of randomly stumbled upon the book The Loss of Certainty by Morris Kline about the foundation crisis, which was an entirely new perspective for me then, and probably the most exciting experience from that time. It was like I stumbled upon an actually taboo territory, a shameful secret that the math, and the wider rationality world swept under the rug while still trying to maintain the "absolute truth" facade against all odds. But of course I later learned about the anti-rationality pomo crusade, ascendant in other quarters, which is probably even worse in the grand scheme of things. An "enlightened" "meta-rational" synthesis of this mess is still far off from any respectable mainstream.
> You read about all those geniuses who proved famous theorems at the age of 12 or whatever, and surely it came easy to them? Or is that just cope and most mathematicians have to work hard at it before they develop both their intuitions and their skills to the point where it feels "natural"?
These seem like important questions! Many people have strong opinions about them, because they think these have fundamental moral and political consequences. As far as I can tell, there is very little empirical research, so the opinions probably aren't meaningful. I don't have one, except that science should be done on this!
> probably the most exciting experience from that time. It was like I stumbled upon an actually taboo territory, a shameful secret
Ooo, that's very cool!
>Ooo, that's very cool!
But what was almost equally surprising to me was that basically nobody else cared about any of that! People were perfectly happy to take part in the respectable facade, entirely dismissing any such taboos. I also never really understood that. Clearly they were smart enough to understand it in principle, but probably you also need a contrarian personality to be fascinated by these topics.
My impression is that a lot of people just really like doing math, and don’t like thinking about what it means. Maybe partly because that’s a quite different activity, and maybe partly because it’s anxiety-provoking if you look into it.
I don’t feel like I fully understand what “liking doing math” is. It seems to be similar to liking doing puzzles like Rubik’s Cube or Sudoku, only much more difficult. I don’t like puzzles at all.
I do like learning math when I feel I’ve got some insight from it. I’m not interested in solving mathematical problems for their own sake.
People are surprisingly different, in many ways!
But isn't it presumptuous of us non-naturals at math to pontificate about what it means to do math for those who actually enjoy doing it? You need the first-person experience to have the full picture...
Thank you for the wonderful discussion, and the great anecdotes. I love the all-seeing Cthulhu-Turing machine! Also looking forward to the metarationality chapters you gestured at!
Your thoughts reminded me strongly of the talk "Mathematics, Morally" by Eugenia Cheng. Here's a relevant excerpt:
"The key characteristic about proof is not its infallibility, not its ability to convince but its transferrability. Proof is the best medium for communicating my argument to X in a way which will not be in danger of ambiguity, misunderstanding, or defeat. Proof is the pivot for getting from one person to another, but some translation is needed on both sides.
So when I read an article, I always hope that the author will have included a reason and not just a proof, in case I can convince myself of the result without having to go to all the trouble of reading the fiddly proof. When this does happen, the benefits are very great. But is it always possible?"
You can read the full thing here: https://eugeniacheng.com/wp-content/uploads/2017/02/cheng-morality.pdf
I have other things to say, but this seemed like a "low effort / high value" thing to mention. (I hate talking like that.)
Thank you! That paper was fascinating. I'm not sure what to think about it... I'll probably be turning it over in my mind for years to come.
I took a graduate-level seminar in category theory with G-C Rota. Pretty much the only thing I remember from it was his saying that there are two kinds of proofs. There's the ones where you go through it and check that every step is correct and at the end you don't know anything (other than that something is true, maybe, but who cares). And there's the ones where you get a picture in your head and you say "oh, I see, so that's why!"
Which seems to be pretty much the point Cheng makes here? Or, at least one of them.
I think that's certainly part of it.
You could take Rota's observation without the context of category theory, and many people might agree with it, without being able to do anything useful with it. To me, the point of Cheng's talk is first that
(1) "mathematical morality" is a meaningful and useful notion,
and then that
(2) it is closely related with category theory.
(1) seems to be more or less Rota's point, and is also what Cheng spends most of her talk building a case for. She then basically asserts (2), accompanied by some comments about the importance of analogies.
A solid argument for (2) would involve plenty of examples and greatly lengthen the talk, but it's essentially the same argument as
(3) category theory is useful for doing maths.
It's just that (1) is of clear and direct relevance to lots of people, and then leads to (2) when examined more closely; while (3) is less obviously motivated and its framing verges on "religious wars" territory, so many people will never bother to look into it seriously.
I am a fan of yours who's mostly content the just being a passive reader. But since I am an academic mathematicians and have thought about these questions a lot, I feel somewhat obligated to comment. Sorry this will be a long response.
On the role of intuition and meta-level thinking in mathematics
It is definitely true that intuition and meta-level thinking is central to how a mathematicians work, and I think teachers should definitely discuss it with their students, and encourage them to think about it. I am, however, quite cautious about how much it can be "taught". I have always avoided learning about other people's intuition because I believe there is probably very few practical methods that applies universally, very much like meta or philosophical thinking does not make one automatically better at concrete tasks. For things I want to do myself, I sometimes deliberately not read about it to avoid "spoilers". To me, the intuition is gained when working on concrete problems, and what is learned is often not generalizable. Often, what I got is a vague sense of confidence or unease depending on the problem. Another reason for my caution is that I have seen people obsessed with building intuition or meta-understanding become mediocre mathematicians, because they think the formalization part as unimportant grunt work.
On the role of foundations in mathematics
There is no foundation of mathematics that is evidently the true one. This is because the methods of formal logic cannot be a universal truth, as the foundation crisis demonstrated. A formal system that includes all the established mathematics, and also does not have (somewhat) weird consequences does not exist, hence we have accepted that mathematics does not represent the universal truth. However, this concern does not make the real number less real for me. There is a set of axioms that describe the number system leading to theorems that seems right, which serve as a good foundation for topics like differential geometry and partial differential equations. To me, the real number axioms feels "right", and it has a solid foundation in set theory. Asking the foundation to not have any weird consequences (and indeed, many people don't think those consequences as weird, just as a matter of fact) is too much to ask.
I do think your professor should have given a better answer on "what is real number". I was lucky to have been taught that real numbers can be defined by a set of axioms, and among which the completeness axiom has a number of different equivalent formulations. They represent different views into how the real number is "complete" or "continuous", and it gave me almost a platonic idea of what real number is.
On epiphanies
I did have a few epiphanies during my career, and it was extremely rewarding. One of my professors describe a magic moment when "something clicks and the student is transformed from a plain apprentice into a master of his/her own subject". Often, this is the moment when you prove your first truly original result.
On teaching how to do mathematics
I am not embarrassed at talking about how I do mathematics, but it's hard to find any particularly interesting thing to say. I have a few practices, but none of them feels that special or mystical. The way I learn things is try to internalize it, trying to come up with different proofs and connecting with different ideas. I found this very hard to teach to students because what "internalize" mean is very personal.
Let me also say that I represent your average non-genius mathematician, so I don't have any insight on how to prove breakthrough theorems. With that said, one really nice thing about mathematics is that we generally understand each other. For example, I don't find Terrence Tao to be an incomprehensible genius (based on his blog, I have not met him in person). His approach seems very reasonable and doable for a non-genius person, just without his level of consistency and power.
Thank you for this thoughtful and extensive comment!
Your practice of not learning about other people's intuitions is very interesting. It would not have occurred to me, but I can completely understand why this could be a good idea. There's a lot of things I do want to do for myself, even though I know there's a well-known solution approach, because ignoring that and doing it my way may lead to better understanding (and occasionally a better solution than the standard one).
And, the emergence of "a vague sense of confidence or unease" is familiar (in other disciplines besides mathematics). It seems important. I wonder if one can increase one's sensitivity to that, so it is more accurate and salient as a guide to how to proceed.
> I don't find Terrence Tao to be an incomprehensible genius
Yes... I wonder how much of that is because he's an outstanding communicator, who makes actually extremely difficult things seems relatively easy.
Somewhat relatedly, I think I remember reading someone saying that part of his special sauce is a willingness to grind through pages and pages of tiresome algebraic simplifications which don't seem to be going anywhere, long after most people would have given up, on the off chance that suddenly something understandable will pop out. That is almost the opposite of intuition! And sometimes it works.
It's sometimes said that people are generally good at, and specialize in, one or the other of those modes. Perhaps part of what makes Tao successful is being good at both?
My next post (scheduled for tomorrow morning) has more reflections on the incident in which I didn't get an answer to my question about the reals in class. I got out my copy of Rudin, and made an interesting discovery!
I'd be interested in your comments on that, if it prompts any interesting thoughts for you.
Thanks for your reply! To add to the discussion about intuition, there is a sense that mathematics can only be learned "by doing mathematics", which roughly means the process of problem solving, proving theorems, developing theories, formulating conjectures etc. I think some intuitions can only be gained by "doing mathematics", and I value those intuitions so much that I don't want it spoiled by preconceptions.
Terrence Tao's blog was very instrumental for my growth during my PhD years. I saw one of the highest regarded mathematicians use an approach that feels replicable and do not require being a genius. My understanding of his approach is after first understanding things on a formal level, then try to further build informal principles and intuition, but then as much as one can, also formalise those intuitions (for example, as alternative formulation of theorems). This reminds me of the higher levels of rationality you mentioned in the QnA (I don't remember the exact wording).
I am by no means implying I can do what Tao does. By being able to understand formal mathematics on an alarmingly fast rate (I heard many stories about this) and then producing synthesis and intuition at a fast rate, he can go to much further heights than most of us can.
There are of course mathematicians who are closer to the "incomprehensible genius", Ramanujan comes to mind. But modern mathematicians tend to comprehensible, maybe it's linked to the increasing teaching duty of mathematicians making them better communicators.
Yes, very interested in the new post!
wow, that comic is gorgeous, thanks for posting it!
"what is a real number, really?" is a great question and a great example of the thing i was trying to talk about on twitter awhile back about the surprisingly specific philosophical commitments you have to make to be okay with doing modern mathematics, which are almost never made explicit and so which students either have to make intuitively by osmosis or not at all. i could go on a long rant about this. for starters, it's quite strange that provably almost no real numbers can be written down explicitly in any way whatsoever, and i think a student would be perfectly in their right to say that this is a bizarre state of affairs foundationally and that it indicates something strange is going on. people really do not want to talk about this!
Well you can do a great deal of mathematics constructively, if you like. It's just a gigantic pain in the ass.
The ideology is far more comfortable, but the process becomes rather grim.
These descriptions the experience of mathematical insight seem very unlike my personal experience.
A possible example of “what is is like”.
Imaginethat you are trying to demodulate a (digital) radio signal. One of the early steps in this is you want to determine when the signal actually starts. If the signal is binary, you can compute a correlation with a sync word at the start:
Sum_0^n a_i . x_{i+t}
… and declare you have found the start when you see a high correlation,
But … what if the signal uses 4 level modulation, and is sending two bits at a time?
So, I try to figure out what the n-ray generalisation of correlation is.
Then, while walking in the park … no, that is just dumb.
The thing that generalises to the n-ary case is the Euclidean distance, not the. correlation:
\Sum_i (a_i - x_{i+t})**2
Now, it makes sense.
That's a cool example!
Mathematical experience is diverse. Which makes it even more important to share many personal stories! If there's only a few, from Great Mathematicians, and one's own experience is different, one might wrongly conclude that one is Not Capable. Whereas, in fact, there are many different ways of being capable (it seems!).
@Sarah Constantin's comment here, and my reply to it, are about that.