PhD in math here with several published papers. And my recommendation is a metaprinciple: enjoy mathematics. Benjamin Finegold said similarly that the secret to chess is to enjoy every move. Personally, I had no trouble in mathematics, ever. And I think the reason for that is that I really enjoy just doing it, writing symbols down, learning about new theories, and even inventing my own.
Not everyone will enjoy mathematics at first sight. But I think at least 50% of that can be explained by the lack of obvious paths to enjoy mathematics. Obviously, most mathematics taught in high-school is not taught as it should be: a cool artistic logical pursuit that has all kinds of fun in it.
So my advice is to really find a mentor who already has found that path and let them show you how to enjoy math.
Believe me, I've tutored a lot of people, many of which initially disliked math and found it difficult. But after a few tutoring sessions, I could see a little sparkle in their eye that said, "hey, this might be cool".
So before you apply logic, studying, and other tedious "productivity" measures to your math learning, make sure you find a way to enjoy it first.
One of the biggest problem of maths education is that they are taught by people who dislike it. They think of maths as eating bitter medicine or training of a complex, rigid skill. The way maths is taught by them is clumsy and authoritarian, and this makes the students either passive or rebellious.
On a side note, recently the government of Manitoba in Canada removed requirement for maths teachers to take university maths courses. This is being pushed strongly by the education departments of university, which shows how much these maths teachers hate maths.
> On a side note, recently the government of Manitoba in Canada removed requirement for maths teachers to take university maths courses. This is being pushed strongly by the education departments of university, which shows how much these maths teachers hate maths.
That is messed up. Harsh. What is also messed up is that to become a math teacher, you have to go to teacher's college. That's one reason why I never became a teacher. I think I would do better than most at teaching (at least based on the comments I got in my student teacher reviews) but spending another two years at school is rather humiliating and costly.
I get what you're saying, but I think confidence is actually more important than enjoyment.
A lot of people confuse enjoying [x] with enjoying being good at [x]. This is why so many students switch subjects later on in life; when a field suddenly doesn't come naturally to them, they seek to play to their strengths elsewhere. Problems occur when they quit too early, and building confidence early on is important for stopping this.
In my experience, when you think you're bad at something, it's almost impossible to enjoy doing it, which makes preliminary mastery actually the first step to enjoyment and therefore downstream success.
Nah I disagree with that. Well, I do think confidence is useful BUT it only gets you so far. There are plenty of things I am "confident" in doing but don't really enjoy and after a while I just left them behind because they just don't provide a lot of enjoyment.
Any tips or anecdotes for us (well, me) about the fun things you'd do with students to encourage that joy?
I am not a PhD, but have done a fair bit of tutoring young people in maths. I feel similarly to you there, and am always on the lookout for new ways to foster that feeling of mathematics being fun and wonderful.
It can be hard. The feeling a lot of young people pick up - of maths being roughly akin to pointless abject suffering - is so strongly rooted in young people sometimes, and can be strongly connected to feelings of inadequacy and shame and so on.
Wrote that reply then clicked on your page to see if there were a blog or anything where you elaborate already on these topics - I've already read a couple of articles from your substack! Oops, hah, I did not realise I was writing to the person behind that.
I'd filed it away in my head as something to sink more time into when I've the emotional space to do so, but essentially I find it quite courageous to go against the grain in a world where it's perhaps harder and harder to do so. Everyone has a hot take, of course, but the hot takes are usually very much within the bounds of acceptable discourse.
Well, I think they key is I try and select topics that I find very fun, even they are a little off the beaten path. I know there's pressure to teach the curriculum, but honestly there are some cool things you can do with students if there's extra time that might get them hooked even if it's not in the curriculum, like:
1. Coloring maps (four color theorem)
2. Drawing curves and then looking for pathological ones (quadratic equations ...but more of an exploratory rather than methodological approach)
3. Infinite series (but just discuss some paradoxes people thought of centuries ago...)
4. Triangles on spheres
...if you don't make some of it extremely rigorous, there's a lot of basic facets of these things like the drawing aspect that can be quite fun. And TBH it makes sense to start there because mathematicians started out exploring these topics just with random calculations and doodles rather than rigorous proofs.
> Believe me, I've tutored a lot of people, many of which initially disliked math and found it difficult. But after a few tutoring sessions, I could see a little sparkle in their eye that said, "hey, this might be cool".
Exactly the same for me. Honestly, the satisfaction of seeing that "sparkle" in the eye of an initially unmotivated or discouraged student is probably among the most fulfilling moments of my professional career.
Great comment, and applies to any activity we do. Seems like in western culture we're told to not have enjoyment, to work hard and grind, and that any bit of enjoyment means you're not trying hard enough and lacking. Much better to have the attitude of allowing the happiness to come from the actions.
I've been told that the secret to exercise, woodworking, playing the piano, whatever is to "enjoy it." But, having or getting the skill to have some successes to "enjoy it" is difficult and not everyone has the ability to get to that level. Just sticking with something or worse, fooling oneself that more effort will lead to good results is annoying to hear if one doesn't have ability. We can't all succeed at everything.
My secret sauce as an undergraduate for all my math courses was solving problems. Solve all the problems at the end of the chapters in the textbook. Find other textbooks in the library and solve all the problems at the end of their chapters.
In graduate school that was expanded: take every chapter of the textbook and rewrite it, filling in all the intermediate steps of every proof, those where the author writes "it follows that ..." or "from which it's obvious that ..."
> take every chapter of the textbook and rewrite it, filling in all the intermediate steps of every proof, those where the author writes "it follows that ..." or "from which it's obvious that ..."
I recently-ish had a read through some of my old fundamental/pure maths notes from Uni, including plenty of proofs. The damned things are littered with steps which were "obvious" to my smug self-satisfied 20 year old self but impenetrable to me reading without much context 20 years later.
I did this but faced enormous problems with transfer/recalling definitions/etc. what helped me improve several fold in approximately exponential value
1) Drill/spaced repetition basic definitions. The Cornell note taking method is convenient to do this while taking notes.
2) Keep a diary of thoughts, things you couldn't solve or did solve. Especially identifying problems, what works or doesn't, why something went wrong. Metacognitive thinking was really useful for transferring problems to solving new ones.
3) A study group involving a lot of us explaining to each other.
I’ll add: go to the TA study sessions. And try to answer questions in class. Even if you have nothing to ask, use that time to do homework, listen to other people’s questions.
Yes, you’ll learn more. Also, TAs will recognize you put in the effort so if you’re arguing for partial points or you’re really close to a cutoff grade they will be more likely to bump you up versus someone they’ve not seen or noticed all semester.
It's totally possible to slog through a chapter of a maths text and feel like we got it. But it turns out our 'understanding' was a facade. We can't apply the concepts in a new situation. Exposing ourselves to feedback via problem sets reveals this.
An lot of brute force memorizing details that everyone else would look up. Biology feeds to medical fields where you have to have details you rarely used memories because you might be doing surgery on someone who is slightly different from typical and have to figure out what that thing in the way is - can you safely move/remove it or not. (I'm not a doctor, real doctors can come up with plenty of much more likely real world examples of situations where you don't have time to look up some detail)
> Solve all the problems at the end of the chapters in the textbook
It really depends on the textbook, isn't it? I find it impossible to solve all the problems in CLRS, for example. Our professor assigned one of the problems about universal hashing, and it took me hours to get the key insights to find the correct proof. I can't imagine how one can solve all the problems given so many competing priorities, except for a few truly talented.
Did you do the few hundred problems before that build up the techniques and theorems?
> given so many competing priorities,
Undoubtedly, the best time to do this was when you were young. The second best time is now. Pick a book and work on a problem or 2 every day. It will likely take 6 months or so but you will learn the material. This is an incredible way to level up in a technical area.
The point is that you gradually get better at it. And by gradually I mean exponentially. If you start with a problem at then end of the book, it'll be very difficult.
I agree with this. In the end it is beneficial with repetition for learning anything, and I think subjects with calculations are partly a craft where you need to get it in your fingers. I teach a subject, it not maths but a (different) engineering subject and what I see is that the students don't buy text books any more (for several understandable reasons), but one of the things they miss out on is a lot of practice problems. This becomes evident in the types of errors they do at the exam, and an apparent lack of system for laying out their problem and solution in an understandable matter for themselves (and me).
More often than the "at the end of the chapter" math problems are not about calculations, but are essentialy mini-problems requiring mini proofs. It's not about 1-1 applying the knowledge that was in the chapter, but rather about creative thinking. Some problems can take hours or days to solve. There isn't much to "get in your fingers".
My issue during my undergraduate days was that the textbook didn't come with solutions, and I really don't know how to unstuck myself when I got stuck (except asking the TA in the office hours).
When I was an undergraduate we had Schaum's Outline Series (do they still exist?). Every chapter came with a bunch of completely solved problems plus another bunch of unsolved problems with the answers. The optimal strategy was still to try and do the solved problems by yourself and then verifying your solution.
Ask your peers. As an ex-prof, one of the most prominent sources of missed potential I see in undergrads is not working with their peers. Like serious 2-6 hours sessions of collaborative problem solving.
You're right, but it depends on your peers. If you're the only one motivated to do extra work, your peers are unlikely to want to spend hours on verifying the solution. Guess it shows the importance of good peers.
Yep, I hated that. Spending an hour on a problem and get a solution you can't verify. It very much affected my motivation on starting on the problem knowing that I wouldn't know if I was right or wrong.
An undergraduate text like Pugh's Real Mathematical Analysis can have over 500 exercises. Trying to do them all on top of your regular workload seems excessive, unless maybe it's an area you really want to specialise in. I also tend to find certain kinds of exercises boring because they appear terribly unmotivated.
On the other hand, I second the suggestion to engage more deeply with the subject material itself: Modify assumptions and see what happens. Can certain proofs be simplified? Try to reconstruct proofs by only memorising certain key details. Try to draw a mental map of a subject and how the different theorems and definitions relate together. Try to implement some proofs in an automated theorem prover, if that's your thing.
500 exercises over 10 weeks is 50 a week, or 10 a day. This leaves you weekends free, and you get the final weeks (typically 5) of the semester left for all those projects that are due at the end of the semester but you can't start early. 500 a day is too much, but you should be able to do 50-100 problems a day in your study time. And others report that when you do this you get good at doing math and so it takes less time.
Sadly I didn't do that. I graduated and do okay, but I encourage everyone to do better than me. As I get close to retirement I need a few people who are still working to build things (and medical treatments) that makes my life better (and in turn take some of that money I saved up over the years for your own life)
Pugh's exercises aren't trivial calculations, they're the proof kind of exercise. If you manage to do 10 a day consistently and still have time for your other courses, congrats.
I also don't think that people who develop new medical treatments necessarily did all the exercises in a pure maths textbook. Being able to prove that continuous functions on a compact set are uniformly continuous probably won't help you fight cancer.
Someone correct me if I'm misremembering, but I think Donald Knuth wrote somewhere that when he'd be assigned the odd-numbered problems, he'd always do them, and do the even-numbered problems as well.
The first thing you have to get used to when moving from school to uni is being utterly lost and defeated.
At the end of high school, I could do everything. I finished my IB exams with huge amount of time to spare, the only thing holding me back was being able to write fast enough. It had been months since I saw a regular curriculum question that I didn't know how to do. Any marks I lost were just trivial errors.
When I got to university, there would be question sheets where I would look at the questions and wonder what it had to do with the lectures I had just been in. As in "I went to this lecture, and I'm supposed to use the information to answer these questions, but I don't even know what the questions mean".
The learning happens when you are doing this frustrating head-bashing.
You read, you read more, you fill a notebook with useless derivations, and eventually you things start to take shape. This could take the entire week's worth of time, just sitting there fumbling about.
The difference is that in uni, the amount of material is so vast you cannot explain it to someone in the time that you have. The students have to pick up some key ideas, and then fill in all the details themselves by pouring hours into it on their own.
> The first thing you have to get used to when moving from school to uni is being utterly lost and defeated.
Very well said, at the university where I studied, there was a pre-semester math repetition course. It was a week long and started with addition of natural numbers. After two days, everything I learned in 13 years of school had been repeated. That was a brutal resetting of expectations. But it made everybody clear, that this is not school anymore. That a different kind of work and focus would be necessary.
> The Germans have aptly called Sitzfleisch the ability to spend endless hours at a desk, doing gruesome work. Sitzfleisch is considered by mathematicians to be a better gauge of success than any of the attractive definitions of talent with which psychologists regale us from time to time. Stan Ulam, however, was able to get by without any Sitzfleisch whatsoever. After his bout with encephalitis, he came to lean on his unimpaired imagination for his ideas, and on the Sitzfleisch of others for technical support. The beauty of his insights and the promise of his proposals kept him amply supplied with young collaborators, willing to lend (and risking to waste) their time.
Interestingly, this guide states that the intuitive understanding of maths is only suitable at the school level but not for the university.
In his recently published book "Mathematica: A Secret World of Intuition and Curiosity", David Bessis argues that the intuition is the "secret" of understanding maths at all levels.
Not sure what conclusion to draw from here, but my (rather dated) experience with university maths tells me that the intuition is a powerful tool in developing the understanding of the subject.
This seems like a contradiction but I don’t think it is. What it’s really saying is that experience is a precondition for intuition.
When a high school student looks at a high school math problem they’re drawing on all of their experience in K-12 math to get intuition for how to solve the problem. When they leave high school to study math in undergrad they struggle because their experience is no longer sufficient. They’re faced with a lot more abstract problems and the demands for rigour are much higher. The problems also tend to operate at higher levels on Bloom’s taxonomy [1] than high school math, something with which the average high school student would have little or no experience. It is this unfamiliar territory where intuition is hard to come by.
After gaining more experience (later undergrad and into grad school and beyond) the intuition starts to come back. But it’s fundamentally a different kind of intuition. In high school math it was often a visual/geometric intuition that teachers were trying to build. In higher math it’s an intuition for abstractions and for the tools you need to attack problems. This is really no different from a programmer looking at a problem and saying “I need a hash map and then this problem is trivial.”
Possible harmonization of the two ideas: the intuition that we go into math at high school level can help serve us at that level of math. We have some idea of geometry-like objects and 2d-calculus like curves from our everyday life
At university level the objects become more abstract, so the intuition we use in normal daily life may no longer apply. New kinds of intuition may develop but it takes work, including lots of time spent with the formal processes and calculations along with reflection on that time, and the active creation of new metaphors to drive the intuition. For example, I still remember a professor using "Ice-9" (from _Cat's Cradle_) as a metaphor for how proving some local property of a holomorphic function on the complex plane made that property true for its global behavior
I think he's saying here that intuition is sufficient for high school math, but not sufficient for college. That's not to say that it isn't necessary, only that it isn't sufficient.
That topic fascinates me. As a kid, a lot of topics felt intuitive, and college became a pit of darkness. Makes me wonder what in our brains makes something feel natural, obvious, with that feeling of playfulness and certainty .. while some times you're drowning in a blur.
In my case I realized I achieved a whole new level when I tried to do all of the proofs by myself. I used to read the whole subject a few times and after that, I would go one definition/proof at a time and first to recall the statement and then try to do the proof by myself. By doing that you indirectly achieve all of what the article says.
A nice thing I realised is that once I did that, almost all of the exercises that were complex before for me, turned out to be straightforward. It was like a cheat code where I almost did not need to do any exercises.
I used to teach at the uni at several levels, and every year I would ask if anyone tried to recall the proofs of the theorems at home. and no one did. They were always shocked when I told then they should do it.
>I used to teach at the uni at several levels, and every year I would ask if anyone tried to recall the proofs of the theorems at home. and no one did. They were always shocked when I told then they should do it.
I try to do this as well. If you combine this with understanding the definitions of the various units you more or less will have the textbook in your head, some assembly required.
This article is missing the "meta" in studying mathematics.
Creative introspection into how one learns begins to really pay off partway through college.
One's relationship to convention becomes as important as one's relationship to technique. Understanding the "whole" of something involves understanding the biases that shape the presentation you're seeing. You'll probably want to shed them.
This applies whether one wants to change math or just learn it. A passive stance, trying to do what others want, is a recipe for frustration.
I’ve been very pleasantly surprised by my recent experience, having signed up for MathAcademy.com after reading about it on HN.
Now in my 50’s, I wanted to relearn high school maths from 35 years ago and I scooted through their Foundations series (now half way through Foundations 3, rapidly accruing like 9000 xp in 9 weeks, 2 hours a day). Planning in 2025 to do 1-3 university level courses with them at a slower pace.
It’s suited my way of learning as an autodidact: enjoyable; measurable; adaptable level of hardness; no hitting of a “wall” or “unmet dependencies”; thorough explanation of problems I didn’t solve.
Perhaps my biggest realisation was that one can learn without needing to document many notes to revise/memorise, because experience and spaced repetition suffices. I’m taking a Xmas hiatus which will be the real test of baked learning.
That hit home. I'm afraid I was one of those lazy math undergrads who struggled with a few of the first year topics, didn't get help or put the hours in and never really recovered. I will maintain I think the teaching was very poor in places (lots of "just trust me" handwashing and "this is obvious so I'll leave it to you to complete" which for an 18 year old frankly sucks). A system that lets you get 30% in "analysis 1" and then just marches you straight into "analysis 2" next semester and expects you to just pull your socks up isn't much of a system to me. Honestly I'm afraid my time at university doing maths was miserable. I should have done something more applied like engineering or CS probably.
Someone once told me "If you like biology at school, do psychology at university. If you like chemistry, do biology. If you like physics, do chemistry. If you like maths, do physics and if you like philosophy, do maths". I should have listened.
I agree that the system is garbage. School ruins math -- but I just want to share that nothing stops you from revisiting math later as a fully actualized adult, without the bullshit time constraints and grade pressure. Math isn't well suited to time pressure -- in my view math is about spending lots of time with ideas and playing with them and taking risks and making dumb guesses, it's moonshot through and through. At its core is play, and time pressure doesn't serve anyone well.
Anyway, if you were interested in it, you can always revisit it, and even try taking a class again if you ever want to. There are lots of great books out there for self-learners, and lots of communities of folks learning together
> "If you like biology at school, do psychology at university. If you like chemistry, do biology. If you like physics, do chemistry. If you like maths, do physics and if you like philosophy, do maths"
This is good advice if the objective is to do well regarding grade results.
If you want to get down to the bottom of things, to understand everything and to solve fundamental questions of science, you might well want to invert the advice:
If you liked biology, study chemistry, for the processes of life are (electro-)chemical processes.
If you liked chemistry, study physics, for the processes of molecules and atoms, their formation and reactions, are physical processes.
The way I heard this phrased back in the day was, "Biology is really chemistry, chemistry is really physics, physics is really math." Never heard psychology or philosophy added to the chain.
> "A good way of seeing how a subject works is to examine the proof of a major result and see what previous results were used in it. Then trace these results back to earlier results used to prove them. Eventually you will work your way back to definitions"
I find the parallels between proofs and programs to be fascinating. We could write an analogous thing for programming:
"A good way of seeing how a sort of program works is to examine one of the popular programs/libraries and see what functions were used in it. Then look inside of those functions and see what functions are used inside of those. Eventually you will work your way back to the lower level primitives."
Sometimes it is weird to see a webpage from my university on here. Usually UH doesn't get a lot of attention for its STEM. At least from my anecdotal experience.
Does someone have experiences with using o1-preview for mathematics? A while ago I tried to use GPT-4o and Claude Sonnet for certain algebraic questions related to probability theory. The models did help significantly (at least relative to my rather limited ability), though they also often produced wrong results and struggled to make progress on harder questions.
I appreciated the article for emphasising memorising definitions and statement of theorems... But not for proofs. For proofs, a general outline would be sufficient.
For proofs, I find it a good idea to memorise (or at least implicitly retain) the reason a result is true. So, yes, an outline, but minus any of the implementation details of the proof. I kind of think every book in the definition-theorem-proof style should really be definition-theorem-reason-proof.
The reason part being essentially a one or two line natural language summary of ‘why the proof works’ — something that is almost always possible and is enlightening and conducive to efficient memorisation, but that for some reason is very rarely written down explicitly.
I think a better word is "motivation" -- why we chose this option at this juncture instead of many other options. Yes, it's a "reason", but "reason" already means something else.
The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
Motivation is important too, but it’s not what I meant. A very simple example would be
Theorem: Every subspace Y of a second-countable topological space X is second-countable.
Reason: Intersecting each set in a basis for X with Y yields a basis for Y.
Proof: [formal symbolic stuff involving open sets and unions, and mentioning cardinality, etc.]
(I’m not claiming ‘reason’ is the best word for this — it probably isn’t. But it’s not the same thing as motivation.)
> The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
One could argue this is not the reason a result is true; it’s the reason we know it’s true. The fact that true statements follow from established truths by logical reasoning is more a property of the formal system (which hopefully is sound and consistent) than it is to do with the notion of truth itself.
It depends if you want to be able to prove new things by yourself or not. If you want to do it, then you definitely need to understand /recall all of the whys of every section of the proof. They are all there for a reason. If you don't, you just want the intuition of why the whole theorem is true.
You should definitely memorize most of the "basic" (and short) proofs in some field you are super interested in. The intermediate and advanced proofs, only the outline is sufficient.
>Step 3. Memorize the exact wording of the definition.
Huh. Any mathematicians who want share their own opinions and experiences about this?
This pretty much goes completely against my experience with other grad school level neuroscience/ML
You don't want to be so familiar with stuff as to make it second nature but NOT from memorization. That, at least an other areas, leads to surface level recognition
Does the author mean internalize and not memorize?
The author really does mean memorize. To engage with pure mathematics, you must know the definitions, since the definitions are the bedrock of the subject. If you don't know the axioms of a topology, how can you check for yourself whether something forms a topological space? Or without knowing the exact definition of continuous, how can you know whether a proof of continuity is correct? Without knowing the definitions, you can't really know mathematics.
To be clear, this does not mean memorizing all the theorems. Getting to know the theorems (and solving problems) is what helps you internalize the subject. Math is the art of what's certain, and knowing exactly what the objects of the subject are is necessary for that. Theorems are derived from the definitions, but definitions can't be derived.
In my experience with a math (undergrad and PhD), I realized I had to know definitions to feel competent at all. In my teaching, it's hard to convince students to actually memorize any definitions — so many times students carry around misconceptions (like that "linearly independent" just means that no vector is a scale multiple of any other vector), but if they just had it memorized, they might realize that the misconception doesn't hold up. Math is weird in that the definitions are actually the exact truth (by definition! tautologically so), so it does take some time to get used to the fact that they're essential.
> the definitions are actually the exact truth (by definition! tautologically so)
It’s easy to forget that non-math people find this — the idea that the definition is its own ‘model’ rather than an approximation of something more ‘real’ — somewhat hard to stomach. Outside of pure mathematics the idea is that mathematics is a tool for (usually lossy) modelling of reality, not a collection of already perfectly well-motivated objects to be studied in their own right.
When you are studying science and technology, and the math theorem doesn't match experiment, the theory is probably wrong (or incomplete, missing factors), so you can discard it or try to improve it.
When you are studying math, and the intuition doesn't match the theorem, the intuition is probably wrong (or incomplete).
Things get a bit messier once you're doing research mathematics — definitions don't just come from nothing, and a good definition is one that serves its theorems. Definitions can be "wrong" (they might be generalizable, they might have unexpected pathological examples, etc.), and it's the result of lots of hard work by lots of mathematicians throughout history that we have the definitions we enjoy the use of today.
But yeah, while studying math, I think it's similar to learning programming — don't blame the compiler for your mistakes, it's a well-tested piece of software.
Yes. It’s slightly disingenuous of me to suggest that any definition is as valid and ‘real’ as any other. Obviously, mathematicians care about some ideas more than others.
Yeah, during department teas you can hear mutters of "interesting" as ideas are exchanged and evaluated.
But, in my last comment I was just trying to temper my previous comment's claim about how important definitions are. At some point you get so used to a definition that even if you don't know a particular formulation word for word, you could still write a textbook on the subject because you know how the theory is supposed to go.
For what little it’s worth, the thing that finally made it click for me was a series of comments on HN that were discussing musical scales.
I don’t have any musical training, but I related it back to the practice and warm up sessions we had before we’d play an actual game in the sports I played as a kid.
Perhaps some explanation like that will get it to click with someone.
I also learned of the existence of soft question tags on Math Overflow and Math Stack Exchange that contained an incredible amount of guidance that I think was never possible in lectures. Sharing links to those websites in the syllabus may be helpful for the odd student that actually looks at the syllabus.
I'm teaching discrete math in January — I'll try the analogy, wish me luck!
As someone who's gone through the mathematical ringer, the analogy doesn't ring true to me, but it does sound pedagogically useful still (my students will be CS majors, so the math will be for training rather than an end). Even at the highest levels the definitions are of prime importance, though I suppose once you get to "stage 3" in Terry Tao's classification (see elsewhere in the thread) definitions can start to feel inevitable, since you know what the theory is about, and the definitions need to be what they are to support the theory.
Personal aside: In my own math research, something that's really slowed me down was feeling like I needed everything to feel inevitable. It always bugged me reading papers that gave definitions where I'm wondering "why this definition, why not something else", but the paper never really answers it. Now I'm wondering if my standards have just been too high, and incremental progress means being OK with unsatisfactory definitions... After all, it's what the authors managed to discover.
Yeah, sorry if I wasn't clear. I 100% agree with definitions, theorems, counterexamples, and proof techniques being incredibly important. Those are the "warm ups" or "scales" or things that need to be repeatedly drilled in my mind before trying to jump into the "game," which, to me, is solving problems.
The reasoning is that, similar to memorization of times tables, being able to recall a definition or a theorem and its context / assumptions "automatically" without needing to use your brain frees you to worry about higher order activities. Being able to apply a theorem over and over again ultimately builds mastery and internalization.
Counterintuitively, mathematicians like being "brain-off" as much as possible -- you want to be able to read a phrase like "closed convex subset of a Hilbert space" and effortlessly think to yourself "oh! there's a unique norm minimizer" -- if you have to piece that together from scratch every time, you're going to have a hard time -- reading papers and learning new fields becomes a dreadful slog, similar to how math in general becomes a slog for kids who don't memorize their times tables.
I think they mean memorize. Most students simply lack the mathematical experience to internalize a definition correctly without simply memorizing it. They will forget an "only if", or accidentally swap two quantifiers around, or conflate two variables that need to be kept separate, etc.
This is okay! They're students on the way to gain that experience. At some point you can and will go over to internalizing, instead. But as advice to students just starting out, memorizing is the way to go.
The author also said to internalise it, but the simple fact is that in a proof-y math test you can't just say "well it should converge if everything is well behaved", the edge cases matter and definitions set the boundaries.
PhD in math here with several published papers. And my recommendation is a metaprinciple: enjoy mathematics. Benjamin Finegold said similarly that the secret to chess is to enjoy every move. Personally, I had no trouble in mathematics, ever. And I think the reason for that is that I really enjoy just doing it, writing symbols down, learning about new theories, and even inventing my own.
Not everyone will enjoy mathematics at first sight. But I think at least 50% of that can be explained by the lack of obvious paths to enjoy mathematics. Obviously, most mathematics taught in high-school is not taught as it should be: a cool artistic logical pursuit that has all kinds of fun in it.
So my advice is to really find a mentor who already has found that path and let them show you how to enjoy math.
Believe me, I've tutored a lot of people, many of which initially disliked math and found it difficult. But after a few tutoring sessions, I could see a little sparkle in their eye that said, "hey, this might be cool".
So before you apply logic, studying, and other tedious "productivity" measures to your math learning, make sure you find a way to enjoy it first.
One of the biggest problem of maths education is that they are taught by people who dislike it. They think of maths as eating bitter medicine or training of a complex, rigid skill. The way maths is taught by them is clumsy and authoritarian, and this makes the students either passive or rebellious.
On a side note, recently the government of Manitoba in Canada removed requirement for maths teachers to take university maths courses. This is being pushed strongly by the education departments of university, which shows how much these maths teachers hate maths.
> On a side note, recently the government of Manitoba in Canada removed requirement for maths teachers to take university maths courses. This is being pushed strongly by the education departments of university, which shows how much these maths teachers hate maths.
That is messed up. Harsh. What is also messed up is that to become a math teacher, you have to go to teacher's college. That's one reason why I never became a teacher. I think I would do better than most at teaching (at least based on the comments I got in my student teacher reviews) but spending another two years at school is rather humiliating and costly.
I get what you're saying, but I think confidence is actually more important than enjoyment.
A lot of people confuse enjoying [x] with enjoying being good at [x]. This is why so many students switch subjects later on in life; when a field suddenly doesn't come naturally to them, they seek to play to their strengths elsewhere. Problems occur when they quit too early, and building confidence early on is important for stopping this.
In my experience, when you think you're bad at something, it's almost impossible to enjoy doing it, which makes preliminary mastery actually the first step to enjoyment and therefore downstream success.
Nah I disagree with that. Well, I do think confidence is useful BUT it only gets you so far. There are plenty of things I am "confident" in doing but don't really enjoy and after a while I just left them behind because they just don't provide a lot of enjoyment.
Any tips or anecdotes for us (well, me) about the fun things you'd do with students to encourage that joy?
I am not a PhD, but have done a fair bit of tutoring young people in maths. I feel similarly to you there, and am always on the lookout for new ways to foster that feeling of mathematics being fun and wonderful.
It can be hard. The feeling a lot of young people pick up - of maths being roughly akin to pointless abject suffering - is so strongly rooted in young people sometimes, and can be strongly connected to feelings of inadequacy and shame and so on.
Wrote that reply then clicked on your page to see if there were a blog or anything where you elaborate already on these topics - I've already read a couple of articles from your substack! Oops, hah, I did not realise I was writing to the person behind that.
I'd filed it away in my head as something to sink more time into when I've the emotional space to do so, but essentially I find it quite courageous to go against the grain in a world where it's perhaps harder and harder to do so. Everyone has a hot take, of course, but the hot takes are usually very much within the bounds of acceptable discourse.
Anyway, more power to you, in your endeavours.
Thank you very much.
Well, I think they key is I try and select topics that I find very fun, even they are a little off the beaten path. I know there's pressure to teach the curriculum, but honestly there are some cool things you can do with students if there's extra time that might get them hooked even if it's not in the curriculum, like:
1. Coloring maps (four color theorem) 2. Drawing curves and then looking for pathological ones (quadratic equations ...but more of an exploratory rather than methodological approach) 3. Infinite series (but just discuss some paradoxes people thought of centuries ago...) 4. Triangles on spheres
...if you don't make some of it extremely rigorous, there's a lot of basic facets of these things like the drawing aspect that can be quite fun. And TBH it makes sense to start there because mathematicians started out exploring these topics just with random calculations and doodles rather than rigorous proofs.
> Believe me, I've tutored a lot of people, many of which initially disliked math and found it difficult. But after a few tutoring sessions, I could see a little sparkle in their eye that said, "hey, this might be cool".
Exactly the same for me. Honestly, the satisfaction of seeing that "sparkle" in the eye of an initially unmotivated or discouraged student is probably among the most fulfilling moments of my professional career.
Great comment, and applies to any activity we do. Seems like in western culture we're told to not have enjoyment, to work hard and grind, and that any bit of enjoyment means you're not trying hard enough and lacking. Much better to have the attitude of allowing the happiness to come from the actions.
This post is quite rude and dismissive to people who enjoy math but struggle with it.
I've been told that the secret to exercise, woodworking, playing the piano, whatever is to "enjoy it." But, having or getting the skill to have some successes to "enjoy it" is difficult and not everyone has the ability to get to that level. Just sticking with something or worse, fooling oneself that more effort will lead to good results is annoying to hear if one doesn't have ability. We can't all succeed at everything.
Why? Can you elaborate what you find so offensive?
I don't think it's offensive, but it may be frustrating advice for people who cannot enjoy math at all.
That might be true but then I don't think people should pursue things they don't enjoy in the first place.
My secret sauce as an undergraduate for all my math courses was solving problems. Solve all the problems at the end of the chapters in the textbook. Find other textbooks in the library and solve all the problems at the end of their chapters.
In graduate school that was expanded: take every chapter of the textbook and rewrite it, filling in all the intermediate steps of every proof, those where the author writes "it follows that ..." or "from which it's obvious that ..."
> take every chapter of the textbook and rewrite it, filling in all the intermediate steps of every proof, those where the author writes "it follows that ..." or "from which it's obvious that ..."
I recently-ish had a read through some of my old fundamental/pure maths notes from Uni, including plenty of proofs. The damned things are littered with steps which were "obvious" to my smug self-satisfied 20 year old self but impenetrable to me reading without much context 20 years later.
Git.
I did this but faced enormous problems with transfer/recalling definitions/etc. what helped me improve several fold in approximately exponential value
1) Drill/spaced repetition basic definitions. The Cornell note taking method is convenient to do this while taking notes.
2) Keep a diary of thoughts, things you couldn't solve or did solve. Especially identifying problems, what works or doesn't, why something went wrong. Metacognitive thinking was really useful for transferring problems to solving new ones.
3) A study group involving a lot of us explaining to each other.
I’ll add: go to the TA study sessions. And try to answer questions in class. Even if you have nothing to ask, use that time to do homework, listen to other people’s questions.
Yes, you’ll learn more. Also, TAs will recognize you put in the effort so if you’re arguing for partial points or you’re really close to a cutoff grade they will be more likely to bump you up versus someone they’ve not seen or noticed all semester.
Brute force exposure to as many problems and solutions as possible is the same way AIs learn.
That strategy in my opinion is not optimal for humans.
What we need to do is develop math resources that can help students learn things analytically & conceptually.
Like how they learn biology.
It's not quite like that.
Maths content is the ne plus ultra of conceptual.
It's totally possible to slog through a chapter of a maths text and feel like we got it. But it turns out our 'understanding' was a facade. We can't apply the concepts in a new situation. Exposing ourselves to feedback via problem sets reveals this.
> Like how they learn biology.
How do they learn biology?
An lot of brute force memorizing details that everyone else would look up. Biology feeds to medical fields where you have to have details you rarely used memories because you might be doing surgery on someone who is slightly different from typical and have to figure out what that thing in the way is - can you safely move/remove it or not. (I'm not a doctor, real doctors can come up with plenty of much more likely real world examples of situations where you don't have time to look up some detail)
> Solve all the problems at the end of the chapters in the textbook
It really depends on the textbook, isn't it? I find it impossible to solve all the problems in CLRS, for example. Our professor assigned one of the problems about universal hashing, and it took me hours to get the key insights to find the correct proof. I can't imagine how one can solve all the problems given so many competing priorities, except for a few truly talented.
Did you do the few hundred problems before that build up the techniques and theorems?
> given so many competing priorities,
Undoubtedly, the best time to do this was when you were young. The second best time is now. Pick a book and work on a problem or 2 every day. It will likely take 6 months or so but you will learn the material. This is an incredible way to level up in a technical area.
Imagine how much knowledge is in CLRS.
The point is that you gradually get better at it. And by gradually I mean exponentially. If you start with a problem at then end of the book, it'll be very difficult.
I agree with this. In the end it is beneficial with repetition for learning anything, and I think subjects with calculations are partly a craft where you need to get it in your fingers. I teach a subject, it not maths but a (different) engineering subject and what I see is that the students don't buy text books any more (for several understandable reasons), but one of the things they miss out on is a lot of practice problems. This becomes evident in the types of errors they do at the exam, and an apparent lack of system for laying out their problem and solution in an understandable matter for themselves (and me).
More often than the "at the end of the chapter" math problems are not about calculations, but are essentialy mini-problems requiring mini proofs. It's not about 1-1 applying the knowledge that was in the chapter, but rather about creative thinking. Some problems can take hours or days to solve. There isn't much to "get in your fingers".
I find that in any creative fields it helps being really good at the fundamental skills so they don't get in the way of your creativity.
You need to get creativity in your fingers. You can get good at solving problems. As those who did much better than me in school proved.
My issue during my undergraduate days was that the textbook didn't come with solutions, and I really don't know how to unstuck myself when I got stuck (except asking the TA in the office hours).
When I was an undergraduate we had Schaum's Outline Series (do they still exist?). Every chapter came with a bunch of completely solved problems plus another bunch of unsolved problems with the answers. The optimal strategy was still to try and do the solved problems by yourself and then verifying your solution.
Ask your peers. As an ex-prof, one of the most prominent sources of missed potential I see in undergrads is not working with their peers. Like serious 2-6 hours sessions of collaborative problem solving.
You're right, but it depends on your peers. If you're the only one motivated to do extra work, your peers are unlikely to want to spend hours on verifying the solution. Guess it shows the importance of good peers.
Yep, I hated that. Spending an hour on a problem and get a solution you can't verify. It very much affected my motivation on starting on the problem knowing that I wouldn't know if I was right or wrong.
An undergraduate text like Pugh's Real Mathematical Analysis can have over 500 exercises. Trying to do them all on top of your regular workload seems excessive, unless maybe it's an area you really want to specialise in. I also tend to find certain kinds of exercises boring because they appear terribly unmotivated.
On the other hand, I second the suggestion to engage more deeply with the subject material itself: Modify assumptions and see what happens. Can certain proofs be simplified? Try to reconstruct proofs by only memorising certain key details. Try to draw a mental map of a subject and how the different theorems and definitions relate together. Try to implement some proofs in an automated theorem prover, if that's your thing.
500 exercises over 10 weeks is 50 a week, or 10 a day. This leaves you weekends free, and you get the final weeks (typically 5) of the semester left for all those projects that are due at the end of the semester but you can't start early. 500 a day is too much, but you should be able to do 50-100 problems a day in your study time. And others report that when you do this you get good at doing math and so it takes less time.
Sadly I didn't do that. I graduated and do okay, but I encourage everyone to do better than me. As I get close to retirement I need a few people who are still working to build things (and medical treatments) that makes my life better (and in turn take some of that money I saved up over the years for your own life)
Pugh's exercises aren't trivial calculations, they're the proof kind of exercise. If you manage to do 10 a day consistently and still have time for your other courses, congrats.
I also don't think that people who develop new medical treatments necessarily did all the exercises in a pure maths textbook. Being able to prove that continuous functions on a compact set are uniformly continuous probably won't help you fight cancer.
Also, it's very unlikely that Calculus will be the only subject you're taking. Other subjects require homework too.
Someone correct me if I'm misremembering, but I think Donald Knuth wrote somewhere that when he'd be assigned the odd-numbered problems, he'd always do them, and do the even-numbered problems as well.
This is not applicable to every book. Try doing this with Casella and Berger and see how difficult it becomes.
Some books can take many many months to finish off like this, and most courses only cover a small percentage of the book.
Solving problems (and trying to invent proofs) is indeed the way to enjoy math.
po-shen loh?
“The proof is left as an exercise …”
The first thing you have to get used to when moving from school to uni is being utterly lost and defeated.
At the end of high school, I could do everything. I finished my IB exams with huge amount of time to spare, the only thing holding me back was being able to write fast enough. It had been months since I saw a regular curriculum question that I didn't know how to do. Any marks I lost were just trivial errors.
When I got to university, there would be question sheets where I would look at the questions and wonder what it had to do with the lectures I had just been in. As in "I went to this lecture, and I'm supposed to use the information to answer these questions, but I don't even know what the questions mean".
The learning happens when you are doing this frustrating head-bashing.
You read, you read more, you fill a notebook with useless derivations, and eventually you things start to take shape. This could take the entire week's worth of time, just sitting there fumbling about.
The difference is that in uni, the amount of material is so vast you cannot explain it to someone in the time that you have. The students have to pick up some key ideas, and then fill in all the details themselves by pouring hours into it on their own.
> The first thing you have to get used to when moving from school to uni is being utterly lost and defeated.
Very well said, at the university where I studied, there was a pre-semester math repetition course. It was a week long and started with addition of natural numbers. After two days, everything I learned in 13 years of school had been repeated. That was a brutal resetting of expectations. But it made everybody clear, that this is not school anymore. That a different kind of work and focus would be necessary.
Related:
How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=26524876 - March 2021 (73 comments)
How to Study Mathematics (2017) - https://news.ycombinator.com/item?id=16392698 - Feb 2018 (148 comments)
Bonuses:
Ask HN: How to Study Mathematics? - https://news.ycombinator.com/item?id=23074249 - May 2020 (31 comments)
Ask HN: How to self-study mathematics from the undergrad through graduate level? - https://news.ycombinator.com/item?id=18939913 - Jan 2019 (227 comments)
Ask HN: How to self-learn math? - https://news.ycombinator.com/item?id=16562173 - March 2018 (211 comments)
Others?
Another bonus:
Ask HN: How to self-learn math? - https://news.ycombinator.com/item?id=16562173 - March 2018 (211 comments)
Thanks! Added above.
> The Germans have aptly called Sitzfleisch the ability to spend endless hours at a desk, doing gruesome work. Sitzfleisch is considered by mathematicians to be a better gauge of success than any of the attractive definitions of talent with which psychologists regale us from time to time. Stan Ulam, however, was able to get by without any Sitzfleisch whatsoever. After his bout with encephalitis, he came to lean on his unimpaired imagination for his ideas, and on the Sitzfleisch of others for technical support. The beauty of his insights and the promise of his proposals kept him amply supplied with young collaborators, willing to lend (and risking to waste) their time.
Taken from Gian-Carlo Rota in The Lost Cafe, a quote I found here http://www.romanpress.com/Rota/Rota.php
Article on Sitzfleisch: https://www.bbc.com/worklife/article/20180903-to-have-sitzfl...
Interestingly, this guide states that the intuitive understanding of maths is only suitable at the school level but not for the university.
In his recently published book "Mathematica: A Secret World of Intuition and Curiosity", David Bessis argues that the intuition is the "secret" of understanding maths at all levels.
Not sure what conclusion to draw from here, but my (rather dated) experience with university maths tells me that the intuition is a powerful tool in developing the understanding of the subject.
This seems like a contradiction but I don’t think it is. What it’s really saying is that experience is a precondition for intuition.
When a high school student looks at a high school math problem they’re drawing on all of their experience in K-12 math to get intuition for how to solve the problem. When they leave high school to study math in undergrad they struggle because their experience is no longer sufficient. They’re faced with a lot more abstract problems and the demands for rigour are much higher. The problems also tend to operate at higher levels on Bloom’s taxonomy [1] than high school math, something with which the average high school student would have little or no experience. It is this unfamiliar territory where intuition is hard to come by.
After gaining more experience (later undergrad and into grad school and beyond) the intuition starts to come back. But it’s fundamentally a different kind of intuition. In high school math it was often a visual/geometric intuition that teachers were trying to build. In higher math it’s an intuition for abstractions and for the tools you need to attack problems. This is really no different from a programmer looking at a problem and saying “I need a hash map and then this problem is trivial.”
[1] https://en.wikipedia.org/wiki/Bloom's_taxonomy
Possible harmonization of the two ideas: the intuition that we go into math at high school level can help serve us at that level of math. We have some idea of geometry-like objects and 2d-calculus like curves from our everyday life
At university level the objects become more abstract, so the intuition we use in normal daily life may no longer apply. New kinds of intuition may develop but it takes work, including lots of time spent with the formal processes and calculations along with reflection on that time, and the active creation of new metaphors to drive the intuition. For example, I still remember a professor using "Ice-9" (from _Cat's Cradle_) as a metaphor for how proving some local property of a holomorphic function on the complex plane made that property true for its global behavior
I think he's saying here that intuition is sufficient for high school math, but not sufficient for college. That's not to say that it isn't necessary, only that it isn't sufficient.
This is related to Terence Tao's notion of the stages of mathematical rigor.
As Tao puts it, the value of intuition becomes much higher in the post-rigorous stage once you have sufficiently developed your technical skills.
https://terrytao.wordpress.com/career-advice/theres-more-to-...
I read the book.
To me, what it says is "intuition can be honed and it is powerful, but hard to pass along to others". Just that.
Bessis actually mentions examples of how intuition and technique complement each other nicely.
That topic fascinates me. As a kid, a lot of topics felt intuitive, and college became a pit of darkness. Makes me wonder what in our brains makes something feel natural, obvious, with that feeling of playfulness and certainty .. while some times you're drowning in a blur.
In my case I realized I achieved a whole new level when I tried to do all of the proofs by myself. I used to read the whole subject a few times and after that, I would go one definition/proof at a time and first to recall the statement and then try to do the proof by myself. By doing that you indirectly achieve all of what the article says.
A nice thing I realised is that once I did that, almost all of the exercises that were complex before for me, turned out to be straightforward. It was like a cheat code where I almost did not need to do any exercises.
I used to teach at the uni at several levels, and every year I would ask if anyone tried to recall the proofs of the theorems at home. and no one did. They were always shocked when I told then they should do it.
>I used to teach at the uni at several levels, and every year I would ask if anyone tried to recall the proofs of the theorems at home. and no one did. They were always shocked when I told then they should do it.
I try to do this as well. If you combine this with understanding the definitions of the various units you more or less will have the textbook in your head, some assembly required.
This article is missing the "meta" in studying mathematics.
Creative introspection into how one learns begins to really pay off partway through college.
One's relationship to convention becomes as important as one's relationship to technique. Understanding the "whole" of something involves understanding the biases that shape the presentation you're seeing. You'll probably want to shed them.
This applies whether one wants to change math or just learn it. A passive stance, trying to do what others want, is a recipe for frustration.
I’ve been very pleasantly surprised by my recent experience, having signed up for MathAcademy.com after reading about it on HN.
Now in my 50’s, I wanted to relearn high school maths from 35 years ago and I scooted through their Foundations series (now half way through Foundations 3, rapidly accruing like 9000 xp in 9 weeks, 2 hours a day). Planning in 2025 to do 1-3 university level courses with them at a slower pace.
It’s suited my way of learning as an autodidact: enjoyable; measurable; adaptable level of hardness; no hitting of a “wall” or “unmet dependencies”; thorough explanation of problems I didn’t solve.
Perhaps my biggest realisation was that one can learn without needing to document many notes to revise/memorise, because experience and spaced repetition suffices. I’m taking a Xmas hiatus which will be the real test of baked learning.
I've been using Math Academy daily for over a year now and have been similarly impressed with how much I'm learning with it.
Congrats on your pace, 9k XP in 9 weeks is impressive!
> Do not let yourself fall behind.
That hit home. I'm afraid I was one of those lazy math undergrads who struggled with a few of the first year topics, didn't get help or put the hours in and never really recovered. I will maintain I think the teaching was very poor in places (lots of "just trust me" handwashing and "this is obvious so I'll leave it to you to complete" which for an 18 year old frankly sucks). A system that lets you get 30% in "analysis 1" and then just marches you straight into "analysis 2" next semester and expects you to just pull your socks up isn't much of a system to me. Honestly I'm afraid my time at university doing maths was miserable. I should have done something more applied like engineering or CS probably.
Someone once told me "If you like biology at school, do psychology at university. If you like chemistry, do biology. If you like physics, do chemistry. If you like maths, do physics and if you like philosophy, do maths". I should have listened.
I agree that the system is garbage. School ruins math -- but I just want to share that nothing stops you from revisiting math later as a fully actualized adult, without the bullshit time constraints and grade pressure. Math isn't well suited to time pressure -- in my view math is about spending lots of time with ideas and playing with them and taking risks and making dumb guesses, it's moonshot through and through. At its core is play, and time pressure doesn't serve anyone well.
Anyway, if you were interested in it, you can always revisit it, and even try taking a class again if you ever want to. There are lots of great books out there for self-learners, and lots of communities of folks learning together
> "If you like biology at school, do psychology at university. If you like chemistry, do biology. If you like physics, do chemistry. If you like maths, do physics and if you like philosophy, do maths"
This is good advice if the objective is to do well regarding grade results. If you want to get down to the bottom of things, to understand everything and to solve fundamental questions of science, you might well want to invert the advice:
If you liked biology, study chemistry, for the processes of life are (electro-)chemical processes.
If you liked chemistry, study physics, for the processes of molecules and atoms, their formation and reactions, are physical processes.
...
The way I heard this phrased back in the day was, "Biology is really chemistry, chemistry is really physics, physics is really math." Never heard psychology or philosophy added to the chain.
> If you like maths, do physics and if you like philosophy, do maths.
Seems like I made the right decision! (I did maths.)
> "A good way of seeing how a subject works is to examine the proof of a major result and see what previous results were used in it. Then trace these results back to earlier results used to prove them. Eventually you will work your way back to definitions"
I find the parallels between proofs and programs to be fascinating. We could write an analogous thing for programming:
"A good way of seeing how a sort of program works is to examine one of the popular programs/libraries and see what functions were used in it. Then look inside of those functions and see what functions are used inside of those. Eventually you will work your way back to the lower level primitives."
Sometimes it is weird to see a webpage from my university on here. Usually UH doesn't get a lot of attention for its STEM. At least from my anecdotal experience.
UH is famous at least in the high temperature superconductivity world.
Does someone have experiences with using o1-preview for mathematics? A while ago I tried to use GPT-4o and Claude Sonnet for certain algebraic questions related to probability theory. The models did help significantly (at least relative to my rather limited ability), though they also often produced wrong results and struggled to make progress on harder questions.
I appreciated the article for emphasising memorising definitions and statement of theorems... But not for proofs. For proofs, a general outline would be sufficient.
For proofs, I find it a good idea to memorise (or at least implicitly retain) the reason a result is true. So, yes, an outline, but minus any of the implementation details of the proof. I kind of think every book in the definition-theorem-proof style should really be definition-theorem-reason-proof.
The reason part being essentially a one or two line natural language summary of ‘why the proof works’ — something that is almost always possible and is enlightening and conducive to efficient memorisation, but that for some reason is very rarely written down explicitly.
I think a better word is "motivation" -- why we chose this option at this juncture instead of many other options. Yes, it's a "reason", but "reason" already means something else.
The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
Motivation is important too, but it’s not what I meant. A very simple example would be
Theorem: Every subspace Y of a second-countable topological space X is second-countable.
Reason: Intersecting each set in a basis for X with Y yields a basis for Y.
Proof: [formal symbolic stuff involving open sets and unions, and mentioning cardinality, etc.]
(I’m not claiming ‘reason’ is the best word for this — it probably isn’t. But it’s not the same thing as motivation.)
> The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.
One could argue this is not the reason a result is true; it’s the reason we know it’s true. The fact that true statements follow from established truths by logical reasoning is more a property of the formal system (which hopefully is sound and consistent) than it is to do with the notion of truth itself.
By Gödel's first incompleteness theorem there are true statements that cannot be proven (without adding new axioms).
> definition-theorem-reason-proof
Along the lines of your own argument: even better would be
reason0-definition-reason1-theorem-reason2-proof
It depends if you want to be able to prove new things by yourself or not. If you want to do it, then you definitely need to understand /recall all of the whys of every section of the proof. They are all there for a reason. If you don't, you just want the intuition of why the whole theorem is true.
You should definitely memorize most of the "basic" (and short) proofs in some field you are super interested in. The intermediate and advanced proofs, only the outline is sufficient.
>Step 3. Memorize the exact wording of the definition.
Huh. Any mathematicians who want share their own opinions and experiences about this?
This pretty much goes completely against my experience with other grad school level neuroscience/ML
You don't want to be so familiar with stuff as to make it second nature but NOT from memorization. That, at least an other areas, leads to surface level recognition
Does the author mean internalize and not memorize?
The author really does mean memorize. To engage with pure mathematics, you must know the definitions, since the definitions are the bedrock of the subject. If you don't know the axioms of a topology, how can you check for yourself whether something forms a topological space? Or without knowing the exact definition of continuous, how can you know whether a proof of continuity is correct? Without knowing the definitions, you can't really know mathematics.
To be clear, this does not mean memorizing all the theorems. Getting to know the theorems (and solving problems) is what helps you internalize the subject. Math is the art of what's certain, and knowing exactly what the objects of the subject are is necessary for that. Theorems are derived from the definitions, but definitions can't be derived.
In my experience with a math (undergrad and PhD), I realized I had to know definitions to feel competent at all. In my teaching, it's hard to convince students to actually memorize any definitions — so many times students carry around misconceptions (like that "linearly independent" just means that no vector is a scale multiple of any other vector), but if they just had it memorized, they might realize that the misconception doesn't hold up. Math is weird in that the definitions are actually the exact truth (by definition! tautologically so), so it does take some time to get used to the fact that they're essential.
> the definitions are actually the exact truth (by definition! tautologically so)
It’s easy to forget that non-math people find this — the idea that the definition is its own ‘model’ rather than an approximation of something more ‘real’ — somewhat hard to stomach. Outside of pure mathematics the idea is that mathematics is a tool for (usually lossy) modelling of reality, not a collection of already perfectly well-motivated objects to be studied in their own right.
More generally and symmetrically:
When you are studying science and technology, and the math theorem doesn't match experiment, the theory is probably wrong (or incomplete, missing factors), so you can discard it or try to improve it.
When you are studying math, and the intuition doesn't match the theorem, the intuition is probably wrong (or incomplete).
Things get a bit messier once you're doing research mathematics — definitions don't just come from nothing, and a good definition is one that serves its theorems. Definitions can be "wrong" (they might be generalizable, they might have unexpected pathological examples, etc.), and it's the result of lots of hard work by lots of mathematicians throughout history that we have the definitions we enjoy the use of today.
But yeah, while studying math, I think it's similar to learning programming — don't blame the compiler for your mistakes, it's a well-tested piece of software.
Yes. It’s slightly disingenuous of me to suggest that any definition is as valid and ‘real’ as any other. Obviously, mathematicians care about some ideas more than others.
Yeah, during department teas you can hear mutters of "interesting" as ideas are exchanged and evaluated.
But, in my last comment I was just trying to temper my previous comment's claim about how important definitions are. At some point you get so used to a definition that even if you don't know a particular formulation word for word, you could still write a textbook on the subject because you know how the theory is supposed to go.
For what little it’s worth, the thing that finally made it click for me was a series of comments on HN that were discussing musical scales.
I don’t have any musical training, but I related it back to the practice and warm up sessions we had before we’d play an actual game in the sports I played as a kid.
Perhaps some explanation like that will get it to click with someone.
I also learned of the existence of soft question tags on Math Overflow and Math Stack Exchange that contained an incredible amount of guidance that I think was never possible in lectures. Sharing links to those websites in the syllabus may be helpful for the odd student that actually looks at the syllabus.
I'm teaching discrete math in January — I'll try the analogy, wish me luck!
As someone who's gone through the mathematical ringer, the analogy doesn't ring true to me, but it does sound pedagogically useful still (my students will be CS majors, so the math will be for training rather than an end). Even at the highest levels the definitions are of prime importance, though I suppose once you get to "stage 3" in Terry Tao's classification (see elsewhere in the thread) definitions can start to feel inevitable, since you know what the theory is about, and the definitions need to be what they are to support the theory.
Personal aside: In my own math research, something that's really slowed me down was feeling like I needed everything to feel inevitable. It always bugged me reading papers that gave definitions where I'm wondering "why this definition, why not something else", but the paper never really answers it. Now I'm wondering if my standards have just been too high, and incremental progress means being OK with unsatisfactory definitions... After all, it's what the authors managed to discover.
Yeah, sorry if I wasn't clear. I 100% agree with definitions, theorems, counterexamples, and proof techniques being incredibly important. Those are the "warm ups" or "scales" or things that need to be repeatedly drilled in my mind before trying to jump into the "game," which, to me, is solving problems.
The reasoning is that, similar to memorization of times tables, being able to recall a definition or a theorem and its context / assumptions "automatically" without needing to use your brain frees you to worry about higher order activities. Being able to apply a theorem over and over again ultimately builds mastery and internalization.
Counterintuitively, mathematicians like being "brain-off" as much as possible -- you want to be able to read a phrase like "closed convex subset of a Hilbert space" and effortlessly think to yourself "oh! there's a unique norm minimizer" -- if you have to piece that together from scratch every time, you're going to have a hard time -- reading papers and learning new fields becomes a dreadful slog, similar to how math in general becomes a slog for kids who don't memorize their times tables.
I think they mean memorize. Most students simply lack the mathematical experience to internalize a definition correctly without simply memorizing it. They will forget an "only if", or accidentally swap two quantifiers around, or conflate two variables that need to be kept separate, etc.
This is okay! They're students on the way to gain that experience. At some point you can and will go over to internalizing, instead. But as advice to students just starting out, memorizing is the way to go.
The author also said to internalise it, but the simple fact is that in a proof-y math test you can't just say "well it should converge if everything is well behaved", the edge cases matter and definitions set the boundaries.
Maybe not the exact wording but the exact meaning...
Mathematical definitions are precise. If you miss one part of a definition, then you cannot actually understand what the definition actually means.
You have to internalize the meaning also, but you must know the definition precisely.
It is also useful to know all alternative definitions, if any exist (they often do).
The biggest obstacle for me was (and is) fear or failure when it comes to solving exercises.
I consider college math is all about abstraction.
It's about relationships. Abstraction is the tool for understanding those relationships.
That is why we have category theory and homological algebra.
https://en.wikipedia.org/wiki/Abstract_nonsense
"Young man, in mathematics you don't understand things. You just get used to them."
- John Von Neumann
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