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How to Study Mathematics (2017) (uh.edu)
661 points by kenny87 on Feb 16, 2018 | hide | past | favorite | 148 comments

By far the best explainer of mathematics I've seen anywhere is 3Blue1Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

These videos do an incredible job of illustrating how to intuitively arrive at an answer by composing many of the parts you need to build a proof for more complex topics.

In a similar vein, I've been enjoying PBS Infinite series. The original host recently stepped aside to finish her dissertation and the new hosts are doing great but still finding their footing a bit - I suggest looking at some of the older videos to get a good sense of the channel.


It always saddens me how mathematicians seem to look down on “intuition”. Maybe higher math as a full-time job is just hard work and stubborn precision but for me, an intuitive, visual look is probably getting me closer to understanding than any cold-hard-facts book does. Not to mention how much easier it is to appreciate the beauty of it.

You can take it from one of the most productive living mathematicians:


I recommend reading the entire post - it's not too long, and links to other great materials - but this is a good relevant quote:

> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.

Mathematians (such as myself) don’t look down on intuition. It is the only way to construct a map through a complex proof.

However what you may have picked up on is we disstain people who insist they have a brilliant idea for which someone should prove they are right. That’s like saying a car engine consists of cyclinders and piston; someone else just needs to do the work to put the pieces together.

> It always saddens me how mathematicians seem to look down on “intuition”.

They do this for good reason. I had to take, as part of my philosophy concentration, a bunch of mathematical logic classes. First Order and Second Order logic, as you might imagine, are pretty simple. The rules make sense in a very intuitive way. I'd get 100 on exams just because I'm good at programming. But when I had to study metalogic, model theory, Henkin Proofs, and Godel's Theorems, that intuition quite literally flies out the window. I guess my point is that most interesting stuff is rarely intuitive.

That is ludicrous. The intuition is just harder to find, harder to grasp. But it is still there because these theorems are logical consequences of the base axioms. And the base axioms are logical consequences of our desire for sensible assumptions.

This is provably untrue and all I need to do is think about how weird stuff gets when increasing dimensionality of "household" shapes like circles and squares. It literally goes against every intuitive fiber of my body.

If you think that coming up with theorems about, e.g. infinitary algebra, is intuitive, then I guess you're just a lot smarter than I am. Infinities always throw a wrench into our intuitions. Something being a "logical consequence" of something else is an incredibly simplistic way of looking at things, not to mention that I'm not exactly sure what that has to do with intuition anyway.

It /becomes/ intuitive with time, thought, and exposure, as you pick up more tools for understanding the problem space.

That 'non-intuitive' aspect of high-dimensional spheres having very small volume relative to a hypercube becomes a point that builds intuition for how those spheres behave, once you're familiar with it. And then you can start throwing out things that seem intuitively wrong, and following intuition towards new ideas.

You just described the opposite of intuition.

Intuition is a part of the mind that makes approximate models based on experience up to that point. The experiences without anything that would help model infinites would make them non-intuitive. Experiences that would help model infinites would make them at least more intuitive. Experience with rational investigation of them going through examples is actually one way to do that. There's been organizations like Marine Corps incorporating intuition into training for a long time where they set out to program it using realistic scenarios in drills that people do over and over until the intuitive processes pick up the patterns/model. It will happen as many people study that topic as well.

The trick is that you need good data to feed it that makes the patterns clear. You might have not gotten a lot of that on the topic you had no intuition of. On some topics, there hasn't been much of an attempt to do that since people expect it to be hard with everyone forced to do the methods that are non-intuitive. Lots of traditions are like that. Then, there's the possibility something can't be intuitively handled at all. I don't know if those exist but they might. That many activities of people doing math seem to get an intuitive boost during their day-to-day work makes me lean towards intuitive possibilities for about everything that can be approximated (esp with heuristics).

A chess master can intuitively tell when mate is three moves away. They weren’t born with that intuition, though. They gained it over time.

You aren't born with all of your intuitions, you learn most of them. This includes mathematical intuitions.

At this point a mathematician might step in and say that you all need to define your terms. You're basing your arguments on two subtly different definitions of the word "intuition."

The differences aren't subtle. I'm using the run-of-the-mill definition you might find in a dictionary[1], what other definition could you possibly be using?

[1] https://www.merriam-webster.com/dictionary/intuition

From that I would choose 2.c, with the emphasis on evident

> the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference

There is such a thing as developing your intuition.

I think your meanings are not qualitatively different.

It depends how you define intuition. Maybe we could define intuition as your personal "model" for making guesses on how things work.

Let's claim that writing a proof is similar to writing a program. You start from a state, your premise, and have your assumptions, input, which you process through a series of steps, maybe taking cases in between (branching) and return an output (hopefully True).

Then considering this correspondence, it will be fine to say that you, as a programmer, can guess the output/behavior/semantics of any program without running it. Similarly, a mathematician would be able to reach the conclusion that the theorem holds, without going through all the details of the proof.

Sure, in most cases you can, and as you get more and more experience and knowledge as a programmer you can say you improve your guesses. But this ends up backfiring with a good probability, when our model skipped a step, e.g. the interaction of two concurrent data structures.

Thus I don't think it is right to say "as you are able to know all axioms and rules of inference, you should be able to achieve perfect intuition". You can know the fundamentals of each layer on your stack and you would find it insane if one asked you to guess the output of any non-trivial terminating program.

Note also that the way of inference (logical consequences) can and does change (e.g. ZF vs ZFC).

() I may be misusing the word model here -- it is usually defined (at least to things I work with) as the interpretation of a theory that satisfies all theorems (inferences). () constructive proofs yes. No flame wars please :)

Working mathematician does derive his ideas from axioms, it is ridiculous perception of mathematical process. And axioms are not logical consequences of sensible assumptions, it is try and fail process, long historical process. Intuition in mathematics has nothing to do with common sense intuition. Respectfully.

In abstract mathematics, one can choose any axioms one wants. To derive all kinds of consequences/theorems.

But respectfully, ZFC came about because of logical paradoxes that couldn't be accepted as consistent.

You can create a new number system and derive all kinds of consequences but the truth is, most mathematicians care more entirely about prime number theory on the naturals that are entirely based on counting.

Most of modern math is based on the natural numbers. You can't remove all intuition. The thread that holds our love for math is also the same one that tells us we are exploring consequences that tell us a dearth about our universe.

You'll find that no such thing is true. Intuition is something you develop over time in mathematics, and mathematicians will often use this intuition to have certain "feelings" about problems e.g. a feeling that a certain conjecture must be true.

There's nothing wrong with intuition, but intuition does not write proofs. It can help you write proofs, but you need more than that.

To pick an example:

Intuition is great for understanding the intermediate value theorem. If the temperature was 40 degrees at 8 am, and is 60 degrees at noon, it must have been 50 degrees somewhere in between.

Why, though? How can you guarantee that there will always have to be a time where it was 50 degrees[1]?

[1] provided that temperature is real-valued and temperature change is continuous, the debate of which I'll leave to the science people

Am I correct in understanding that it's not that intuition is invalid, but that it's insufficient for communicating a proof to another mathematician.

What I believe to be true is that a solid intuitive proof will be sound, but not necessarily transcribable until formalized. Formalization of the proof is essentially the matter of making a proof communicable.

Putting it one more way: purely-intuitive proofs would be just fine if one could share one's mental state with another.

What do you think?

The first senctence:

> Am I correct in understanding that it's not that intuition is invalid, but that it's insufficient for communicating a proof to another mathematician.

is close to correct. It's a very useful tool, but it can sometimes lead you into incorrect assumptions.

However, the rest of it incorrect. Math, and in particular, probability and statistics, are full of things that seem intuitive and easy at first glance, but are actually a bit more nuanced.

Take for example, the Monty Hall Problem. It's probably familiar to most here, but essentially you have three doors. Behind one, is a car or some other desirable object, and behind the other two are goats or something undesirable. Select a door, and you get whatever is behind that door. What's the probability of getting the car in this scenario?

Easy enough right? It's just 1/3. However, if Monty Hall opens one of the doors after you select a door, he always reveals a goat, and he offers to let you switch your door now, should you switch or keep your door?

What's the probability that you get a car if you switch? Is it 1/2 or 1/3? What's the probability that you get a car if you stay? 1/3?

Well, the somewhat surprising answer is that you have a 2/3 probability of winning the car if you switch, and a 1/3 probability of winning the car if you stay.

Often things will break down in the limits as well, so if you try to apply your finite dimensional intuition to something that corresponds to an empty object or an infinite number of objects, you'll find that you're often wrong.

Not who you're asking, but I think that is incorrect. Intuitive explanations are how mathematicians usually communicate proofs to each other in person, when discussing mathematics. The receiving mathematician is convinced of the proof's validity if their own intuition and experience tells them that the nitty-gritty details can be formalised correctly.

>purely-intuitive proofs would be just fine if one could share one's mental state with another.

No, I have had purely intuitive proofs myself that turned out to be wrong.

Intuition is not proof. We all have many intuitive ideas that are not actually true!

what the hell kind of mathematicians have you been talking to, if they look down on intuition?

> It always saddens me how mathematicians seem to look down on “intuition”.


3Blue1Brown makes great videos and offers good explanations. However, particularly with his latest Fourier Transform video, he not only gives the incorrect explanation, he makes it unnecessarily and overwhelming complicated. It is still better than the linked circles approach, but the Fourier transform can be explained more accurately in around two minutes and in three if you were to include the the definition of frequency.

3Blue1Brown is better than most but far from best.

What errors did he make and could you link s better explanation?

He has a convoluted (no pun intended) center of mass explanation that the equation does not describe, admits that this is incorrect, and then goes onto another incorrect explanation of a strange ellipse where there is another center of mass.

I've never been able to find a good explanation online, only in some older textbooks. I've begun to fear that there are few people who are able to read equations while the others interpret the explanations of those people, where then those same old explanations gradually mutate as they propagate without reference to their source. This Stanford lecture is the best I could find: https://www.youtube.com/watch?v=1rqJl7Rs6ps

If you could name those older textbooks I'd be very grateful...

"Harmonic analysis in phase space" by Folland who illustrates the Fourier Transform with physical applications.

"The Analytic Theory of Heat" Joseph Fourier who explains the realization at length.


What I still don't understand is why there isn't a Wikipedia for explorables.

3b1b's videos are excellent, but if they were interactive, open-source and in a similar model to Wikipedia, people could contribute and improve upon content that is already great.

There are great examples of what is possible with explorables, see Bret Victor, D3.js, distill.pub, observablehq etc. Just no encyclopedic model yet.

> "What I still don't understand is why there isn't a Wikipedia for explorables."

write one and maybe include that fact as a resume / interview item :)

All his visualizations are open source.

I nice companion project for it would be a tool for creating these 'explorable explanations'. Anyone know if any good attempts have been made at that yet?

To add to your list, I recently saw an HN submission where a PhD student made a markdown-derivative with some explorable features, not sure what it was called though.

The closest I know of are the widgets in Khan Academy

Could you please specify which part of his explanation is incorrect?

Make the video please, I’d love to get your take.

Who's the best?

Also the book Burn Math Class.

That book got me back into math. Using functional equations to teach concepts is very powerful. You get to say I want these properties and derive things like the exponential function.

Something I would add :

Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today?

In my mathematical education I noticed that I had a more pleasant time and felt more motivated to learn the material when the teacher gave us this kind of background story. Since most teachers don't do so today, the student typically needs to get on the Internet to do their own research.

This happened almost through all my formal education. The knowledge was just presented, as something that was obvious. There was very little discussion of motivation, or what problems was person/people trying to solve. I understand that there is so much knowledge to cover during high school and college, but IMHO, this causes students to develop mindset, that you either see solution instantly, or you are just not smart enough to crack the problem.

I think Bourbaki are partly to blame for this state of affairs. Vladimir Arnold had a lot to say on the subject: he was a big fan of keeping the intuition and motivation for development in the subject clear.

At times he went a bit overboard, but he makes some valid points in this lecture: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

I don't know if I can agree with that. I bought every book Serge Lang produced. He was a member of Bourbaki. His Linear Algebra book is far better at providing intuition and motivations than the rest. Yet it is also rigorous. Lang is #1 for Mathematics books in my world.

Fair - I don't doubt that there are some great educators in the Bourbaki school. But as Arnold mentions in his lecture I do think mathematics lost something when it embraced formalism so fully.

For me, the definition "A group is a set of transformations on an object such that..." is so much more enlightening than "A group is a set G together with a binary operation * such that..."

Only yesterday I was trying to learn about differential forms. Most of the notes I found online introduce the wedge product in a deeply unhelpful way, by listing some axioms that it satisfies and deducing results from the axioms. It takes hours of work to understand why those specific axioms were chosen. For me - and maybe I'm wrong here - that's Bourbaki's formalist approach in a nutshell.

It was only when I found Terry Tao's notes [0] and Dan Piponi's notes [1] that I could actually see the use of differential forms. It's an unfortunate state of affairs for the discipline that in order to learn about X, you have to google "X intuition", since it's not given to you as a matter of course.

[0] https://terrytao.wordpress.com/2007/12/25/pcm-article-differ... [1] https://github.com/dpiponi/forms/raw/master/forms.pdf

I totally agree. A lot of times you have to search for intuition rather than it being provided. Rote memorization is useless because mathematical research requires the intuition to give your mind the right direction to go in.

Got to love a piece that begins "Mathematics is a part of physics" !

i realize the entire speech is a troll, but it successfully made me very angry. i'm not a physicist, and have no interest in physics, yet I do need mathematics.

so it's always been irritating to me how much introductory or "applied" texts bias themselves towards physics and engineering. like how a lot of multivariate calculus/introductory real analysis texts pretty much limit themselves exclusively to three dimensions and teach notation to match.

it even plagues more advanced books; i'm struggling through Villani's text on optimal transport and nearly every example is from physics even though the theory was developed for economics and has extremely broad applications across many disciplines. half of the "motivating examples" i can't even make sense of what they're talking about.

as much as I struggle with the hyper-abstract bourbaki-style stuff, at least I have a hope of understanding it without needing to dip into an irrelevant discipline where i have no background or historical context.

As a former physicist it also made me angry that they used geometry, when teaching us about multi-dimensional algebra. Professor would tell us that vectors are orthogonal to each other, but not telling us what it actually means is, that changing one of them doesn't affect other. I mean, yes, in hindsight, this information was there, however, when you are just starting to learn something, you have a problem filtering, what is really important and what not.

And for example, a real revelation with regard to infinity came, when I read (or heard somewhere, it was long time ago) that infinity can sometimes be few millimeters or even micrometers. Up until then I always imagined some really large number, but at that point I realized, you need to put problem before you need to put problem before you into perspective.

...and in Austria the education was almost the exact other way round - the historic context and the motivations and problems causing the search for a solution was given great importance.

I'm not sure it really takes much time or effort to convey enough of the background: I still remember my high school math teacher telling the class, in passing, that Galois was killed in a duel at age 20.

30+ years later that remark is still having an effect in that when I took my kids to the top of the tour montparnasse I made them look down at the cemetery to point out that Galois is down there (somewhere).

> I still remember my high school math teacher telling the class, in passing, that Galois was killed in a duel at age 20.

I think that's actually an example of the problem. Galois's politics really had nothing to do with the problem he was trying to solve. Historical features are thrown in as random info with no point. The point of making mathematicians real people is so that students can say "Oh! That's how they came up with that problem! That's why they solved it this way! I could have done that!"

Rather instead, we further mythologize the individuals. You don't need to join radical political organizations and get yourself killed in your 20s to be a great mathematician.

This is a really good point I think! Up until the last few months, I've avoided doing any math with my brain that I'm not required to do (so, no brain math since college). Just recently I've been picking up some books about the history and philosophy of math like Fermat's Enigma and Everything and More (I'm a sucker for DFW, so this was a no-brainer when math was back on my radar).

Now every evening I find myself saying out loud, "why the hell didn't the math teachers explain it like this." For example, we all learn about the Pythagorean Theorem, but why didn't we learn about the Pythagorean Brotherhood[1] a little bit? And feel just a little bit of the excitement that Pythaogoras did when he found an exposed wire of the universe and figured out how to work it.

Fun fact: Pythagoras had to pay his first pupil to listen to him. No one gave a shit, probably not too different than how kids don't give a shit today. But the kids we're teaching don't even get paid, they're told their true reward is something intangible or some vague math-fu. All this while they have a black hole for their mind in their pocket where at least the fall is comfy.

[1]: https://en.wikipedia.org/wiki/Pythagoreanism

I am surprised you would say that about David Foster Wallace. I went to school with him from kindergarten until 6th grade and 10th through 12th grade, in Urbana, Illinois. Math was never his strong point. However he wrote manic funny stories that would make you laugh so hard you would start to cry. That is why you have to limit yourself to 20 Pages a day at most.

Well I think the reason I like his math book is because it's his typical style on a different topic, so it's a more relaxed read and not something like a hard math book. The first words of the book are something like "Unfortunately this is an introduction you actually have to read" which if you've read much DFW you just kind of settle in right away.

He does mention in the book that the only math classes he had any success with were AP classes. I think it's dedicated to his teacher. But really it makes no attempt to disguise that it's mostly the math angle of his uneasiness with the world and yep I agree with you on the 20 pages a day.

> Find out where the mathematics you're learning comes from : who first developed it? what problems where they trying to solve? why were they trying to solve those problems? what problems does it solve for us today?

For some of us, this is incredibly useful advice re: math study habits. I also figured out that once I had the context and history, it was much easier for me to learn and apply.

The 'Why' lends to the 'How'. I agree. I usually like to know why even in things that are not math before attempting to apply or understand them.

I had such a problem with this. At my college, the professor I had for my entire calc sequence once said, in response to a question about applications, "My degree is in pure math. I don't care how it's applied."

Thankfully(?) I was taking physics at the same time, so I did get to see how double and triple integrals and differential equations could be applied. But it frustrated me to no end that the person who was driving my math education cared so little about helping us understand it.

That's a poor professor of calculus. They should have been teaching real analysis instead. (In pure math, "Real" means "divorced from reality") Maybe a fine mathematician.

I think the real means real numbers

Yes, exactly. Real numbers aren't real.

Im not sure that real in the context of math means anything other than the real numbers or (at a stretch) existence. I agree with you. And no one talks about this but a priori i dont see why even the natural numbers have anything to do with reality either. "Counting" some objects seemingly would require a notion of the object and some way to determine whether each part of reality was or was not the object which already seems fairly abstract and removed from the world. I think certain parts of math just seem more intuitive to certain people

Annoyingly they've also laid claim to 'proper' and 'normal'.

Do you know any books or websites that provide this information? I also feel more inspired and motivated when I know what problem they were trying to solve with that particular piece of mathematics.

Memorization is so underrated. Memorizing the fundamentals and having them available for instant recall is hugely valuable, especially when trying to grok a new concept.

I generally buck the standard advice and memorize first, before trying to understand. Understanding is much easier for me if I can easily hold everything in my working memory.

I wouldn't personally find advice like this useful for me. I suspect it varies with "personal style". I grasped mathematical concept easily from an early age and I never made an effort to memorize things.

If anything, I saw those coming from less-advanced math founder on advanced math (arithmetic to algebra, algebra to calculus, calculus to advanced subjects) because they attempted to deal with the subject based on memorization rather than grasping the basic point.

Generally, I did wind-up committing a lot of content to memory but it was and is much easier, even "effortless", when I knew the reason for each part of the content.

Of course, language is slippery and we might be talking about exactly the same thing. So I wouldn't give blanket advise - each person has to find their own learning style (or styles, if it varies by discipline).

Rote Memorization is awful. Memorizing by practicing so much that the details become "muscle memory" is the foundation of progress in learning.

A basketball player has memorized how to shoot a free throw, not by cramming by rote, but by practicing.

The best basketball players start by spending years and years just playing the game all the time, as recreation.

The equivalent for mathematics is solving problems that you find personally relevant and interesting and which are at the limits of your current abilities. NOT just doing piles of rote calculation. Often solving a hard problem requires a big pile of intermediate calculations, but those a tool in service to some meaningful purpose, not an end in themselves.

Forcing students to work through thousands and thousands of trivial calculation exercises is horrible pedagogy (and a very inefficient use of time).

It’s like doing nothing but shooting drills and getting quizzed on the NBA rule book for years before ever trying to play a game.

>> The best basketball players start by spending years and years just playing the game all the time, as recreation.

No, they start by joining a club where they get professional training, from a very young age. Just playing the game all the time won't make you any better at it. There are millions of children that spent significant part of their childhood playing basketball for fun, and never went past amateur level.

This is just a conclusion from Ericsson famous paper[1]: you need DELIBERATE practice, not just recreational playing.

Playing for fun will just develop bad habits, which you then will have to unlearn, so recreational playing might actually be an obstacle to learning.

[1] http://www.nytimes.com/images/blogs/freakonomics/pdf/Deliber...

As you say, the key to skill improvement is playing deliberately and getting meaningful feedback in a tight loop, ideally at the edge of your abilities.

But there are many aspects to any complicated skill, including tiny subskills, combination of various subskills, and at a higher-level meta skills like knowing when to change strategies, etc.

The problem with only sport games as a mechanism for feedback is that (a) there’s not necessarily a good way to correct and try again right away if you want to practice one particular thing, (b) situations that arise are not homogeneous enough to compare feedback across trials, (c) there are periods of waiting in between direct practice, (etc.?).

As a result, people who are training to be basketball players spend only like half their time playing realistic games, and the rest of their time on a variety of drills and workouts including general aerobic and strength training, practice running sideways and jumping, shooting / passing / dribbling drills, practice setting screens / rebounding / driving / inbounding the ball, larger-scale coordination drills, and so on. These let the players focus their conscious attention at one level at a time, and repeat a situation with analytical feedback in between.

But those who improve fastest are the ones who figure out what they need to work on (or have a mentor/coach to figure out what they need to work on) and then get lots of feedback about that particular area, in an intentional way. This can even be done within the context of a realistic full game by choosing what to direct attention/focus to.

But you’re missing my basic point, which is that all the skills in basketball are situated within a context. It’s obvious how they fit into playing a game. Students start with the high-level context (by watching / playing full games), and then break it down into small pieces they can work on separately or jointly, and then frequently practice putting those pieces fully back together in the context of the game. I doubt you’ll find anyone seriously training to play basketball who doesn’t play in a semi-realistic game-like situation at least once per week.

In classroom education, often the high-level context is completely missing, and students end up memorizing / drilling on a pile of atomized facts and trivial very low level skills without ever practicing combining those in larger settings, without ever getting a very fast or effective feedback loop going, and without ever having any motivation for going through the practice.

In the case of mathematics education, someone being taught 1-on-1 by a skilled tutor can move I’d say about 4–10 times faster than a student only watching lectures, doing completely independent book reading / trivial homework exercises / quizzes, and then getting back graded homework/quizzes a week afterward. That student can be assigned dramatically harder problems, can be helped along by limited (but well placed) hints, and can be given direct feedback/corrections at every level (from basic arithmetic computations through higher-level strategy to meta skills like strategy selection/evaluation). This is something that can’t be done in basketball because basketball is a group activity; you can’t watch a single player go through the game, and pause/backtrack everyone to the beginning of a situation whenever that player makes a mistake or suboptimal play.

I grasped mathematical concept easily from an early age and I never made an effort to memorize things.

That was sometimes a problem for me, if I picked up something quickly I often found it harder to retain, whereas if I had to work hard to grasp a concept then I would remember it better.

I think people tend to conflate "memorization" with "cramming". You're right that memorization is underrated. Are you familiar with the 'method of loci' (i.e. memory palace). It's an amazing technique.

I did that approach, too... up until my real analysis class. Then we'd have something like 50 theorems [edit: per test], and I couldn't remember them all. But I noticed that two or three theorems were used to prove the other 47, and so I memorized those. They gave me enough to survive the tests.

I agree, but I still haven’t found a way to make that process more enjoyable or less fatiguing.

I found that grinding through the problems, over and over again, while using the textbook as a reference, resulted in the facts becoming committed to memory. I think it worked in two ways. First was the obvious repetition. Second, it created "slots" in my brain that were receptive to related material. There may have been an emotional factor as well: Feeling "on top" of the material made it easier to learn.

Proper mental indexing is even more so. You can not memorize everything, but you can certainly index almost everything.

That is why we can have databases that do not fit in 'memory'. Do you see the analogy here?

Luckily we have brains that expose this interface quite readily with intuition. Learn the logic, and the partial indexes, the foreign keys, etc..they all follow. Reason before RAM. It is often much faster to derive data than to store it, and look it up. This applies to CPUs and to our brains. Our brains have instruction cache, and often instructions are better use of memory than their final executed output.

I always found 'memorization' incidental- after step 1 and 2 in What should you do with a definition?, step 3 was already done.

Some books on proof, theorems/axioms, set theory, epsilon/delta/continuity/limits/differentiability, natural/rationals/reals/countability etc before heading into your first proof based LA or analysis sequence:

- Kevin Houston "How to Think Like a Mathematician"

- Keith Devlin "Intro Mathematical Thinking"

- "How to Study as a Mathematics Major" Lara Alcock (for some reason, she/Oxford Press has 2 books with seemingly identical content under Math Major and Math Degree titles)

Even more useful as a first step in my opinion is to think about problem solving per se. E.g. see:

Mason, Burton, & Stacey, Thinking Mathematically

Pólya, How to Solve It; Mathematics and Plausible Reasoning (2 vols); Mathematical Discovery (2 vols)

Schoenfeld, Mathematical Problem Solving

Larson, Problem-Solving Through Problems

My favorite is Burn Math Class: And Reinvent Mathematics for Yourself

Seems well liked but whats is up with the cover? Most people who are into mathematics aren't zealous rebels bent on burning the world. I feel like the marketing director for the book screwed up big time. I won't judge the book by its cover though.


I actually don't remember the cover. The title is provocative but adequate. You could rephrase it as "Could you reinvent modern mathematics if it disappeared?" - but then probably less people would buy it.

Huh? where did you get "zealous rebels bent on burning the world" ?

I bailed on high school math, thinking I'm math dumb.

In my late 20s I decided to try again, but jumped straight into calculus. And at first regretted that decision. However, I got lucky by stumbling upon this book:


It "reads" like a book, with the ideas given context. I had an "ok" connection with Algebra, and the book explained the rest well enough for me.

In school, the textbooks were loaded with symbols, but not enough description -- I guess they relied on bored teachers making minimum wage to do that part. I went to a school with poor academic showings (but connections to state superintendent of ed got them a grant for football facilities).

Coincidentally, this book goes well with the technique described here:


What was the name of that 30-something woman who had never studied mathematics before, but then finished a degree in quantum physics in two years? Her name has been posted on HN at least two times before. I guess what she did is exactly how you should study mathematics if you want to become got at it.

Think you're talking about Susan Flower and two of her articles:

If Susan Can Learn Physics, So Can You - https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgli...


So You Want To Learn Physics... - https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-l...

On the one hand, her account is very inspirational to people who might think that learning physics is out of reach or they're too old to do it. But on the other hand, her case is very out of the ordinary -- I think a lot of us took longer than 1.5 years to get through undergraduate physics.

According to Susan's account, it took her 1.5 years to go from no college-level math to a full college-level physics curriculum. At the school I studied at, the Calc series (I, II, and III) takes 1.5 years. The intro physics (I and II) takes a full year, and the upper physics take at least a year (Mechanics I & II, E&M I&II, Relativity I&II, Quantum I&II, Lab I&II). (And you had to take intro physics before higher-level physics, and most courses had dependencies.)

That's approximately 45 credit hours. If you took 18 credits a semester, that would take at least 2.5 years. And those are all tough courses -- no 3 credit gen ed "filler" courses. Even at 15 credits a semester, that's still 3 years.

So, to get through all of those courses in 1.5 years would be like learning about 30 credit hours / semester. And add in the physics electives, like thermo, waves, statistical mechanics, quantum field theory, additional math courses, etc... that's a lot on top.

I guess my point is that if Susan's account is discouraging, note that a lot of people take far more than 1.5 years to get through undergraduate physics.

Sadly (given her immense passion for physics), she ended up in software (doing SRE).

Hey I am in SRE. At one point last year there were 7 ex professors sitting within 15 feet of my desk including me (and I did a PhD in theoretical computer science at the same school as this math professor whose article we are reading). I don't think it is an underrated career.

Barbara Oakley?

Maybe not who your thinking of, but similar story.

See: https://barbaraoakley.com/books/a-mind-for-numbers/

She also has a great Coursera class called learning how to learn

I left math after college for software engineering, but reading A Mathematicians Lament recently re-kindled my love for it. It is tragic how intuition, technique and mental models are left out of modern mathematics education and writing.

It occurred to me that while I learned in college how to show, using Galois theory, that quintic equations are not generally solvable by radicals, I had no idea how Galois theory really relates to the process of finding roots. So I went back and derived the quadratic formula, the cubic formula, and sketched the quartic formula to see how the process used the ideas formalized by Galois theory and where it breaks down. I've tried to write the result up in a motivated and understandable way, instead of like a math textbook: https://alexcbecker.net/mathematics.html#the-quadratic-equat...

I studied a lot of math, pure and applied, taught it, applied it, published research in it, etc. so developed some ideas relevant to the OP.


> To the mathematician this material, together with examples showing why the definitions chosen are the correct ones and how the theorems can be put to practical use, is the essence of mathematics.

Is good, but more is needed.

(1) Plan to go over the material more than once. The early passes are just to get a general idea what is going on.

In such passes, for the proofs, they are usually the near the end of what to study and not the first.

(2) When get to the proofs, for each proof and each of the hypotheses (givens, assumptions), try to see where the proof uses the hypothesis.

Next, try to see what are the more important earlier theorems used in the proof. So, sure, in this way might begin to see some of how one result leads to or depends on another and have something of a web, acyclic directed graph, of results.

And try to see what are the core, clever ideas used in the proof.

(3) For still more if you have time, and likely you will not, can use the P. Halmos advice, roughly,

"Consider changes in the hypotheses and conclusions that make the theorem false or still true."

(4) But, note that to solve exercises or apply or extend the theory, need some ideas. So, where do such ideas come from? In my experience, heavily the ideas come from intuitive views of the subject.

So, my best suggestion is to try to develop some intuitive ideas about the material. Definitely be willing to draw pictures, maybe on paper, maybe only in your head.

In the end, a solution or proof does not depend on intuitive ideas, but finding a solutions or proof can make use of a lot in intuitive ideas.

For research, most of the above applies, but IMHO there are more techniques needed.

The Franklin method from yesterday still applies.


Also a lot of books these days have wolfram mathematica code and it works surprisingly well even for some more abstract parts of math, To get a good intuition.

The greatest book review of all time, sort of similar approach: https://www.amazon.com/review/R23MC2PCAJYHCB

Put the book down when you get to the example or proof of a theorem, keep all the theorems and examples before it in your head, and see if you can reason it out yourself. Can't promise that your head won't hurt afterwards

Those prices seem comical. Rudin passed nearly 8 years ago. Who is this benefiting? America is off its rocker.

Google the title + PDF

This is more "How to study pure mathematics", when the aim is to understand how the theory is build, so that you learn to contruibute to the theory by discovering and rigorously proving your own theorems.

I don't think I've ever seen a "How to study applied mathematics". How do applied mathematicians, physicicst and engineers (who apply mathematics to real world problems) study mathematics, when they use it as a tool? How much or little emphasis do they give to proofs and theorems?

Some personal thoughts: the key for me at least, is a bit like the hammer-nail mentality. An software engineer for example has a (small) mathematical toolkit (a hammer), sometimes forcing your problem at hand into the shape of nail is sufficient to solve it.

A dump example, consider the problem of detect if two date ranges overlap (for a calendar app perhaps). By recognizing a date range, DateRange(a, b) as an interval on the real line: [a, b] or a set {x | a <= x <= b}, the problem becomes find if two sets' intersection is empty. Set {x | a <= x <=b } and {x | c <= x <= d}'s intersection is {x | (x >= a and x >= c) and (x <= b and x<=d)} which simplifies to {x | max(a, c) <= x <= min(b, d) }. Therefore, the set is empty iff max(a, c) > min(b, d).

Translating the problem into mathematical structures here (intervals), and manipulate them with math tools (inequalities) would count as applied math I guess. By having a repository of math knowledge and an eager attitude to look for opportunities to apply them, one can find success in solving practical problems. In this context, rigor and formal proofs are less important and math intuition is paramount.

An real world example I like to motivate myself with is Richard Feynman and connection machine[1], a story about Feynman using partial differential equations to model on boolean circuits. I do believe even with a limited set of mathematical tools, by trying them on every problem you encounter, you can still get good results. Something like, if one squint hard enough, you can find a monad anywhere :)

[1] http://longnow.org/essays/richard-feynman-connection-machine...

All mathematics gets applied sooner or later, no?

Maybe, but “applied mathematics” is conventionally more similar to “how to apply mathematics” than it is to “mathematics that is applied”. Studying how to apply mathematics is qualitatively different in many ways than studying pure mathematics.

Ah. I genuinely did not know that. In Scotland we did not have Applied Mathematics (as a school exam subject) so I guess I just assumed it was about physics problems (there was an A-level with that name in England). It never came up again in my Mathematics career..


A related thread from two days ago for those who missed it:


One thing they don't talk about is how you decide what to study. Before spending a lot of time on a particular topic in math, you have to decide whether it's worthwhile studying it at all.

It's apparently just assumed that you're taking a course so the decision is made for you.

I'm willing to bet that a course in post-secondary school is the context for the vast vast majority of people studying mathematics.

One thing I figured out on my own that I wished I had realized sooner (which I'm trying - so far unsuccessfully - to impress on my 14-year-old son) is that, when reading math books, they follow a similar pattern. They describe a concept, show an example problem fully worked, and then discuss the ramifications of that concept, followed by another concept, followed by a fully worked problem, etc. I started making it a habit to try to work the example fully-worked problem by myself, based on the description that preceded it, before reading through the author's work. I was amazed how much better I was able to understand what he presented, and how much better I did working the exercises in the chapter afterwards.

Ha, Dr. Blecher's webpage on HN! That's great.

I did my PhD in math, in large part because I enjoyed his class on advanced linear algebra so much, and later on real and functional analysis.

This is good. I think two principal components should be emphasized when studying mathematics.

1. Proper preparation. There are textbooks at even the graduate level which have no formal prerequisites and which are largely self-contained. Technically speaking, someone with no prior background but a strong mathematical maturity could tackle these, but it might take them an inordinate amount of time to really grasp the material. For example, if you understand things like mathematical induction and proof by contradiction, you can learn analysis before you've been exposed to calculus, or category theory without abstract algebra. But it's far from ideal because you'll probably need to go over the same material several times and struggle with it.

Furthermore, even the same subject within "advanced" mathematics can have wildly different depths of coverage depending on the author. Pinter's A Book of Abstract Algebra is probably approachable for anyone currently reading this comment, or even high school students. Dummit-Foote is a step beyond that, and appropriate for undergraduates who are already immersed in a math degree. But Lang or MacLane-Birkhoff would be significantly more challenging without first building up to them.

Sometimes this is not just a question of depth, but also of pedagogical style. You can get a lot of satisfaction by learning analysis from Rudin for the first time, but it's really a rough go of it if you're not prepared for the terse definition->theorem->proof->remark->definition->theorem->proof->remark style of writing. On the other hand, Tao's Analysis I and Analysis II are much more approachable (similarly, some writers, like Halmos or Munkres, are praised for their exposition in introducing otherwise complex material).

Ideally someone looking to study a subject should introspect about whether or not they are prepared for that subject overall. Once they've confirmed they are, they should read the first 10 pages of five or so well-recommended textbooks on the subject at their level, then choose to stick with the one that has the most approachable exposition style for them.

2. Proper study. When studying any given textbook (or videos, lectures, etc) it's really important to understand that mathematics is an active discipline. You cannot learn it by reading it. The process that has worked for me is the following: first, read through a chapter without taking any notes. Do so quickly, but not quite so quickly as skimming. When you come across things you don't know, compartmentalize them a bit and keep moving forward to the end of the chapter. The idea is to let the chapter's new material percolate a little before you begin actively tackling it.

Next, start over at the beginning of the chapter and write down every single definition and theorem as you read. Before reading the author's proof of any given theorem, try to prove it yourself for at least 10 minutes. Then compare your work to the author's, and copy their proof meticulously in order to learn the method. Continue on to the end of the chapter.

Finally, there will probably be anywhere between 5 - 20 exercises at the end of the chapter. Solve a meaningful fraction of these exercises, and don't look up the solutions to any of them until you've struggled with them for a good half hour or so (each). When you do look up the solutions, make sure you check multiple proofs for the same exercise so you can understand how the chapter's material can be applied in different, flexible ways.

Mathematics has always exemplified a central belief of mine, which is that humans learn under conditions of optimal struggle. Even though it feels like being mired in hopeless complexity while you're struggling to complete a particularly difficult problem, you're actively learning the subject by doing it. But it's a question of efficiency. You want to aim for a subject and a presentation of that subject which is difficult enough to be just out of your current capabilities, but not so difficult that you can't follow its exposition.

This comment should be way higher up.

Notation and syntax are areas that I have struggled with in higher level mathematics. Can anyone recommend a guide or resources useful to understanding things like converting set theory, sequence and series style problems into equations and solving them?

Number theory's as good a place to start as any. You'll learn proof techniques and see a lot of notation in contexts that require varying amounts of background knowledge. It's feasible that an undergrad intro class would be up your alley if you're a software dev.


Bill Shillito's lecture series for Project Polymath is by a mile, the best introduction to all the notation necessary for higher level math. It requires absolutely no prerequisite knowledge.

Introduction to Higher Mathematics: https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6Kz...

If you find he talks a bit slow, change your playback speed to 1.5x. Enjoy! :-)

I always strugle with memorizing part. I am never able to reproduce word for word what I’ve learned or read, but I can put it into context and show my understanding pretty well. Here, with math theory, there is no alternative. I failed exams so many times only because of that vocal reproduction of learned theory that required word for word knowledge. I am having math exam on integral theory in 2 days and here I am sitting and lookin at screen. :)

> with math theory, there is no alternative.

You can do the exact same thing you mentioned with math. Theorems aren't arbitrary; there's a way they're derived. If you learn how the theorem comes to be, even if you can't remember exactly, you can often remember enough to come up with the theorem again.

I think the way you learn is the ideal. Knowing the context behind the facts you learn gives them worth, and the redundancy of being able to derive the fact from the context also makes it better integrated with the rest of your knowledge and easier to remember in the long term.

I totally agree with you, but I feel like with math theory there is always good chunk of knowledge that needs to be somehow inserted in your memory as is. That’s the most irritating part for me personally. If i find a way to overcome it or make it more accessible, I think I will enjoy math theory much much more. :)

I'm afraid it's just practice. Lots and lots of solving problems until the definitions, methods, and techniques are burned into your mind.

Once you have the background knowledge, everything is a lot simpler. Learn whether each theorem has an easy proof or a hard proof. If it has an easy proof, you can forget it (deriving it when you need to; an example is the contraction mapping theorem [1]). If it has a hard proof, you can forget it and look it up when you need to.

[1]: https://www.patrickstevens.co.uk/how-to-discover-the-contrac...

> I failed exams so many times only because of that vocal reproduction of learned theory that required word for word knowledge.

That sounds odd for college-level math courses (note that I am just a first year engineering undergrad and therefore possibly know nothing). Do you have oral part in your exams?

Yes. Talking with professor in front of you, and trying to pull everything out of my head is the hardest part. There I just don’t have enoigh time to sit down think about something and try to prove theorems on my own. And I’ve been on 2 faculties, more or less the same story on both. Forst had harder written test and assignments, second on which I’m currently has brutal oral part. I leave satisfied when i got the exam right, but the process of preparing it is fairly boring and fatiguing.

I wonder if anyone here has a background in instructional design and what they think of this tutorial.

Do you need a background in instructional design to have an opinion on this? ;-)

No. Not what I'm saying. I'm asking for an option from someone with particular experience. That doesn't invalidate the options of others, snowflake.

Drop the "snowflake". Gratuitous personal insults don't belong on HN. You have a valid point without it. Leave it at the valid point, without the personal dig.

Anyone who claims to be offended by an honest question on behalf of some unnamed party is a snowflake, in my opinion.

Feel free to hold that opinion. But saying it is counterproductive. It leads at least some of your readers to dismiss you as a jerk, rather than hearing your message.

[Edit: Oh, yeah. It's also against HN site guidelines. Saying such things repeatedly can get you banned.]

When someone misinterprets a rather clear communication and claims to be offended by the result, I'm going to point that out. If that offends you, that's your problem.

Or, look at it as a life-hack: Don't waste your time protecting the feelings of a non-existant group of people.

It should emphasize more that you have to read very slowly, and often re-read the same section.

This text should be required reading for ANY university-level math course. It made me go from "ugh, not math again" to "yay, I'm going to learn something new!" in a matter of days.

This advice also applies to more advance mathematics at the high school level, such as AP calculus BC or even linear algebra.

Source: have followed this technique indirectly for the past few months

One of the best explanations of the subject that I've seen.

If you can't read the page:

    window.document.body.style['max-width'] = '650px'

I cannot stress the importance of working problems yourself enough. There is literately no substitute.

Mathematics for the million is a great book if you are looking to reignite your passion for math.

So basically.. Mug up everything and hope you have exactly the same thought process as the course instructor.

This is a disgrace.

In any case. Definitions vary from book to book so mugging them up ad verbatim would hardly help.

Love it

I've always wanted to learn mathematics but have no idea where to start, I have no formal education in anything.

I tried the Ivan Savov books and they seem great but I have no way to judge the quality of any resources.

Where do I start as a complete beginner? I want books not online resources like Khan etc.

I would eventually like to get into Geometry but I have no foundation whatsoever, just basic algebra.

For the very basic precalc foundations this one was great: https://www.amazon.com/gp/product/0760706603/.

For easing into higher maths I recommend this one on proofs: https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat....

Also if you're concerned about quality the MAA reviews math books and publishes a basic library list for undergrad maths: https://www.maa.org/press/maa-reviews.

Measurement, by Paul Lockhart, may be helpful to start with. It carefully and clearly derives mathematical principles from basic ideas, in a very approachable way.

Here's how it describes its content:

Part One: Size and Shape

In which we begin our investigation of abstract geometrical figures. Symmetrical tiling and angle measurement. Scaling and proportion. Length, area, and volume. The method of exhaustion and its consequences. Polygons and trigonometry. Conic sections and projective geometry. Mechanical curves.

Part Two: Time and Space

Containing some thoughts on mathematical motion. Coordinate systems and dimension. Motion as a numerical relationship. Vector representation and mechanical relativity. The measurement of velocity. The differential calculus and its myriad uses. Some final words of encouragement to the reader.

I have absolutely no mathematical background. https://www.reddit.com/r/mathbooks/comments/1ehupq/totally_d...

Here people are saying you need a solid pure math background. I skimmed through it and it gets complex very quickly. Around page 58 with the formulas I get lost.

Thanks though, I think I'll be ready for this after Gelfand and Kiselev. Can't wait.

The web site https://betterexplained.com/ has physical books that you can purchase "Math, Better Explained: Learn to Unlock Your Math Intuition". The explanations on that website are excellent, I imagine the book is the same.

You could try : What Is Mathematics? An Elementary Approach to Ideas and Methods https://www.amazon.com/dp/0195105192

I find the online courses the most entertaining and stimulating. Many are available for free, like on Coursera, MIT.edu. Some offer real classes, where you follow live classes I believe. You cannot start when you want, you have to follow the program.

On Udemy you can find nice courses as well for a good price. They have discounts all the time, each course for $10-20. Never pay the full price. See the reviews, and you can always take a risk for that price, plus they have a 30 day money back guarantee.

Start with:

* Algebra, Gelfand

* Geometry, Kiselev

Thanks for this, everyone should take the time to review this material

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