Self-Taught Mathematician; an Impossibility?

https://medium.com/@artagnon/self-taught-mathematician-an-impossibility-67d9b6893710

by adamnemecek • 7 years ago

53 67 7 years ago

Doyen• 7 years ago

This article brings to mind the movie "Good Will Hunting", specifically the mention of Srinivasa Ramanujan (https://en.wikipedia.org/wiki/Srinivasa_Ramanujan). I disagree that it's difficult to teach yourself mathemitcs. As with all things, success in self-directed teaching depends on the aptitude of the learner combined with their desire to learn.

adamnemecek4• 7 years ago

I really wish Hollywood never talked about mathematics ever again.

pm90• 7 years ago

Why is that?

I agree its somewhat disingenuous, and not representative of how mathematics is really done (most of which regular people would probably find pretty un-exciting and mundane). But it does increase visibility about those things. Instead of John Nash being someone who only mathematicians/academics knew of, his name is now (somewhat) more better known, isn't it? A book and a movie.

To be honest, I grew up in a pre-internet, pre-wikipedia world, and without movies like those I maybe would have never heard/known about these folks and what they did.

warent2• 7 years ago

Oh, come on. Hollywood can tell a compelling story to inspire people to get into fields they wouldn't have considered otherwise, even if only as amateurs and hobbyists. A lot of kids watching Iron Man are probably excited to grow up to get into machine learning and robotics.

kd0amg• 7 years ago

Can you offer any empirical evidence that it is not difficult? The vast majority of people I've encountered who are entirely self-taught beyond typical high school (or college freshman) material have been quite bad. They repeatedly make fundamental errors of the sort I would expect to be swiftly corrected in undergrad coursework, even in topics they (claim to) have studied for years.

I don't think I've met anyone who learned to write a good proof without significant human instruction. That is probably something to be expected because much of what makes a "good" proof is social expectations, but I see a lot more cases of unjustifiable leaps of logic than excessive, obvious detail.

empath75• 7 years ago

What does it mean to ‘learn math’? I feel like there’s a value to learning enough math that you understand the general concepts and vocabulary and having an intelligent conversation with a mathematician about, for example, how group theory might apply to some problem you’re working on, without having an actual working knowledge of group theory yourself.

I feel like it’s relatively easy to learn a lot _about_ math, as a self learner, without knowing a lot about how to _do_ math.

I’m mostly set taught, and I know the notation, I know the vocabulary, and I can follow along with a lot of math papers without too much of looking up terminology and I’ve adapted things I’ve read in math papers into working code, but I’m also well aware of my limitations —- I don’t know how to evaluate whether any of these papers are valid, I don’t have any sort of working knowledge of the tools of proofs to be able to do anything outside of fairly elementary calculus, etc. I know about higher math, but I wouldn’t say I know higher math. But even without that working knowledge, I wouldn’t say that what I know is valueless.

Koshkin6• 7 years ago

Difficult or not, people - programmers, engineers, scientists - keep learning mathematics all the time. True, it may be difficult if you have little background, but for the vast majority of people with a high school diploma it should be no problem, especially given the amount of materials available.

warent2• 7 years ago

I agree that teaching yourself mathematics is possible, but I don't think Ramanujan is a good example. He was a prodigy with something about his brain that was simply wired for mathematics and patterns. Solutions came to him as divine visions with little formal rationale or proof.

soVeryTired• 7 years ago

Ramanujan was also (relatively) bad at actually proving theorems, and worked mainly from intuition. He got a bit of culture shock when he moved to Cambridge and was expected to document his logic in the laborious detail that mathematics expects.

chaboud• 7 years ago

Spoiler alert: As with almost all headlines of this form, the answer to the question is "no". The author seems to be telling us that something they did is hard, but they did it anyway... Because.... Humble genius?

I've found math to be one of the easiest subjects to self learn (though I'm not saying that I've learned it or anything else to any noteworthy degree). The body of inexpensive literature is vast, and education in the field requires less practical support than, say, chemistry, physics, or engineering. I haven't gotten around to laser-induced plasma metal crystal deposition in my garage, but I do have bookcases of read and pending math books in the living room.

dvt• 7 years ago

This is a great point, and I think it shows why people don't study math in their spare time. It's the same reason why most people aren't body builders with 5% body fat: it takes a tremendous amount of hard work.

Not only that, but the work isn't glamorous or particularly creative. In my teens, I always though I was "bad" at math -- I never did the homework, didn't do the supplementary exercises, had an "intuitive" understanding that wouldn't translate when exam time rolled around, and I consistently got C's and F's.

In college, I decided to take things more seriously and I remember in Calc I-III doing literally every single problem in the books. If I missed any problem, I would do it over and over again until I got it right. "Miraculously," I started getting A's in math classes.

imranq• 7 years ago

Sadly Calc iii is not “real” math, just an applied version of what newton created 300 years ago. Real math starts at analysis and abstract algebra. It is extremely inefficient to learn this kind of math completely by yourself since there is a specific ‘math intuition’ that is hard to gain without a great teacher.

trentmb2• 7 years ago

but I do have bookcases of read and pending math books in the living room.

That doesn't really tell me much.

GuB-42• 7 years ago

Reading the article, the issue doesn't seem to be that math is particularly hard compared to other STEM fields.

Rather, to be productive in maths, you need a really high level. Before your PhD, chances is that you can't do something meaningful in the field. It means that your degree is the only proof of your level.

So the idea is : you can teach yourself maths but in the order to become a working mathematician, you need to do it all the way up until the end, with little help besides books.

Contrast with programming, after a few weeks, you will be able to write working software, maybe get an internship or even a low level job. You can then work your way up. You don't have to be at the top just to start working or make meaningful contributions.

cosmodisk3• 7 years ago

Even a monkey can write a simple program, that's easy. However, to come up with something unique in computer science field, can easily take decades of study and research.

cosmodisk3• 7 years ago

Math is the purest form of art humans can produce.I'd say one can learn a fair bit of it on his own and be relatively knowledgeable,but..There are many many things in math that are not just mind boggling,but also so difficult that unless you can singlehandedly crack problems that haven't been solved for centuries, you'd need a support from someone more senior. Those more seniors most likely will be found at good universities teaching and researching the subject.

artiste• 7 years ago

"Math is the purest form of art humans can produce."

Curious, what makes you say this? Not that I strictly disagree.

Koshkin6• 7 years ago

Math is no art, its structure and logic severely limits what one can imagine or do.

colossal3• 7 years ago

I think it's incredibly disingenuous to call yourself a self-taught mathematician when you already have a masters in physics.

matz1• 7 years ago

It's not mathematics degree, fair enough.

throwayEngineer2• 7 years ago

Math is being able to do logic with numbers.

I've tried creating my own math as an engineer, it's possible, but I also know every attempt I've made failed at finding something new.

My point is, we learned a lot about numbers in engineering school.

colossal3• 7 years ago

Yes but physics != engineering. You really need very minimal higher math to do engineering work. As far as I know, at my school the furthest engineers go is a course in vector calculus, ODES, baby linear algebra and simple probability. None of which would be really be considered higher math. In contrast, physics needs complex variables, Pdes and most students take at least a group theory class.

cosmodisk3• 7 years ago

Math,queen and servant of science..

Koshkin6• 7 years ago

Undergraduate physics gets by with surprisingly little, if any, advanced math, so it is only natural when one realizes how inadequate one's knowledge of math actually is.

colossal3• 7 years ago

Have to disagree with this. I'm a math undergrad but a good chunk of my algebra class this semester was physics students. Additionally, the author said they have a graduate degree. As far as I know (which is not very far) modern physics makes considerable use of algebraic constructions such as tensors and lie algebras, so I'd imagine the author would be familiar with these concepts, which puts them lightyears ahead of any self-taught mathematician.

neilv• 7 years ago

A recent HN discussion got into some of the difficulty of self-taught mathematics, with emphasis on the Good Will Hunting level: "P = NP Proofs: Advice to claimers (rjlipton.wordpress.com)" https://news.ycombinator.com/item?id=19716303

hardwaregeek• 7 years ago

Not a very interesting article imo, but the topic of self teaching mathematics is a pretty interesting one. There's a few challenges with self teaching math. First, math is inherently about communication. If you cannot communicate your proof, you're not doing math–you're doing mysticism. For instance, the supposed proof of the ABC conjecture by Mochizuki is not verifiable because other mathematicians have not been able to read and vouch for it in its entirety. More mundanely, if you have a proof for the first Sylow theorem in your head with hand waves and loose intuition based arguments, that's extremely different from a proof with tight logic, well defined lemmas and carefully detailed phrasing. Basically if you're self teaching, you have to do problems and have their solutions verified by another mathematician with a degree of rigor.

Second, it's quite easy to convince yourself that you understand a topic. I don't know what it is about math, but there's so many people, myself included, who fall into the trap of thinking they understand a topic, when in fact they don't at all. For instance, there's a multitude of people who can recite verbatim the quadratic formula, or the power rule for differentiation. But ask them to explain why this is true, or give them variants on the power rule (what if we redefine derivatives to be based on multiplication? How does the power rule change?), and they won't be able to give you satisfactory answers. This goes back to my first point; the only way to truly understand math is to do problems and have them checked.

theonemind• 7 years ago

Math tends to bug me because "proving" theorems seems to come from convincing arguments, but we have enough computation power at this point that we should reduce them to mechanical/machine verification and then just run them and know.

With enough rigor, you don't need to "convince" anyone--a machine can verify it.

BigFish124• 7 years ago

To be rigorous you would need to prove that the machine verified it correctly. Good luck doing that. Also theorem proving is only good for basic rudimentary maths, and not abstract state of the art math research.

artagnon4• 7 years ago

I agree that it's necessary to do exercises in order to understand the subject, but I disagree that you need to have them checked by someone else -- in many cases, you should be check it for yourself, honestly.

Beyond the junior undergraduate level, computation-style problems go away, and you're just left with exercises that ask you to prove a certain something. It takes a long time to build up a sufficient degree of rigor, but it happens eventually -- I did go get some of my proofs checked by serious Mathematicians at some point, and it turned out that my intuition wasn't far behind.

_hardwaregeek3• 7 years ago

The main reason you'd want proofs to be checked by someone else, at least initially, is to get a feel for writing clear, rigorous and tight proofs. Much as a novice programmer should have their code reviewed, a novice mathematician needs to have their math reviewed for common mistakes and pitfalls.

agumonkey• 7 years ago

Weren't there people in history who managed a sound level of self testing ? I understand it's super hard to be non biased and not claiming victory when you think you did a few checks.

jqgatsby• 7 years ago

Wait, "redefine derivatives to be based on multiplication"? That sounds really interesting, could you please expand on that?

_hardwaregeek3• 7 years ago

It's a pretty interesting topic: https://en.wikipedia.org/wiki/Multiplicative_calculus

sunstone• 7 years ago

If self taught mathematics was impossible mathematics itself could not exist. Any new mathematics is by definition self taught.

rocqua2• 7 years ago

Self-taught mathematician, not mathematics.

Those who come up with new results tend to have had a lot of previous instructed study.

salty_biscuits2• 7 years ago

So many weird jarring points in this article. Self taught, but masters in physics? Takes 10 years to produce original research? Lots of PhDs graduating without publication? I'd say it might take 10 years to develop a taste for interesting research problems but it isn't at all unheard of for undergrads to get a paper for some research project they did in final year. I think it would be really rare for someone to get a PhD without a publication these days.

ghufran_syed2• 7 years ago

Most undergrads who get a paper published are not publishing math papers.

This stackexchange answer is relevant

https://academia.stackexchange.com/questions/48399/how-many-...

Also this quote from a math professor, linked to in that SE link: "Many people get a PhD in mathematics before having a single accepted paper (I did), and if they have an eminent advisor who goes to bat for them, having no papers need not be much of a strike against them in the postdoctoral market."

salty_biscuits2• 7 years ago

"Most undergrads who get a paper published are not publishing math papers."

Err, I am not sure how to respond here. I said "it wasn't unheard of", which means when I went through I heard of a few bright undergrads getting publications in maths. It definitely happens. My personal experience says it would be extremely rare to get a phd now without a publication (again, in maths). It was different in the past. Hard numbers would be better than anecdata I guess.

jondubois• 7 years ago

I enjoy the pure logical aspect of maths, but really dislike the notation/abstraction aspect.

I think that if we could rewind time and humanity had to reinvent math from scratch, it would probably end up very different from what we have today; most of the underlying logic and ideas would be the same, but the abstractions, notations and representations of those ideas would be completely different (and possibly better/simpler and more logical).

When it comes to code; whenever something is kept a certain way because of historical reasons, that's called technical debt and it's a bad thing.

In maths, basically all of the abstractions and notations are the way they are because of complex historical reasons. Math never gets refactored.

_hardwaregeek3• 7 years ago

I disagree, there's been quite a few attempts at refactoring math. Whether it's the Hilbert Program, which attempted to formalize the foundation of mathematics, the Cartesian plane as a way to describe functions in space or the modern reformulation under homotopy type theory, mathematics has gone through some serious refactoring. Even something as simple as algebra has gone through extreme changes in notation and in phrasing. Try comparing a copy of Euclid's elements to your standard geometry textbook. Or a proof of the Galois Correspondence Theorem in an Algebra textbook to the original in French.

Also, I highly doubt that math would significantly simpler or "logical" (a rather peculiar word to use, as math is inherently about logic, so what would more logical logic look like?). Math is inherently about communication and is therefore inherently subjective. People have a million different ways of writing math because they have a million different ways of speaking. And depending on who you ask, one way may be more simple or more logical than another.

rocqua2• 7 years ago

The dy / dx notation has a very deep meaning from differential geometry. With some other notational things, there is a deep reason behind it.

Other times, notation is bad and it is refactored. The thing is, mathematics by its very nature is a lot of highly coupled 'code bases'. So any time you try to refactor, you end up with sprawling rewrites or a boundary with some translation method.

This is different in coding because there is less coupling, and the translation methods (Foreign function interfaces / shims) can be automated.

daturkel• 7 years ago

Do you have any examples of things due for a refactor? As another commenter mentioned, mathematical notation and language have evolved heavily over time.

When you talk about abstractions, what in particular do you mean? Math is in many ways about describing objects through abstraction.

trentmb2• 7 years ago

Math isn't context free. Neither are most/all human languages. Yet we communicate succesfeully (within some margin of error).

artagnon4• 7 years ago

Actually, the view that Mathematics was "discovered" and not "invented" is quite popular among Mathematicians. There is an area of research called the Foundations of Mathematics, which investigates this question.

It's untrue that Mathematics does not undergo any refactoring -- in fact, it does: even in a field as new as Algebraic Geometry, the "classical" textbooks which build up the subject using ideals and varieties are very different from the modern ones which start from Scheme Theory.

Moreover, Category Theory shows that there is a beautiful underlying structure in all fields that can be extracted out into a new field. For example, the concept of adjunctions existed much before Category Theory was invented -- CT came along and showed that similar structures exist in Algebraic Topology, Algebraic Geometry, Differential Geometry, and other fields.

BigFish124• 7 years ago

Definitions and notations are not discovered, they're invented. I think that's what OP was talking about.

credit_guy• 7 years ago

An impossibility indeed.

When learning anything you need feedback. In many cases you don't need feedback from humans, and then self-teaching is possible: if you learn chess, you can see if you win more games and if your ELO goes up; if you learn to draw, you can appreciate quite objectively if the apple you drew today looks more realistic than the one you drew one month ago. If you learn programming, you can see if your web page renders or not, or if your sql query returns duplicate records. In machine learning, you can see your AUROC score, is it improving? Is your kaggle ranking breaking the top 100, or you are stuck to 5000 ?

In math, what you deliver is proofs. It's very difficult to "debug" your own proofs, as logical lapses that you might have had when you put together the proof are very easy to persist when you do the self-verification. Sure, mathematicians have developed all sorts of heuristics to flesh out bugs in proofs, but these heuristics come with experience, so for an aspiring self-thought mathematician, this is Catch-22 situation.

What about Ramanujan, or maybe Gauss, or some other historical figures? I don't have a very good explanation, I'm inclined to say they are the exceptions that prove the rule. What I can say for today's day and age is that if someone says they know a self thought mathematician, my reaction would be "extraordinary claims require extraordinary evidence".

Koshkin6• 7 years ago

The possibility of making a mistake does not prevent one from learning a subject on one's own as deeply as one wants. In fact, discovering a mistake is an achievement and a great learning experience in its own right.

artagnon4• 7 years ago

I will concede that I have some friends who are Mathematicians who could provide me _some_ feedback. Besides, I did end up contacting a few serious Mathematicians at various points to get feedback on my exercises -- it was just at extraordinarily low levels, compared to coding.

fspeech• 7 years ago

Let's start with Boolean algebra and throw in same basic set theory. Surely that is not too hard to handle for someone able to program. Indeed it is not hard to imagine that our programmer could write some program that can systematically spit out Boolean formulas of set relationships that are tautologies, a.k.a. math theorems. In theory her program could conceivably produce all possible theorems. The problem is that only vanishingly small portion of the produced theorems are of any interest to another person. So the hardest part is actually to know/find what is interesting and relevant.

Koshkin6• 7 years ago

Let's start with Boolean algebra and throw in same basic set theory.

I know where you are coming from, but yawn. Learning something that has practical applications is much more enlightening and fun (besides being useful). Linear algebra is one example.

graycat• 7 years ago

All mathematicians who get very far after formal schooling, especially in research, are essentially necessarily "self-taught".

At least at one time the math department at Princeton stated that graduate courses were introductions to research by experts in their fields, that no courses were given for preparation for the qualifying exams, and students were expected to prepare for the qualifying exams on their own.

A standard remark is -- "Learning mathematics is not a spectator sport.".

For learning math, teachers, courses, recommended text books, in high school and in good math departments in colleges and universities are necessary at least for a good start; else students will too often drift off into nonsense. So, such guidance, direction, motivation, feedback, environment, explanations, seminars, etc. are from helpful, ..., to crucial.

Still, in the end, especially after formal classes, mathematicians are essentially always "self-taught".

In K-8, the teachers were all females, really liked how the girls worked and behaved, and treated me like dirt. I learned enough anyway -- Dad really good at education monitored my progress and was satisfied.

But I was not a usual good student. Ninth grade was algebra, and it was a dream for me. Still in most ways I was still not a usual good student, but I learned the material, mostly on my own, well and got sent to a math tournament. The 10th grade with plane geometry was a big turning point: The teacher was the most offensive person I ever knew, and the subject was a total dream for me. So, no way did I want the teacher to have any credit for my learning and just ignored her, slept in class, refused to admit doing any homework, etc. In fact, I was likely the best math student she ever had: I solved a few of the hardest problems in the main part of the book then turned to the more difficult supplementary problems in the back and solved them ALL, never once missing one. I started college at a cheap place I could walk to. The math class they had me in was beneath what I'd done in high school, so a girl told me when the tests were and I showed up for those. The prof said I was the best math student he'd ever had. But starting in my sophomore year I was going to a good college with a quite good math department and didn't want to fall behind. So, I got a calculus book, taught myself, and started on sophomore calculus at in the good department. So, I've studied freshman calculus, taught it, applied it, learned much more in advanced calculus, ..., and published research, but never really took freshman calculus!

So, from the ninth grade on, I've heavily taught myself. For my Ph.D. dissertation, I identified the problem in industry, was chatting with a math prof about something else, mentioned my problem, got three words of advice, and when my plane landed I had an intuitive start on my dissertation. In my first year I took a terrific course in pure math, and independently in the following summer turned my intuitive solution into a solid one. I got essentially no direction on the dissertation research -- was "self-taught".

So, being "self-taught" is important. Still, it is crucial to have the guidance, feedback, etc. of a good college math department for at least some of the time.

And if want to have a good career in academic math research, then likely it is really important to learn from some of the best research profs, listen carefully for hints of good directions at research seminars, etc.

BigFish124• 7 years ago

If you went through a math undergrad and some grad courses, you are not self-taught.....

Self-Taught Mathematician; an Impossibility?

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