
Self-Taught Mathematician; an Impossibility? - adamnemecek
https://medium.com/@artagnon/self-taught-mathematician-an-impossibility-67d9b6893710
======
_hardwaregeek
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.

~~~
artagnon
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.

~~~
_hardwaregeek
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.

------
credit_guy
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".

~~~
artagnon
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.

------
salty_biscuits
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_syed
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-...](https://academia.stackexchange.com/questions/48399/how-many-papers-
should-a-phd-student-in-math-try-to-publish-before-graduating)

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_biscuits
"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
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.

~~~
artagnon
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.

~~~
BigFish12
Definitions and notations are not discovered, they're invented. I think that's
what OP was talking about.

------
Doyen
This article brings to mind the movie "Good Will Hunting", specifically the
mention of Srinivasa Ramanujan
([https://en.wikipedia.org/wiki/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.

~~~
kd0amg
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
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.

------
neilv
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](https://news.ycombinator.com/item?id=19716303)

------
fspeech
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.

~~~
Koshkin
> _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.

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

~~~
rocqua
Self-taught mathematician, not mathematics.

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

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

~~~
Koshkin
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.

~~~
colossal
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.

~~~
artagnon
Let's just say that there was very little emphasis on Math in my Physics
Masters, which in itself was very poorly taught, and leave it at that.

------
chaboud
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.

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

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

------
graycat
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.

~~~
BigFish12
If you went through a math undergrad and some grad courses, you are not self-
taught.....

