Outliers exist. But have any good mathematicians been self taught in the last 50 years?

It'd be simple to grab a curriculum and start from point A (as in, the method is simple - not the content), taking that as the order.

More thoughts and discussion on this would be extremely interesting.

I've seen others say that linear algebra is a prerequisite of any kind of maths maturity goal.

I think, personally, maths is an incredibly important field and there's a feeling that some degree of comfort with it will help me in more ways than I can imagine. That is, however, an opinion formulate after some self study based on a great interest in the field.

It fascinates me that even things we take for granted, such as addition and multiplication, has entire subfields that have generalised the structure of those operations over those types of sets (in what's taken for granted: addition and multiplication over the set of integers). What's even more fascinating is that this work has had an impact on other areas of deep interest e.g category theory and Haskell.

And yes I agree. It is glorious that the binary operations that seemed so natural to us formed the basis for groups, rings, and fields. Associativity, commutativity, distributivity of two related operations are quite the pearls :-) Its also interesting as a programmer and from a "properties of functions" perspective to study magmas and semigroups and groupoids..etc..

Knowing more about abstract algebra in terms of special properties of functions has led me to believe that most programmers are missing out on some of the beauty of functions by merely implementing them and not so much understanding the common properties and classifications of them.

to give a personal anecdote, i often commented with my fellow graduate students that our first two years in graduate school were spent merely getting to the 1950s in terms of mathematical technology. most people who graduate with a bachelor's in math only know math up until the late 1800s and early 1900s at best.

mathematics is the hardest intellectual activity i have done, and it has made my job as an engineer and software developer much, much easier. the ability to abstract yet get down and diry with details is something math beats out of you.

    (\Ax |: P => Q) === {x | P} \subset {x | Q}

It's important to not be afraid to admit you do not understand something. It's an opportunity to learn after all! And there are some of us who are voracious learners.

The point I was making is that mathematical literature is rich enough that it's possible for people with no formal training to engage with a topic and learn something from it. Maybe they could take cracks at interesting proofs. There's nothing in the field that says you cannot practice mathematics without a degree or license.

What I do disagree with is that the power of mathematics is the sole domain of the priesthood. I believe that anyone motivated enough can work their way through the material. Formal training would benefit those people, for sure, but it's not a prerequisite for enjoying, using, and engaging with mathematics.

If your argument is "you can have fun pretending to know maths all by yourself" then fine. But that's not valuable to anybody.

However if we broaden the scope of contribution we can see people finding applications for theory in industry as useful. It is useful to revisit pedagogy in order to bring more people into mathematics. Even people who take the role of Martin Gardner and bringing mathematics to laypersons is a good contribution -- it gets people excited about the developments of mathematics!

What I have learned has improved my life and made my work better. I think that's useful.

update

I have an inkling as to how vast mathematics is. I realize I'm not going to be publishing any papers and most people who are self-taught will not either. It's not a goal of mine besides.

However I'll still take a stab at the Erdos-Sekeres conjecture of the convex n-gon from time to time. I realize someone else will solve it but what's the worst that can happen? It's fun and enjoyable and sometimes I wish there were more people who found mathematics as enjoyable as I do. And that it was more accessible.

This is an interesting contrast with my experience with learning programming, where, it doesn't matter how much formal education / training you get, you're still going to have to do a massive amount of self-learning before you can hope to achieve excellence.

All good programmers are self-taught. No good mathematicians (currently) are. I wonder why this is. Probably boils down to what 'math' is trying to accomplish vs. what 'programming' does.

I think that may be because, in mathematics, it's hard to "follow the thread". For example, while programming, you can take a program (or a technique) and start reading the code, following links on Wikipedia, and you will eventually catch on. That is, you can do a top-down approach without many problems.

However, doing that in mathematics is way harder. It's not easy to take a paper, read it and start searching for concepts you don't know in Wikipedia: they usually require other base concepts and ideas which are not easy to understan. Sometimes those concepts are not even mentioned becauase the author just expects every mathematician in the field to know them, but to the layman it just looks like magic.

So in maths you need a bottom-up approach, and that's hard to do alone. You need someone to tell you what to study next and why, teaching you the applications and the related fields. Doing it alone is way harder than in the university.

So I'm a self-taught programmer, and this attitude annoys the hell out of me.

1. EVERYTHING requires self-learning to be more than mediocre. In fact, the more I learn about other fields -- engineering, medicine, law -- the more I realize those fields probably actually require MUCH MORE self-directed learning than programming. It's just that this self-learning is much harder to do in isolation without at least a college education. E.g., try teaching yourself any of the traditional fields of engineering with only what you learned before 10th grade. You'll find it's much harder than teaching yourself programming. Whereas you can easily land a developer position with things that the average pre-calculus 11th grader can pretty easily teach themselves.

2. College-educated people who didn't start programming until 18+ ALSO do a lot of self-learning. The difference between them and me (or anyone else who self-taught as a kid) is that they could learn in 1-2 years during their 20s what it took me 4+ years to learn in my pre-teens/teens. Which there's absolutely nothing wrong with. I called it reading tutorials and messing around at night while they called it reading a textbook and doing course projects/homework assignments. I called them high school friends and IRC budddies while they called them TAs. But the actual content of the work and the amount of external direction provided is about equivalent.

Just because programming is simple enough that you can learn how to do enough to get a well-paying job before hitting 16 doesn't mean that people who go through a college degree aren't also doing a ton of self-learning alongside lectures.

If you read my statements carefully, I said exactly the same thing.

The main thrust of my post was that all good Xs are self-taught, for all values of X.

That's the primary point of disagreement between us.

Lots of self-teaching happens within the confines of formal education. And almost all learning that happens outside of formal education is done via some form of instruction or another (books, tutorials, video lectures, etc.)

You can't really make a living off of being a self-taught mathematician, whereas you can by knowing how to program, and run your own business.

The only way to make a living as a mathematician is to into academia, so that why you see all mathematician in academics. Basically it isn't that academia produces mathematician, although it helps, but that it attracts mathematicians. Just as, say, Y-Combinator attracts good startups.

I would guess the more applied math you care about / the closer to software your math is, the better you can do as a self-taught. I actually know of one guy who transitioned from an engineering ph.d. to teaching applied math doing self-taught / working on fluid dynamics in aerospace, but I expect the examples of someone doing this in more abstract / pure areas are extremely few.

Businesses interface socially with other entities (individuals, governments, corporate customers). That social interface will always be 99% report/client service/cursory analytics and 1% hard science domain expertise. That's just the nature of business.

It could be different in a government lab, a think tank, or a boutique consultancy, and it's definitely different in organizations like prop trading firms that do not interface with outside entities in the way that almost all businesses do. But all of those things also place a much higher value on employing people with significant math maturity and hard science expertise.

Did I mix up greater-than, less-than again at work? Well, I'm getting opposite behavior. The errors make themselves apparent. I can play around with it, try things, and get basically real-time feedback as to whether I'm doing it right. What the hell is a monad? Well, here's a problem that is basically crying out for a monad solution.

I've often wondered if automated theorem provers (Coq and the like) could be useful for this. Sure, it requires learning Coq (or Agda or ACL2 or whatever), but it seems that it might be worthwhile just as a way to sanity check one's work - somewhat akin to the way a compiler "checks your work" when programming.

Mathematics has developed so extensively that there's going to be a base corpus of knowledge one generally has to acquire before they're at the caliber of being able to publish. Take a tenured topologist and place him in at a conference where people are presenting their findings in an entirely different field and he'll more often than not struggle to keep up. Combined with the fact that higher level education is a lot easier to attain than it was say, 50 years ago (and definitely 100 years ago) allowing a larger percent of population who those who want to enter into pure mathematics to do so, it's fairly understandable why you won't see many 'good' self-taught mathematicians. (Though, I argue that recluses like Perelman who go off the grid for years at a time after being inspired by something like Ricci flow to solve Poincare are the analogues of yesteryears self-taught mathematicians, as they are no longer collaborating with institutional academia.)

The first hump is understanding a branch of mathematics. Once that is done one is likely to be able to self learn other branches. Can that first hump be done except in rare cases? I don't think so.

[1]http://matt.might.net/articles/phd-school-in-pictures/ Getting published in the J. of Top. is being on the arc (perhaps, even deforming the disc in R^2 heh heh). Being able to move laterally s.t. your findings substantial enough they are accepted into the J. of Alg. Geo. requires a requires such an absurdly large lateral movement along the arc that it's a feat analogous in difficulty to self-teaching oneself up until say the 1920s.

I've known one such person. He skipped Math 55 at Harvard as a freshman because it was too basic for him.

He's such an extreme outlier though that there's almost nothing to learn from his case other than to marvel at it.

I'm a mathematician and have read about temporal logic. What beautiful proof of weak fairness do you mean here?

Interest piqued. Do you have a good link for an intro to this?