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Yup. He's probably just unaware of how difficult and vast math actually is. Reminds me of comments I've read from programmers talking about learning 'advanced mathematics': Linear Algebra. Lol.

As someone who has done research in the area, I'd argue that a lot of work on temporal logics is at about the same level of difficulty as a better-than-average undergraduate linear algebra course. Certainly none of the standard results about various temporal logics even come close to approaching the difficulty of a first or second year graduate mathematics course.

No, not completely unaware. I'm glad I've reminded you of something you find entertaining. I hope you discourage more people from learning something awesome.

Hes just being real. Ive been self learning group theory, and cardinality/ordinals, surreal numbers, VN universe, shit is hard. Foundation is key. In order to be profound one needs tools. Unlike programming, dedication to tightening ones toolbelt first, helps immensely later. Order in math progression is way more significant than programming by trial and error.

It'd be interesting to know what the optimal order of progression would be.

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'd say it really depends on what one wants to specialize in. You could do a ton of calculus and it wouldn't help all that much in some other areas of math. Yet, calculus is a pillar for some things. Ramanujan used integrals everywhere in his work on infinite series. I don't think there's a very straight forward path. Math is a very diverse subject.

I was intrigued because there is some indication of a rough but important ordering to the topics in formal maybe antics education. I was interested in seeing what that order was.

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.

Right, in fact the interaction between multiplication and addition is one of the most fundamental forms of chaos due to combining recursive operations. One could say, addition is repeated counting. Multiplication is repeated addition. And the interaction between the two points a finger at the relationship between space and time.

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.

Does anyone have like, a map or presentation or article or something that will give me an intuition of how vast maths is?

just go to the local university library and look at the math section. :P

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.

Define 'good'. I've seen people in their early 20s go from one branch (say, combinatorics) to another branch (say, algebraic geometry) without taking any formal courses [one could argue this is equivalent to going from being a great neurosurgeon then 6 months later publishing papers at the forefront of pancreatitis research]. They end up out performing post-docs who've spent a decade solely in that field, often within months. In my mind that's just as impressive as the Ramanujans and all (who was 'discovered by Hardy' but still had quite a base line of formal training prior to being invited to England).

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.)

Going from combinatorics to algebraic geometry with formal training in the former but not the latter doesn't count in my opinion. Having formal training in pure math makes one more able to self learn other branches. I have a hard time believing that one can go from undergraduate calculus/linear algebra to algebraic geometry in anything more frequently than than extremely rarely.

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.

No one is asserting that someone taking 100 level calculus is going to self-teach => publishing in algebraic geometry. I agree that your first hump correctly describes the first barrier. I assert that a lateral movement in 'sufficiently different' subsets of mathematics will provide an additional hump within the space of knowledge, quantified by a magnitude of eh maybe > 2/3s minimum. Going from comb. to alg. geo. is amazingly hard; hard enough, I suggest, that it's the equivalent of modern day 'self-teaching', given the large corpus of knowledge one has to attain[1]. I assert it'd take at least ~2 years for one to move from combinatorics to an understanding of say, Sheaf Theory circa 1950 (up to Serre) unless you're one of those rare guys like Tao who can jump from field to field (ugh pun not intended).

[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.

The person I responded to seems to think that mathematics is not difficult to engage with outside of academia. You seem to be under the impression that even making a lateral move post professional training is amazingly hard. It appears that you agree with my first post that grasping second year graduate level mathematics does require formal training for all but a very few.

Yes they are. this whole thread is about mathematics being accessible to people outside academia. the greatgrandparent poster posted that they could follow mathematics despite having skipped university in a thread about mathematical maturity and being accessibility outside of formal training. someone pointed out that you could cross disciplines but this isn't what we were talking about as this builds on top of a classical training in university.

Certainly some that were active in the last 50 years, though it's hard to think of any that were trained in that time-frame: Gelfand and Ian MacDonald are the first pair that come to mind, though Gelfand had some great mentors and Macdonald did an undergraduate degree.

I did not know that about MacDonald. Thanks for pointing it out. Do you agree, though, that such examples are rare? The original premise is that in math it appears self taught is not really a viable route for all but a very small few.

If I'm not mistaken, Paul Lockhart was also self-taught. If I recall correctly, he met Ernst Strauss at some point, who introduced him to Erdos who then somehow managed to get him admitted to the graduate program at UCLA (he dropped out of undergrad). He had published before being accepted to graduate school.

> I have met people who claimed to be self taught in mathematics and it was obvious that they didn't really understand the concepts though they were absolutely convinced they knew what they were talking about.

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 have programmed professionally but that was a long time ago. In my limited experience I believe one advantage programming has over mathematics in terms of self learning is that in programming one is more likely to know that a program works/doesn't work versus knowing whether your conception of an abstract definition or theorem is correct. It seems overall easier to believe a false notion in math than in programming. I might be wrong in this perception. I think your last sentence is absolutely correct.

This is the problem I had learning math in school, and why I believe I'm doing okay as a self-taught programmer. I always thought I was doing okay until I got my graded test back.

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.

In my limited experience I believe one advantage programming has over mathematics in terms of self learning is that in programming one is more likely to know that a program works/doesn't work versus knowing whether your conception of an abstract definition or theorem is correct.

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.

Insanely useful. I believe it should be state of the art. Even model checkers are vastly more useful than nothing.

Simple profit incentives.

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.

As someone who just finished a ph.d. in (non-applied) math I would agree profit incentives are a large percentage of the reason - in programming, you can learn enough basic skills to make money, and then smoothly transition that to higher skill / profit levels just through the experience you gain doing more actual programming. In math, you can't really make money with a lower level of math skill (your options of making money at a lower math level are maybe accounting or teaching, and neither of these provides a transition to deeper math skill levels). Thus your option to increase your math skill is to tough it out through 5 years of grad school making about minimum wage, and then another three years of a postdoc, making a 1/3-1/2 of the average software developer salary, and then you can start making more money if you are lucky enough in the problems you choose, and work hard enough.

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.

I think this is it. To be good at mathematics you need a considerable amount of time to sit and do mathematics. The only way someone is going to fund this as part of a job is if you're already a mathematician.

Just chiming in to add another vote that this is the explanation. In most business settings, business types don't like mathematical maturity or really even care unless it is fundamental to the bottom line -- and even then, it's usually only fundamental to one tiny corner of the product or business, and after that everything else is reporting software, client analytics, customer support, websites, etc.

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.

>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.

I think this highly depends on what you mean by self-taught. Probably a majority of good mathematicians are "self-taught" in the sense that they learned most of the things they know from reading books or thinking on their own rather than taking classes. What you won't see commonly is someone who has a day job but becomes really good at mathematics on the side. This is probably because of the sheer amount of training it takes to become what is considered "professional-level" in math. It takes quite a bit of background before you can even hope to do something that hasn't been done before. The same is not true for programming.

> ...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.

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.

> 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.

You stated ...No good mathematicians (currently) are [self-taught]

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.)

> I've never encountered anyone who understood typical second year graduate level mathematics without formal training.

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 not making any claims that I'm a self-taught prodigy. I know what I know and what my limitations are. I can sit through ONAG for fun, have worked my way through Concrete Mathematics and delve into TAOCP when I come across interesting topics, and work out the proof to

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

I'm not kidding myself that I can stand with the best of them and work on bleeding edge problems. The sphere packing problem sounds really cool but it would probably take me quite a while to work my way up to really understanding it. But I believe it's possible if I don't fall into the trap of self-deception and work out the requisite material before digging into it. "Convince yourself," and all. That's what is so amazing for me.

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.

The point here is that people who do what you say very very rarely contribute to the field of mathematics in a meaningful way. You can "learn something" all day long every day for the rest of your life. Mathematics is vast. But that doesn't mean you're learning something that is both useful and not already known.

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

Am I to understand that "engaging with mathematics" is to mean that the only useful contribution anyone can make is furthering the state of the art? If that's the case then yes I can see why it's so rare.

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.

Srinivasa Ramanujan

50 years? No.