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Not knowing your level of mathematical insight and knowledge makes it hard to know if your belief about your mathematical talent is a self deception. I've neve

Not knowing your level of mathematical insight and knowledge makes it hard to know if your belief about your mathematical talent is a self deception. I've never encountered anyone who understood typical second year graduate level mathematics without formal training. I know such people could exist. I've just never met any. 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. Outliers exist. But have any good mathematicians been self taught in the last 50 years?

Nimitz14 10y ago
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.
- agentultra 10y ago
  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.
  - goldenkey 10y ago
    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.
    - jwdunne 10y ago
      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.
      - goldenkey 10y ago
        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.
        jwdunne 10y ago
        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.
        goldenkey 10y ago
        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.
- throwawaysocks 10y ago
  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.
- sullyj3 10y ago
  Does anyone have like, a map or presentation or article or something that will give me an intuition of how vast maths is?
  - nikofeyn 10y ago
    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.
iheartmemcache 10y ago
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.)
- yequalsx 10y ago
  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.
  - iheartmemcache 10y ago
    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/ 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.
whynotqat 10y ago
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.
- yequalsx 10y ago
  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.
  - jinfiesto 10y ago
    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.
vinceguidry 10y ago
> 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.
- yequalsx 10y ago
  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.
  - lawpoop 10y ago
    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.
  - mindcrime 10y 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. 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.
nilkn 10y ago
> 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.
agentultra 10y ago
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.
- Retra 10y ago
  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.
  - agentultra 10y ago
    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.
petegrif 10y ago
Srinivasa Ramanujan
- sn9 10y ago
  50 years? No.

Nimitz14 10y ago

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.

agentultra 10y ago
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.
- goldenkey 10y ago
  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.
  - jwdunne 10y ago
    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.
    - goldenkey 10y ago
      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.
      - jwdunne 10y ago
        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.
        goldenkey 10y ago
        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.
throwawaysocks 10y ago
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.
sullyj3 10y ago
Does anyone have like, a map or presentation or article or something that will give me an intuition of how vast maths is?
- nikofeyn 10y ago
  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.

iheartmemcache 10y ago

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.)

yequalsx 10y ago
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.
- iheartmemcache 10y ago
  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/ 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.

whynotqat 10y ago

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.

yequalsx 10y ago
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.
- jinfiesto 10y ago
  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.

vinceguidry 10y ago

> 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.

yequalsx 10y ago
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.
- lawpoop 10y ago
  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.
- mindcrime 10y 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. 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.

nilkn 10y ago

> 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.

agentultra 10y ago

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.

Retra 10y ago
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.
- agentultra 10y ago
  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.

petegrif 10y ago

Srinivasa Ramanujan

sn9 10y ago
50 years? No.