My observation is that many programmers, especially those who have come of age by working in startups, tend to value ability and sometimes experience over formal education. This is a result, I believe, of noticing that they can outperform many people who have a classical education, and also seeing that many of the people to whom they look up also do not have much in the way of formal education.
However, I truly believe that Mathematics is a discipline that is very hard to engage with outside of formal education - or at least nobody has really found a great model for doing so yet.
Learning Mathematics in a classical, structured way really does change the way you think. I notice a substantial difference even between those of my colleagues who entered industry straight after their Masters or even Bachelors, and those who completed Doctorates or even held postdoctoral positions.
In my opinion, it is this lack of mathematical maturity that makes the switch from general programming to scientific programming more challenging than the converse.
I disagree that mathematics itself is difficult to engage with outside of academia.
There are some academic mathematicians who think mathematics has to be hard because it was difficult for them to understand. There are others for whom sharing their insights is more important than searching for new ones. I think that the set of mathematicians together have done a rather good job in the 20th century of producing mathematical literature and pedagogy that laypersons and the polymaths can understand.
I skipped university when I was young due to circumstance. I still had a love for maths but couldn't pursue formal training. When I came around to programming as a career instead of odd-loner-hobby it was as you say. I was quite impressed with my ability to code circles around CS graduates who, for all their theory and training, weren't prepared for "coding in the real world."
I never stopped studying mathematics. I've recently been going through temporal logics. If only more programmers were aware of the beauty of the proof of weak fairness. And its implications in system specifications.
Where I can see formal training being useful is exposing you to subjects you were not aware of in short order. My experience has been more akin to the mansion metaphor. Where I encounter limitations or difficulties in my programming I look to maths for the right tool to simplify the task and ensure it is correct.
So I agree about "mathematical maturity." I just don't believe it is exclusively acquired in an academic setting. I think some people have an inclination or awareness that pushes them to think this way.
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?
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.
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.
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.
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.
> 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.
>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.
> ...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.)
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.
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.
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.
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.
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.
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.
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.
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.
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.
Programming has an instant feedback loop -- you know almost immediately if what you wrote works or doesn't. Math is more abstract, and doesn't have that. As humans, it's much easier to learn something on your own when you have immediate feedback. I think that's why programming is an outlier for people who are self taught.
> I've recently been going through temporal logics. If only more programmers were aware of the beauty of the proof of weak fairness. And its implications in system specifications.
I'm a mathematician and have read about temporal logic. What beautiful proof of weak fairness do you mean here?
How do you get the feedback necessary to check your learning? With programming you have techniques to check your work at some level. But that's not the case with math.
This is an issue I'm having- eventually there are no solutions manuals, and the people that can (or are willing) to answer my questions become few and far between on math.stackexchange.com
(warning: studied Math in college, so don't know how that biases me in this discussion)
I wonder if some of this is because of the number of jobs in industry for each discipline? There are a great many jobs available which nurture CS skills but I can't quite think of a job besides academia off the top of my head where that lecture in Toric Varieties I took would come in handy.
Maybe its that there is a career/financial incentive to go deeper into CS but not mathematics?
I think math is "easier" to learn while it still holds a clear practical value, up to calculus and linear algebra. Past that, when you start to enter the world of "pure" mathematical s, it can be muh more difficult. The practical value of earlier subjects allows a student or what have you to draw connections between what they already know and this new concept. Something like group theory however is more difficult to find a "use" for, outside of solving problems in your text book.
There are some good texts out there for learning by yourself. I currently own 7 or so Dover Books on Mathematics, and I could understate my appreciation of them.
My experience was the opposite. The more practical Mathematics was too easy to hand wave through so no one ever bothers with proof. But then when something goes wrong or you're not sure whether a particularly complicated calculation makes sense, you're SOL without a real formalism to go on.
I didn't really believe in calculus except on a case by case basis until after my first analysis course.
I think the only reason the less formal mathematics courses seem easier is because the thing being studied is also dead simple and the calculations are easy. But that approach doesn't scale.
Actually, group theory is not a very good example of a theory that has no or few practical applications; see https://en.wikipedia.org/wiki/Group_theory#Physics. In general, you would probably be surprised to learn how much of the modern mathematics (including category theory) has already made its way into theoretical physics and other sciences.
Sub-disciplines of mathematics don't blow up and become cornerstones of education if they don't have practical applications. Analysis, Topology, Algebra, Abstract Algebra, Linear Algebra, Statistics, Geometry... If something doesn't have a major practical use, it'll probably be named after somebody specific and studied in relative obscurity.
Number theory has always had practical use. It used to just be called "Arithmetic", and it didn't become something else until that something else had demonstrated practicality.
The sort of number theory that mathematicians studied for centuries had no practical use until the development of cryptography and computing in the latter half of the 20th century.
That's why Hardy famously used it as his example of mathematics done with no consideration or hope of there ever being a practical application.
I'm talking about education. Mathematicians work on useless stuff all the time. They don't teach it as part of their core curriculum.
Either way, Hardy was wrong. Number theory became relevant because of the work of people who thought it could be. There was hope for a practical application, even if Hardy couldn't see it.
I've seen some examples that don't seem to agree with that. Programmers often don't need advanced math, but the ones that do, such as video game engine developers, seem to get fantastically good at it.
You need a lot of time to master any skill, and few jobs provide that for mathematics. At least, not with any diversity in problems. The degree provides some years of dedicated effort.
Out of curiosity, a graduate or undergrad applied math degree? Contemplating embarking on a similar path myself in a few years after I have the requisite background courses.
Undergrad, 10 years, one class at a time. I swore off night academics afterwards, but... now I'm feeling the itch again to do a math graduate degree. My wife isn't pleased. ;-)
How much difference is there between those who entered industry after their masters and those who hold doctorates? I personally haven't encountered much of a difference as a masters holder (spent 4 years in graduate school though, 2 to finish the masters, and 2 in a math PhD program before leaving), but I also haven't worked with many PhD holders.
My experience: not all masters degree holders are created equal.
There's a big difference between people who leave a funded phd program early with a masters, and people who enroll in a terminal masters program. Especially if the terminal masters program was a "professional", non-thesis program.
But there's much less difference between people who leave a Ph.D. early with a masters and those who stick it out.
mathematics is unconstrained by any desired artificial 'engagement' requirement
engagement is as easy as acting on and developing an interest
and i encourage everyone to engage with mathematics
your attempts to create an artificial toll or make a case for one, especially in lieu of your argument being unsolicited from the contents of the linked article, is suspicious at best, and wholly detrimental at worst
> very hard to engage with outside of formal education - or at least nobody has really found a great model for doing so yet
this seems like contradictory logic.. you appear to be denying 'informal' students from using the same model you advocate from these 'formal' sources
also i think you need to flesh out your definitions a bit..
what precisely do you mean by: formal education, outperform, general programming, scientific programming?
you also seem to be setting up a logical fallacy in your attempt to define your thesis of 'mathematical maturity'
> my colleagues who entered industry straight after their Masters or even Bachelors, and those who completed Doctorates or even held postdoctoral positions
are you comparing a ~20 year old at their first job to a ~30 year old who spent thaer twenties in academia? have you tested your hypothesis by comparing others of similar time spent on the subject but lack receipts for the money they paid into the academic institution? are there anomalies present in your investigations?
your argument seems to lack any substantial scrutiny, and this would seem to me to be the defining element of some such concept of an interest in mathematics 'maturing': devotion to rigor
I did not downvote your comment, but to me your comment seems to be building up a bunch of straw men, peppered with negativity. It also doesn't clearly communicate anything relevant that I can understand.
The original commenter is noticing a difference of culture between the startup community and the academic mathematics community: the stereotypical attitude of startup folks is "establishment be damned," but that attitude doesn't seem to fare well when it comes to mathematics because the process of learning mathematics is so incremental. Disruption and the hackathon mentality won't help you learn math (in the author's opinion).
Meanwhile you're nitpicking definitions that seem to me to already have satisfactory informal definitions with no need for that level of scrutiny. It doesn't add to the conversation in a meaningful way. What I can glean from your comment is that if you have the right attitude toward math, then none of the original commenter's points hold. And I think everyone agrees with that. It's just that the people being discussed don't have the right attitudes, so it seems mildly pointless to say "Well if only they had the right attitude!" The same statement applies to most discussions about human behavior.
I sincerely hope this helps you communicate your views better in the future.
my desire to respond was instigated by this quote from the gp:
> However, I truly believe that Mathematics is a discipline that is very hard to engage with outside of formal education - or at least nobody has really found a great model for doing so yet.
my views:
mathematics is unconstrained by any desired artificial 'engagement' requirement
engagement is as easy as acting on and developing an interest
and i encourage everyone to engage with mathematics
I have a great personal story that highlights how long the journey of understanding mathematics is.
I took linear algebra my freshman year of college. It was the non-math major course, so it didn't require proofs. I got an A+ in the class. Not just an A, an A+. I was able to obtain such a high grade by taking tons of practice tests, and since the actual tests were basically mildly veiled calculations, I just had to map the question to the right calculation. So for instance, if after a little interpretation, I figured out that the question was asking for me to calculate the singular value decomposition of a given matrix, I would mindlessly compute, check my algebra, and move on.
However, it was very clear to me by the end of the course that I didn't really understand what the heck linear algebra was about.
Five years later, I started a job as an algorithmic trader. One of the first things my boss wanted to do was to do a Principal Component Analysis (PCA) of bond price movements. This is a very common thing to do. I didn't know what PCA was, but I read a short paper he gave me and I was able to grok it. After reading that paper and actually performing the PCA (which by the way was basically one line of R code), I finally came to understand the core essence of linear algebra, which is the idea of linear transformations. I was able to connect the equation Ax=lambdax to the geometry of what an eigenvector meant. Through a little more reasoning, I realized that every real matrix corresponded to a linear transformation of that space via a rotation, a reflection, a stretching, a shearing, etc. At that point, all of the mindless calculations I had been doing half a decade earlier instantly clicked, and I was enlightened.
This was literally half a decade later after I "aced" my linear algebra class. I know that it seems absolutely ridiculous that I could "score so well" in a math class yet so clearly miss the core idea behind the entire class, but that's been my experience with math for as long as I can remember. You start by doing the calculations and just getting comfortable with them. Some arbitrary time later, you have an insight and suddenly everything is so crystal clear and trivial that you wonder how you could even not have understood it before.
Oh, and even to this day, I don't understand what singular values actually are. Something to do with a mapping from the row space to the column space, blah blah. I'm sure if I spent an hour to read about them and picture the geometry, I could figure it out, but I just haven't gotten around to doing it.
Linear algebra is the sort of topic where it seems like you can always go deeper. I also got an A in my linear algebra class in college. When I got to grad school for math, I took the first year graduate course on algebra and saw linear algebra in terms of algebraic things like modules and representations. I decided that I hadn't really known linear algebra before, but that I did then. Then I took differential geometry, which involves studying infinite collections of vector spaces parameterizes by points on a manifold. I realized I still hadn't know linear algebra, but after that I certainly did. Then I took a functional analysis class, where we did infinite dimensional linear algebra, where a lot of the finite dimensional theory has analogues but everything is more complicated (e.g. instead of finite bases and dot products, you consider things like Fourier analysis and Hilbert spaces).
The same realization that I don't know linear algebra came when taking a Lie algebras class, and again when learning homological algebra, and probably a few other times as well. There are certainly lots of areas of math that I've never explored that take linear algebra in some other direction (for example, I don't have any idea what the applied math guys do with....).
It's really an amazingly vast subject, especially considering that it's usually just thought of as a tool used to study more advanced topics.
I've been learning (trying to learn) linear algebra by studying Gilbert Strang's "Introduction to Linear Algebra" book. There are some really mind bending sections in which he explicitly treats a concept like linear regression in terms of geometry and then algebra and then calculus and then probability and then ties them all back together.
What's really struck me is when I dive into a Wikipedia rabbit hole of linear algebra, following links for terms I don't know. I start on the Topology page, read the introduction section of 10 articles that start with "x is a generalization of y" and somehow end up back on the Topology page. Shallow exposure to the sheer volume and conceptual depth of stuff like topology and algebra has really made me respect modern math.
I've had a similar experience several times. I feel like in college I was mentally much faster, but lacked insight (and probably curiosity).
There have been several times when I've picked up an old math or CS textbook and read something I never really fully understood in college -- it almost instantly makes intuitive sense to me now -- 20 years later.
There's an old saying that college is wasted on the young... in many ways I believe it is true.
This is also largely a consequence of the fact that most linear algebra courses are computational by nature. They'll ask you to compute lots of things as a means of assessment. Work out this determinant. Find the eigenvalues of that matrix. "Matrices are grids of numbers". Take this matrix and write it in terms of this other basis. I only really perceived the impact of linear transformations being vector-space-structure preserving mappings long after I'd seen their analogues in the form of homomorphisms, homeomorphisms, diffeomorphisms etc. Nobody told us that determinants are just the alternating k-tensor on real k-space (up to a constant factor). At some level, this is because most students taking linear algebra don't have the mathematical maturity to stomach a course in finite dimensional vector spaces. I was indignant when I found out that a matrix was not just a grid of numbers, but rather a manifestation of a linear transformation, with respect to a particular basis.
I only really started to understand linear algebra when I was forced to in a differential geometry class. The opening chapters were a review intended to fix my university's notoriously broken linear algebra training. As I said, it comes with the territory. You can't design a course based on Halmos' FDVS and expect students coming in, that is, students who've scraped through calculus 1, to manage. So, the recipe book / cookbook style abounds. Granted, there were inklings of mathematics in my linear algebra course. I don't think anyone really appreciated it however. It's hard to grok "vector space over a field" when you've never been introduced to the abstract concept of a field.
When my second year stats lecturer told me that a determinant of a 2x2 matrix was an area, I almost didn't believe him.
Fun exercise I was told about just the other day. Every invertible matrix with integer coefficients has determinant +-1. I would never have known how to solve that after my linear algebra course.
I guess daniel-levin meant to say "invertible matrix with integer coefficients whose inverse also has invertible coefficients". The fact isn't too hard to see:
* If M has integer coefficients, then det(M) is an integer.
* det(inv(M)) = 1/det(M)
* Since M has integer coefficients, det(M) is an integer
* Since inv(M) has integer coefficients, det(inv(M)) is an integer
* So det(M) and 1/det(M) are both integers, so det(M) is either 1 or -1
Math programs in general suck at telling you why a particular technique is useful in the real world, or more importantly, how to model. Lots of ways to solve an existing model, but very few touch the much harder task of building a model in the first place.
Students and parents are demanding that their education fees generate employable graduates, not creative or intelligent ones. This trend will probably continue as long as education keeps getting more expensive.
In general, mathematicians are bad teachers for introductory Math courses (shouldn't be very surprising, experts are often bad teachers for introductory courses). Too bad universities think they are the most prepared for that.
I had a similar experience. Not only I passed linear algebra without understanding linear spaces, but I passed analytic geometry without learning any geometric concept that I didn't know.
I was randomly thinking about logarithms and it just "clicked" that what a logarithm really tells you about a number is it's order of magnitude relative to some base.
I had never seen it explained that way, but once I thought about it it seemed so simple. It's funny, programming has actually helped me understand math. It helps me to look at a formula or function as a programming function. I try to read maths like I read source code. I ask the same question: what dynamic system is this jumble of symbols trying to represent? How does an object behave as it moves through this system?
Once I began to think about math like that, a lot of things began to "click" for me.
Somewhat ashamedly, I have a 1st Class BSc Honours in Mathematics with much the same experience. It's fairly easy to fake an understanding of mathematics if you can understand that the exams paper will be a variant past exam papers.
I think this is why I enjoyed physics classes more then math classes. The formulas we were taught had a visible effect. I could watch the values transform to something I could physically confirm at each step. As a result the formulas became more obvious and therefore more memorable.
This reminds me of the book "The Perfect Wrong Note". The book is focused on learning to play music but the principles it teaches apply to learning just about anything. The core message to not be afraid of mistakes during practice. Little kids fall over when they learn to walk, you'll have moments of confusion learning new things. There's a time to get things done well, like playing at a recital or releasing production code. There also needs to be time to practice and part of practicing is the expectation that there will be mistakes.
I feel the same way when solving tasks in my day job. The thing I tell to young people learning programming/tech that I hope they don't get frustrated easily, because they will spend every day of their life feeling rather stupid and confused, never knowing when will they discover a solution for a particular problem.
This is something that was a great source of stress early in my career.
I always thought that in the end, one has got to like facing these hard problems every day on the job in order to thrive in this field (I am in tech too). This is something I try to detect in candidates in job interviews I conduct.
I noticed people can be roughly split in two categories:
-- ones who spend a couple of years dealing with this sort of job activities, then want out by any means, on to something different;
-- others enjoy it more and more as they gain expertise, start having more and more fun -- get more expertise, and with results (which necessarily come up), the right to pick and choose the subjects and so the hard problems that come with them.
I happen to belong to the second category (and it has been a while I stay in the same domain), and I don't believe someone in that category would feel stupid and confused. it rather feels like investigation every time.
Though over the years, I've become wary of the stress always associated with being put on such issues -- a last-minute demo for a trade show that doesn't work, a customer who has escalated to upper management, a delivery which is hopelessly late... -- with all attention of the management attracted. I keep telling myself that a soldier has got to participate in war campaigns, not be doing paperwork at the headquarters, and literally force myself in last years.
But I don't recall I have ever felt stupid or confused, even in the beginning of the career.
Indeed, but the more experienced I get the harder/bigger problems I work on, so that balances it out. I still feel rather stuck on a daily basis, but I accept the situation and try to deal with it with a calm focus.
A mathematician was walking home from campus one day, and as he walked he was pondering a particularly thorny problem. At one point, he snapped out of his reverie and looked around and realized that he had no idea where he was. He saw a young boy playing with a ball in a yard, and figured maybe the boy could tell him the way home. So he says to the boy, "young man, do you know where Prof. So-and-so lives?"
The boy looked at him and said, "Dad, what's wrong with you?"
My favourite quote about Mathematics is from John von Neumann: "In mathematics you don't understand things. You just get used to them." - This quote highlights precisely why I ended up choosing software engineering over maths.
I'm just not very good at applying processes/methodologies which I don't fully understand.
For example, I wasn't very good with linear algebra until I was able to visualize the equations in my head. For example, now, when I think about the equation 'f(x) = ax^2 + bx + c' - I can see that this represents the set of all possible quadratic equations and I can roughly visualize what that looks like on a cartesian plane (well it would turn the whole plane black because there would be an infinite number of graphs). Then if I choose any three points on that crowded cartesian plane, I can visualize that among this infinite set of curves, one of them passes through all three points. Thinking about it in that way allows me to make sense of Gauss-Jordan Reduction and other mathematical processes related to linear algebra.
Programming is much easier for me because I can visualize the results instantly on a computer - I don't need someone else to explain it to me. Any uncertainty can be quickly resolved by simply running some code.
> "In mathematics you don't understand things. You just get used to them." - This quote highlights precisely why I ended up choosing software engineering over maths.
Haha probably the most unintentionally funny and ironic comment I've read in awhile. Given more than 50% of software projects fail, which is worse than chance, I think it's safe to say the no one really understands software engineering either. Like math, they just get used to doing some things that seem to work better than others.
>Any uncertainty can be quickly resolved by simply running some code.
Until you have to interact with a black box of someone else's code. You can only be certain that the data you sent that particular time works, not that all possibilities of valid data work.
I guess with programming, you can never get 100% certainty when it comes to correctness of your code - Unless you perform formal analysis which is impractical for most use cases.
You might not succeed at 100%, but you can remove "classes of error" by adopting particular styles or techniques backed by formal analysis. That's one of the biggest appeals of compiler technology - it can encode an understanding of patterns proven to detect failure, and in so doing lower your resulting bug count.
Does anybody know a good guide on where to begin? Resources are not the issue here, but usually the overwhelming amount of information regarding all those topics and areas of mathematics. I was always very interested but got discouraged rather quickly, even after a semester at university. So far my favorite access to math was through philosophy.
Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, with a view not to rush through it, but to expand your mind through each problem.
Step 4: Pick any of these books that interest you the most and do the same:
- Calculus by Spivak
- Algebra: Chapter 0 by Paolo Aluffi
- Linear Algebra Done Right by Axler
By then you should have enough mathematical maturity to know what to do next.
I'd also recommend Hubbard & Hubbard for a beautiful and beginner-friendly mix of algebra and analysis.
My preferred starter kit is Rudin plus Halmos or Axler, but treating Rudin as a summary. So a helper would be needed, like Counterexamples in Analysis. This is what Math 55 used to do.
I started with Hubbard & Hubbard. It is truly wonderful (and based on Spivak's other calculus book), but I couldn't imagine learning it without either significant background or a formal class. Even with a 2 semester class devoted to it in college, I ended up not understanding the most advanced concepts (eg. differential forms) until years later when it finally came up again in graduate-level courses. Of course, it was really an excuse to learn to think mathematically, not to learn calculus on manifolds.
I don't really recommend Rudin for a true beginner at all (unless, by "as a summary," you mean not really digging into the proofs themselves, in which case any good analysis book will do). Rudin will always try to take the most elegant route to the theorem, regardless if that route goes anywhere near where the rest of the text has been. The result, for me, has been that many of his proofs seem to just meander about for a little while until, at the very end, you arrive at the theorem. It's a bit like driving to work on auto pilot, and just as disconcerting to me.
Caveat Emptor: Aluffi and Axler are texts meant for vastly different levels of maturity. The former for first-year graduate students, and the latter for first/second-year undergraduate students.
I don't find calculus and lin alg very interesting, school ruined those for me forever I guess. I am mostly interested Logic and Information Theory but I guess the latter doesn't go with a good calculus base.
Thanks for the links, I will read the book of proof and try the exercises, though mostly studied those topics already.
Where to begin depends on what your goal is, mine was to be able to complete Sussman's SICM book (and later, his Functional Differential Geometry book).
The book I used to do this was Advanced Calculus by Loomis & Sternberg because it covers classical mechanics, potential theory, differentiable (Banach) manifolds, differential equations, (multi)linear algebra, fundamental theorum of calculus and the Fourier transform. The exercises are not very difficult compared to a lot of other texts (Spivak's Calculus) so this is an accessible math book as I don't have any formal math training.
I also liked the Mathematical Preliminaries crash course in the Art of Programming Vol 1 because it led me to looking into probability which has turned out to be an infinite rabbit hole of discovery. (a grad students highly opinionated list) https://www.amazon.com/gp/richpub/listmania/fullview/1F85VWN...
The Second Edition of Courant & Robin's 'What is Mathematics?', revised by Ian Stewart is good. This is from the preface:
" In short, it wanted to put the meaning back into mathematics. But it was meaning of a very dif- ferent kind from physical reality, for the meaning of mathematical ob- jects states "only the relationships between mathematically 'nndefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. "
As an added bonus, the book also covers mechanics (Physics 101), which is really important to understand to build up you general modelling skills. (Physics is all about coming up with math models to describe reality.)
I read a lot of math in ML papers, but I don't do any exercises, and maybe that's why I am not sure of myself. A very gradual collection of problems pertaining to linear algebra, calculus and statistics would be great. Something like Khan Academy, but the key point is to be gradual, so as not to terrify me before I get through it. Ideally such a training ground could guide students through problems based on exercise logs collected from other students, by creating an optimized curriculum. Anyone knows where to look?
Perhaps Incandescence, though it's less about mathematics and more about a society working out the physics of its environment. His website has more maths on it than I recall being in the novels.
a) get started on something - if you can't decide, roll the dice and pick something at random
b) ignore everything else until you are done
c) repeat
Don't worry that at your pace it will take ages - very soon you will develop an idea how to rank what you should look at next. If you don't, go back to a).
Obviously it's useless to think about it too much when your knowledge about a subject consists mostly of holes and gaps, so first gather data (a).
I can't quite see how concrete your question is, but try Khan Academy. Follow the suggested order if you have no preferences.
The last math class I took was in high school and I dropped out of college because my CS degree required me to do a metric fuckton of math, which I hated at the time.
A few years ago I got interested and started working on Integer Factorization. It is not that likely that I will solve that problem but in the past few years I have learned a lot more math by attempting to solve that problem that I learned in college!
Moral of the story: Just have fun and try to solve some problem(s).
Which you won't find out unless you start - somewhere, anywhere. How are you going to find out when you just sit there and think about something you know nothing about? Wait for sudden enlightenment? Math is one off the easiest subjects to get started, soooooo many starter books and by now even great online courses (again: Khan Academy, for the basics, probably many more). So much guidance. There is so much out there - free and easily accessible thanks to the Internet, the problem isn't "how do I get started" but "what resource do I use to get started".
As an undergrad who recently became serious about math (thanks to its importance in areas I am interested in) this is very inspiring. I have been trying to grok mathematics for some time and sometimes being too frustrated with problems I can't handle. I have experienced the phenomena of giving up on something and coming back to it and finding it trivial. This is exactly what I needed.
"What’s much more useful is recording what the deep insights are, and storing them for recollection later. Because every important mathematical idea has a deep insight, and these insights are your best friends. They’re your mathematical “nose,” and they’ll help guide you through the mansion."
I can't think of a better argument to goad programmers into writing code comments, especially for elegant code.
For some reason, programmers think good code should be self explanatory. It doesn't have to - if your insight is outside the flow (or the argument) of your code.
I see a lot of Java code written like this; if the programmer's job is to multiply two numbers, they add number a number b times, and say comments are not needed!
What a great blog post! Totally agree with him. And, personnaly, it is why i'm having so much fun doing maths. Everytime it's a new exploration, a new challenge. My best math teacher I had was seeing math with this philosophy in mind and his class was like discovering new lands every time. I'm sure that if we explained in a way that failing a math problem is as normal and challenging that failing a Mario Bros Level, more people would be in peace with it.
> Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark...
What are you supposed to do if you like math and the idea of grokking it, but you also have a job and a family and can't afford to spend six months contemplating each room in the mansion?
> What are you supposed to do if you like math and the idea of grokking it, but you also have a job and a family and can't afford to spend six months contemplating each room in the mansion?
Learn to come to grips with the idea that the universe isn't always going to support your mutually exclusive preferences?
My observation is that many programmers, especially those who have come of age by working in startups, tend to value ability and sometimes experience over formal education. This is a result, I believe, of noticing that they can outperform many people who have a classical education, and also seeing that many of the people to whom they look up also do not have much in the way of formal education.
However, I truly believe that Mathematics is a discipline that is very hard to engage with outside of formal education - or at least nobody has really found a great model for doing so yet.
Learning Mathematics in a classical, structured way really does change the way you think. I notice a substantial difference even between those of my colleagues who entered industry straight after their Masters or even Bachelors, and those who completed Doctorates or even held postdoctoral positions.
In my opinion, it is this lack of mathematical maturity that makes the switch from general programming to scientific programming more challenging than the converse.