Will Our Understanding of Math Deteriorate Over Time? | Hacker News Reader

Will Our Understanding of Math Deteriorate Over Time?

blog.computationalcomplexity.org

144 pointsyummyfajitas71 comments

“In mathematics and theoretical computer science, we read research papers primarily to find research questions to work on, or find techniques we can use to prove new theorems.”

This is why figuring out an elegant, concise, and powerful set of mathematical models which apply to multiple domains, and then devoting effort to simplifying, organizing, and explaining those ideas in an accessible way is so important.

Incentives for researchers are mostly to push and prod at the boundaries of a field, but in my opinion mathematical ideas are only of marginal value in themselves; more important is the way they help us understand and interact with the physical universe, and for that building communities, developing effective languages and notations, codifying our understanding, and making it accessible both to newcomers and to outsiders is the most important task for a field, and perhaps for our society generally.

Just like with software projects or companies, the most “success” comes from helping a range of other people solve their problems and extend their abilities, not from making technically beautiful art projects for their own sake (not that there’s anything inherently wrong with those).

Perhaps more generally, while theorem proving has overwhelmingly dominated pure mathematics and related fields for the past 80–100 years, and has been an important tool since Euclid, theorem proving is only one way of approaching the world, and in my opinion is a mere tool, not an end in itself. Just like simulation is a tool, or drawing pictures is a tool, or statistical analysis is a tool.

I like this bit from Feynman: https://www.youtube.com/watch?v=YaUlqXRPMmY

But sometimes the connection between "beautiful art project" and "practical tools" is totally unexpected. We often invest time in projects that seem simply like "beautiful art", and then much later stumble upon something practical.

I think a great example of this is cryptography. The foundations of it come from number theory (prime numbers, modular arithmetic, elliptic curves), but the subject of number theory, before the advent of computing, was possibly the most useless kinds of mathematical 'art' that could have existed. I imagine it was the mathematical equivalent of frolicking in the fields.

Mathematicians explored Fermat's little theorem starting in 1640, but they didn't do it because they knew it'd be useful several hundred years later in RSA. They did it simply because math is worth exploring in itself.

Even if you don't subscribe to the idea that we should pursue math for math's sake, history shows us that it's very difficult to know what parts of math will be useful to humanity, especially hundreds of years later. Since people work best on what they find interesting, mathematicians should continue exploring the topics that most interest them, because we really can't say with any certainty what will prove useful (or even essential) to future generations.

That's a popular meme but its mostly false. See http://lesswrong.com/lw/4kt/the_value_of_theoretical_researc...

The vast majority of "useless" mathematics really do turn out to be useless. In the rare exceptions, there's not much evidence that doing the work beforehand is actually an advantage. E.g. Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC.

Stuff like prime numbers have eaten up millions of brain hours of highly intelligent people. I remember thinking it was weird that so many project Euler problems were about prime numbers. And I looked up what the applications of them were and couldn't find anything significant beyond cryptography.

And they seem to have been chosen for cryptography simply because it was a well studied problem with certain properties. Not because cryptography inherently needs prime numbers and would be impossible without centuries of previous work studying them.

> Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC.

That's the worst example you could find, because Einstein didn't develop the mathematics for general relativity. He relied on the math invented in the XIX century for non-Euclidian geometry. If nobody had though about such a "sillY' geometry with "no practical value" it would probably take much longer because the necessary results would be out of the reach for Einstein.

He figured out that spacetime may be non-Euclidean before he was aware that the math had been extensively studied. Then he learned about the preexisting work.

It's true it probably would have been much harder for him to work out the math on his own. But he and/or others eventually would have done it.

Riemann's contribution is overlooked far far too often. The early non-Euclidean geometries were spaces of constant curvature - spherical and hyperbolic - and Riemann brought the idea of a manifold, and the notion of having a geometry that changes as you move around the space. And he did it in a fantastic lecture with only one equation in 1854, a good 50 years before special relativity.

Einstein was also definitely familiar with the work of Helmholtz, who did some fascinating work on non-Euclidean geometry in the context of ophthalmology: Lenses change the amount of curvature we perceive in space (think of fish-eye lenses), and provide a great jumping off point for the notion that the universe might not be as flat as it appears.

The Dover book 'Beyond Geometry' collects a bunch of the major papers in non-Euclidean geometry leading up to relativity, and is a fantastic read.

> Stuff like prime numbers have eaten up millions of brain hours of highly intelligent people

I think the idea that brilliant minds have been 'wasted' on prime numbers is nonsense. Don't 'highly intelligent people' have the right to pursue what interests them, and even disregarding that, won't they do their best work on problems that interest them?

Even further, is learning anything that is not practical or useful a 'waste'? Certainly not. Calculus might not be of the utmost importance career-wise for an aspiring musician, but learning it helps us think in new ways.

> The vast majority of "useless" mathematics really do turn out to be useless.

That's fine! So long as we strike gold every once in a while (cryptography, which is pretty essential to the internet functioning as anything more than a bulletin board), math is doing it's job.

>And I looked up what the applications of them were and couldn't find anything significant beyond cryptography.

Cryptography is a pretty big deal, though. You can't run a modern economy without it.

That is nonsense. We have have no idea what math will turn out to be useful in the future. To say that it has already turned out to be useless presupposes that we already know all uses we might put it to in the future, which we clearly don't.

Well empirically most math developed in the past turned out to be useless up until now. Are you suggesting that it will suddenly become useful in the (near) future? Are the past few centuries not enough time for you to generalize from?

I think it is very plausible that a large portion of mathematics that is not very useful presently could be very useful for problems we have yet to tackle. As science progresses it will be less and less able to make grand unified theories and increasingly focus on the manifold particular. I can imagine much of math being useful only for problems we haven't even identified yet, like algebraic topology being used to study social dynamics, or engineering problems at strange scales. Even beyond that I think it may be the case that much of mathematics will become useful for reasons unforeseen. Unknown unknowns always seem to be where new science pops up.

I think you are needlessly generalizing the personal argument of the LessWrong post to apply to general epistemology. As a person who is a utilitarian, you cannot fully justify studying pure math. However, at a societal level, you do need critical mass in terms of enough people working on math for practically useful insights to emerge. So 'don't do pure math' (or art, or music, or theoretical CS) is good as a personal goal for someone with utilitarian aspirations, but is inappropriate 'public policy'.

Consider this: a vast majority of mutations are useless, but for this reason, if there were no mutations at all, and mutations somehow willed themselves out of existence, then there would only be primitive lifeforms on earth.

>But sometimes the connection between "beautiful art project" and "practical tools" is totally unexpected. We often invest time in projects that seem simply like "beautiful art", and then much later stumble upon something practical.

That's like saying that we should randomly start drilling holes in the ground because sometimes we will strike oil.

people arguing for it usually ignore the silent evidence of research that lead nowhere and also, more importantly, the potential research accomplishments those people could have acheived if guided to work on different problems.

>That's like saying that we should randomly start drilling holes in the ground because sometimes we will strike oil.

Unless you have better tools for locating the oil, that's not a bad strategy.

I would argue that if you didn't have better tools for locating oil or determining viable research areas you shouldn't be digging/funding it in the first place because unless you have a high probability of finding something (and we've left the times of early science when there was a bunch of low hanging fruit) the cost of failed attempts will outweigh the benefits of the success stories.

Your broader point - that we're a lot more likely to find useful things using some guidance as to where useful things are likely to be than in proceeding randomly - is important.

A crucial difference between digging for oil and doing math, however, is the nature of the externalities. In either case, you're burning some work that could be spent somewhere better, but with oil you're left with a hole that you probably want not to be there and there's no good way to put it back. In both drilling and math, "drilling" helps us refine our methods. In the case of math exploring more of the ramifications of our axioms also helps raise our confidence that they're not subtly inconsistent.

And of course, math is generally less expensive than an oil well.

I don't know where the cost-benefit analysis puts work on math when we don't yet see practical application. And I think that's often over-romanticized. However, I do think there are a lot of reasons we should expect the analysis to come out more favorably than for drilling random holes.

> This is why figuring out an elegant, concise, and powerful set of mathematical models which apply to multiple domains, and then devoting effort to simplifying, organizing, and explaining those ideas in an accessible way is so important.

This reminds me of this Von Neumann quote about the importance of mathematics having an 'empirical source':

—

I think that it is a relatively good approximation to truth—which is much too complicated to allow anything but approximations—that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality" it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely I'art pour I'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical.

In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the re-injection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.

"I'art pour I'art" should be "l'art pour l'art".

I'm only nitpicking because I recently started compiling a list of interesting quotes and intend to save this one :)

You wrote the same quote twice.

I had to read it over and ended up pastiing it into IPython before I realized that.

Is this some kind of affirmational statement, kind of a counter to "ceci n'est pas une pipe"????

It basically suggests an intrinsic motivation in something. The pipe statement suggests that there is a difference between an actual thing and a representation of it.

The real shame is that academia funds a lot of research, but not a lot of scientific "scholarship". We should have academics dedicated to high quality exposition of advanced material. You don't see much of that beyond the never ending rewrites of basic calc and algebra textbooks.

If you haven't already, take a look at Bill Thurston's paper "On proof and progress in mathematics"

http://arxiv.org/abs/math/9404236

Your comment could very well be its abstract.

This is a very good and thought-provoking essay for a short blog post, and I have already shared it in a Facebook community heavily populated by professional mathematicians (where the moderator, with a Ph. D. in math from Berkeley, has given it a thumbs up). Thanks for sharing.

I really like the overall point of the post that mathematics once known can be forgotten or neglected, and mathematics written up for mathematics journals can be difficult to understand. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998):

"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.

". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.

"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."

Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.

http://www.amazon.com/gp/product/0387982892/

I think this is an interesting perspective to think about. I certainly remember a long time where I felt this way—that calculus was the entire reason for learning mathematics and the rest are a little silly or outdated. It took a long time for me to catch on to why the simple, silly stuff is where all of the fun is.

Today, calculus feels boring and dead to me. Obviously useful, but a mere tool instead of something greater. I spend my time thinking about things like topology where I work really hard to think about what it means for things to be close to one another and nothing more.

A younger me would not have understood. Which is a little scary.

Are you excluding things like 'continuity' from calculus because sure, you get topologies on discrete spaces and such, but on the other hand: manifolds.

I'm being a little poetic and talking a little limitedly about calculus. Obviously it's got big connections elsewhere, but my perspective on it has shifted a lot.

> Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus.

I doubt one's "outgrowth of certain branches of math" is the reason math as taught to non-math majors is a castrated mess it is. It's probably the market forces that reject real analysis, abstract algebra or anything at that level or higher. It's the same reason "some programming language du jour > fundamentals of CS, IRL".

>"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."

While I do agree, we have to remember why most math classes actually exist: to teach calculus to physicists and engineers, and, as my stepfather's undergraduate advisor once said, "to keep the children from running in the halls".

(For the mathematician's extremely self-centered view of "children" as "anyone who has yet to ace two semesters of real analysis".)

I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus, and found that, once I get past the applied stuff, I really like the approach of building up calculus from its foundations in real numbers (taken as Dedekind cuts), limits (Cauchy-convergent sequences), the set-theoretic construction of functions, and the construction of topological and metric spaces "from scratch". But I can tell that I like it because, deep down, I have the mind of a theoretical computer scientist (which is what I like to be when I'm not writing firmware), which is a kind of mathematician. I appreciate that someone has to teach the applied classes to the people who aren't going to kvetch about "how can I trust that works!?" and who demand to just get their math over with as quickly as possible.

[1] -- http://www.amazon.com/Mathematical-Analysis-Undergraduate-Te...

> I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus

Do you have recommendations for other books? I stopped at multivariable calculus as well. For what it's worth those yellow Graduate Texts in Maths books feel like reading TaoCP or CLRS; I'm looking for more approachable textbooks. I feel like I'm not even up to the 1800s, math-wise, not even up to Gauss.

Are you comfortable with proving math statements? If not, How to Think About Analysis by Lara Alcock is an amazing intro to Analysis. http://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/01...

There are hundreds of good math textbooks to recommend, it really depends on your interests.

For a broad overview at an undergraduate level, with a great job explaining the context of various mathematics topics, these Russian books from the 50s, Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrentiev, are pretty fun. Amazon link to the one-volume Dover reprint (but I’d recommend finding a used three-volume hardback copy): http://amzn.com/0486409163

Or check out John Stillwell’s Mathematics and its History: http://amzn.com/144196052X

If you're going to work with Cauchy sequences, why bother defining the reals as Dedekind cuts? You can define them as Cauchy sequences of rationals.

Pugh mostly just wanted to be able to state the theorem that a sequence has a limit iff it is Cauchy convergent.

But the proofs survive because they are proofs; if they don't communicate the proof of the result then they have failed and should not be accepted by journals. At the extreme end, machine-checkable proofs are in standard, documented formats; an alien reading them in ten thousand years should still be able to understand what's going on, at least if they understand the notation and the axioms.

And codes does what code does. Given some binary executable, an alien reading it far in the future should be able to understand what's going on, at least if they understand the architecture. :P

I've written a few machine checked proofs, and there's really two ways that I've seen, either writing it for the next human to read, or just enough that the checker accepts it. The latter makes free use of tactics like `crush`, which brute force solutions out of current assumptions, exploring the search space automatically. That's really convenient, but can make reading the proof very un-enlightening.

Chlipala, if you're on here, I want to be the second person to say: `crush` is the least communicative tactic I have ever seen. Where the hell is `unsafePerformIO` when I want some reporting on exactly how you crushed my proof goals!?

Ludwik Fleck's "Genesis and development of a scientific fact" goes very much in the line of "[science] only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new." (written pre-WW2; it served as an inspiration for Khun). Its most eye-opening example is the history of [the concept/knowledge/science/... of] syphilis, from ancient to modern times.

PDF (of print from 1979): http://www.evolocus.com/Textbooks/Fleck1979.pdf

Integrate concise and effect explanations into the relevant Wikipedia articles and you at least give future generations a good head start on understanding these things.

Most of the Wikipedia articles on technical subjects, and especially on mathematical topics, are terrible as introductory exposition. They are jargony, highly technical, and self referential. They usually contain much that is irrelevant, and they almost never properly explain the context for an idea.

The main problem is that Wikipedia articles are tiny and atomic, so it’s difficult to synthesize and organize ideas into a coherent story. The culture of Wikipedia frowns on the kind of exposition found in textbooks or lectures. And perhaps most importantly, no one is responsible for either individual articles or sets of related articles in a field. Working within those confines is not the best way to spend your time if the goal is to give future generations a leg up, in my opinion.

If you want to learn about mathematics, even a mediocre textbook is nearly always better than the relevant Wikipedia pages. The Wikipedia pages are then useful later, as a reference, for people who already understand their content.

The Wikipedia pages are then useful later, as a reference, for people who already understand their content.

Which, if I'm not wrong, is exactly the intent of an encyclopedia. It's a reference work.

If the only people capable of understanding what you are saying are the people that already know it, saying it is quite useless.

Reference material or not.

If I (still) knew it, I wouldn't need reference material. But I agree, if I never knew it, then reference material is... not quite useless, but quite useless to me at the moment.

A reference is a brief reminder and links to primary sources. We don't hate cars because they aren't boats.

I think language and symbology is at the core of why they are so impenetrable.

One major sin is taking new concepts and ideas and putting the primary discoverer's name on them. Such names yield no clue as to the interpretation or application of the idea itself.

Another problem is the symbols used in certain mathematical texts. Everyone who uses them treats them like they're universally understood, but in reality the syntax and meaning of the symbols can and frequently are recycled and reused across disciplines and even theories in the same discipline. You have to be close to the 'in-group'. Like reading other people's code where operators have been overloaded, it's like learning a new language every time you want to dig into a cool new maths paper.

I don't actually have any good solutions to these problems. I would guess there are lots of lessons to be learned from the history of Chinese characters, though. They have thousands of unambiguous symbols which _can_ be learned by non-natives and which _do_ give an appreciable degree of cross-lingual intelligibility among languages that use them.

I was extremely frustrated with my machine learning class because of this. The notation used was hazy. Also if you ask me, probability theory should just be squashed and the ideas reformulated with new more consistent syntax / symbols, because the entire thing is so damn inconsistent and disorganized at present.

>Also if you ask me, probability theory should just be squashed and the ideas reformulated with new more consistent syntax / symbols, because the entire thing is so damn inconsistent and disorganized at present.

How do you mean? Probability doesn't even have that much complicated symbolism... although I do wish we would teach in probability courses how to translate between random-variable "distributed according to" notation and actual density functions. As in, I wish I knew how to do that.

> Most of the Wikipedia articles on technical subjects, and especially on mathematical topics, are terrible as introductory exposition.

Yet they are trivial(and often more helpful) as compared to the arcane material in a math text.

Wikipedia is a horrible way to learn math. At most it works as a way to get initial pointers for literature. In most specialized fields, don't expect wikipedia to provide any understanding of mathematics beyond a summary of formulas without much explanation of what they are.

Wikipedia has the same definitions that are in your textbook, but often provides much more readable proofs as compared to the very short and "elegant" ones or unwieldy monstrosities in your usual textbooks. At least that's true for textbooks at the level of, say, Dummit&Foot's Abstract Algebra or Baby Rudin.

Of course. Most modern mathematicians aren't fluent with half the material in (the ~100 year-old text) Whittaker and Watson "A Course of Modern Analysis". This was standard material even 60 years ago. You can get a PhD in mathematics today without once seeing an elliptic function, because computers are good enough at numerically solving the problems they were once used to solve symbolically.

How many people know how to multiply two numbers expressed in Roman numeral format without reference to an algorism (not a typo!) or other methods based on Hindu-Arabic numerals?

How many people are fast at computing fifth roots without recourse to computational tools such as Hindu-Arabic numerals?

Are those things math or arithmetic?

> multiply two numbers expressed in Roman numeral format without reference to an algorism (not a typo!)

Thanks for making me waste my entire afternoon on wikipedia.

On the other hand, most modern mathematicians know stuff that would blow Whittaker and Watson's socks off.

Much mathematics is obsoleted. For example, there was a lot of incredibly difficult mathematics for finding areas under curves, which was all completely obsoleted with the discovery of the fundamental theorem of calculus. Nothing of value was lost.

Individual pieces of mathematics come and go from general awareness, but the overall trend is definitely one of increasing, not decreasing, understanding.

To make it readable, paste this in the console:

    document.querySelector('.date-outer').style.backgroundColor = 'white';

4real. What a color scheme!

Whatever about Mathematics, this is certainly true of Computing. Well understood ideas are continually being reinvented, frequently badly. New programming languages and frameworks spring up like mushrooms and everyone wants to jump on board the next big thing.

Nowhere is it more true that those who don't know the past are condemned to repeat it.

Most people use math less and less (even your average cashier will have issues if the machine isn't working).

Will Myron Aub give us the feeling of power back?

http://downlode.org/Etext/power.html by Isaac Asimov on just this topic.

I am a cashier. Never use math. I doubt anyone outside academia and parts of industry ever uses math in its proper sense.

Even outside of "proper" academic math, it does seem like there is a worryingly large innumerate population.

It may be that there are few industries that require it professionally, but just because one's profession doesn't require it doesn't mean that one can't, or doesn't, use it. Consider, for example, Fermat, Young, and legions of other, perhaps less well known, amateur mathematicians (for example, Garfield (http://en.wikipedia.org/wiki/Pythagorean_theorem#cite_ref-Ga...) and Napoleon (https://en.wikipedia.org/wiki/Napoleon%27s_theorem))!

I read the SA article the blog refers to and I couldn't decide if that particular colossal theory on symmetry was just an isolated incident or that "deterioration" is really happening to many disciplines/theories of math. It is certainly an obvious fact that things become popular and then eventually forgotten and then sometimes brought back. There is also different levels of understanding: breadth vs depth. I recall at one point there was concern of the opposite. That is too much depth and not enough breadth (the above theory is depth problem as many mathematicians know of the theory just not the exact proof).

I still think the unpublished problem ie "publication bias" is a bigger issue which I suppose is somewhat in similar vain. Supposedly google was working on that.

The theorem the SA article is talking about - CFSG, the Classification of Finite Simple Groups - is somewhat special in that respect. Lots of things in math fall out of fashion and get forgotten, often whole subfields. CFSG is different because the theorem itself is so basic and important that it's not likely to be forgotten in any foreseeable future. But its proof is so long and complicated that it's not even clear that there's one person who understands all of it, and the heap of details is not organised well enough for someone to just study it from books/articles without the help of people who lived through proving it back in the 70ies.

Suppose there just isn't enough interest in the younger generation of mathematicians to study the proof, even if the old guard are able to organize it better before they retire. Then we may reach a situation in which CFSG will still be used as a proved theorem and not a conjecture - because it's so powerful and important in many fields of math - but its proof will be lost to collective memory. I'm not sure, but I think that state of affairs might be without precedent.

(Here's a quote from Gian-Carlo Rota's _Indiscrete Thoughts_ on forgotten and rediscovered math:

"The history of mathematics is replete with injustice. There is a tendency to exhibit toward the past a forgetful, oversimplifying, hero-worshiping attitude that we have come to identify with mass behavior. Great advances in science are pinned on a few extraordinary white-maned individuals. [...]

One consequence of this sociological law is that whenever a forgotten branch of mathematics comes back into fashion after a period of neglect only the main outlines of the theory are remembered, those you would find in the works of the Great Men. The bulk of the theory is likely to be rediscovered from scratch by smart young mathematicians who have realized that their future careers depend on publishing research papers rather than on rummaging through dusty old journals.

In all mathematics, it would be hard to find a more blatant instance of this regrettable state of affairs than the theory of symmetric functions. Each generation rediscovers them and presents them in the latest jargon. Today it is if-theory, yesterday it was categories and functors, and the day before, group representations. Behind these and several other attractive theories stands one immutable source: the ordinary, crude definition of the symmetric functions and the identities they satisfy.")

How are categories equivalent to group representations?

I agree. To some extent new generations invent notations that subsume existing ideas, but I don'tthink they are claiming to reinvent symmetric functions.

Talking about the finite group classification, there was a project which achieved to formally prove it correct in Coq for the odd case: http://www.msr-inria.fr/news/feit-thomson-proved-in-coq/

However, having a Coq proof does not mean someone human understand the proof, and the software can evolve in incompatible versions unable to recheck the proof.

I really wish people would explain why they downvoted me on hackernews. Its not that I really care about the points its that I want to know what I did wrong. Like if I accidentally offended someone or it just wasn't an interesting comment or observation or that it is completely wrong statement.

Your response is excellent anatoly and I appreciate it.

Exactly, this unfortunate "saturate and move on" can be observed in pretty much any area.

Ah, I remember a lecture course where we classified all groups of order up to 1000.

Joy.... "Like having your brains smashed out by a slice of lemon wrapped around a large gold brick."

Not sure why the downvotes, it was an interesting course: a lot of tricks to show isomorphisms / reasons why certain groups couldn't exist. It just went up to order 1000, which was a lot of cases (30 lectures + exercises worth).

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