"In solving the traditional game, the researchers have also solved 21 of the 156 three-move openings, leaving a crack of hope for humans..."

At the Mineshaft.

IYKWIM.

Also, I'm really not sure what this means:

"It's a computational proof," Dr. Schaeffer said. "It's certainly not a formal mathematical proof."

Well, what is, then?

Something like the proof of the four-color theorem, where a space of possibilities were computationally explored and checked, instead of something general?

You really are just trawling the comments for posting material, aren't you?

2: He means that the proof relied on computing the result of a lot of cases, probably an intractable task for a human.

There has been some debate about the validity of such every since the four color problem was solved in just such an exhaustive way.

There are really typically two related arguments. One is the level detail required in a mathematical proof. The other is the reliance of a proof that relies on reduction to a large number of cases that are necessarily computed by a machine.

3: I still don't understand what the real difference is, though.

Consider the mathematician who uses something like Coq to help her formalize, or check, a particularly hairy and complicated proof.

Consider that, under some circumstances, a "formal mathematical proof" is pretty much a shorthand for proving what you would discover if you exhaustively enumerated and checked all possible cases (in situations such as this, where the number of cases is finite-but-very-large).

We're not talking about, you know, infinities here. We're squarely in the realm where constructive and classical mathematics agree -- all we're differing on is, how much abstraction and shorthand is necessary before it's "mathematical" rather than "computational."

And I guess I'm confused because I thought most mathematicians had gotten over this kind of thing, a lot time ago. Computational proofs, when properly done, are mathematical proofs, for all intents and purposes. It makes no difference whether a computer or a human manipulates the symbols, it's just symbol manipulation. A proof is a proof, regardless of the prover, right?

OTOH, IANAM, so please correct me if I'm wrong.

6 "lot" s/b "long".

So the thing about a proof like the four-color-theorum proof is that it can't really be checked for accuracy by a human in the same way that a fully-human-generated proof can.

That is, suppose there's a subtle logic error in the code that generates the reams of special cases in the four-color proof, and so like .0001% of them have a logical fallacy in them. Since humans can only viably check the code, even if you're the paramount grandmaster mathematician and have looked over that code to your satisfaction, you've really only become satisfied by something one step away from the actual proof.

That said, that doesn't mean that the four-color theorum isn't a mathematical proof, it's just not an easily verifiable mathematical proof. Like arthegall, I'm puzzled by what distinction Dr. Schaeffer is drawing.

I mean, presumably one could write a proof-checker (separate from the original proof-finding program) that could check the proof of the first one, and find those exceptionally rare errors.

The beauty, in general, of a proof is that it's much easier to check than it is to find in the first place.

(Or so we think, otherwise P=NP, and all that, right?)

It could be, based on what the article says, that there is a formal proof that all of the cases that the machine examines are solved by Chinook, but that it doesn't look at every board position, and that there is no formal proof that the board positions it does not look at are impossible to get into via normal play.

There are lots of things which we basically know to be true, but don't have a formal proof for -- Fermat's Last Theorem until like 1992 or so, right? -- so it's possible that the CS people looked at this and said, "C'mon, it's obviously impossible to get into this board situation, so we aren't going to develop a solution for this board situation," but never bothered to actually proove that you can't get into that board situation.

Well isn't the whole point of a proof, in a way, that it's supposed to be a sort of argument that saves you from checking all possible counterexamples one by one? And doesn't a computational proof in essence go through all the possible counterexamples mechanically, on your behalf, and rule them out?

6 I can't speak generally of course, but I think for the most part people have `gotten over it' in the sense that they accept the ability to prove something this way. Some will nevery like it, but then again some people adamantly dislike 200 page proofs (I'm looking at you, A. Wiles) on principle too. Lot's of people dislike this sort of approach because it is complex and somewhat opaque, not so much (I think) because there was a computer involved. If the computer was neccessarily involved, it is so much worse.

I'm pretty sure in the quote he was making that sort of distinction.

I bet whether or not one is comfortable with computer-assisted proofs as being "enough" of a proof - or feels defensive about it - is somewhat of a generational distinction.

I should note I haven't read teh article, and should shut up until I do that, probably.

I was going to say, "obviously someone needs to go read the article."

I don't understand y'all's confusion (although I don't know shit about math). Isn't there a difference between proving something by showing that certain characteristics of what's being discussed compel or exclude certain outcomes, as opposed to checking each possible outcome?

13: It's partially that, but I can't imagine a mathematician who wouldn't believe it was very real improvement if you could hand them a 2 page proof of the same thing.

17: Sure, a real improvement for the mathematician. But not (necessarily) the dinstinction between "is it maths" or "isn't it."

Anyway, I wanted to say, I love the last line of the paper's abstract.

"Solving a game takes this to the next level, by replacing the heuristics with perfection."

Kick it up a notch, yo.

This math goes to 11.

These checkers were solved by a proof ... of unknown origin.

From what I know about similar efforts in creating mathematically based go-playing programs and "solutions" to go (or its subspaces), Epoch's 10 is pretty much it exactly. Although, I guess it's possible that the methodologies used differ because of large gap in problem space size between checkers and "real" go, although most of the significant attempts at solutions for go are all well bellow 19x19.

There's nothing like getting an enormous Coq proof to go through.

Worthless. They started from the openings? Good grief.

6 piece tablebases

...seem to be a couple terabytes here. For those who don't know, that is like KNP vs KBP.

So is the allegation that the computer caused Dr. Tinsley's cancer?

22: Enormous Coqs generally go through anything.

That's a good summary of why people look sideways at computer proofs. (Plus, you have to prove that the compiler is correct, yadda yadda, lthough the 4-color map theorem has been proved using Coq, so that's eliminated. Coq: is there anything it can't do?)

The last I heard about the four-color mapping theorem during an undergraduate seminar was that someone had improved on it enormously and it is now nearly checkable by hand. (The number of cases has been reduced from the low four digits into a few hundred.) Also, amusingly, the reason this is a hard problem at all is because when Ringel and Youngs solved for the general case (the Heawood conjecture, which weirdly enough is wrong in the sole instance of the Klein bottle), you end up dividing by zero in the case of the plane or the sphere.

24: Yes, Brock, with MathRays.

A friend of mine is writing a dissertation on the stuff arthegall's on about.

He just finished a completely formalized proof of Euler's formula, though not using Coq.

Coq: is there anything it can't do?

Simultaneously provide extensional equality and canonicity?

The Times article specifically affirms what Epoch said in 10.

Even with the advances in computers over the past two decades, it is still impossible, in practical terms, to compute moves for all 500 billion billion board positions. Instead, the researchers took the usual starting position and then looked only at the positions that would occur during the normal course of play.

Seems like I've seen this story before.

Plus, you have to prove that the compiler is correct, yadda yadda

Yes, but interesting how you never have to "prove" that the humans' algorithms are correct. Knuth had a nice little monograph where he contrasted implementing and debugging TeX with publishing a "largish" mathematical proof.

They have a truly marvelous proof of the proposition that Chinook cannot lose which this blockquote is too small to contain.

I don't know shit about math, but doesn't introducing hardware involve allowing error into your proofs? I mean, sure if you reproduce a proof in a hundred different machines the chance of one hundred identical hardware errors is negligible, but it's non-zero. If I were to go into mathematics, it would be because numbers as we conceive them are as pure and eternal as it gets, and fucking about with physical systems would ruin the romance. Besides the sentimental value of pure cognition, there's the value of being absolutely fucking sure that if there is a flaw in your proof, there is a flaw in your reasoning. Being able to judge my reason against an absolute standard appeals to me. Having to worry about fucked machines dilutes that.

I don't know shit about math either, but it seems to me that the risk of human error in this sort of thing must be greater than the risk of machine error.

34: I don't know shit about math

Given your recent classist bitching about cultural literacy, I am shocked, shocked, sir!

The kind of error you're imagining could be caused by hardware, but it would have to be broken in a fairly rare and unique way to introduce errors that would return bad results and not otherwise cause the program to crash out entirely. Or just badly designed, but even the original Pentium bug was really, really rare in practice.

Seriously: stop wanking on about the pureness of cognition and the romance of pure numbers, and learn some fucking math. And physics.

Wait, I should elucidate: for anything other than a processor error, the hardware would have to be broken in a fairly specific way. If it was introduced by a flaw in the processor (again, like the Pentium bug), this would show up pretty much immediately if the code was run on a different platform.

Sometimes the term "negligable, but non-zero" means "the odds are very low, but it's conceivable that it could happen." There are other times that the term "negligable, but non-zero" means, "actually, zero."

This is one of the latter times.

It's "negligible".

w-lfs-n, don't make me blow off my parents just so I can come to the Unfogged meetup this weekend and punch you.

I can make myself available to be punched some other time. When's good for you?

Wait. If it's going to be an especially arranged sort of Punch and Tea occasion, first we must find a venue that serves basil aioli.

Seriously though, 38 nearly made me injure myself. Epoch is awesome.

Ms. Rockette, of
36: Given your recent classist bitching
, I honestly have no idea why you dislike me so. You seem, in your response to others, like a perfectly affable person. Why should I become the focus of your righteous anger? What have I done to offend you? God knows I'm an asshole, but as far as empirical evidence goes, you're not supposed to find that out until later. I apologize for any offense given, intentional or otherwise.

Your Humble Servant,
Foolish Mortal

P.S.
Please stop calling me "fucking stupid"

43: I can't speak for LR, and I don't recall what you were saying in that thread --- but I read LR's comment as pretty straightforward J.P. Snow style frustration at the irony of claimin g ignorance of maths and/or science on the one hand, and commenting on the decline/lack of education/understanding/literacy on the other. May have been mistargeted though.

Many kudos to 43, which is the most good-humored move I've ever seen in a low-level blogular animosity (real or imagined). Wish I were so clever.

That's C.P. Snow, you illiterate dullard.

46: heh, you're right. I must be sleepy. On the other hand, going after typos and misattributions is the purview of halfwits and wankers. Unless it's really funny.

44: Yep, pretty much. Throw in a dash of my own occasional frustration at my own deliberate restraint from making similarly fatuous science snob pronouncements in the face of a commentariant that seems heavily weighted towards those in the humanities and that can occasionally be really provocative on the subject (but mostly that I value and enjoy precisely for that demographic difference), and a very large helping of coming from a really blue-collar background in which basically every single member of my extended family are either scientist/mathematician/engineers, or have absolutely no college education at all. I take classist bullshit, particularly classist bullshit veiled in humanities snobbery, quite personally.

44: I wasn't commenting on the decline of anything. I was trying to express the ideal that a non-mathematician imagines of mathematics, If you'll re-read my comment I think you'll find my emotional reaction to the intersection of the romantic ideals an undergrad has of mathematics with how math is actually done: nothing more, nothing less. I'm not fool enough to denigrate mathematics just because I don't know linear algebra. I was only pining, for romantic purposes, for the age when math was an ideal, unrealized outside the mind.

basically every single member of my extended family are either scientist/mathematician/engineers ...

I put it to you that a family that contains a large number of scientists, mathematicians and engineers is not really blue-collar at all, unless you are using some definition of blue-collar I am not familiar with.

Humanities snobbery is annoying, though.

50: Want a detailed breakdown of who went to college and which fields/jobs they went into, and which went into blue-collar jobs and unions basically straight out of highschool, if they went at all?

What I meant to say, and I think actually said if properly unwound, is: "my family are mostly blue-collar, but of those of thus who did go to college (non-plurality) are all in nerd fields of some kind". Not "almost all of my family are in nerd fields".

re: 51

Yeah, I was mostly being cheeky. I hear a lot of people describing themselves as 'working class' or 'blue-collar' when it takes a massive stretch to define them as anything other than solidly middle-class, so I am predisposed to a certain degree of snark in the area.

I think actually said if properly unwound.

Also, I don't think my reading of your sentence was wrong. 'All of x are P or Q' certainly does not imply, 'Most of x are Q (and a very small number are P)'.

If you'd like to mess around with a Java app designed to explain the proof you can do it here.

It seems correct that the reason he called the proof "computational" is that they found a guaranteed win or draw from every possible starting move, but didn't prove it for each board position.

Okay, so I've read the paper at least twice now (well, scanned it, followed by a closer re-read) and I think 54 isn't quite correct.

They call it a 'proof' throughout. There's never any talk of 'mathematical' vs. 'computational proofs' -- they either call it a proof or a computational proof. They do this, as far as I can tell, because this proof is computer- (as opposed to human-) derived. There's one paragraph, on the first page, where they mention that there exist "skeptics" who may not accept proofs that are not human-derived, but that's as much as they get into it.

The issue that Sifu mentions in 54, the fact that it's a proof for the starting position (but not for every position reachable through normal play) means that this is a proof that "checkers has been weakly solved" (that's the first sentence of their fourth 'graf).

They review three possible options for the solution of a game in the first paragraph: ultra-weak solution, weak solution, and strong solution. For an ultra-weakly solved game, the perfect play result is known, but not the strategy for achieving it. In a weakly-solved game, the perfect-play result is known, and the strategy for achieving it from the starting position is also known (that's what they show here, for checkers). For the strongly-solved game, "the result [is] computed for all possible positions that can arise in the game."

So it's a proof that checkers is weakly solved (and that the perfect-play result is a draw). There are essentially two errors that could arise here. The first possibility is that they've proved it in the wrong way, using incorrect or unsound mathematical strategies. But they don't deal with that here because that's not really a possibility -- they're relying (as I read it) on two well-known search strategies whose correctness has been proved on their own. That is to say: they're relying on a fair bit of literature in the AI/search community (and game theory too, I suspect).

The second possibility of error is that the algorithms/search-strategies are ideally correct, but incorrectly implemented. Or that data was corrupted, either in storage or transmission, etc. They devote one paragraph on the third page (titled, "Correctness") to this possibility. Basically, they say that they "took great care" to verify computational results and to implement consistency checks. They also suggest (with a sort of hand-wavy argument, although I haven't read the supporting reference) that even if an error had entered in at some point, "it likely does not change the final result.... [a] vanishingly small [probability]."

It's actually a really fun paper to read, and not hard to follow (even for the not-incredibly-mathematically-sophisticated). And Sifu's kind link also lead me to this paper by two of the authors, which looks kinda cool in its own right.

Okay, so I've read the paper at least twice now (well, scanned it, followed by a closer re-read) and I think 54 isn't quite correct.

Yeah, I got that sense when I read the paper last night, but didn't think I could sum it up particularly accurately. Are you talking about the paper from Science or the older one (which only addresses one opening)?

The one from Science, "Checkers is Solved," 19 July 2007.

The carbon-based chauvinism exhibited in this thread is appalling.

Lunar in 36:

Seriously: stop wanking on about the pureness of cognition and the romance of pure numbers, and learn some fucking math. And physics.

Lunar in 48:

frustration at my own deliberate restraint from making similarly fatuous science snob pronouncements in the face of a commentariant that seems heavily weighted towards those in the humanities and that can occasionally be really provocative on the subject.

soubzriquet in 44:

I can't speak for LR, and I don't recall what you were saying in that thread --- but I read LR's comment as pretty straightforward J.P. Snow [sic] style frustration at the irony of claiming ignorance of maths and/or science on the one hand, and commenting on the decline/lack of education/understanding/literacy on the other. May have been mistargeted though.

I take these criticisms seriously. I haven't studied any math since highschool calculus and consider myself innumerate. I know almost nothing about statistics, and I think that I could remedy that pretty easily. (I never took statistics in college, because the professors were always ranked 1 or 2 on a 5 point scale.)

I'm not smart enough to be a mathematician, but I really wish that I could learn more math and real economics. One of the difficulties I've always had was that there seemed to be very little interest in teaching math in a serious way to non scientists. There were courses in calculus for economists--useful tools, and now there's the mathematical equivalent of rocks for jocks, but I've never known how someone like me who's basically just intellectually curious but doesn't plan to pursue a scientific career could learn some real science and math.

Bostoniangirl: Notwithstanding my little J.P. vs. C.P. brain fart, I think he had some good points that really haven't been followed up enough, or at least not generally enough.

However, I also really understand your frustration with courses (although I'm uncomfortable with the `not smart enough' premise). I think there are really several problems. One is that there is that some things simply can't be reached without enough background --- background that is in general pretty poorly served by the primary & secondary mathematics curriculum. Another is that introductory level mathematics has mostly been cast as service courses for hard and soft sciences and engineering.

However, there is also a deep cultural issue within mathematics. You'll often hear an engineer, say, tell you that they took `almost as much math as a math major' or a hard science student talk about how much math they had to take. What these people often don't realize is that many mathematicians wouldn't consider most of these courses `real' mathematics courses. There has been a system of service courses that keeps the math departments in good stead with the dean and other disciplines (mostly). As the curriculum has been forced to adapt to the needs of these other departments, many are perfectly willing to write off all these students as a needed but ultimately unimportant. That way the real energies can be concentrated on a smaller group of insiders, particularly upper division math majors (and preferably honours).

I'm not saying this is universal even at the level of departments, let alone individual faculty. But it is there, and doesn't help your situtaion at all.

I'd love to teach a course (or two, really we'd probably need a full year) aimed at upper division humanities students and the like (or grad students, even better). Not because they would need prerequisites, but because of intellectual maturity and reasonable study habits. We'd cover a bunch of useful mathematics and try and develop some idea of how to think in this mode. I'm junior enough that I don't have any ability to introduce this, but I have floated it before. Invariably the response is that this is a great idea that would be very difficultt to implement.

I've found Roger Penrose's new popular maths/science text quite interesting and entertaining. My high school level maths was good [top few % in the country, etc] and I hyave one year of university level maths, but my maths is extremely rusty in some basic areas and non-existent in others. So the early maths orientated chapters of his book have been quite entertaining to me. Not worked through anything like all of it yet, though.

At some point I really need to get my physics up to speed. I have taught philosophy of science courses, ad my 'advanced layperson' science knowledge is OK, but I would not be competent to teach a more specialist philosophy of physics or philosophy of maths course, and that seems, to me, to be a lack that I need to rectify.

59:

There are some good books and good writers about math; Ivars Peterson writes topical articles and has written some books. is a high-level overview of modern math, demanding reading, but doesn't call for prerequisites; originally a Britannica article, I think. Motivation is missing from most math books/ courses, which is frustrating.

In my (limited) experience, lay science literature is in better shape than lay mathematics is. I'm not exactly sure why this is, but the popular interest in, say, physics books seems to be healthy, so that might explain some of it.

I think one of the important issues that BG is alluding to is that there are several areas of maths that a decent, basic, working knowledge of would benefit most people. It's nigh impossible to get them all in one place, and few people outside the discipline are going to take the time involved to get it out of a half dozen courses or more....

soubz, some of what you're talking about is in the realm of what is "useful." I'd certainly place my desire to learn statistics in that area. But some of what I'm talking about is a bit purer and unlikely to be covered in any service course for economists.

I'd even be interested in a history of science way in a kind of history of math course--looking at the original proofs of various developments. I wish that I knew linear algebra and had a serious grasp of multivariable calculus, but that's not what I mean by math exactly. I'd really liek to learn about number theory, for example. When I said that I'm not smart enough for it, what I meant is that I'm never going to get a Ph.d. in pure mathematics from Princeton. I'd love to learn some of the stuff that's taught to the true insiders (and I think that some of them really are Platonic in the way that they view the world), but I'd never be good enough to get the attention of the people who teach that stuff.

This is a really neat general overview math book, but it's from '50s India, so its missing a lot of the newer stuff.

It is crazy to me the way math is taught; so much of it is so widely useful, and yet those are exactly the classes most people have no ability to take. I've found the upper division math classes I've taken quite useful and accessible, but a lot of them are closed off to me because I haven't done the six course series in formal proofs. Is that important? Could be, could be. But it's hard for me to find the time to do it, not being a math major.

Why won't somebody just teach me the useful parts of network theory, dammit?

BG in that case you might like that book I linked.

Amir Aczel wrote a very neat book about the development of the zero, but that's extremely light on actual mathematics.

63: by the way soubz, thanks for not calling me on my execrable hedonic calculus yesterday.

64: gotcha. Some such history of math courses exists, by the way, but it's pretty inconsistent. Really, you are probably talking about two different courses (or streams). I think there is a need for both.

There is a lot of neat stuff that you can do to no terribly practical end. All kinds of things are glossed over on the way to more `important' things, partially because they are difficult to absorb. If first year calculus students had any idea how weird the real numbers are, we'd never get out of chapter 1 (or rather, that's why this stuff doesn't show up in those texts).

So by all means, yes. I'd love to teach that course too. Basics in logic & the nature of proofs. A little number theory, a little modern algebra, some linear algebra. Play around with set theory a bit, and some combinatorics, probably. Heck, we could probably do an entire course that never strayed from nominally `high school' topics, but was really quite challenging.

I've long said that my preferred major would be "Math for Stoners." Light on the foundations, heavy on the infinity and weirdness.

Concepts of Modern Mathematics by Ian Stewart cannot be recommended highly enough.

68: I've actually joked with a friend about offering a course sometime called "infinity and other weirdness"

But BG, it sounds like you may be looking for something more like Foundations and Fundamental Concepts of Mathematics by Howard Eves. I'm not really sure, though--I haven't read the entire thread.

69: Stewarts stuff is pretty good, yes.

One of the real problems with lay literature on this topic is that you pretty much cannot learn mathematics by reading about it. Beyond a fairly superficial point, you actually need to work things out yourself.

This works fine if you are motivated & disciplined enough to work your through a book by stopping regularly and actually doing it on paper, but I'm guessing many people need more structure (and feedback) than that. Hence the value of a course, if you could take it.

This course looks pretty nifty, although I'm not sure why it would be shunted into "Math for Teaching."

When I was in college, I"m pretty sure that there was a special multivariable class for Econ people, 21a and 21b were aimed at science concentrators, but taht the real math people took something Math 25, and there might have been an extra hard version of that called Math 55.

74: It's pretty typical to separate calculus streams from analysis streams. Or sometimes everyone takes roughly the same intro courses, but the math people do an additional analysis stream, 2-3 courses or more. It also makes sense if you have enough students in service courses to tailor the examples to their fields ... econ examples for econ students, physics examples for physics students, whatever. May or may not be taught by different departments, and may or may not involve many tedious and disagreeable curriculum meetings.

75: yeah, your latter example is how they do it here. Everybody (in science and math, at least) takes the same classes through linear algebra, and then the math majors seperate off. Reminds me, I have to sign up for that probability course for fall.

72: This works fine if you are motivated & disciplined enough to work your through a book by stopping regularly and actually doing it on paper, but I'm guessing many people need more structure (and feedback) than that. Hence the value of a course, if you could take it.

On the more practical side of things, in terms of getting enough math to be able to take an econ or science class, basically everything I.M. Gelfand ever wrote is quite good for this, if you are motivated. Short, consists almost entirely of examples and problems, with concise, clear explanations instead of "do this arcane procedure for no reason".

75: I don't think that it was an extra course. I think that the really great people were tracked differently. 19a seems to be a new course. Most of my friends took 21a. (My highschool taught AB calculus. I really wish that BC had been offered.)

Here are the courses that I was talking about:

There are a number of options available for students whose placement is to Mathematics 21. For example, Mathematics 19a,b are courses that are designed for students concentrating in the life sciences, chemistry, and the environmental sciences. (These course are recommended over Math 21a,b by the various life science, environmental science, and chemistry concentrations). In any event, Math 19a can be taken either before or after Math 21a,b. Math 19b requires some multivariable calculus background, and should not be taken with Math 21b. Math 19a teaches differential equations, related techniques and modeling with applications to the life sciences. Math 19b focuses teaches linear algebra, probability and statistics with a focus on life science examples and applications. Mathematics 20 covers selected topics from Mathematics 21a and 21b for students particularly interested in economic and social science applications.

Mathematics 23 is a theoretical version of Mathematics 21 which treats multivariable calculus and linear algebra in a rigorous, proof oriented way. Mathematics 25 and 55 are theory courses that should be elected only by those students who have a particular interest in, and commitment to, mathematics. They assume a solid understanding of one-variable calculus and a willingness to think rigorously and abstractly about mathematics, and to work extremely hard. Both courses study multivariable calculus and linear algebra plus many very deep related topics. Mathematics 25 differs from Mathematics 23 in that the work load in Mathematics 25 is significantly more than in Mathematics 23, but then Mathematics 25 covers more material. Mathematics 55 differs from Mathematics 25 in that the former assumes a very strong proof oriented mathematics background. Mathematics 55, covers the material from Mathematics 25 plus much material from Mathematics 122 and Mathematics 113. Entrance into Mathematics 55 requires the consent of the instructor.

I did once read an MIT undergrad application by a guy who had elected to stay in highschool and do normal highschool stuff like cross country about which he wrote his application essay. He was fairly talented in the humanities. (His English teacher described him as a genius, and she said that she knew that she was a young teacher, but he was really extraordinary.) The professors he was working with at UNH said that they expected him to do graduate work in mathematics after a couple of semesters. He'd already presented stuff at conferences. Sigh. I wish that I were that smart.

73: My guess it that it's a fairly lightweight course, not something that would fit into a rigorous math curriculum.

soubz, when you get tenure and I'm rich with plenty of leisure time, maybe you could be my private tutor.

sounds like a pretty good plan!

Concepts of Modern Mathematics by Ian Stewart and Foundations and Fundamental Concepts of Mathematics by Howard Eves are both very good and readable.

It is my goal to have an entire shelf full of Dover books. I am currently far from my goal.

Get a smaller shelf?

Occasionally I run across a CP Snow novel in used bookstores. I've heard they aren't very good.

BG: I'm going to let you in on the secret of learning mathematics, one that I had to learn the hard way. When you read a book that you don't understand, you have a choice between blaming yourself for not being smart enough and blaming the book for not explaining it well. Always blame the book

There's a good history of mathematics book in Springer's UTM series. By Stillwell, maybe?

74, 78: So you went to the big H? The curriculum for the first year math course (23, 25, 55, etc.) looked pretty interesting when I visited a friend there back in fall of 2002. It actually sounded a lot like the course that soubzriquet talks about in comment 67.

That sort of mix, of rigorous but still not obscenely abstract proofs (some basic number theory, then linear algebra dealing with vector spaces, then finally some basic set theory/logic and perhaps some basic real analysis) always seemed like a great intro for most intelligent people into what some higher-level math actually looks like. And I think those were the areas that helped me most later on, in terms of providing logic frameworks and methods of building upon existing knowledge that I'm still using to this day even as most of my precise theory knowledge decays.

BG: From what I've heard, course 55 is particularly rough. Apparently only about 40-50 people try taking it each year, with most dropping out. By the end, there are 20-25 people left, and the only ones doing well were the people who medalled at the International Math Olympiad back in high school. My buddy, who was one of the guys from my high school who had competed in the top national competitions our last couple years, considered it a complete non-possibility. Some of those courses and competitions are invaluable for teaching you just how smart some damn people are. The kid in that link also knows 7 languages.

Soub, are you a math prof somewhere? I think I've missed it if you've ever hinted at your location.

Soubz lives in Houston.

More effort needs to be spent on making math communicable.

88: But it's already vectorized!

ideal mathematician

The stuff quoted in the link in 90 is excerpted from this book.

Soub is an ideal mathematician, I believe. I personally flew too close to the sun and my wings melted.

I didn't really like that book, but the quoted dialogue is oh-so-true.

68/70: probably lightweight for an actual course, but lay readers interested in "infinity and other weirdness" may enjoy Kaplan's The Art of the Infinite. I did.

Some of those maths books mentioned above look interesting, thanks.