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January 14, 2004

History of Math


I have a question. This is supposed to be a culture blog; why don't we ever talk about what may be the most significant, or at least most accomplished, field of human intellectual endeavor: mathematics?

I’ve been reading Carl B. Boyer’s “A History of Mathematics” and it’s been quite an eye-opener. Granted, just following along with the examples is, to put it mildly, stretching my brain a good deal. (Before I go any farther, let me freely admit to being a math lightweight. I never went beyond calculus. I like to think I could do more advanced math, but realistically I know that it would only happen if someone was holding a gun to my head to give me the, uh, necessary motivation.) So keeping in mind that I’m a highly superficial kind of guy, math wise, I thought I’d share some observations from the first few chapters.

Does it ever strike you as weird that you read about a lot of guys like Steven Pinker, who study how people acquire language, but nobody who studies how you (or the rest of humanity, for that matter) acquired the ability to do math? They may exist, I grant you, but I don’t run across them. What’s with that?

And in evo-bio theory, is it culturally or biologically significant that symbolic representations of numbers came so much earlier than symbolic representations of speech? ‘Cause they sure did:

…in Czechoslovokia a bone from a young wolf was found which is deeply incised with fifty-five notches. These are arranged in two series, with twenty-five in the first and thirty in the second; within each series the notches are arranged in groups of five… Such archaeological discoveries provide evidence that the idea of number…antedates civilization and writing…for artifacts with numerical significance, such as the bone described above, have survived from a period of some 30,000 years ago.

Regrettably, the discussion of the fascinating prehistory of math is quite short, as either so little is known or, possibly, Mr. Boyer has so much ground to cover in one book that he can’t dilly-dally in the Stone Age. So we move rather quickly to ancient Egypt.

I was immensely gratified to learn that Egyptian mathematical ideas were, to put it kindly, sort of idiosyncratic. I remember in high school thinking that the most intimidating thing about math is how logical and systematic it all is. I mean, just about any math textbook makes me feel like a scatterbrained moron as it carefully works its way from one topic to another, in an unbroken flow of tight logic. So when I realized that the Egyptians wanted to take one-third of a number, they first found what two-thirds of the number was and then divided it in half, I was tickled pink.

Also, for reasons known chiefly to themselves, Egyptians seem to have taken a dislike to any fractions (except, oddly, 2/3) that weren’t inverses of whole numbers, i.e., 1/2, 1/3, 1/4, etc.:

Egyptian hieroglyphic inscriptions have a special notation for unit fractions—that is, fractions with [1 in the numerator]…Such unit fractions were freely handled in Ahmes’ day, but the general fraction seems to have been an enigma to the Egyptians…Where today we think of 3/5 as a single irreducible fraction, Egyptian scribes thought of it as reducible to the sum of the three unit fractions 1/3 and 1/5 and 1/15.

The Egyptians also had an interestingly relaxed notion about handing out mathematical rules of thumb:

A serious deficiency in [Egyptian] geometery was the lack of a clear-cut distinction between relationships that are exact and those that are approximations only. A surviving deed from Edfu, dating from a period some 1500 years after Ahmes, gives…the rule for finding the area of the general quadrilateral [:] take the product of the arithmetic means of the opposite sides.

Come on, you have to admit, it would be greatly cheering to be able to point out even one dunderheaded mistake like this in your own math textbook. But no, the math powers-that-be have edited all this good stuff out. In any event, the best part of Egyptian math may have been its vocabulary:

[In Egyptian texts on algebra, the] unknown is referred to as “aha,” or heap. Problem 24 [of the Ahmes Papyrus] calls for the value of heap if heap and a seventh of heap is 19.

I'll bet you could get little kids to do math easier if we asked them to calculate 'heaps' rather than 'unknowns.'

Anyway, the next chapter moves to the ancient inhabitants of Mesopotamia, who seem to have had their act much more together, mathematically. In fact, they were so darn slick that it made me realize that mathematics is really the only area in culture that can legitimately be considered avant-garde. I mean this in the sense that mathematicians often appear to be living in the future relative to ordinary humanity. Take one example from around 2000 B.C.:

Many problem texts from the Old Babylonian period show that the solution of the complete three-term quadratic equation afforded the Babylonians no serious difficulty, for flexible algebraic operations had been developed. They could transpose terms in an equation by adding equals to equals, and they could multiply both sides by like quantities to remove fractions or the eliminate factors. By adding 4ab to (a-b)² they could obtain (a+b) ² for they were familiar with many simple forms of factoring.

Does it ever strike you that there’s something weird about the fact that Babylonian scribes could easily solve problems that the average American of today has no idea how to handle? I mean, 4000 years have passed and society still hasn’t caught up! (How many millennia must pass before ordinary mortals begin to grasp, say, advanced ring theory? Don’t tell me, I’d rather not know.)

In Chapter 3 on the math of classical Greece, I was rather intrigued to realize that mathematics can also elucidate the character of a whole civilization. The Greeks seem to have invented the notion of ‘indirect’ proofs. (Okay, I grant you, they seem to have also invented the whole notion of proofs period, but I'm trying to make a point here.) Mr. Boyer gives as an example of such an indirect proof a theorem from the work of Eudoxus, possibly the greatest of Greek mathematicians. He proved (among other things) that the areas of circles are proportionate to each other as squares of their diameters:

Let the circles be c and C, with diameters d and D and areas a and A. It is to be proved that a / A = d ² / D ². The proof is complete if we proceed indirectly and disprove the only other possibilities, namely, a / A < d ² / D ² and a / A > d ² / D ² [Emphasis added.]

It’s hard to put this into words, but the fact that the Greeks appear to have typically resorted to such ‘indirect’ proofs clarifies something about their simultaneously hyper-rational and yet often emotionally overwrought culture. It’s as if their rationality wasn’t really in the driver’s seat, but it was darn useful in allowing them to justify what they wanted by making it seem highly logical, after the fact.

Anyway, I’m plowing on into the chapter on Hellenistic developments, which is supposed to be the real golden age of Greek mathematics; this stuff is really fascinating. Have you ever spent any time since graduating from our Lousy Ivy College looking at the history of math?



P.S. Thanks so much to Jaquandor for the explanation on how to get superscripts into HTML. My initial failure to use superscripts, however, forced me to rely on the notion of writing "[squared]". This constituted a little demonstration of another one of Mr. Boyer's main points, which is that awkward notation has turned out repeatedly to stymie progress in many areas of mathematics.

posted by Friedrich at January 14, 2004


"Let the circles be c and C, with diameters d and D and areas a and A. It is to be proved that a / A = d [squared] / D [squared]. The proof is complete if we proceed indirectly and disprove the only other possibilities, namely, a / A d [squared] / D [squared]. [Emphasis added.]"'s exactly paragraphs like that that made me want to skip geometry class in highschool. I studied last year for the standardized examination one must take for a doctoral program. It forced me back into looking at math in a way I have not done since college. (Quick---what's the formula for the volume of a cylinder? The one fact that stuck is that the length of a hypotenuse of a triangle is equal to the square root of the sum of the squares of the other two sides. Know what that means? It mean I know as much math as the Scarecrow in "The Wizard of Oz."). It also caused smoke to start pouring off my brain. I did just fine on the test, almost solely through the English and Logic portions.

But I agree----I DO like it that somebody calculated one-third by figuring out two-thirds and then dividing in half! :)

Posted by: annette on January 14, 2004 5:17 AM

nobody who studies how you (or the rest of humanity, for that matter) acquired the ability to do math?

Check out The Math Gene (which is a very poor name, as mathematics ability is a polygenic trait) and related books, like The Number Sense.

Also, the thing is that "math" gets really close to "IQ". Studying that is charged for a number of reasons (as we cover thoroughly if you've ever hit our blog). But...

check this

Where does the drive come from? Researchers are just beginning to understand that there are differences in the functioning of the brain's neural circuitry that appear to differentiate prodigies from their ordinary peers. Neuroscientists have learned more about human gray matter in the past 10 years than in all of previous medical history combined, partly due to the advent of sophisticated technology such as a functional magnetic resonance imaging (fMRI) scanner, which measures blood flow to different segments of the brain, revealing which parts "light up" during various mental activities. The only fMRI scanner in the Southern Hemisphere can be found in Melbourne, where American psychologist Michael O'Boyle has been scanning the brains of young people gifted in mathematics.

He's making some startling discoveries. O'Boyle found that, compared with average kids, children with an aptitude for numbers show six to seven times more metabolic activity in the right side of their brains, an area known to mediate pattern recognition and spatial awareness—key abilities for math and music."It turns out those quantitative explanations don't fit. They're doing something qualitatively different."

Lots more on that if you're interested. I'll try to put together a follow up post to this one.

Posted by: godlesscapitalist on January 14, 2004 5:37 AM

People say that all the time---that "math ability" and "IQ" are closely related. How to explain my scores then? My IQ always tests out above average, and my math score on my standardized test was just in the fiftieth percentile. Right in the middle. I think people sometimes confuse "math ability" and "critical thinking". I scored in the ninety something percentile on the "critical thinking" portion of the standardized test. What would make the neurons in someone's brain light up that way?

Posted by: annette on January 14, 2004 6:02 AM

How to explain my scores then?

Correlation need not be 1.

Posted by: godlesscapitalist on January 14, 2004 6:11 AM

Hmmm. that might have come across more tersely than I intended.

All i'm saying is that in any population there are outliers. But there is absolutely no question that high mathematical ability is strongly correlated with high IQ.

more here and here.

Posted by: godlesscapitalist on January 14, 2004 6:14 AM

Well, even I know enough math to understand that answer! :)

Posted by: annette on January 14, 2004 6:15 AM

I never went beyond calculus

God, I never even got that far. There should've been a scandal at our school, given that the head of the bloody maths department was teaching the bottom class (the one I was in) and no one in the class got above 60 percent in final exams, with most students scoring well below 50.

Posted by: James Russell on January 14, 2004 7:17 AM

To do superscripts for the numbers 1, 2, and 3, you use special character codes, as listed here. There is also a SUP tag, but I've never used it, so I don't know if it works in all browsers or not.

Posted by: Jaquandor on January 14, 2004 7:23 AM

Given that I had a flaming arguement last night with my middle school age daughter about the standard notation of ordered pairs when graphing the slope of a linear equation, I am finding this post amusing this morning.

My understanding of what the Greeks had over the Egyptians in terms of math is that the Greeks had the concept of "0" as a mathmatical quantity while the Egyptians didnt. However, that doesnt seem to have bothered them when it comes to engineering, as the Pyramids demonstrate.

My other understanding is that math, or at least the basic operations of arithmatic, orginated in the need for recordkeeping in trade and in the need to maintain some sort of calendar for whatever reason. Perhaps the scratches in groups of 5 (equal to the number of fingers on one hand) were some sort of permanent record of either days or items given or received in trade. As I recall, written language orginated from a need to communicate about trade and commerce as well.

Also, I am a bit hesitant to buy into the thought that IQ and mathmatical ability are totally related since the whole concept of IQ has changed in the last few years. The kids looked at in the study may light up the moniters of an MRI machine when doing math or whatever but that is only one side of the whole picture of IQ. Giftedness might be a better term. Just a thought.

Posted by: Deb on January 14, 2004 9:35 AM

godlesscapitalist mentioned the work of Stanislas Dehaene: The Edge has his "What Are Numbers, Really? A Cerebral Basis For Number Sense" and a discussion of the article by George Lakoff, Steven Pinker, Jaron Lanier, Mark Hauser, and others. The Edge is a particularly excellent source, with regular contribuions by all the above as well as people like Denial Dennett, Richard Dawkins, Reuben Hersh, Esther Dyson, Danny Hillis, and Sherry Turkle.

Posted by: Mike Snider on January 14, 2004 9:53 AM


Regarding "advanced ring theory" and catching up to the Sumerians--the reason most people don't know advance ring theory isn't because it's so difficult; it's because it's not very useful in day to day affairs. (I, much to my surprise, have a bachelors in mathematics, and I didn't need to learn any real amount of ring theory to get it.)

It's true that the Sumerian scribes could solve quadratics--but they were a very small fraction of the population. I'm betting that the vast majority had trouble with simple arithmetic--if they were even aware it existed.

Posted by: Will Duquette on January 14, 2004 11:18 AM

"Math"? What's that?

I pretty much define math-deficient myself. The subject never spoke to me the way it seems to speak to some people. It always seemed abstract, featureless, and of no intrinsic interest, although I always enjoyed it when teachers would show "Donald in Mathemagic Land." All those Golden Sections seemed like fun, even if I couldn't figure out what was supposed to fascinate me about them.

Interesting to learn a bit about the history of math. Even there, though, I marvel at my limitations. I sometimes find I can find my way into a subject I otherwise have little aptitude for (science and econ are two examples) by first approaching it via the history-of. It's a human activity! It's got a history! People do these things! -- my brain seems able to hook into some subjects that way.

I wonder how far I could make it with a history of math, though. I forget what a quadratic equation is and don't really want to be reminded. I don't even like the language of math -- it hits my brain the same way that diagrams of circuitry do, which is to say it just doesn't speak to me, and I switch off. I'm assuming here that I might be capable of following if only I could keep myself awake, and that might well be pure self-delusion.

Hey, a propos of more or less nothing, I recall getting involved in a skirmish at some blog or other. People were going on about how calculus should be required in school, how it's good for the brain, etc. And, silly me, I dove in and tried to make the case that it's possible to have a rich and rewarding life with nothing more advanced than basic arithmatic (using myself as an example, of course). Boy, did I get stomped on. I was the enemy of education, a proponent of lowering standards. Higher math (IMHO, anything beyond "dividing" qualifies as higher math) is good, and first-class brain training, and etc etc. I quickly tried to make the case that ballet is good training too but I'm not sure it should be a required course then dove for cover. Does anyone else feel this way? That even algebra and geometry (which I semi-enjoyed and did semi-OK at, by the way) should be optional instead of required? I mean, I haven't put either one to any kind of use since I left those classes. What was the point?

To the teeny extent I manage a thought or two about math, it mostly has to do with the old conundrum: what do mathematical relationships represent? Is it some brain trick? Is it embedded in the nature of reality? When you think about music, for instance: why do the tones and patterns that we (and apparently most) people find interesting and comprehensible have such simply-stated mathematical relationships? Coincidence or not?

I remember reading a Richard Feynman book. In it, he wrote about what math meant to him as a kid. I could be misrepresenting him here, but as I recall it, he found the mental world of math reassuring, freeing -- a kind of secure playground where his thoughts and mental horsepower could really romp. Boy, the "math and numbers" world never struck me like that. Did you (or anyone else here) ever experience math that way?

Interested to hear more about the history of math, and your reflections about it.

Posted by: Michael Blowhard on January 14, 2004 11:49 AM

"But there is absolutely no question that high mathematical ability is strongly correlated with high IQ."

Isn't this because if you have any sense at all you can work your way through math?

If you are studying like a fiend to prep for a French lit test, there's a chance you might see or hear something on the bus that will cast a certain element (father-daughter relationships for instance) in a new light. This doesn’t happen with math. Not in the same way. Once you get it, you got it. (At least until the test.)

Further on the topic of correlation: I am surely not the only Blowharder with a friend or two who prefers the comfort of simple, unvarying mathematical logic to the messiness of life to the point of being dismissive of, even downright hostile to, non-pun humor and other lively exchanges.

This type is, of course yin-yanged, with those people who insist that their problems with simple addition prove they inhabit a lofty, somehow more human plane.…

The Math Gene is a fun book and, according to people who care about these things, there is considerable work on how people come to do math. (There’s also some work on how dogs do math, but it’s bogus.)

Posted by: j.c. on January 14, 2004 12:07 PM

"Such archaeological discoveries provide evidence that the idea of number…antedates civilization and writing…for artifacts with numerical significance, such as the bone described above, have survived from a period of some 30,000 years ago."

I don't know whether the book you reference talks about it, but the first example of written language we have (from Sumeria) seems to be directly derived from a system of tallying goods for commerce. Thus, mathematics is (arguably) the father of the written word.

As to Michael's question about the value of algebra and geometry to a general education:

I consider the modes of thought required for these two subjects (which are pretty radically different; many students do much better or worse in one than the other) to be fundamental tools of reasoning. Algebra requires the ability to identify and apply a general rule from situations not obviously related to that rule. Geometry is the most basic (and pure) form of deductive reasoning. Both skill sets are useful generally, even if the math does not prove of use again.

I consider these to be at least as useful, for example, as a knowledge of Shakespeare. The latter provides a limited (though somewhat distilled) knowledge about human nature, a few phrases to drop in conversation, and some information about how to tell an interesting story.

FWIW, I consider the most useful subjects in a general primary and secondary education to be mathematics (through geometry and practical probability and statistics, though probably not calculus), spelling and grammar, typing, rhetoric (both persuasive writing and public speaking), government, and a basic cultural literacy.

As might not be surprising, I do find the certainty and formality of mathematics to be reassuring, but partial differential equations are an abomination.

Posted by: Doug Sundseth on January 14, 2004 12:36 PM

Doug's reminding me of some research (ahem, more like browsing and stumbling) I did about the origins of writing years ago. The first records, at least as of when I looked into it, of written thingamajigs, weren't just math-y, they were administrative and commercial -- tallies of deals, deliveries, stockpiles, etc. Which, for a brief moment as a dreamy ex-English major, I found distressing -- I wanted to think that it would be myths, stories, heroes, archetypal things. Instead: makin' a living.

Learning that gave my view of the arts a shove in what I now think is a better direction. These days I generally find it useful to think of the arts less as a driving, originator force than as something people (often zany people) do with stuff that other fields have come up with.

FWIW, of course.

Posted by: Michael Blowhard on January 14, 2004 12:56 PM

Theo Gray, one of the co-founders of Wolfram Research, has an provocative piece here on what students should be able to do with math, among other things. The key sentence: " The most profound engine of civilization is the inability of a larger and larger fraction of the population to do the basic things needed to survive. Many people fail to realize this."

Posted by: Mike Snider on January 14, 2004 1:07 PM

I agree. And, like I say, I've never found any math beyond basic arithmatic necessary for my survival. In a country where some kids evidently get out of school unable to do basic arithmatic, let alone write a passable paragraph of English, (let alone understand the fundamentals of economics or science or history), it seems to me a bit silly to debate whether kids should be required to take calculus, and ...

But I'll shut up now.

Posted by: Michael Blowhard on January 14, 2004 1:16 PM

I'm a bit hesitant to add to the reading list, but I can't resist recommending Mario Livio's The Golden Ratio. (I'd find a purchase link for you but I don't know how to get the link to appear in comments ... I'm so simple-minded.) Dollars to doughnuts it makes your socks roll up and down.

And since Deb mentioned zero, I have to add that Fibonacci is responsible for introducing the concept of zero into Greek civilization. He learned it from the Arabs, who learned it from the Indians. I could go on and on about the importance of zero (and the importance of Fibonacci generally) but darn it I actually have work to do at work today, so it'll have to wait for the next opportunity. There are a few interesting links at my website but you'll have to look around for them.

Posted by: Dente on January 14, 2004 1:18 PM

If anyone's actually interested in seeing the links on my page, they're in the Nov. 18th entry.

Posted by: Dente on January 14, 2004 1:41 PM

unable to do basic arithmatic, let alone write a passable paragraph of English

Since I'm a smartass, it's "arithmetic" :)

As for the utility of mathematics and calculus and so on...if you want to do engineering or science of any kind, you need to know a good deal of math.

Re: math & IQ...I think many people who are good at math are math chauvinists. That is, they believe that if you are very good at math, you are probably also a good writer - but not vice versa. I think there's some truth to this, at least in the sense that a research mathematician could do a competent job as a journalist or editorialist whereas the typical journalist would be at his wit's end as a mathematician.

Posted by: godlesscapitalist on January 14, 2004 1:48 PM

Well, I have to admit that I was fairly disastrous in math in school. That may have had more to do with HOW math was presented than anything else, since I did pretty much dope out pi on my own in order to compute surface areas and figure distances on maps of imaginary planets I had created. But anything involving a teacher droning on at the front of the classroom and me in a desk towards the back, and I zoned out. Ninth Grade Algebra was as close to flunking a subject outright as I came in High School and it was a narrow escape at that.

Just to reassure myself that I have some brain cells left, I tried my hand at solving that sample Egyptian problem. Finding the value of "heap" is pretty nifty in itself, but "aha" as the word for the answer is even cooler. And I did solve the problem, which surprised me for its sophistication -- the answer ("Aha!") isn't a round number but involves a fraction. Pretty smart cookies, those Egyptians.

Posted by: Dwight Decker on January 14, 2004 1:50 PM

"...a research mathematician could do a competent job as a journalist or editorialist..."

Yikes! I don't think so. I DO think there are math snobs (not that the math gene isn't valuable or that it doesn't represent "smarts") who think ANYONE can do that English and History stuff (let alone the perceptive-enough-to-notice-nuances-in-what-is-going-on-around-them stuff). If they believe that (and not everyone "good at math" believes that) then they typically are the type who are horrendous at it.

Posted by: annette on January 14, 2004 2:59 PM

Feeling a bit impish here...

Godless, your math friends think that "being a journalist" has to do with "being a good writer"? Not that a lot of journalists aren't competent writers, and avoiding all discussion of whether journalists are generally good writers or not ...

But how about: a nose for a story? An ability to negotiate the journalism world? A feeling for hooks, angles, and human interest? An audience sense? Let alone the ability not to be too bugged by having to move from topic to topic without ever being able to make thorough sense of what you've encountered? A liking for the scene? A preference for "what's hot" over "what's not"?

Not being remotely defensive -- I'm no good at much of the above either. Just chuckling about the way some otherwise sensible people imagine that succeeding in the softer/artier occupations has to do with how well you can perform the specific task (ie., "writing"). In my experience, success in these kinds of fields has much more to do with how well you can negotiate the life and manage the intangibles. Temperament, desire, energy, connections and luck are career-makers or career-breakers in these fields much more often than talent is.

In my experience, the people who tend to succeed as journalists aren't the best writers or even the best reporters. They're the ones who are best at ... playing the journalism game, whatever that happens to consist of in a given place at a given time. Same with movies, tv, bookwriting, art, etc etc. Talent and skill at the superspecific task are necessary -- nearly everyone who's a professional in these fields is good enough. But beyond a certain competence level, differences in superspecific talent mean next to nothing.

Posted by: Michael Blowhard on January 14, 2004 3:05 PM


Actually, you know more math than the Scarecrow in "The Wizard of Oz." He mis-quotes the Pythagorean theorem at the end of the movie.

By the way, the Egyptians had their reasons (however zaney) regarding getting 1/3 by first going to 2/3 of a number and dividing by 2. They knew you could get to 2/3 by taking 1/2 and adding 1/6. Since halfing and doubling were essential to Egyptian practical math (they had big tables of halves and doubles) they apparently found it easier to do it that way. Or else they really didn't understand much about what they were doing and just repeated what other people had told them! In one papyrus a scribe spends a lot of time going through a complicated formula to show that if you want to divide a loaf of bread between 10 men, each of them should get 1/10th of the loaf! I dunno, maybe the pyramids really were designed and built by aliens!


Thanks for the links. I'll try to come up to speed. I may get back to you later for more stuff.

Mr. Russell:

Surely by now you should know that being placed in the lowest group meant that the school had already performed triage on its students and decided that you were doomed, hence, didn't waste serious instruction on you. Getting the department head as your teacher should have been a clear tipoff.


Thanks again for the tip!


Actually, the Babylonians (or Mesopotamians?) had a rudimentary place system, which, along with a generally superior mathematical notation, is why they seem to have run rings around the Egyptians. I'm guessing the Greeks learned most of their math from the Babylonians (they certainly learned lots about warfare from that source, so they were obviously in touch.)

I'm not a big expert on the ins and outs of I.Q. tests, but I will say that my 'folk psychology' always assumed that prowess in math was a pretty good indicia of high intelligence, as math seems dependent on abstract reasoning, which seems to be central to what people mean when they talk about intelligence. Of course, that's kind of a circular argument...!

Mr. Snider:

Thanks for your resource recommendations.

Mr. Duquette:

I wasn't trying to say that the average Babylonian was in advance of the average American, just that the mathematical elite of a 4,000 year old civilization was still ahead of the average man-on-the-street today. Of course, I'm sure our mathematical elite represents a far, far larger fraction of the population than in Babylonian days. Still, the persistance of the 'elite' nature of math raises questions. Is it that a little mathematical leavening will raise the entire social 'loaf'? I.e., that the vast majority of humanity didn't then and doesn't now need to know anything more than the most rudimentary mathematics, say, enough to buy groceries? Maybe.

Michael Blowhard:

I guess my point here was intended to suggest that a historical approach may work to ease people like you and I (i.e., people who are generally curious about culture but not math enthusiasts) into the study of this subject. Also to point out that while we may not be math-o-maniacs, if we are attempting to evaluate human culture the contributions of mathematicians in many ways are not only as compelling as those of the arts or literature or science, but possibly even more compelling. I guess it seems to be that you can't discuss human thought or ideas or philosophy without spending a fair amount of time on math.

Posted by: Friedrich von Blowhard on January 14, 2004 4:00 PM

A delightful post! It sounds like a fun book; I'll have to find myself a copy.

Micheal: You asked if anyone else found math to be a "playground." Well, I did and still do. In fact, I regret having let my math knowledge slide so much because I've forgotten too much to play many of the mental games I used to play. I have always been fascinated by numbers and their relationships; in some way, numbers appeal to me the way characters in a novel do. I'm not much of a visually-oriented person, but numbers give me the kind of direct, immediate aesthetic pleasure most people get from painting and sculpture.

One more thing: I was puzzled by your comment in your original post connecting the Greek fondness for indirect proofs with Greek emotionality. I don't get the link. Moreover, I would have thought that their fondness for indirect proof would have suggested the quality their neighbors most associated with Greeks: their deviousness, their sneakiness. It makes sense, doesn't it, that the descendents of the creators of the Trojan horse would come up with indirect proofs for geometric theorems?

Posted by: John Hinchey on January 14, 2004 4:53 PM

See...none of you geniuses answered my question, fer all your high falutin' math talk! The answer to what is the formula for the volume of a cylinder is:

pi (my keyboard doesn't have the symbol)r (squared) h or area of base times the height.

I also know this from my test studying. And if you didn't have that memorized, you wouldn't have shot the lights out of the score on the math side of the test yourselves! However, you may be able to run Microsoft without EVER needing to know that! I think "math thinking" and "memorization of math formulas to answer test questions" are two different things.

Posted by: annette on January 14, 2004 5:34 PM

I was a "cool kid" in middle school. I copied math homework and in grade 8 it was discovered that I couldn't do long division. My math phobia continued for a while in high school until one day when I went to kumon with a korean friend. I told my dad about it, although I was still under the strange impression that math isn't useful for anything, so I wasn't too enthusiastic about doing it. Fortunately my dad ingeniously decided to pay me money for each level I completed.

Kumon has checkpoints, unlike public school math, where just passing is often good enough and failure is ignored if you're not perceived to be a serious student. Kumon lessons aren't perfect, but they build on previous knowledge and force you to practice, whereas public school often emphasizes shortcuts and silly metaphors. Public school teachers do recognize that students have different levels of knowledge and motivation and so do sometimes give students extra attention or divide up the class using that criteria, but by then it's usually too late, it seems to me.

I learned a lot more math doing Kumon and working with Schaum's books than public school math, but looking back, they all suffer in different degrees from an overemphasis on mechanical procedure and calculation without instilling more general knowledge.

Posted by: Shai on January 14, 2004 6:21 PM

just to say something on topic, one can spend days reading this history of math page.

annette says:

"I think "math thinking" and "memorization of math formulas to answer test questions" are two different things."

right, I was browsing through my George F. Simmons Calculus and Analytic Geometry text and found this:

"Serious mathematical problem solving is mental activity on the highest level. Even exceptionally able students, who are confident they have the necessary imagination and intelligence may derive addictional comfort from the possession of a method. Such a method was derived by George Polya out of his own vast experience as an eminent creative mathematician and the foremost mathematics teacher of his generation. Polya's method consists of four simple principles that will be recognized at once as only common sense: (1) understand the problem; (2) devise a plan; (3) carry out your plan; (4) look back on your work and learn"

Learning how to think about math is a little bit like learning how to write well. People who are good at it won't be able to tell you as a novice what exactly to do, sort of like teaching the funnel method is a horrible way to teach expository writing. Simmons instead relies on tips and (sometimes not so) clear explanations of the topic at hand. Following the method he introduced above, he writes this for solving maximum-minimum problems:

"(1) Understand the Problem. Begin by reading the problem carefully, several times if necessary, until it is fully understood. It is a sad fact of life that students often seem driven to start working on a problem before they have any clear idea of what it is about. Take your time and make your efforts count.

(2) If geometry is involved -- as it often is -- make a fairly careful sketch of reasonable size. Show the general configuration. For instance, if the problem is about a general triangle, don't mislead yourself by drawing one that looks like a right triangle or isosceles triangle. Don't the hasty or sloppy. You hope your sketch will be a source of fruitful ideas, so treat it with respect.

(3) Label your figure carefully, making sure you understand which quantities are constant and which are allowed to vary. If convenient, use initial letters to suggest the quantities they represent, as A for area, V for volume, h for height. Be aware of geometric relations among the quanities in your figure, especially those involving right triangles. Write these relations down in the form of equations and be prepared to use them if needed.

(4) If Q is the quantity to be maximized or minimized, write it down in terms of the quantities in the figure, and try to use the relations in Step 3 to express Q as a function of a single variable. Draw a quick informal graph of this function on a suitable interval, perform little thought experiments in which you visualize the extreme cases, and use derivatives to discover details and thereby solve your problem."

that sort of thing.

Posted by: Shai on January 14, 2004 7:08 PM

Here's a question -- a real one, not a blog-style rhetorical one. I know that the Blowhard brothers and many comments-dwellers often recommend products from the Teaching Company or other groups that try to give you the benefit of a good education with less of the boredom. Does anyone have any such adult-education products to recommend for math? If, say, hypothetically, someone wanted to learn calculus more easily as an adult, can anyone recommend how to do so?

Posted by: williamsburger on January 14, 2004 7:47 PM

Does it ever strike you as weird that you read about a lot of guys like Steven Pinker, who study how people acquire language, but nobody who studies how you (or the rest of humanity, for that matter) acquired the ability to do math?

Just a guess...

Former literature majors vastly outnumber former mathematics majors, so perhaps it is the case that people who read serious books for pleasure are more likely to have an interest in language than in mathematics; and, therefore, the market for books about acquiring language skills would be larger than the market for books about acquiring mathematical skills?

(Please note that I'm speaking generally and hypothetically here, and that I'm certainly not saying that people who enjoy mathematics don't read seriously!)

Posted by: Steve Casburn on January 14, 2004 10:09 PM

Actually, as fascinating as the Greeks, Egyptians, Indians can be, I think the modern history of mathematics is even more interesting than the ancient stuff. Besides what had been inherited from antiquity, mathematics in late medieval and early Renaissance Europe was an arcane art. If you made a new mathematical discovery, for example, an algorithm to solve a certain kind of equation, you didn't publish the result: you kept it secret, passed it from one generation to another, and challenged other mathematicians to do what you couldn't. This only really changed when Girolamo Cardano published _Artis magnae sive de regulis algebraicis liber unus_ (The Great Art). Standard notation, i.e., all those crazy formulas you had to memorize in school, is actually a way to make mathematics publicly accessible.

There's also Descartes' invention (with a lot of help from others) of analytic geometry (using a coordinate grid with algebraic numerals instead of open Euclidean space), the related invention of calculus (where the unusual and counterintuitive uses of zero and infinity would have driven the Greeks nuts), finite math and probability, the tightening of standards of proof (which were really quite sloppy until quite late), set theory and foundationalism, Russell's Paradox, Godel's Incompleteness Theorem (which nicely states that for any mathematical system of a certain complexity, there will be true statements that can't be proved), game theory, and so on.

In one sense modern math has tended towards greater and greater abstraction: ring theory is a terrific example of this, taking rules we know from arithmetic and seeing if/how they apply to more abstract structures. But it's also moved us away from the fixed, logical-deductive structure of classical arithmetic and geomtry and towards a kind of reasoning that's more nuanced, probabilistic, and capable of engaging with the world in more complex and ultimately more useful ways. Understanding modern mathematics, how it's changed over time, and how it differs from classical mathematics is essential for understanding how our world and our modes of relating to it have changed as well. In short, math can teach you what it means to be modern (or postmodern).

And on top of that, if it helps you build a bridge, calculate your mortgage payments, or win at poker, then all the better.

Posted by: Tim on January 15, 2004 3:26 AM

Annette: Nobody needs to memorize the formula for the volume of a cylinder. You simply reason it out: it's a solid figure, its area is the same at every height, therefore its volume must be the height times that area, or πr²h. It's true that you need to know the formula for the area of a circle, which isn't obvious, but once you do you're home free.

Posted by: Aaron Haspel on January 15, 2004 11:13 AM

Nobody named Aaron needs to! Or those with the math gene. Look, I freely plead guilty to being a math moron. But you do have to at least memorize the area of a circle.

Posted by: annette on January 15, 2004 11:34 AM

williamsburger: you may want to check out either Saxon Math or Jacobs Math products. Both are widely used in the homeschool community and both go up to at least trig and possibly calculus. They are also designed to be self teaching and in my perusal of the Saxon Algebra I book I found the concepts much easier to grasp without a teacher than the standard publishing compasny one my kid is using in school. Just a thought.

Posted by: Deb on January 15, 2004 11:46 AM

My point, which I guess I made somewhat elliptically, is that mathematical reasoning looks a great deal like ordinary reasoning.

Euclid has in many ways been a disaster for math instruction, because it gives students the idea that geometry, and math in general, is an enormous frozen edifice and that you proceed from one theorem to the next only by deduction. In fact you solve math problems by analogy, by induction, by examination of border cases, by all the same means for solving any other kind of problem. Then, having solved the problem, you write the proof and present the result. The geometry student sees only the proof and never grasps how it was arrived at, which is a tragedy.

The best writer on mathematics as it's actually done is George Polya. How to Solve It can, and should, be read by complete tyros; the two volumes of Mathematics and Plausible Reasoning are for people with a little more background.

Posted by: Aaron Haspel on January 15, 2004 11:50 AM

Godless, your math friends think that "being a journalist" has to do with "being a good writer"? Not that a lot of journalists aren't competent writers, and avoiding all discussion of whether journalists are generally good writers or not ...

But how about: a nose for a story? An ability to negotiate the journalism world? A feeling for hooks, angles, and human interest? An audience sense? Let alone the ability not to be too bugged by having to move from topic to topic without ever being able to make thorough sense of what you've encountered? A liking for the scene? A preference for "what's hot" over "what's not"?

We seem to assume here that "good writers" are the people who craft the best sentences. That's seldom true. In fact, all of the qualities you've described are the hallmarks of good writing. People who have "a nose for the story" and "a sense of audience" are often evaluated as terrific writers, even if their sentences aren't exactly according to Hoyle. How else to explain the enduring reputation of Thomas Wolfe -- or Tom Wolfe, for that matter?

As for math, I suspect Aaron has found the problem: The reason we don't talk about math is because most of the math we see is strictly low-level stuff. When you get into numbers theory and so forth, mathematics becomes as creative an endeavor as poetry.

But let me also add that we talk about math all the time at 2blowhards. Economics -- the grim science of who gets what -- has a strong mathematical component. When we talk about quantifiable amounts, we're talking math. So math isn't as neglected as we might have thought. It's just slightly hidden from view.

Posted by: Tim Hulsey on January 15, 2004 3:10 PM

Typo in previous post: Second paragraph was supposed to be italicized as well.

Posted by: Tim Hulsey on January 15, 2004 3:11 PM

I always found myself a disappointment where all this formal-logic-type stuff was concerned -- math, logic, chess... Here I was, a bright kid, but with no aptitude for that stuff at all. (Big mystery to me is how I managed to do OK on math SATs -- actually better than on the English SATs. Maybe the math SATs only tested arithmatic back in those days...) I always felt that any self-respecting bright kid should be able to take off into that kind of super-abstract thinking. But symbols and diagrams (let alone words like "volume" and "polygon") have never spoken to me. My mind switches off, and might well not be capable of following even if it could stay switched on.

As far back as I can remember I wanted everything explained to me in clear English -- I could really get indignant about it. It struck me as self-evident that everything worth knowing could be delivered in clear English, so why weren't people doing that? One outcome was that I've always been pretty good at getting people to explain themselves. Oddball talent, but there it is. I've got (perhaps mistakenly) lotsa confidence in my brains and my ability to understand .... if only you'll put it in plain English. I don't know if that's in fact true (can higher math, for example, really be explained in plain English?). But I remain convinced it's true anyway.

Most of the time people seem amused that someone's interested and trying to understand, though I have run into a few scientists who were wildly paranoid about anyone who didn't measure up where math was concerned. The rest of us were evidently idiots, or perhaps it was just that they were convinced we'd misrepresent them ...

Shameful admission: I often like reading good expositions of philosophers more than the philosophers themselves. How many of them wrote such fabulous prose that they were worth wrestling with for that reason? Otherwise, my feeling is: "Would someone please explain what this is about in clear English?" [Sound of foot tapping impatiently.]

What was your experience as a brainy kid like? Where were were you strongest? I always did pretty well in everything, which probably means that I've got an adequate-enough, well-balanced brain; my good grades and test scores probably just mean that I had a pushy mother ...

Posted by: Michael Blowhard on January 15, 2004 4:01 PM

"I dove in and tried to make the case that it's possible to have a rich and rewarding life with nothing more advanced than basic arithmetic (using myself as an example, of course)." Getting back to this - I am mathematically challenged enough to be unsure of what you mean by "basic arithmetic." It does seem that a bright society would endeavor to make sure its rug rats were forced to learn enough cyperin' and science to understand stats and how to negotiate through swamps of data.

Do John Gardner's math books count as history? All I remember is enjoying them in a lost way...

Posted by: j.c. on January 15, 2004 5:28 PM

Funny, what a load of...stuff is beaten into your head - the formula of volume of cylinder popped up immediately - teacher, teacher, don't you see my raised hand?
Aaron in his explanation reminded me of the method my father would teach me - don't try to memorize it, apply logic [any miniscule resources of it you might scramble]. I remember, somewhere in 8th grade, when I would struggle with particular calculus problem trying to apply algorithm from immediately preceding lesson and not succeeding, my dad in 10 min. would find three different ways to solve it - applying that exact method - and being creative person he is. (Alas, our parents' expectations - and our inadequate performance!)Calculus in 8th grade? Well, don't think Russian schools were raising geniuses - simply last grade in school was 10th, not 12th like here.
I remember also having many truly joyful moments when I'd find an elegant solution in my mind w/o textbook direction. Thing of pure beauty, math - like ice architecture...
Well, not for long - although I managed to graduate high school with practically all "A"s and 1 math year in college was a song, 2nd required some effort, and 3rd was near fiasco.(I remember on final exam -was it on Matrixes?- my professor Mr. Pak looking with disgust at my paper and saying - But this is trivial... And what did you say you are studying here for?Engineering? Why don't you get married instead?)

Interesting though, that now I recall that in high school math classes we were actually given a sense of the history of math, like biographical facts of Lobachevsky's, Eichler or the story of Marie Sklodovsky-Curie Nobel prize. Not that I remember finest details of it now, of course, but that was a highlights of the boring lessons - for me, at least.
My homework scenes with my dad actually came to bite me in the ass - my son seem to inherit my dad's ability, got 800 on math on SAT and I, having 2 degrees, shamefully can't follow even on his Stuy math level...

Posted by: Tatyana on January 15, 2004 7:43 PM

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