I didn't have time to skim this, let alone read it, but each chapter is coherently summarized both at the beginning and the end. Also, the pictures have good captions. It's a pity to reward this clarity by remembering only scraps.
Still, these are the scraps I have:
It's astounding how long the repetitive, empirical, pre-deductive* mathematical tradition lasted with very little change: more than a millenium, easily, of what has to have been person-to-person transmission. It is not really astounding that surveying was a living tradition that long, but it's surprising in hindsight that there was so little innovation when the techniques were so laborious. This is, I imagine, partly due to legal precedent not wanting innovation, as in the law now; but perhaps it's the necessary 'long tail' to an exponential rate of change. (Dyed linen string, from wild flax, seems to be at least 30,000 years old; possibly one of the things that got us through the Ice Ages.)
There were clearly numerate female workers, and this is often followed up by comments that they may have always worked for women, not for men. But look, that means there were a steady supply of women who owned and ran farms and other businesses (as is clear from some of the records).
The image on the cover is of a woman, a goddess of measurement --fans will recognize the image as 'carrying a 1 and a 0'. Well, no, she's holding a rope and a stick, because geometry came first and good measurement was one measure of a good state.
Officially, one of the things that didn't change was prices and work rates and workers' wages, and many of the surviving records seem to be back-calculating one of these from others. I couldn't tell if the evidence was that wages were actually changing under the table, or unchanging and immiserating various parts of the population, or what.
Spreadsheets got invented in cuneiform! Rather late in its span, and much of the innovation may have been in a very few offices, but still. Row totals checked against column totals, and explanatory comments in some cells. Very clear even in the pictures of the clay tablets.
back to that 'pre-deductive' dig; the last chapters take on the Accepted Belief that all was rote memorization of slightly wrong formulae until the Greeks axiomatized and brought light. I was really skimming at this point, and it seems to be a delicate and contentious argument, but it looked to me as though there was -- very close in time -- the beginnings of innovation among the last of the workers in cuneiform, and evidence that no-one heard what the brilliant Greek dilettantes were doing for a while. Perhaps perhaps it was an age in which innovation was bound to happen, because the exponential curve was ticking upwards (on what process? Trade? Climate change? Accumulation of experience? What?) and it would happen everywhere, as with the calculus.
Find in a Library: Mathematics in Ancient Iraq
The introduction argues that mathematical illiteracy is the current equivalent of the Civil Rights struggle in the 1960s. The next several chapters are a narrative of the Civil Rights struggle, by one who was there and remembers people who were murdered for it, just to make clear how serious an analogy he's making. The end and heart of the book are a thick, much-quoting description of the development and principles of the Algebra Project. This last not only teaches more kids more math faster than most schools, certainly poor schools with minority students, manage; but it does so by getting them to love math, to play with it, to demand more algebra classes -- sometimes to demand that the uninterested teachers just get out of their way. The students have to care, and then the teachers, and then the administrators (some of whom clearly saw this as yet another insurrection).
It sounds enviable, and raises test scores, and may be teachable (more by Each one teach one than by professional seminars). It sounds exhausting and exhilarating. It also doesn't seem to be growing very quickly, if I understand their website correctly; I hope that's only places they are currently teaching new people, not all schools using the system. The NSF and a fair cut of professional mathematicians support them.
And it rises out of formal philosophy as well as lived philosophy -- Moses wrote his doctoral thesis on "the history and insights of's philosophy and math, and one of Quine's insights turned out to be of direct relevance and importance to the teaching of school mathematics." That would be the 'regimentation of ordinary discourse', or, getting from the idea of quantity to the idea of vector (on the T; truly, the importance of public transportation is hard to exaggerate) to algebra.
Find in a Library: Radical Equations
Evolutionary Theory: Mathematical and Conceptual Foundations, by, tries to not use any more math than is needed. This only holds off the PDEs until the case of multiple alleles at one locus, p. 24.
Theoretical Ecosystem Ecology: Understanding Element Cycles,and , is PDE-free all the way to a concept of substrate quality, page 37. Those aren't bad, actually; the persnickety equations are often just ODEs.
Theoretical... compares its results to quite a lot of empiric data, considering that it's a short book on mathematics. Also, there's some odd, odd Nordic poetry (in translation) and some worked exercises; very useful, and I must get back to it after this quarter's class in PDEs.
Somewhere else asserts nitrogen:food-energy:water::carbon:fossil-fuel-energy:air. In the soil, it's more like potential and ?metabolic? energy, but they're still wonderfully linked.
ISBN: 0521580226 (Theoretical...)
LCCN: QH 344 A35 1996 (Theoretical...)
ISBN: 0878937021 (Evolutionary...)
LCCN: QH 366.2 R523 2004 (Evolutionary...)
Subtitle: Making the Right Connections
The game Hex, which can be explained in a paragraph and played on (say) bathroom tiles, is quite hard despite being perfectly deterministic; like Go. (Not that I can beat the Java applet version; maybe after I read the book, which is mostly about strategy with a lot of alternate versions of the game, including Hex on a torus.)
John Nash was one of the two simultaneous inventors, so in English the game was called "Nash" for a while except when played on those bathroom tiles, when it was called "John".
Odd that two people should have come up with such a simple game at once. (The other is Piet Hein.) Browne points out that it has a little to do with the four-color problem, too, in a metaphorical, simplifying way. It's more directly descended from game versions of maintaining or destroying network (circuit) connectivity.
There are little exercises-for-the-reader in the beginning of the book, some dating from the Victorian heyday that produced questions one could illustrate; but as the history progresses more of the book is about the people than about the increasingly abstruse problem. It does all wind towards the political or philosophical question that the long, computer-calculated proof produced; to quote Wilson, half the mathematicians at a conference
...could not be convinced that a proof by computer was correct...[half] could not be convinced that...700 pages of hand-calculations could be correct.
Those are not exclusive opinions, grumps the empiric.
Did discrete math look like one body of inquiry before computers? Was it called something else, or did it suffer from the simplicity with which many of its problems can be stated? I wonder only because, on laughably cursory examination, the discrete section in the math library is short and shiny. Maybe the aged classics are in the computer science library. (Not an explanation I often try, that last sentence. I wonder what the oldest book in the CSci library is.)
A Beginner's Guide to Discrete Mathematics,, has nothing explicit about map coloring but, of course, lots of simple graph theory, Hamiltonian cycles, Boolean circuits. Discrete Mathematics: Elementary and Beyond, , , , does mention the four-color theorem (and lots else, including more crypto). The prose in the latter is distractingly perky and humorous, and it's a bit more mathematical and maybe slightly less aimed at CSci than the former. Both provide the puzzles I missed in Wilson.
Shorter versions of the four-colors-suffice proof are already appearing, but that's not where the glory is.
ISBN: 0691115338 (Four Colors Suffice)
ISBN: 0817642692 (A Beginner's Guide...)
ISBN: 0387955852 (Discrete Mathematics:...)
From Matilde's comment on Invisible Adjunct, I was led to a history and summary of the algorithm used to match residents to hospital slots. Nice to have these things clear - or "Clearing", as the UK system for getting into college seems to call it. The algorithm is easy to understand; the old and basic example is stated in terms of boys proposing to girls who keep an eye out for better engagements. Still, startling to find so blunt a summary of old-fashioned sexual mores and politics:
Gale and Shapley also showed that the match achieved in this manner has a remarkable property: It is "boy-optimal" and "girlpessimal," meaning that each boy is matched to the best girl he can get in any stable matching, while each girl ends up with the worst possible guy. (I leave this as an easy exercise for the reader's morning commute.) Of course, the corresponding algorithm that has the girls proposing achieves the opposite, prompting some reflection on real-life dating conventions.
Another exercise is to show that it's possible for those on the side that's not proposing to "game the system." By lying about her preferences, a girl can do better in the male-proposing algorithm than she would otherwise.
I need to tidy up whatever is preventing this blog from having several categories for one post, because it's not all that often I can categorize something at once as math and 19th c. fiction.
Locksley Hall gave me my title.