Neuclid on the Block

How many blows does it take to demolish a wall with a hammer? It depends on the wall and the hammer, of course. If the wall is reality and the hammer is mathematics, you can do it in three blows, like this:

α’. Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.
β’. Γραμμὴ δὲ μῆκος ἀπλατές.
γ’. Γραμμῆς δὲ πέρατα σημεῖα.

1. A point is that of which there is no part.
2. A line is a length without breadth.
3. The extremities of a line are points.

That is the astonishing, world-shattering opening in one of the strangest – and sanest – books ever written. It’s twenty-three centuries old, was written by an Alexandrian mathematician called Euclid (fl. 300 B.C.), and has been pored over by everyone from Abraham Lincoln to Bertrand Russell by way of Edna St. Vincent Millay. Its title is highly appropriate: Στοιχεῖα, or Elements. Physical reality is composed of chemical elements; mathematical reality is composed of logical elements. The second reality is much bigger – infinitely bigger, in fact. In his Elements, Euclid slipped the bonds of time, space and matter by demolishing the walls of reality with a mathematical hammer and escaping into a world of pure abstraction.

An ancient fragment of Euclid's Elements (Book II, 5) (c. 100 A.D.)

An ancient fragment of Euclid’s Elements (Book II, 5) (c. 100 A.D.)

Consider his first ὅρος, or definition: “A point is that of which there is no part”. Taken literally, that definition is meaningless. If something has no parts, it’s nothing: it doesn’t exist. Definition two is equally ridiculous and equally impossible: “a line is a length without breadth”. How can you have length without breadth? He’s at it again in definitions five and six:

ε’. ᾿Επιφάνεια δέ ἐστιν, ὃ μῆκος καὶ πλάτος μόνον ἔχει.
ζ’. ᾿Επιφανείας δὲ πέρα τα γραμμαί.

5. A surface is that which has length and breadth only.
6. The extremities of a surface are lines.

Neither points nor lines nor surfaces, as defined by Euclid, can be found in the real world. Physically speaking, these concepts are impossible. It’s worth asking, then, how human beings ever came to think of them. It wasn’t by direct experience, but it was by abstraction from experience. From the imperfect, three-dimensional points, lines and surfaces of the physical world, early mathematicians abstracted the concepts of points without parts, lines without breadth, and surfaces without height. Euclid was summarizing and refining the work of his predecessors and we don’t know how long early mathematics and its concepts were in development. Without the discovery of lost manuscripts or the invention of a chronoscope or time-machine, it may remain impossible to say. But that early mathematical abstraction clearly depended on language, although language is not, in itself, an explicitly mathematical medium. Human beings cannot exist without language, which is a sine qua non of our species. But we can exist very happily without explicit mathematics and even explicit numbers:

The Pirahã language and culture [in the Amazon region] seem to lack not only the words but also the concepts for numbers, using instead less precise terms like “small size”, “large size” and “collection”. And the Pirahã people themselves seem to be surprisingly uninterested in learning about numbers, and even actively resistant to doing so, despite the fact that in their frequent dealings with traders they have a practical need to evaluate and compare numerical expressions. A similar situation seems to obtain among some other groups in Amazonia, and a lack of indigenous words for numbers has been reported elsewhere in the world. (Language Log, viâ Information Processing)

However, all languages offer their speakers the ability to abstract from reality and to create ideas and images that are physically impossible. Indeed, the essence of language is abstraction, because language is symbolic. A word is not what it represents. And because words are not realities, they can be combined and altered in ways that realities cannot. Beanstalks and clouds, boys and thumbs, hair and towers are all realities, and stand in more or less fixed relation of size, position and so on. But once those realities are abstracted into words, those fixed relations become fluid. Beanstalks that reach the clouds; boys no bigger than thumbs; hair that hangs the full height of a tower. All of these are impossible, but can work within the imaginary world of a story. They obey an internal logic, if not the logic of reality. I suggest, then, that those early definitions in Euclid were following the example of the folk-tales and myths that preceded them. Myths are full of distortions of size, speed and strength. Euclid was, in a sense, applying mythic techniques to points, lines and surfaces.

But there is an important difference between myths and maths: something that might be called a Neuclid on the block. A myth can be pictured in the mind; Euclid’s abstractions cannot. Think of the transformations Zeus underwent in pursuit of beautiful girls: he turned himself into a shower of gold, a bull, a swan, and so on. We can picture each and see its logical consequences. Euclid’s abstractions cannot be pictured like that. Try it. A point without parts? A line without breadth? I can’t picture them. I’m not even sure I can understand the concepts. But I can understand their logical consequences and implications and follow some of the uses Euclid makes of them as, using bricks of axiom and logic, he builds his vast mathematical temple. The abstractions of “Zeus-as-shower-of-gold” and “line-without-breadth” have something in common and the latter abstraction, I suggest, relies on the mental abilities underlying the former. Human beings had to evolve language before they could create mathematics. But “line-without-breadth” is not merely abstract: it is unimaginable. And that is the new thing: the butterfly of mathematics emerging from the chrysalis of language and taking flight into a world inaccessible to language.

And still inaccessible to many human beings, however intelligent they are. Although all brains are mathematical engines, whether they belong to physicists or to poets, they run on implicit mathematics and geniuses of language are not necessarily competent at maths. Or even aware of its existence. Have poets like Homer or novelists like Dickens been aware of the mathematical nature of their creations? There seems little sign of it. This disconnection is even more striking in music. Leibniz remarked that music is an exercitium arithmeticæ occultum nescientis se numerare animi – “a hidden arithmetic of the soul, which knows not that it calculates”. A symphony is a giant mathematical structure, but that symphonic master Beethoven (1770-1827) “never learned his tables and has left us with a bill scrawled with ‘7 + 7 + 7 + 7 + 7 + 7 + 7 = 49’” (Little Wilson and Big God, 1987, Anthony Burgess, “One”, pg. 60). Whether, with the right education, Beethoven could have been a great explicit mathematician, as well as a great implicit one, is an interesting question. But I don’t think it was a coincidence that he, a world-historic genius of music, had the same mother-tongue as Karl Gauss (1777-1855), a world-historic genius of mathematics. Nor do I think it a coincidence that Europe has given birth to the richest and most complete systems of both music and mathematics. Human beings can survive without explicit mathematics and did so for many thousands of years. This raises an important question: Why and when did explicit mathematics begin to emerge?

That is partly a genetic question, in my opinion. There are certain psychological prerequisites for the creation of mathematics, amongst them high intelligence. And these traits must occur in a significantly large fraction of a population. According to proponents of HBD, or Human Bio-Diversity, which seeks biological explanations for political and cultural patterns, high intelligence is not evenly distributed among all human groups. There is a simple reason: because the genes responsible for high intelligence are not evenly distributed either. The white mathematician and HDBer John Derbyshire warmed up for his expulsion from National Review in April 2012 by offending the black mathematician Jonathan Farley:

John Derbyshire, a columnist for the National Review, wrote an essay last week implying that black people were intellectually inferior to white people: “Only one out of six blacks is smarter than the average white.” Derbyshire pulled these figures from a region near his large intestine. One of Derbyshire’s claims, however, is true: that there are no black winners of the Fields medal, the “Nobel prize of mathematics”. According to Derbyshire, this is “civilisationally consequential”. Derbyshire implies that the absence of a black winner means that black people are incapable of genius. In reality, black mathematicians face career-retarding racism that white Fields medallists never encounter. Three stories will suffice to make this point. (“Black mathematicians: the kind of problems they wish didn’t need solving”, The Guardian, Thursday 12nd April, 2012)

I don’t think Farley makes his point and he’s wrong about Derbyshire’s implication. Derbyshire does not imply that black people are incapable of genius, rather that they are capable of less of it. Which is indeed “civilisationally consequential”. While attempting to prove that white racism has suppressed black genius (but not Chinese or Indian), Farley says this:

The second story involves one of the few black mathematicians whom white mathematicians acknowledge as great – or, I should say, “black American mathematicians”, since obviously Euclid, Eratosthenes and other African mathematicians outshone Europe’s brightest stars for millennia. (Ibid.)

That last claim is true, but disingenuous or dishonest. Geographically it’s right, genetically it’s wrong. Euclid, Eratosthenes et al might have lived in Egypt, which some would call part of Africa, but they were not genetically African. Instead, they were post-African, descendants of human beings that had migrated from Africa. Some of those migrants evolved higher intelligence in the new environments they entered. The wheel was invented outside Africa. So were writing and mathematics. In fact, writing was invented in Mesopotamia to a proto-mathematical end: to record the number and type of livestock and other possessions. Spoken symbols manipulated by instinct became written ones manipulated by conscious choice. In short, air became ink. Or marks on clay, in those early days. How important is that for mathematics? I think it must be very important: writing must be one of the foundations of maths. Another foundation must be, well, foundations. That is, architecture: the creation of substantial structures in permanent and semi-permanent materials like stone, clay and wood. Architecture demands design: the symbolic representation and manipulation of lengths, areas, loads and so on.

Underlying both accounting and architecture was, of course, agriculture: cities were supported by fields and flocks; scribes kept accounts of the flocks and fields. Mesopotamia, the cradle of agriculture, writing and architecture, was also the cradle of mathematics. A symbol for zero, or for a proto-zero, was even in use there. This was another hammer-blow against the wall of reality. Zero is both very familiar and very strange. Think about it: a symbol for nothing. But then “nothing” is a symbol for nothing: like the words “cat” or “Cthulhu”, the word “nothing” is an acoustic or graphical pattern pointing at an image or concept. But languages are much more than collections of symbols: they contain rules for the ways in which symbols can be combined and altered. One way to alter a symbol is to negate it: all languages contain the concept of negation and can express the idea of a-thing-that-is-not-so. “There is no fruit on this tree.” “I didn’t catch a fish today.” Human beings must have been expressing ideas like that for many millennia and negation is an essential part of logic, which is an essential part of maths. If X, then Y. If not Y, then not X.

But from “if X, then Y”, it does not follow that “if not-X, then not-Y”. Logic has subtleties and traps and although all languages have ways of expressing logical conditions, logic itself was not explicitly studied and formalized until the early centuries before Christ, when men like Aristotle (384–322 BC) defined and analysed some of its terms and propositions. Logic, abstraction, negation, counting: these were prerequisites of full mathematics. The first three are dependant on language, but the ability to count and to compare one quantity against another is not unique to human beings:

Monkey math: Zoo baboons shed light on the brain’s ability to understand numbers

Opposing thumbs, expressive faces, complex social systems: it’s hard to miss the similarities between apes and humans. Now a new study with a troop of zoo baboons and lots of peanuts shows that a less obvious trait — the ability to understand numbers — also is shared by man and his primate cousins. “The human capacity for complex symbolic math is clearly unique to our species,” says co-author Jessica Cantlon, assistant professor of brain and cognitive sciences at the University of Rochester. “But where did this numeric prowess come from? In this study we’ve shown that non-human primates also possess basic quantitative abilities. In fact, non-human primates can be as accurate at discriminating between different quantities as a human child.”

[…] Cantlon, her research assistant Allison Barnard, postdoctoral fellow Kelly Hughes, and other colleagues at the University of Rochester and the Seneca Park Zoo in Rochester, N.Y., reported their findings online May 2 in the open-access journal Frontiers in Psychology. The study tracked eight olive baboons, ages 4 to 14, in 54 separate trials of guess-which-cup-has-the-most-treats. Researchers placed one to eight peanuts into each of two cups, varying the numbers in each container. The baboons received all the peanuts in the cup they chose, whether it was the cup with the most goodies or not. The baboons guessed the larger quantity roughly 75 percent of the time on easy pairs when the relative difference between the quantities was large, for example two versus seven. But when the ratios were more difficult to discriminate, say six versus seven, their accuracy fell to 55 percent. (Monkey Math, 3rd May, 2013)

Brains, whether molluscan or mammalian, are mathematical engines: that is, they are mechanisms for processing sense-data about size, quantity, weight, speed, colour and so on. Intelligence is a measure of the speed, efficiency and accuracy of this processing. Monkeys are more intelligent than molluscs and men are more intelligent than monkeys. This says something about how the math-mechanism, or the brain, differs between these groups. But in only one group, human beings, can the math-mechanism run on explicit numbers: “the human capacity for complex symbolic math is clearly unique to our species.”

But some human groups are clearly better at maths than others. This says something about brain-differences and those brain-differences say something about different paths in evolution. Some groups have a long history of agriculture and some don’t, for example. Agriculture created a new environment, with new benefits and new costs. Inter alia, numeracy is beneficial to farmers and innumeracy is costly. So are genes for numeracy favoured among farmers? It seems likely:

32:13 And he lodged there that same night; and took of that which came to his hand a present for Esau his brother; 32:14 Two hundred she-goats, and twenty he-goats, two hundred ewes, and twenty rams, 32:15 Thirty milch camels with their colts, forty kine, and ten bulls, twenty she asses, and ten foals. (Book of Genesis)

Numbers are very important in the bronze-age pastoralism of the Old Testament, which is based on Mesopotamian culture and myth. Farmers need to be able to count and when that numeric ability is put to other uses, its agricultural roots are still sometimes clear:

4,729,494 occurs as a coefficient in the famous cattle problem attributed to Archimedes. The problem concerns the number of the cattle of the sun, which were divided into 4 herds of different colours, milk white, glossy black, yellow and dappled. 8 conditions then describe the numbers of bulls and cows in each herd. The text is actually ambiguous; it is unclear whether a certain number is to be made square or merely rectangular. If it has to be a square, then this equation appears:

t^2 – 4729494u^2 = 1

… Amthor calculated that the least solutions to this equation are:

t = 109,931,986,732,829,734,979,866,232,821,433,543,901,088,049
u = 50,549,485,234,315,033,074,477,819,735,540,408,986,340.

and that in this case the total number of cattle is a number of 206,545 digits, starting 7766… This number has recently been churned out by computer [before 1986], of course, taking a mere 46 and a bit pages of printout. It is unlikely that Archimedes could have found such a solution, though he may well have known how to solve this type of equation in principle, and he was interested in very large numbers. (The Penguin Dictionary of Curious and Interesting Numbers, 1986, David Wells, entry for 4,729,494, pg. 187)

But would Archimedes have been interested if he had not been descended from farmers, from a long line of men whose ability to have children depended on their ability to count and compare? Note too that Archimedes, like other Greek mathematicians, was not only interested in very large numbers: he was interested in infinity, or the concept of counting without limit. Infinity is another abstraction, another escape from the physical world. Human beings can’t truly understand it, but they can prove things about it. For example, they can say with certainty that prime numbers never end: there is an endless supply of numbers that cannot be evenly divided by any number but themselves and 1. This is another astonishing thing. Here is Euclid discussing the primes and striking more hammer-blows at reality:

prime_proof_Greek
Οἱ πρῶτοι ἀριθμοὶ πλείους εἰσὶ παντὸς τοῦ προτεθέντος πλήθους πρώτων ἀριθμῶν. ῎Εστωσαν οἱ προτεθέντες πρῶτοι ἀριθμοὶ οἱ Α, Β, Γ· λέγω, ὅτι τῶν Α, Β, Γ πλείους εἰσὶ πρῶτοι ἀριθμοί. Εἰλήφθω γὰρ ὁὑ πὸτῶν Α, Β, Γ ἐλάχιστος μετρού μενος καὶ ἔστω ΔΕ, καὶ προσκείσθω τῷ ΔΕ μονὰς ἡ ΔΖ. ὁ δὴ ἤτοι πρῶτός ἐστιν ἢ οὔ. ἔστω πρότερον πρῶτος· εὐρημένοι ἄρα εἰσὶ πρῶτοι ἀριθμοὶ οἱ Α, Β, Γ, ΕΖ πλείους τῶν Α, Β, Γ. ᾿Αλλὰ δὴ μὴ ἔστω ὁ ΕΖ πρῶτος· ὑπὸ πρώτου ἄρα τινὸς ἀριθμοῦ μετρεῖται. μετρείσθω ὑπὸ πρώτου τοῦ Η· λέγω, ὅτι ὁ Η οὐδενὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός. εἰ γὰρ δυνατόν, ἔστω. οἱ δὲ Α, Β, Γ τὸν ΔΕ μετροῦσιν· καὶ ὁ Η ἄρα τὸν ΔΕ μετρήσει. μετρεῖ δὲ καὶ τὸν ΕΖ· καὶ λοιπὴν τὴν ΔΖ μονάδα μετρήσει ὁ Η ἀριθμὸς ὤν· ὄπερ ἄτοπον. οὐκ ἄρα ὁ Η ἑνὶ τῶν Α, Β, Γ ἐστιν ὁ αὐτός. καὶ ὑπόκειται πρῶτος. εὑρημένοι ἄρα εἰσὶ πρῶτοι ἀριθμοὶ πλείους τοῦ προτεθέντος πλήθους τῶν Α, Β, Γ οἱ Α, Β, Γ, Η· ὅπερ ἔδει δεῖξαι. (Στοιχεῖα θ’κ’)


prime_proof_Roman
The prime numbers as a whole are more numerous than any assigned set of prime numbers. Let A, B, C be the assigned prime numbers. I say that the full set of prime numbers is more numerous than A, B, C. For let the least number measured by A, B, C be taken, and let it be DE [Prop. 7.36]. And let the unit DF be added to DE. So EF is either prime, or not. Let it, first of all, be prime. Thus, the set of prime numbers A, B, C, EF has been found, and it is more numerous than A, B, C. And so let EF not be prime. Thus, it is measured by some prime number [Prop. 7.31]. Let it be measured by the prime number G. I say that G is not the same as any of A, B, C. For, if possible, let it be one of them. And A, B, C all measure DE. Thus, G will also measure DE. And it also measures EF. So G will also measure the remainder, unit DF, despite being a number [Prop. 7.28]. This is absurd. Thus, G is not the same as any of A, B, C. And it was assumed to be prime. Thus, the set of prime numbers A, B, C, G has been found and it is more numerous than the assigned set of prime numbers, A, B, C. And this now stands shown. (Elements, Proposition 9.20)

That proof depends on logic and concepts like negation, but proof itself, like points without parts, was another Neuclid on the block. Mesopotamian mathematics was practical and conducted for worldly purposes, like the Egyptian, Phoenician and Hebrew mathematics it influenced. The Babylonians knew by experience that the hypotenuse on a right-angled triangle always seemed related to the other two sides by the rule a^2 = b^2 + c^2. The Greeks proved that this is true of all possible right-angled triangles. That’s why the proof is called Pythagoras’ theorem. The Babylonians also discovered by experience that the circumference of a circle is a little over three times its diameter. They bequeathed their discovery to others:

In the Old Testament, I Kings 7:23 implies that π is equal to 3. The Babylonians about 2000 BC supposed that π was either 3 or 3 1/8 [= 3·125]. The Egyptian scribe Ahmes, in the Rhind papyrus (1500 BC), stated that the area of a circle equals that of the square of 8/9 [sic] of its diameter, which makes π equal to (16/9) squared or 3·16049… (The Penguin Dictionary of Curious and Interesting Numbers, 1986, David Wells, entry for π, pg. 48)

Here is the π-passage in the Old Testament:

7:23 And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about. (The First Book of the Kings)

The Greeks, by contrast, knew that π was much more elusive, something to be caught in a finer and finer net:

Archimedes, by calculating the areas of regular polygons with 96 sides, determined that π lay between 3 10/71 = 3·14085… and 3 10/70 = 3·142857… Archimedes also discovered more accurate approximations to the value of π. This last value is 3 1/7 or 22/7, known to generations of schoolchildren. It is also the best approximation to π using the ratio of two numbers less than 100. In binary π = 11·0010010000111111011… This can be rounded to the repeating decimal 11·001001001…, which is equal to 3 1/7. (The Penguin Dictionary of Curious and Interesting Numbers, 1986, David Wells, entry for π, pg. 49)

Archimedes was interested in numbers for their own sake, not for their practical importance in agriculture or architecture. But their practical importance, as translated into material success and offspring, presumably underlay the genes that allowed him to conduct pure mathematics of such an abstract and impractical kind. Pure mathematics is not a path to material success and offspring, but there may be populations and sub-populations that have undergone a kind of mating-for-mathematical-ability. If success as a merchant or scholar leads to relatively more offspring, the average intelligence of a population will rise and with it the percentage of mathematically gifted individuals. This may have been true of the Ashkenazi Jews in Europe and the Brahmin, or priestly castes, in India. Ashkenazi Jews have supplied disproportionate numbers of mathematicians, physicists and chess-masters and have the highest average IQ of any substantial racial group.

IQ, or the intelligence quotient, is a measure of g, the general factor of intelligence. Although it is controversial to express intelligence as a single number or factor, g is both predictive and statistically robust. There are strong correlations between success at apparently different cognitive tests, perhaps because g expresses the mathematicality of the brain: its power, speed and efficiency as a mathematical mechanism. All activities, at root, are mathematical, from poetry and music to cosmology and gambling. Higher intelligence tends to lead to greater success in any given activity, perhaps because higher intelligence means, more or less explicitly, greater mathematical ability and so greater ability to process the mathematical or crypto-mathematical data involved in composing a poem or symphony or theorizing about the origins of the universe. But mathematics can demolish the walls of the universe and escape not simply reality but imagination. Using maths, mind can defeat more than matter: it can defeat itself. Euclid’s hammer is mightier than a million suns.


Note: The Greek text of Euclid is taken from a pdf edition of Euclid’s Elements of Geometry (2007-8), edited by Richard Fitzpatrick, and the English translation is adapted from Fitzpatrick’s.

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