Here’s Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math – Alex Bellos

Thoughts: It’s been more than a year between when I finished this and when I finished converting my sticky notes into typed-up notes, but I remember quite enjoying Here’s Looking at Euclid. Nothing particularly deep or astonishing, but a good deal of good fun.

(The notes below are not a summary of the book, but rather raw notes - whatever I thought, at the time, might be worth remembering.)

Bellos, Alex. 2010. Here’s Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math. Free Press.

Chapter Zero: A Head for Numbers

• 3-4: discusses the research of Pierre Pica, who did field work studying the language of the Munduruku in the Amazon. Their counting system has words for natural numbers up to five
  • j: to what extent do examples like these lend credence to the Linguistic Determinism hypothesis?
  • cf Cohn 2019, noting that “the tendency to subjectively metricize intermediate [pulses]… may be linked to human capacities for numerosity, as 3 is the point at which pre-verbal subitization begins to attenuate, eliding into mental systems informed by higher-level mental functions of verbalization and calculation”
• 5-6: the Munduruku (with their lack of counting words) tend to experience number logarithmically - if asked to put a mark on a line indicating the number of items of a thing, marks corresponding to 2 and 3 will be much further from each other than those corresponding to 8 and 9.
  • 6: “It turns out that the logarithmic approach is not exclusive to Amazonian Indians. We are all born conceiving of numbers this way. In 2004, Robert Siegler and Julie Booth at Carnegie Mellon University in Pennsylvania presented a similar version of the number line experiment to a group of kindergarten pupils…, first graders… and second graders. The results showed in slow-motion how familiarity with counting moulds our intuitions. The kindergarten pupil, with no formal math education, maps numbers out logarithmically. By the first year at school, when the pupils are being introduced to numbers words and symbols, the graph is straightening. And by the second year at school, the numbers are at last evenly laid out along the line.”
• 17: Bellos cites a study that “found a strong correlation between a talent at reckoning and success in formal math. The better one’s approximate number sense, it seems, the higher one’s chance of getting good grades. This may have serious consequences for education. If a flair for estimation fosters mathematical aptitude, maybe math class should be less about times tables and more about honing skills at comparing sets of dots.”

Chapter One: The Counter Culture

• 37: “the irregularities of Western number words… Are of extreme interest to mathematical historians. The French for eighty is quatre-vingts, or four-twenties, indicating that ancestors of the French once used a base twenty system. It has also been suggested that the reason why the words for ‘nine’ and ‘new’ are identical or similar in many Indo-European languages, including French (neuf, neuf), Spanish (nueve, nuevo), German (neun, neu), and Norwegian (ni, ny) is a legacy of a long forgotten base eight system, in which the ninth unit would be the first of a new set of eight. Number words are also a reminder of how close we are to the numberless tribes of the Amazon and Australia. In English, ‘thrice’ can mean both three times and many times; in French, trois is three and très is very: shadows, perhaps, of our own ‘one, two, many’ past.”
• 43: j: lots of interesting stuff about abacus. Should learn more about doing calculations by abacus at some point.

Chapter Two: Behold!

• 52: The pythagorean theorem holds not only for the areas of squares set against the sides of a right-angle triangle, but also for semicircles, pentagons, etc. - j: presumably, for any given 2D shape

Chapter Three: Something About Nothing

• -91: Bellos describes the Vedic vertically and crosswise or “cross-multiplication” method of multiplication.
  • 91: it’s “faster, uses less space and is less laborious than long multiplication. Kenneth Williams told me that when he explains the Vedic method to school pupils they find it easier to understand.”

Chapter Four: Life of Pi

Chapter Five: The X-Factor

• 129: Log tables are useful because multiplication can be converted into addition plus a few lookups into a log table. If you add the logs of two numbers, the sum will be the log of the product.
  • 131: if you have a line with numbers marked out along a log scale, and want to multiply two numbers, use a compass to measure out one of the numbers, then slide the compass along the line until the first end is on the mark for the second number. The second end of the compass will be on the product of the two numbers
  • 131-132: this is the principle by which slide rules operate, except slide rules have two rulers marked with log scales that slide past one another, obviating the need for the compass.

Chapter Six: Playtime

• 154: “to date, no one has found a sudoku that has a unique solution with fewer givens than 17.”
• 159: Book mentioned: semiotician Marcel Danesi’s 2002 The Puzzle Instinct
• Another book of puzzles: Claude Gaspard Bachet’s 1612 Amusing and Entertaining Problems that can be Had with Numbers, (very useful for inquisitive people of all kinds who use arithmetic)
• 170-171: discussion of Martin Gardner, something of a hub in the world of recreational mathematics, and has written books on math and other topics

Chapter Seven: Secrets of Succession

Chapter Eight: Gold Finger

Chapter Nine: Chance is a Fine Thing

• 223: “the difference between insurance and gambling in casinos is that in casinos you are (one hopes) gambling with money you can afford to lose. With insurance, however, you are gambling to protect something you cannot afford to lose. While you will inevitably lose small amounts of money (the premium), this guards you against losing a catastrophic amount of money (the value of the contents of your house, for example). Insurance offers a good price for peace of mind. ¶ it follows, however, that ensuring against losing a non-catastrophic amount of money is pointless. One example is ensuring against the loss of a mobile phone…. On average you will be better off if you don’t take out insurance, and instead buy yourself a new phone on the occasions when you lose one. In this way, you are ‘self-insuring’ and keeping the insurance company’s profit margins for yourself.”

Chapter Ten: Situation Normal

• 262: Book mentioned: Nassim Nicholas Taleb’s The Black Swan, in which he posits “we have tended to underestimate the size and importance of the tails in distribution curves. He argues that the bell curve is a historically defective model because it cannot anticipate the occurrence of, or predict the impact of, very rare, extreme events—such as a major scientific discovery like the invention of the Internet, or a terrorist attack like 9/11. ‘the ubiquity of the “normal distribution” is not a property of the world,’ he writes, ‘but a problem in our minds, stemming from the way we look at it.’”
• French physicist Gabriel Lippmann, in a letter to Henri Poincaré: “everybody believes in the [bell curve]: the experimenters, because they think it can be approved by mathematics; and the mathematicians, because they believe it has been established by observation.”

Chapter Eleven: The End of the Line

• 281: Book mentioned: David Foster Wallace’s Everything and More (on Georg Cantor)