Multi-Disciplinary Reading – Book Reviews

Alex in Numberland, Alex Bellos, 2010 – This book is a love letter to math and it explores math through the history of math and its progress. It also pays its due to Indian contribution to math, be it Vedic texts and their obsession with numbers large and small, Brahmagupta’s Zero or Madhava’s calculus and Ramanujan’s extraordinary intuition with numbers. For most of our evolution we had no numbers and when we did, we couldn’t count more than 5 and just used “more” or “a lot” for any number above 4 or 5.

I have been thinking philosophically about our obsession with numbers and how it feels satisfying while also being vacuous (like our CAGR / XIRR math). It is pointless to measure most of these. If you are doing well, it stands out so much making a compute unnecessary other than for intellectualising. I see some of these now as a conflict between the intuition (internal, single-player, infinite) and the intellect (external, multi-player, finite)

My notes –

A head for Numbers

• Ethnomathematics – study of how culture/religion influenced mathematics

• Munduruku (tribe in the Amazon) language has no tenses, plurals and no words for numbers beyond five. Working system of words and symbols for numbers is only ~10000 yrs old at most. Counting anything, be it fruit or children was considered ludicrous by most tribes

• Staying in a forest for long, one forgets sense of numbers, time and space

• We are taught numbers are evenly spaced (linear). Our perception of numbers though is logarithmic. We perceive less spaces between larger numbers (space between 10-20 is not same as space between 80-90 – we think in percentage increase 100% vs 12% intuitively!)

• We perceive ratios more intuitively than we do counting. Which tree has more fruit or which tribe has more people is answered without “counting”

• A tree 100m away from us and another a 100m behind that tree is not perceived the same (the far one appears shorter). We are however taught they are same (linear) removing this intuition

• Our perception of time as well is logarithmic, hence time passes faster, the older we get (have oft wondered this), yesterday feels a lot longer than whole of last week

• We use numbers mainly with quantity (cardinality or counts) and order (ordinality)

• Ancient Indian numbers used -, =, ≡ and + for 1, 2, 3 and 4 (Very close to the numbers we use today)

The Counter Culture

• Shepherds use 4 pebbles in their pocket to represent 80 sheep (base 20). Bases simplify how we count and throughout history, we have used 5,10 and 20 as bases. Without sensible bases, numbers become unmanageable

• Base 12 (duodecimal arithmetik) is considered more natural and versatile – 12 is divisible by 2, 3, 4 and 6 while 10 only by 2 and 5

• Sumerian cunieform has symbols for 1,10, 60 and 3600 (mix of base 10 and 60). Babylonians used the sexagesimal (base 60) system from Sumerians and made advances in astronomy and time – hence we use 60 seconds to the minute and 3600 to the hour (Attempt was made to decimalise time in 17th century – each day having 10 hours, with 100 minutes each with 100 seconds – 10k secs in a day but it failed)

• Leibniz fell in love with binary and felt the 1 and 0 represented being and nothingness. I Ching also uses 64 symbols (Fu Hsi) that’s close to modern binary

• Abacus was invented to count but was more useful with arithmetic. People used to using Abacus can perform math much faster by visually processing arithmetic (soroban) than using pen and paper which uses natural language that’s more tedious

Behold

• Square of a number n is the sum of the first n odd numbers (4^2 = 1 + 3 + 5 + 7 = 16). Intuition in squares made of pebbles visual

• Pythagorean brotherhood was a health camp, brotherhood and an ashram (math religious cult). Pythagoras discovered that length of vibrating string when halved increased the pitch by an Octave. Finding order in everyday things was their religious awakening

• Pythagorean brotherhood is the model for several occult secret societies, including freemasonry

• Right-angled triangle with sides with ratio of 3:4:5 was known as Egyptian triangle

• Euclid’s Elements was a magnus opus of pedantry and rigour Nothing assumed, except few basic axioms and everything followed logically from them

• Since The Elements, logical reasoning has been the standard for all human enquiry

• Platonic solids are perfectly symmetrical and there’s only 5 of them (tetrahedron, cube, octahedron, icosahedron, dodecahedron)

• 4 ways to define centre of a triangle – orthocentre, circumcentre, centroid, midcircle (lookup definitions). Euler proved that these always lay on the same Euler line

• Tessellate – cover a plane so that no region is uncovered (with repeating regular polygons like square, triangle or hexagon). Tesselation thus produced were periodic (repeating)

• Non-periodic tesselations which for a long term were believed to be impossible can be produced by penrose’s dart and kite (fascinating)

• 3-D versions of penrose’s tiles is present in quasicrystals. Again, crystals were assumed possible only from platonic solids (Mosaics in Iran, Iraq and Turkey had penrose patterns 5 centuries before it was discovered in the west)

• Hinduism used geometry to illustrate the divine – Sri Yantra, made with 5 triangles pointing down and 4 pointing up (structure described in a long poem) is hard to construct

• Origami is now the cutting edge of maths. Protein folding, arterial stents, robotics, solar panels and satellites have applications for origami

Something about Nothing (Zero)

• Lalitavistara sutra expresses numbers higher than a koti (crore). 100 koti is ayuta, 100 ayuta is kankara and so on until tallakshana which stands for 10^53 (Entire universe measured in meters and then squared would be around 10^53)

• There were several systems above like dhvajagravati and dhvajagranishamani that could count all the way up to 10^421 (Every atom in the universe 10^80, multipled by planck time 10^43 parts to the second would be 10^140 unique configurations to the universe since its beginning – still smaller than 10^421)

• The ancient sanskrit texts also have numbers going all the way down to size of a carbon atom. Buddha is said to have been an expert in numbers large and small – a metaphysical obsession perhaps in groping towards the infinite

• In contrast, Greek culture did not have the Indian hunger for numbers and had their largest number as myriad or M which stood for 10,000

• MMCDLI is larger than DCLXXXVIII which goes against common sense. Neither Romans, Greeks or Jews had a symbol for zero

• Vedas contain numbers from Dasa (10), Sata (100) all the way up to Parardha which stands for a trillion. Indian astronomy was way ahead because its astronomers and astrologers had the vocabulary for these large numbers

• Brahmagupta showed in 7th century how shunya behaved towards other number siblings. (Fascinating stuff based on fortunes, debts and shunya)

• The less maths was tied to actual things, the more powerful it became

• Modern decimal system from Indian system with 10 numerals, place value and all-singing, all-dancing zero was brought to Europe Fibonacci in 12th century (Liber Abaci) through the Islamic world

• Long division and long multiplication were the technological novelty of the 13th century

• Zero cannot be annihilated or destroyed. It means nothing, it means eternity. The conceptual leap happened in a culture that accepted the void as the essence of the universe

Life of Pi (Geometry)

• Many mathematicians are poor at arithmetic. Ability to calculate rapidly has no correlation with mathematical insight or creativity

• Ancient civilisations realised that the ratio of circumference to the diameter of a circle was always constant (pi)

• Archimedes loved to grapple with problems in the real world unlike Euclid who liked to deal in abstractions

• Though the origin of calculus is debated between Leibniz and Newton. A version of it was invented in the 14th century by Indian mathematician Madhava

• 35 digits of pi is sufficient to calculate circumference of earth to the centimeter. 1947, we had 808 digits to pi. 1949, ENIAC took 70 hrs to calculate it to 2037 digits. The numbers obeyed no obvious pattern. Today pi is known till 31.4 trillion digits

• When people use “lowest common denominator” to mean something basic and unsophisticated, they actually mean “highest common factor” (LCD is usually big while HCF is small)

• Numbers that cannot be expressed as fractions were “irrational”. Hippasus who proved their existence was drowned at sea by the Pythagorean brotherhood

• Pi was present in places that did not involve geometry, from pendulum swings, to distribution of deaths in population to probabilities in coin tosses

• Ramanujan formula calculated pi with remarkable speed and was a industrial strength pi making machine (fascinating formula! how the hell did he come up with it?)

• Digits in pi are pre-determined but still mimic randomness very well

• Reuleaux triangle (Similar concept to non-circular wheels – height from floor to top remains constant) and Watt’s drill can drill square holes

The X-Factor (Algebra)

• Descartes’s La Geometrie introduces algebraic notation used today. He used alphabets from the end for unknowns (x, y and z). The printer ran out of y and z as french used them extensively and so chose “x” instead for unknowns

• Ease in stating a problem has no correlation with ease in solving it (Eg. Fermat’s theorem)

• Logarithms turn the complicated process of multiplication into the simpler process of addition (Division was subtraction, square roots, division by 2 and cube roots by 3)

• Richter scale uses logarithms, so an earthquake of scale 7 is 10x higher in amplitude than one with 6

• Peter Roget invented thesaurus due to his OCD of making lists to deal with his mental illness. He also invented log-log scale that helped with calculation fo fraction of power like 4^3.5

• Descartes integrated algebra and geometry with his Catesian co-ordinate system (thus was born a way to visualise abstract notions)

Playtime (Puzzles)

• Rhind papyrus contained the numbers 7, 49, 343, 2401, 16807 – probably first known occurence of geometric progression. Illustrates the counter-intuitive growth when a number is multiplied by itself a few times

• Sudoku was invented by Kaji in 1980s but it took off only in the late 90s when it was published in newspapers (Led to 700% rise in pencil sales in UK and top 6/50 books were sudoku related in 2005)

• Puzzles have always given rise to and grown mathematics. A bridge crossing puzzle from Russia led to the invention of Graph theory by Euler

• Tangrams, Rubik’s cube, sudoku and fifteen puzzle – 4 internationally crazed math puzzles. Rubik’s cube also involved some clever engineering

Secrets of Succession (Combinatorics)

• Every even number above 2 can be expressed as sum of two primes (Goldbach’s conjecture). Its as yet unproved though no even number has been found to disagree with it. Primes are scattered unpredictably on the number line

• Persistence of a number – Number of steps it takes to get to single digit when the digits are multipled with each other Eg. 88 → 8 x 8 = 64 → 6 x 4 = 24 → 2 x 4 = 8 (Persistence is 3). Persistence of large numbers is surprisingly small – because they invariably have a zero somewhere, collapsing under their own weight

• Recaman sequence generates numbers that look like a garden sprinkler when plotted (Music generated with these sequences sound chilling like a horror movie soundtrack)

• Prime that can be written as 2^n – 1 is a Mersienne prime

• GIMPS – Great Internet Mersienne Prime Search. There’s a linear relationship between number of digits in prime and computing power when plotted in log plot

• The concepts of continuity and discreteness are not completely reconcilable (Zeno’s paradox for eg.)

Gold Finger (Fibs and Golden ratios)

• Golden ratio – Ratio of A+B : B is same as ratio of A : B. This led to Greek fascination with phi (1.618) and their discovery of it in the Pentagram and subsequent worship

• For most flowers number of petals is a fibonacci number (Even when its not, like a 4-leaved clover, the avg. is)

• Golden ratio is approximated by the ratio of consecutive fibonacci numbers (1, 2, 1.5, 1.667, 1.6, 1.625, 1.615, 1.619…)

• Nautilus shells and several spiral galaxies are logarithmic spirals – self-similar and never growing out of shape as they grow bigger

• Phi is omnipresent. Peregrine falcons descend on prey in logarithmic spirals. Plants arrange leaves to the golden ratio (of angles) so each leaf gets sunlight

• Cig. packs, credit cards and books often have length and breadth in golden ratio as we find it most appealing (Even the original iPod)

Chance is a fine thing (Probability)

• Slot machines in casinos make $25 billion post payouts (2.5x total movie tickets sold)

• “The book on games of chance” by Cardano was so ahead of its time that it was only published 100 yrs after his death

• Randomness was not seen as randomness but as an expression of divine will (Caeser’s time)

• Invention of probability was the root cause of the decline of superstition and religion

• Pascal’s Wager – If there’s the slightest chance that God exists, a non-believer has nothing to lose by believing in him

• Every bet in Roulette has a negative expected value. Craps on the other hand is the best deal (Expectation-wise)

• Core concept of building gaming machines is expected value and the law of large numbers (IGT builds these)

• IGT’s slot machines can be cherry dribblers or high-volatility – small prizes more often or large prizes rarely (payout rate remains same but plays with emotions differently). The sole objective is to keep the user playing because longer he plays, more he loses

• Insurance is a negative expectation bet often taken to protect something someone cant afford to lose. Actuarial tables and how slot machines are designed aren’t very different

• Insuring against losing a non-catastrophic amount of money is pointless (Don’t buy insurance against household electronics for eg.)

• The belief that jackpot is “due” (because it hasn’t been in awhile) is gambler’s fallacy. Slot machines feed on it. True randomness has no memory of what came before. Truly random iPod shuffle felt less random, so Apple had to tweak it to be less random to make it appear more random

• Even the most miserly slot machines have a payback percentage of 85%, lotteries have 50% or less

• Gambler’s ruin, is the eventual return to zero in a random walk on an iterative bets with long run negative expectations

• Maximising wealth requires minimising the risk of losing it all. With small edges and judicious money management, huge returns can be achieved

• Thorp’s “Beat the Dealer” was the first ever “quant” book, followed by his “Beat the market” for financial securities

Situation Normal (Statistics)

• The ability to describe the world in quantitative rather than qualitative terms changes our relationship with our surroundings

• Poincare noticed the distribution of sizes/weight of bread and saw the bell curve. Even before Poincare, Quetelet noticed it in frequency of murders in population

• Maxwell and Boltzmann’s kinetic theory of gases heavily relied on Quetelet’s statistical thinking to explain pressure of gases

• Reading Pascal’s triangle diagonally gives a fibonacci series. Ancient Indian texts also had versions of Pascal’s triangle

• The bell curve is ubiquitous because we look for it actively and we often choose what serves our interests

• Regression and correlation were big breakthroughs in modern scientific thought

The End of the Line (Non-Euclidean Geometry)

• Hyperbolic geometry or non-Euclidian geometry came about in 19th century on assuming Euclid’s carefully laid out rules as False

• “The Elements” was the bible of mathematics, so assuming fifth postulate (parallel postulate) to be false was untenable and yet thats what gave rise to a new form of mathematics and subsequently a new understanding of the world

• Riemann’s lecture of 1954 dealt with positive (spherical) and negative curvature (hyperbolic) of space and integrated Euclidean and non-Euclidean geometry

• Parallel lines in hyperbolic space get further and further apart from each other

• Hyperbolic surfaces maximise area while minimising volume (Pringles chips). Some plants and corals expose large hyperbolic surface for nutrition

• One geometry cannot be more True than the other, just merely more convenient – Poincare

• Cantor figured out that infinity could come in various sizes (countable infinities and Hilbert hotel)

• After Riemann and Cantor, mathematics umbilical cord with our experience of reality was cut-off

I thought this would be a daunting read but it was similar in structure to ‘The Joy of X’ and I enjoyed it as much as I enjoyed Strogatz’s book. I haven’t even captured a tiny portion of mathematical, anthropological and historical knowledge contained in this book. Read this even if you hated math in school. I was contemplating taking up serious math few months back and this book only made that resovle stronger. 11/10