Multi-Disciplinary Reading – Book Reviews

17 equations that changed the world, Ian Stewart, 2012 – This book is an ambitious attempt to compress the most impactful equations across math, physics, statistics, fluid dynamics, information theory, finance etc. into a ~300 page paperback. It unfolds in increasing orders of complexity, just as we discovered them over centuries. You can sort of see how pythagoras theorem evolved for our need to build things and logarithms for our need to compute astronomical distances and calculus for our need to compute orbits of “heavenly” bodies and the law of gravity as a natural evolution of it.

There’s a sort of a paradigmatic shift with the wave equation which for the first time uses second derivatives. What started as a study of vibration of violin string has had applications far and wide in earthquakes, light, heat and waves in general. The imaginary number i and the Euler’s formula for Polyhedra were abstract for the time they evolved in but had far reaching implications in quantum mechanics and topology much later. Roots for the normal distribution were present in Pingala’s work before 200 BC but such is our fragmentation of geography, culture and ideologies that until recently progress was always bound and hence lost.

The 19th century worked out complex mechanisms of fluid dynamics, thermodynamics and complex nature of waves – most of this involved integrals, derivates, partial derivatives (most developed just a century or two earlier) and abstract concept of fields. 20th century brought us relativity, uncertainty, quantum mechanics and the Shannon’s ground-breaking information theory which in a way laid the base for the information revolution. For all the phenomenal work before it, Black-Scholes has perhaps broken more than it has fixed but if past is any indicator, as things become more and more virtual, we might be in for some surprising things a century or two from now.

My notes –

• Stephen Hawking’s publishers apparently told him every equation in ‘The Brief History of Time’ will reduce sales by half So they finally included only E = mc^2 and sold 10 million copies

• The equal-to sign (=) was invented in 1557 (which is so recent in the scheme of things) to avoid repetition of ‘is equal to’

• Pythagoras Theorem – Tells us that the three sides of a right-angled triangle are related (5th Century BC). The abstraction of the relationship into an equation linked geometry with algebra and inspired trigonometry. Was used in navigation, surveying and also inspired Einstein’s theory of space, time and gravity

• Pythagorean triplets are 3 numbers that satisfy the equation – Eg. 6-8-10 (36 + 64 = 100), 5-12-13 etc.

• Babylonians knew about 3-4-5 triangle, a thousand years before Pythagoras

• Erastothenes around 240 BC calculated the size of earth using sun’s shadow and elementary trigonometry

• Triangulation – measure few distances and angles and work out the rest – two sides and subtended angle can get us opp side. Two angles and a side can give us the other two sides etc. Was used routinely in surveying, for eg. height of Mt. Everest. Currently we use satellite and GPS instead of this

• Limitations in Euclidean geometry and Pythagoras theorem due to curvature of space, lead to development of a separate geometry for curved spaces (developed by Reimann and eventually led Einstein to postulate that gravity curved space)

• An ant confined to the surface of a curved space may never perceive the curvature (just as we don’t). Test for Pythagoras theorem is useful to test curvature.

• Logarithms (Napier and Briggs) – Gave us an easy way to multiply numbers by adding numbers instead before the advent of calculators in the ‘80s and lead to the invention of the slide rule

• Logarithm is the inverse of exponent. So if a^x = y, then log y = x (logarithm to the base a)

• Logarithms also helped with square roots and cube roots. (sqrt(x) = x^(1/2) = 1/2 log x)

• Logarithms were crucial to the estimation of astronomical distances by simplifying the multiplication (Few months effort to few days)

• Radioactivity and half-life calculations use logarithms to estimate

• Human perception is proportional to the logarithm of the stimulus (Eg. decibels). Its the reason why we can detect a 50g difference in 100g vs 5 kgs. (This has lot of psychological applications in terms of how we say, perceive signal vs noise and is extremely useful in investing)

• Calculus (Newton) – To find instantaneous rate of change of a quantity (say speed), calculate how its value changes over a short time interval and divide by that time and then let that time be arbitrarily small.

• Calculus is the basis of most mathematical physics and Newton invented it (disputed with Leibniz) as a precursor to his laws of motion. Its used in calculation of tangents, areas and in volumes of solids etc.

• Differential calculus is the computation of slopes and integral calculus that of areas and volumes

• Calculus philosophically sees God as a detached creator who stood back from his creation and left its to its own devices

• Energy as a concept is a convenient fiction that helps to balance the mechanical books

• Newton’s Law of Gravity – Expresses the force of gravitational attraction between two bodies in terms of their masses and the distance between them. It led to accurate prediction of planetary orbits, eclipses etc. and led to development of satellites, hubble, interplanetary probes, GPS etc.

• Aristarchus of Samos put forward the heliocentric theory in 270 BC but it fell out of favor and wasn’t revived for 2000 years in the West (West drowned itself in organized religion). Aryabhatta meanwhile put forth a mathematical model for heliocentrism in 499. West took to heliocentrism under Copernicus, Kepler and Brahe in the 16th century

• Imaginary numbers – The idea that sqrt(-1) can exist as an abstract entity led to the creation of complex numbers. It led to improved understand on the way we understood such things as waves, heat, electricity and magnetism and the basis for quantum mechanics

• e^(i * pi) = cos pi + i sin pi = -1 (i unites e and pi!)

• Euler’s formula for Polyhedra – (F – E + V = 2). The number of faces (F), edges (E) and vertices (V) of a solid are not independent but are related. Led to the development of pure mathematics of topology (study of surfaces, knots, links etc.)

• Normal Distribution – The probability of observing a certain value near mean is high and it reduces further away from the mean. How rapidly it reduces is captured by the standard deviation. Widespread applications in clinical trials, various testing, census etc.

• Pascal’s triangle is the very basis for modern statistics was known to the West in 16th century whereas it was found in Chandas Shastra written by Pingala somewhere between 500 – 200 BC

• Pascal’s triangle can be seen as binomial coefficients of two-variable expressions of the kind (p+q)^n

• IQ is a statistical method for quantifying specific types of analytical problem-solving but may not imply intelligence (cleverness, intelligence and wisdom are not the same)

• Wave Equation – The acceleration of a small segment of vibrating violin string is proportional to the avg. displacement of neighboring segments (strings will move in waves). It helped us generalise our understanding towards sound, heat, light, water waves and also to improve our understanding of earthquakes. (Modeling a violin string led to a lot of breakthroughs in everything)

• Hearing is nothing but detection of compression in the air and seeing the detection of waves in electromagnetic radiation

• Pythagoras found that two strings of equal tension that had lengths in simple ratios like 3:2 or 2:1 produced unusual harmonic notes (sin x, sin 2x waves superposed). A string half in length plays an octave higher

• Western music uses 1-4-5 progression a lot. Here the 4th and 5th are one-quarter (4:3) and one-third (3:2) down from the root note and the octave halfway (2:1) down (C D E F G A B C → C F G makes the 1-4-5 and the last C the higher Octave root). How we mathematically arrive at these notes as (3/2)^n is something I have never come across before

• Brain adjusts ear’s response to incoming sounds, so what we consider harmonious has a cultural dimension (Perhaps why its hard to listen to music from diff cultures for this reason)

• Fourier Transforms – Any pattern in space and time can be thought of as superposition of sinusoidal patterns with different frequencies. Used in finding structure of large molecules like DNA, digital compression, cleaning up old audio tracks, analysing earthquakes etc.

• Buildings have their natural frequencies of vibration. Earthquake proofing is making sure building’s vibrating frequencies are different from the earthquakes (to avoid resonance)

• Navier-Stokes – It is Newton’s second law of motion in disguise as it provides the acceleration of a section of fluid given the pressure, stress and internal body forces. Gives us an accurate understanding of how fluids move. Led to understanding of aerodynamics, faster and quieter subs, F1 cars, advances in understanding of blood flow in arteries and veins etc.

• Maxwell’s equations – Showed that Electricity and Magnetism are related – A spinning region of electric field creates a magnetic field at right angles to the spin and vice versa. Led to the understanding that light was an electromagnetic wave

• Faraday was working class and lacked formal education so developed theories based on analogies and conceptual machines (built the first motor and generator). Maxwell helped with the math (While it’s called Maxwell’s equations, this has the same problem as pointed by Taleb’s Lecturing birds to fly. Its the technology that came before the science)

• Maxwell borrowed concepts from fluid dynamics to come up with the concept of “field” (Faraday called it ‘lines of force’). A field was an invisible fluid

• Radio waves as longer wavelength electromagnetic waves was predicted by Maxwell. It was verified by Hertz but discarded as useless. Radio was later invented.

• Carrots are good for eyesight was a myth propagated during WW-I. The British were using radar to detect German bombers but propagated the myth as disinformation

• Second Law of Thermodynamics – The amount of disorder in a thermodynamic system always increases. Better steam engines, efficiency of renewables etc. And the theoretical construct of ‘heat death of the universe’

• Heat measures changes but temperature measures states (heat transfer is possible only when fluids have diff temperatures – zeroth law of thermoD)

• Living systems borrow order from the environment and pay it back by making the env. even more disordered – Schrodinger in ‘What is Life?’. (This runs contra to the opinion that life reduces entropy – its explained nicely why its only an illusion that it does)

• Relativity – E=mc^2. Arguably the most famous equation there is. So matter contains energy equivalent to its mass multiplied by square of speed of light – in other words, gargantuan. Changed our view of space, time, matter and gravity. Indirectly led development of nuclear weapons but mainly satnav, gps, blackholes, big bang

• Special relativity vs general relativity. Former takes into account space, time and matter while latter includes gravity as well (Einstein’s book “Relativity” is a surprisingly lucid read even for people with no knowledge of physics)

• Schrodinger’s Equation – Matter not as a particle but a wave, describing how it propagates. Fundamental to quantum mechanics. Radically revised notions of matter at small scales

• Particles as wave functions led to several philosophical questions about the universe (as with Schrodinger’s cat). If you can’t observe an entire wave function, does it really exist?

• Information Theory – How much information does a message contain? It established limits on the efficiency of communications. Applied in statistics, AI, cryptography, meaning in DNA sequences and all digital communications

• Shannon quantified the amount of information a message contained. He also identified that efficiency for a given rate of error detection and correction is the ratio of length of coded message to the original

• The formula for information theory looks very much like entropy in Boltzmann’s approach to thermodynamics (base 2 vs natural log and -ve sign only diff). So entropy can interpreted as “missing information”

• Chaos Theory – How a population of living creatures changes from one gen to next when there is limits to growth. Probably the simplest of equations that generates deterministic chaos – what appears random. Apparent randomness hides underlying order. Uses in weather forecasting, population dynamics in ecology, quake modeling, space probe trajectories etc.

• What’s deterministic can be unpredictable, as any uncertainty in initial conditions can grow exponentially fast

• Black-Scholes – How price of a derivative contract changes over time, so that when the price is correct, it carries no risk. Gave rise to derivative markets and increasingly complex financial instruments

• Simplest derivatives arose in 18th century Japan at the Dojima rice exchange. Although the amount of rice was limited by what farmers could grow, there was no limit on the number of contracts that could be issued

• The time evolution of position of a particle w.r.t a origin in brownian motion is similar to diffusion equation – probability spreads like heat. Bachalier modeled option markets with Brownian motion (Theory of Speculation).

• Bachalier’s work was flawed since it predicted extreme events were extremely rare that it would never happen but financial markets are fat-tailed (indicates increased levels of risk). While Bachalier’s brownian motion assumed exponential decay in risk, financial markets follow power-law curve (x^-a) – the difference could be as high as 1 in a million to 1 in a 100

• By ‘95, virtual economy (investment, currency trading, complex wagers) had overtaken real economy (based on production of real goods). Virtual economy deliberately confused real and virtual -like arbitrary figures in accrual accounting vs cash basis accounting

• Black-Scholes equation is from the world of mathematical physics – where quantities are infinitely divisible, time flows continuously and variables change smoothly which doesn’t apply to financial markets. It also assumes perfect information, perfect rationality, market equilibrium and the law of supply and demand (it has all the pitfalls of efficient-market hypothesis)

The notes don’t do justice to the equations. It wouldn’t be a stretch to call it a capsule of the most important scientific progress we have made. It is a noble cause, especially with the way human knowledge is fragmented and boxed into disciplines and rendered inaccessible for a seamless rumination across them. It doesn’t assume much from the reader, so please by all means go ahead and pick it up and give it a shot, even if you hated math/phy and equations in school. 10/10