The author is having a lot fun with mathematics and words! The book is full of little things like:
• (In a discussion about log, to clarify he meant ‘natural log’) You would use natural log if you are a mathematician, or if you have e fingers.
• (After a discussion on Pythagorean theory) Pythagorean thoughts are part mathematics, part philosophy, part mental illness. You need to read the details on the three parts and decide which is which.
• (Why don’t popular mass market mathematics book give equations or details about the proof?) It’s like you give pre-teens a book on reproductive health and where babies come from, the book tends to wax lyrics about birds & bees, and leave out the hydraulic stuff of how babies are actually made … Similarly, if a mass-market math book included equations, it would ‘shock your modesty’. – Almost fell off my chair upon hearing his analogy between math equations and modesty, unapologetic fun.
Could not get enough of this math fun after the first reading, and immediately jumped back to complete a second read before I could write this review 🙂
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He discusses one math topic at a time, but there are some general themes revisited throughout the book. Below are long details about these general themes (towards the end is a list of topics by mathematicians):
He shared lots of ideas about math education – current math instruction could be off-putting, if we just try to drill quadratic formula into students without the intuition, or gives an assignment of 30 definite integral without explaining how this stuff would ever be used. On the other hand, he is not a radical ‘reformist’ who wants to exclusively use calculator, and abolish any practice on hand calculation. He’d say you can’t write a sonnet if you have to reach into a dictionary for the spelling of every word. Practicing math question is like building strength and skills via weightlifting or running zigzag preparing for soccer. Weightlifting and running zigzag do not seem as fun as actually playing the soccer, but you can’t go far without these practices. (I like all his analogies above.)
He critically points out the ‘cult of genius’ problem – students are turned off whenever they saw someone else advancing ahead of them and thinking themselves are not talented enough – math students look up to better math students, who look up to the IMO winning ones, who look up to mathematicians with a few good theorems, who look up to mathematicians with a lot of good theorems, who look up to Field medalists, who look up to the dead. No one looks into the mirror and say let’s face it I’m a better Mathematician than Gauss?! ( Could not get enough of his math fun 🙂 .
People think praising someone ‘hard working’ is because you can’t honestly praise them to be ‘smart’. Unfortunate result: people would fall off the math track because ‘I’m not talented in math’. He corrects ‘Genius is to make things happen’. He laughs – people don’t get turn off and stop studying English just because someone did AP English better.
With all this rethinking about ‘math losing students to medicine’ sprinkled throughout, he acknowledges a worse problem, is that other fields do not have enough math-literate students. The world would be better if we get math-literate politicians, social policies, doctors, everyday citizen, etc.
The book is full of his math-loving confession. He debunks the popular myth ‘math melodrama’ in which a fiction writer could easily explain a plot by hand-waving ‘math drove the person crazy’. Also add, to the contrary, math calms down his nerve — like meditation, math puts him in direct contact with the universe, “it’s here before you, and it would be here after you”. (Isn’t his passion for math touching!)
He talked about the other popular myths that math progressed with a few lone geniuses that had no contact with other people at all (he blamed movies like A Beautiful Mind for such misinformation). To the contrary, math progressed with collaborative effort with individuals weaving together efforts to tackle problems.
He talked about Theodore Roosevelt’s praising of the person who gets his hands dirty and belittling of the critic who only sits in the library – and rebut that mathematicians in the library contributed far more to the world than acknowledged in Roosevelt’s populist speech. He’d say Abraham Wald, who figured out the way to armor the planes in world war two, never did a practical work, contributed critically to the war effort, a critic that counts.
I like his four quadrants for math, simple/complex vs. shallow/impactful.
• Simple & shallow are 1+1=2, it’s so intuitive that we barely talk about this.
• Complex & shallow are hand calculation of 12-digit numbers multiplying each other, it’s tedious and time-consuming, but it won’t shed much light on things even if we put effort into such exercises.
• Complex & impactful are ideas like Fermat theorem, which deserve books of their own.
• Simple & impactful are ideas introduced in his book, like
o Really old ideas on how to proximate the pi (and he holds it’s meaningless to rote memorizing pi to the hundreds of digits since such memory does not shed light on any math ideas).
o The law of large numbers (vs. the wrong approach of average numbers) – the individual plays do not ‘correct themselves’ or ‘even out’, but they dilute significance of any deviations in small samples.
o The battle on Statistical significance test (should be called statistical detectible instead of significant – and a tangent that math names are frequently existing words preloaded with other meanings – words with baggage)
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He threw in opinionated commentary about other disciplines:
• Social science as haruspicy (entrails reading).
• (After introducing ‘regression to the mean’ problems affecting social science, medicine, etc) He’d say but mathematicians won’t be fooled by this.
• File drawer problem (or stock broker problem) of the replication crisis. His line – scientists are not malicious villains like a stock broker trying to swine money away from unsuspecting victims, but they are fooling themselves into thinking they have found an effect.
• He loves to discuss voting, from Condorcet’s research, to modern day rank ordered voting in Vermont, to Bush-v-Gore 2000, to Nate Silver’s voting models, back to Condorcet’s personal life, belief in progress and rational thoughts, and end of life after French revolution. He needs to update this book after 2016!
• His likens the formal letter-reading in Justice Scalia, is like Cantor’s formalization of mathematics…
• He talked Voltaire, as member of a group buying French government issued bond-lottery (with a mathematician worked out the odds in their favor of course), got very rich from the positive expected value bond-lottery for the rest of his life. Then he said, you thought Voltaire got rich from writing sketches and essays? ‘Then, as in now, that’s not the way to get rich.’
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Reading this book is like reading these … why I like audiobooks
This book discusses impactful ideas and explaining things clearly like “Thinking, Fast and Slow”, is funny like Steven Colbert. It makes mathematics accessible to mass market reader.
Its popularization of a quantitative field does not rely on generating anecdotes about himself (not the Feynman type), is an equal-opportunity attempts to shine lights on many different mathematicians, like the Freakonomics franchise or Sophie’s World (except the author actively conducts frontier research on the subject, and doesn’t have a fictional shell).
I listened to the audiobook, and his fun voice reading was full of passion for math, clearly beat the Freakonomics series – yes, no one expects Econ guys to be top-notch voice-acting, but Ellenberg also beats Dubner who makes podcast for a living.
The other advantage of audiobook – you get to hear his pronunciation of mathematicians’ names (Morgenstein anyone?) … He also throws in German/French sentences from mathematicians (with full emotions), fun!
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Lastly,
Mathematician enjoy talking about mathematicians! I remember from school days, a math professor talks about G.H. Hardy and Srinivasan Ramanujan, with eyes gleaming for the discover and loss of math talent.
Ellenberg also talks about mathematicians in this book, of course G.H. Hardy and Srinivasan Ramanujan, and many more:
• Abraham Wald (airplane stories, his strong preference for theoretical work, and stubborn refusal to work on applied projects),
• Hotelling (sharp observation on regression to the mean, a hobby to play an entire game of poker in his mind with himself, with his brain keeping all scores, and generating the random sequence of cards obeying probability calculations from observed cards?!) — I regret to only know Wald and Hotelling from lemma and distributions back in the days, their stories are infinitely fun and motivating!
• Pascal (probability and religion – expected utility from religious belief WOW!),
• Poincare,
• Fermat (and theorems filling on book margins),
• David Hilbert (his axiomatic approach to Scalia’s formalism approach to law, his rational belief that led him to hold modern beliefs such as defend hiring female professors saying university is not a bathhouse, his failure to see Nazi for what they are in his later life, his fashion choice to wear socks with sandals lead subsequent generations of mathematicians to adopt this clothing option more than the general public!)
• Bertrand Russell (how he easily dismantled Cantor’s entire book with his paradox, set of all sets that are not members of themselves),
• RA Fisher (significance testing, Bayesian approach, smoking habit and his failure to see smoking evidence for lung cancer, his own daughter admitting not to have sensibility to other people’s feelings),
• Karl Pearson & his son Egon Pearson (confidence internal, how they dueled w/ Fisher),
• Claude Shannon,
• Terence Tao,
• Yitang Zhang (his proof, and stories of him working at subway),
• Buffon’s needle problem (noodle problem, words are fun!) — regret only knowing him as a naturalist before