This book will not put you to sleep
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I came across a book called This Book Will Put You to Sleep. The authors promise that it's so boring that reading it cures insomnia. To my horror, one of the chapters was titled, "Some Mathematical Theorems Summarised".
To combat the stereotype that math textbooks are boring, I found some choice excerpts from the ones on my shelf.
] I've been taking out more and more books from the University Library. By the time I graduate, I hope to fit all 9 million of them in my room.
Numerical Methods that (Usually) Work1 describes the joys of writing code:
Do you ever want to kick the computer? Does it iterate endlessly on your newest algorithm that should have converged in three iterations? And does it finally come to a crashing halt with the insulting message that you divided by zero? These minor trauma are, in fact, the ways the computer manages to kick you and, unfortunately, you almost always deserve it! For it is a sad fact that most of us can more easily compute than think — which might have given rise to that famous definition, "Research is when you don't know what you're doing."
Numerical Recipes2 likes extended metaphors:
For many scientific users, fourth-order Runge-Kutta is not just the first word on ODE integrators, but the last word as well. In fact, you can get pretty far on this old workhorse, especially if you combine it with an adaptive stepsize algorithm. Keep in mind, however, that the old workhorse's last trip may well be to take you to the poorhouse: Bulirsch-Stoer or predictor-corrector methods can be very much more efficient for problems where very high accuracy is a requirement. Those methods are the high-strung racehorses. Runge-Kutta is for ploughing the fields. However, even the old workhorse is more nimble with new horseshoes.
That textbook also describes the two options for solving boundary-value problems: Relaxation methods and shooting methods. It then provides this guidance:
Until you have enough experience to make your own judgment between the two methods, you might wish to follow the advice of your authors, who are notorious computer gunslingers: We always shoot first, and only then relax.
Introduction to Metric and Topological Spaces3 knows me too well:
It is quite likely that the reader has at some time taken a rectangular strip of paper, twisted one of the shorter sides through \(180^{\circ}\), stuck it to the other short side, and gone to work on the resulting Möbius band with a pair of scissors for the entertainment of friends.
Curved Spaces4 makes snarky comments about the UK's railroad network:
…no open neighbourhood of \(P\) is homeomorphic to an open disc in \(\mathbf{R}^{2}\). When the author was an undergraduate in the UK, this was known as the British Rail metric; here \(O\) represented London, and all train journeys were forced to go via London! Because of a subsequent privatization of the UK rail network, the metric should perhaps be renamed.
Complex Analysis5 likes poetry:
A common cause of distress for students of complex analysis is the sudden appearance of a plague of Cauchy Theorems, having several variant hypotheses and a similar variety of conclusions, but all begin derivable from each other. At times like this it may be advisable to seek consolation elsewhere than mathematics: perhaps among the poets. Rudyard Kipling's In the Neolithic Age makes the point admirably:
There are nine and sixty way of constructing tribal lays
And - every - single - one - of - them - is - right!It is much the same with the Cauchy Theorems
The Road to Reality6 suspects that its readers are a little tired of trig:
In particular, the basic relation \(\mathrm{e}^{a+b}=\mathrm{e}^{a} \mathrm{e}^{b},\) when expanded out in terms of real and imaginary parts, immediately yields the much more complicated-looking expressions (no doubt depressingly familiar to some readers)
\begin{array}{l} \cos (a+b)=\cos a \cos b-\sin a \sin b \\ \sin (a+b)=\sin a \cos b+\cos a \sin b. \end{array}
Introduction to Quantum Mechanics7 discourages murder:
On the other hand, a causal influence that propagated faster than light would carry unacceptable implications. For according to special relativity there exist inertial frames in which such a signal propagates backward in time—the effect preceding the cause—and this leads to inescapable logical anomalies. (You could, for example, arrange to kill your infant grandfather. Think about it \(\ldots\) not a good idea.)
And it promotes good old-fashioned virtues:
I'm not talking about any fancy quantum phenomenon (Heisenberg uncertainty or Born indeterminacy, which would apply even if we knew the precise state); I'm talking here about good old-fashioned ignorance.
Data Analysis: A Bayesian Tutorial8 collects data from one city in particular:
The prior pdf, \( \operatorname{prob} (H \mid I)\), on the far right-hand side of eqn \((2.1),\) represents what we know about the coin given only the information \(I\) that we are dealing with a 'strange coin from Las Vegas'. Since casinos can be rather dubious places, we should keep a very open mind about the nature of the coin
Gilbert Strang also has Vegas on the brain. In the final chapter of his linear algebra textbook,9 he sheepishly admits:
That is a strange way to end this book, by teaching you how to play a watered down version of poker, but I guess even poker has its place within linear algebra and its applications.
As Elvis (didn't) sing:
Oh, there's black jack and poker and the roulette wheel
A fortune won and lost on ev'ry deal
Use a matrix that's skew-symmetric and real
Viva Las Vegas, Viva Las Vegas
Footnotes:
p. 245. By Forman S. Acton
pp. 712, 755. By William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery.
p. 65. By W. A. Sutherland
p. 4. By P. M. H. Wilson
p. 202. By Ian Stewart and David Tall. On the back of the textbook, Stewart is described as "an honorary wizard of the Discworld's Unseen University."
p. 96. By Roger Penrose
pp. 452, 456. By David J. Griffiths and Darrell F. Schroeter.
p. 15. By D. S. Sivia, with J. Skilling
p. 440. Linear Algebra and Its Applications