Adventures in mathematical reasoning

Preface to 'How the other half thinks: Adventures in mathematical reasoning'

A
AAmong my reasons for writing these pages was a wish to open the mathematical habit of mind to people who have never met real mathematics or enjoyed it. I want the striking discoveries shown — but the reasoning behind them shown even more. On that count the book parts company with most writing on mathematics for a wide readership. In some such works the careers of vivid mathematicians fill the pages; in others the stress falls on where mathematics is applied; in others again its methods are set out on the assumption that the reader already handles algebra with ease.
B
BMy hope is that these chapters help close the famous divide between the two cultures — the arts on one side, the sciences on the other, or the intuitive right side of the brain against the analytical, number-handling left. Mathematics, the book tries to show, is not confined to calculation and analysis, for intuition is at work in it too. That divide is one any of us can shrink, since none of us comes near to using all that either half of the brain can offer. For our range, I point to a structural engineer who paints, an electrical engineer who sings opera, an opera singer who has published mathematical research, and a mathematician who writes short stories.
C
CNow and then a hard piece of music holds a passage of real beauty that is still so easy a newcomer could play it, and mathematics is the same. Some results in higher mathematics ask no specialist training of the reader — no algebra, geometry or trigonometry. They call instead for common sense and, at the very most, a little everyday counting, such as knowing that two odd numbers add to an even one. Each of the book's eight chapters shows this, and not one step in the argument lies beyond an ordinary reader's grasp. Because the reasoning leans on nothing harder than the simplest counting, anyone who opens the book can share in doing mathematics and come to know a way of thought at once logical and guided by intuition.
D
DEven so, a reader with no mathematical background can go a surprisingly long way into the reasoning. What the book sets out are just the details that bring the mathematical cast of mind alive — a patient, step-by-step working-through, broken by testing and by sudden insight. These pages are to be turned far more slowly than those of a newspaper or a novel. Keeping pencil and paper to hand often helps, so that the reader can check each claim and work small calculations of their own.
E
EBooks meant to make a field of science clear to the general public have been written by many other researchers, who had little choice but to leave the mathematics out, though it is the very thing their theories rest on. That leaves the reader a mere onlooker; for across much of science the fitting language for fine detail is mathematics itself, whether the topic is the swelling universe, the particles below the atom, or the chromosome. A theory's broad shape can be conveyed without symbols, yet once some corner of the physical world is finally grasped, the account of it reads like a page from a mathematics textbook.
F
FThroughout the writing, two groups of readers were always in my mind. The first are people who liked mathematics until some sour experience — usually around the fifth grade — drove them from it; the second are the enthusiasts, for whom fresh material waits on every page. A further group is served too: those who simply want sharper powers of analysis. Many professions, law and medicine among them, demand long stretches of exact reasoning, and each chapter lets the reader practise holding to a lengthy, closely argued line of thought. Two testimonies show that mathematics can build that capacity.
G
GOne doctor put it this way: 'The rigorous, analytical habit of thought that mathematics instils was fine preparation for medical school. A doctor meets a problem that must be picked apart in full before a remedy appears, much as in mathematics.' A lawyer reached the same view: 'I came to law with nothing behind me — not a single course in political science — yet I did well at one of the finest law schools. For that I largely thank mathematics, and above all what working with proofs taught me about taking a tangled principle apart. A lawyer schooled in mathematics can master legal doctrine as others rarely can.' I hope you will feel, as I do, the pleasure of watching plain, even naive questions open onto striking answers, and of seeing discoveries made for their own sake prove unexpectedly useful.

Question 1 - 8

Which section contains the following information?

a mention of books written for readers presumed to know little about mathematics

an explanation of how this book is unlike the usual work on mathematics

first-hand accounts from people who gained from mathematics

instances of individuals combining talents not usually found together

a reference to the varying emphases of mathematics books aimed at general readers

the different pace this book demands compared with everyday reading matter

an assertion that every part of the book can be followed by anyone

identification of the several kinds of reader the book is written for

a mention of books written for readers presumed to know little about mathematics

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