4 1 A Taste of Things to Come

cian Israel Gelfand, and had a chance to do some mathematics with

him.

Once I mentioned to Gelfand that I read his Functions and

Graphs; in response, he rather sceptically asked me what I had

learned from the book. He was delighted to hear my answer: “The

general principle of always looking at the simplest possible exam-

ple”. “Yes!” exclaimed Gelfand in his usual manner, “yes, this is my

most important discovery in mathematics teaching!” He proceeded

by saying how proud he was that, in his famous seminars, he al-

ways pressed the speakers to provide simple examples, but, as a

rule, he himself was able to suggest a simpler one. 1

So, let us look at the principle in more detail:

Always test a mathematical theory on the simplest possible

example...

This is a banality, of course. Everyone knows it;

no one follows it. So let me continue:

...and explore the example to its utmost limits.

This book contains a number of examples pushed to their in-

trinsic limits. See, in particular, Section 2.6 and the discussion of

Figure 2.11 on page 40 for some examples from the theory of Cox-

eter groups and mirror systems. What could be simpler than that?

But it is even more instructive to look at an example from Func-

tions and Graphs.

What is the simplest graph of a function? Of course, that of a

linear function,

y = ax + b.

But what are the simplest non-linear elementary functions? The

apparent answer is quadratic polynomials. Well, Functions and

Graphs suggests something different. The simplest non-linear func-

tion is the magnitude, or absolute value, y = |x|.

✲ ❅

❅

❅

x

0

y✻

y = |x|

Indeed, it allows

• easy plotting and interpolation;

therefore almost