NOTES 19

• I usually include a brief description of mathematical results

which deal with mathematical analogues of the conjectural neu-

rophysiological mechanisms.

• In the later parts of the book, I will more and more frequently

venture into the discussion of possible implications of our find-

ings for our understanding of mathematics and its philosophy.

• My conjectures are frequently outrageous and sketchy. I have

no qualms about that. The aim of the book is to ask questions,

not give answers.

David Corfield,

aged 10

I very much value a “global” outlook at mathematical

practice (in recent books best represented by David Cor-

field’s Towards a Philosophy of Real Mathematics [16]), but,

in this book, I prefer to concentrate on the “microscopic”

level of study.

Quite often the mathematics discussed or mentioned in

this book is very deep and belongs to mainstream mathe-

matical research, either recent or, if we talk about the past,

of some historic significance. I believe that this is not a co-

incidence. Mathematics is produced by our brains, which

imprint onto it some of the structural patterns of the in-

trinsic mechanisms of our mind. Even if these imprints

are not immediately obvious to individual mathematicians,

they are very noticeable when mathematics is viewed on

a larger scale—not unlike hidden structures of landscape

which emerge in photographs made from a plane or a satellite.

Notes

1

SIMPLEST

POSSIBLE

EXAMPLES. Of course, simplest, in the relative

sense, examples can be found at every level of mathematics. Here is one

example, due to Gelfand: the simplest non-commutative Lie group is the

group of isometries of the real line R; it is the extension of the additive

group

R+

by the multiplicative group { −1, +1 }. Its representation theory

is a well-known chapter of elementary mathematics, namely, trigonometry;

however, the connection between representation theory and trigonometry

is not frequently discussed. But this is not the simplest possible example

of a simplest possible example, and its discussion will lead us beyond the

scope of this book.

However, it would be useful to record one consequence of the relation

between representation theory and trigonometry: the formula for matrix

multiplication is more fundamental than almost any trigonometric for-

mula. We shall return to that later; see page 189.

2ANALYTIC

FUNCTIONS. A function f(x) is analytic at x = a0 if we can

write f(a0 + z) as a power series

f(a0 + z) = a0 + a1z +

a2z2

+

a3z3

+ · · ·