xii Preface

Why am I earnestly concerned with such ridiculously simple

questions? Why do I believe that the answers are important for

our understanding of mathematics as a whole?

We cannot seriously discuss mathemati-

cal thinking without taking into account

the limitations of our brains.

In this book, I argue that we can-

not seriously discuss mathematical

thinking without taking into account

the limitations of the information-

processing capacity of our brains. In

our conscious and totally controlled

reasoning we can process about 16

bits per second. In activities related

to mathematics this miserable bit rate is further reduced to 12

bits per second in the addition of decimal numbers and to 3 bits in

counting individual objects. Meanwhile the visual processing mod-

ule of our brains

[211, pp. 138 and 143].) We can handle complex mathematical con-

structions only because we repeatedly compress them until we re-

duce a whole theory to a few symbols which we can then treat as

something simple, also because we encapsulate potentially infinite

mathematical processes, turning them into finite objects, which we

then manipulate on a par with other much simpler objects. On the

other hand, we are lucky to have some mathematical capacities di-

rectly wired into the powerful subconscious modules of our brains

responsible for visual and speech processing and powered by these

enormous machines.

As you will see, I pay special attention to order, symmetry, and

parsing (that is, bracketing of a string of symbols) as prominent

examples of atomic mathematical concepts or processes. I put such

“atomic particles” of mathematics at the focus of the study. My po-

sition is diametrically opposite to that of Martin Krieger who said

in his recent book Doing Mathematics [61] that he aimed at

a description of some of the work that mathematicians

do, employing modern and sophisticated examples.

Unlike Krieger, I write about “simple things”. However, I freely

use examples from modern mathematical research, and my un-

derstanding of “simple” is not confined to the elementary-school

classroom. I hope that a professional mathematician will find in

the book sufficient non-trivial mathematical material.

The book inevitably asks the question, “How does the mathe-

matical brain work?” I try to reflect on the explosive development

of mathematical cognition, an emerging branch of neurophysiology

which purports to locate structures and processes in the human

brain responsible for mathematical thinking [159, 171]. However, I

am not a cognitive psychologist; I write about the cognitive mech-

anisms of mathematical thinking from the position of a practicing

(See easily handles 10,000,000 bits per second!