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!
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