Preface xiii

mathematician who is trying to take a very close look through the

magnifying glass at his own everyday work. I write not so much

about discoveries of cognitive science as of their implications for

our understanding of mathematical practice. I do not even insist on

the ultimate correctness of my interpretations of findings of cogni-

tive psychologists and neurophysiologists. With science developing

at its present pace, the current understanding of the internal work-

ing of the brain is no more than a preliminary sketch; it is likely to

be overwritten in the future by deeper works.

Instead, I attempt something much more speculative and risky.

I take, as a working hypothesis, the assumption that mathematics

is produced by our brains and therefore bears imprints of some of

the intrinsic structural patterns of our minds.

a close look at mathematics might reveal some of these imprints—

not unlike the microscope revealing the cellular structure of living

tissue.

Mathematics is the study of mental ob-

jects with reproducible properties.

I try to bridge the gap between

mathematics and mathematical cog-

nition by pointing to structures and

processes of mathematics which are

sufficiently non-trivial to be interest-

ing to a mathematician, while being

deeply integrated into certain basic

structures of our minds and which

may lie within reach of cognitive science. For example, I pay spe-

cial attention to Coxeter Theory. This theory lies

of modern mathematics and could be informally described as an

algebraic expression of the concept of symmetry; it is named af-

ter H. S. M. Coxeter who laid its foundations in his seminal works

[336, 337]. Coxeter Theory provides an example of a mathematical

theory where we occasionally have a glimpse of the inner work-

ing of our minds. I suggest that Coxeter Theory is so natural and

intuitive because its underlying cognitive mechanisms are deeply

rooted in both the visual and verbal processing modules of our

minds. Moreover, Coxeter Theory itself has clearly defined geo-

metric (visual) and algebraic (verbal) components which perfectly

match the great visual/verbal divide of mathematical cognition.

One of the principal points of the book is

the essential vertical unity of mathemat-

ics.

However, in paying attention to

the “microcosm” of mathematics, I

try not to lose the large-scale view

of mathematics. One of the principal

points of the book is the essential ver-

tical unity of mathematics, the natu-

ral integration of its simplest objects

and concepts into the complex hierar-

chy of mathematics as a whole.

at

the very heart

If this is true, then