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
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