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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