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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