The Brain and Mathematics Dr. Shun-ichi Amari Group Director, Brain-Style Information Systems Group Combining the "brain" and "mathematics" may sound a little strange to some people, but mathematics is really a spin-off of the human brain. When looking back upon the history of humanity, we tend to realize that no one probably consciously resorted to mathematics during our early evolution and primitive development stages. Only when we entered the so-called "entrance exam race," was mathematics recognized as something advantageous in our struggle for survival. Up until then, mathematics was simply a minor abstraction within human society. Nevertheless, we did begin to develop mathematics, and as a result our brains have evolved to such an extent that we are now able to address even the most abstract forms of mathematics without much difficulty. Mathematics is often referred to as the "queen of science." If this metaphor is pertinent, this greatest product of the brain will definitely help to open up new channels in elucidating the mysteries of the brain. That is why we started the Research Laboratory for Mathematical Neuroscience. Looking back upon the scientific scene of the 20th century from a little broader perspective, we find the startling development of physics, which took the initiative in the development of science itself. The theories of relativity and quantum mechanics emerged and overthrew conventional ideas of space and time. These theories soon became basic paradigms for every sphere of science and enabled physics to establish a dominant position in the world of science. The achievements within the realm of physics were then developed into many of the cutting-edge technologies that underlie todayﾕs modern society. Mathematics, by comparison, was extremely abstract and seemingly stood in isolation from the other branches of science. Yet there are clouds already on the horizon for the dominant fields of physics and mathematics. Take the case of biology. Naturalistic approaches to observation and the classification of species diversity were dominant in the first half of the 20th century. However, the latter half of the century saw the advent of molecular approaches through which we were able to address the principles common to all creatures. These molecular approaches have developed so rapidly that we are now able to decipher genetic information and explain the functions of proteins, or even address the brain system itself. In many ways, the 20th century was when the age of biology really began to flourish. But aside from biology we should also take note of the information science and technology that emerged in the latter half of the 20th century supported by new computer and telecommunications technologies. The Internet, for example, has now linked individuals and societies in every corner of the world and is on the brink of developing a new information-oriented civilization. With these rapid changes, we find the age of biology and information technology slowly replacing that of physics and mathematics. We are now in the 21st century - a new millennium. In the future of science, further integration will occur instead of the conventional specialization or diversification. In particular, more importance will be attached to identifying the true nature of humanity or society itself. And when it comes to humanity, the brain is uniquely suited for better understanding what characterizes the essence of humanity. The brain is the most complicated system that life has developed or evolved. It is made up of neurons, which use biomolecular functions in a sophisticated manner, that operate within an amazingly intricate system of networks. With this structure, it functions to control and process data for individual organisms and, for human beings, it is the dwelling place of the mind and spirit as well. So we find the brain as an interface between life and information. To elucidate the brain system, therefore, it is essential to integrate both biological and informational points of views. Recently, it has been argued that it is important to use post-genome bioinformatics in the analysis of protein functions, the assertion being that this will provide us with both biological and informational perspectives. This is true, but at the same time we should be aware of the necessity of applying it to brain science as well. In brain research, we ought to not only construct theory-based computational models and information-processing models based on experimental studies, but also develop experimental paradigms to verify those models. To construct and elucidate theoretical models of the brain, it is essential to integrate many different approaches, such as informational, mathematical, physical and engineering. In addition to others, I would also like to emphasize mathematical approaches. They have aided us in clarifying the principles embedded in the advanced systems that underlie todayﾕs computer technologies. However, these principles of computers are entirely different from the computational principles of the brain and are relatively simple by comparison. Why is it that the brain has been able to attain such remarkable information-processing capabilities? This question must be answered by explaining the theoretical possibility of the brainﾕs nonlinear, highly parallel, multi-degree-of-freedom system utilizing its network of neurons. I am most interested in studying this possibility and constructing a new mathematical theory based on it. This cannot be accomplished using existing mathematical approaches which remain isolated in their ivory tower. It will require a new form of mathematical science that is closely linked to the other fields of science. We at RIKEN BSI will be playing a central role in the establishment of new mathematical theories to facilitate the development of these new approaches - it is what I dream of.