Geometries of Nature, Living Systems and Human Cognition

The following sections are included:

• Contents

• Foreword

• About the Contributors

“It seems reasonable to think that general relativity and quantum theory are intrinsically incompatible and that, rather than merely developing technique, what is required is some fundamental breakthrough in our understanding of the relationship between spacetime structure and quantum process.“

Edward Witten, Physics and Geometry, 1987.

We first give a broad (but brief) introduction to several of the main themes encountered in this work: arithmetic geometry, noncommutative geometry, quantum physics and string theory, prime number theory and the Riemann zeta function, along with fractal and spectral geometry. We then explain how string theory in a circle (or in a finite-dimensional torus)—considered from the point of view of noncommutative geometry—can be used as the starting point for a geometric and physical model for the Riemann zeta function ζ = ζ(s) (and other arithmetic L-series). In particular, we contend that the classic functional equation satisfied by ζ, corresponds to T-duality in string theory. The latter, a key symmetry that is not present in ordinary quantum mechanics, allows one to identify physically and mathematically two circular spacetimes with reciprocal radii. Furthermore, we suggest that the Riemann Hypothesis may be related to the existence of a fundamental length in string theory.

This essay may serve as a relatively nontechnical introduction to the beginning of a theory presented in a book in preparation by the author and entitled In Search of the Riemann Zeros. In the latter monograph is developed, in particular, a theory of fractal membranes—viewed as suitable quantum or multiplicative analogues of the author's fractal strings. The notion of fractal membrane—interpreted heuristically as a noncommutative (adelic) infinite dimensional torus, in the spirit of the present discussion—may help in the long-term provide a unified geometric, algebraic and physical framework within which to understand number-theoretic zeta functions and their associated arithmetic geometries.

In some field theories we have the striking feature that we have two different spectra: There is the spectrum of particle-like solitons and the spectrum of the actual particles. From this idea that in certain field theories the two spectra may be inter-related or inter-reliant emerge the particle-soliton duality. This phenomena is surprising and deep and occurs elsewhere in field theory and has had important applications in supersymmetric field theory and in superstring theory. It is similar to, and linked with, the T-duality. In this paper we investigate the foundation and the development of this duality.

“By doing mathematical physics on topological structures, all sorts of new invariants and a new way of thinking about topology has emerged. (…) The knot or link as an embedding in three dimensional space is a whole topological form, and it has long been the aim of topology to treat this form all at once.”

Louis H. Kauffman, Knots and Physics, 1997.

This paper is an introduction to the landscape of knot theory and its relationships with statistical mechanics, quantum theory and quantum field theory. The paper is a self-contained introduction to these topics.

Our macroscopic world abounds in knotted and linked rings. But even at the nanoscopic scale, works in molecular topology have shown that Nature builds DNA molecules, which form actual knots and links between themselves. Qualitative similarities may be observed between forms of the microscopic world, as well as between those of the macroscopic world; and by extracting the structures, it is possible to study their dynamics thanks to topological methods and techniques. This chapter is aimed to show that knots and links are ubiquitous scale-independent objects carrying a tremendous amount of precious information on the emergence of new forms and structures especially in quantum physics and in living matter. In fact, knotting and unknotting are “universal” principles underlying these forms and structures. This study is focused at showing that differential geometry and topological knots theory can be used notably (a) to find invariants for closed 3-manifolds related to quantum field theory, and (b) to describe 3-dimensional structures of DNA and protein-DNA complexes.

“The exploration of topological space is the exploration of the last great frontier – the human mind. These space have been created by humans for the purpose of understanding the world in which we live. But ultimately they lead to an understanding of our mind, for it can only be understood in terms of its creations. Topological spaces, then, are at once a form of art and a form of science, and as such they reflect our deepest intellect.”

J. Scott Carter, How Surfaces Intersect in Space, 1995.

Biologists and social scientists often carry out their research in ways that are quite different from those used by physical scientists. Their results are often different as well; the nature of their subjects makes it less likely that they can produce the firm and generally quantitative predictions that are standard in physics. As mathematics is being applied more and more outside the physical sciences, a new methodology is appearing that better reflects the nature of these other subjects. This is happening only slowly, chiefly because the relevant work is generally seen only in its context of trying to solve a particular problem rather than as a contribution to applied mathematics as such. The aim of this paper is to encourage progress by describing some results that have already been obtained, and by discussing explicitly some of the issues that arise.

We try to describe and explain how are perceived motion and three-dimensional space, and how human beings perceive and control bodily movements. Reviewing a wealth of research in neurophysiology and experimental psychology, we argue for a rethinking of the traditional separation between action and perception, and for the division of perception into five senses. In our view, perception and cognition are inherently predictive, functioning to allow us to anticipate the consequences of current or potential actions. I assert in this paper that the brain is a biological simulator that predicts by drawing on memory and making assumptions. This interpretation of perception and action allows us to focus on psychological phenomena largely ignored in standard texts: proprioception and kinaesthesis, the mechanisms that maintain balance and coordinate actions, and basic perceptual and memory processes involved in navigation. We combine geometrical considerations of the components of perception and action with synthetic concepts borrowed from psychophysics and neurosciences to discuss the sensory receptors that enable us to analyze movement in space.

Mathematics stems out from our ways of making the world intelligible by its peculiar conceptual stability and unity; we invented it and used it to single out key regularities of space and language. This is exemplified and summarised below in references to the main foundational approaches to Mathematics, as proposed in the last 150 years. Its unity is also stressed: in this paper, Mathematics is viewed as a “three dimensional manifold” grounded on logic, formalisms and invariants of space; we will appreciate by this both its autonomous generative nature and its effectiveness. But effectiveness is also due to the fact that we re-construct the world by Mathematics; we organise knowledge of space and language by Mathematics, and give meaning by it to their structuring. But, what is “meaning”, for us living and historical beings? What does “mathematical intuition” refer to? We will try to propose an understanding of these crucial aspects of the mathematical praxis, often disregarded as “magic” or as beyond any scientific analysis. Finally, some limits of the remarkable, but reasonable effectiveness of Mathematics will be sketched, in particular in reference to its applications in Biology and in human cognition.

The following sections are included:

• Deductions, foldings and the brain

• A formal language to describe proofs

• Unfolding

• A geometrical view of the unfolding

• Cyclic structures

• Cycles and spirals

• Conclusions

• Bibliographical guide

• References

The mathematical community has become aware that 20-th century epistemologies such as Bourbaki's structuralism are not satisfactory. In particular, it appears that mathematicians should take signification problems, which have been long neglected, systematically into account. This is obvious for what concerns educational questions, but the recent writings of such an eminent and influential mathematician as Alexandre Grothendieck emphasize that dealing with significations, that is not only with formal or computational tools, is necessary even for research and for the internal progress of science. Husserl's phenomenology is, among other things, a theory of knowledge that deals with such problems. We show in the present article that modern mathematical theories, such as category theory, combine remarkably with Husserl's philosophy to create simultaneously a new image of mathematics and new approaches to classical philosophical concepts and distinctions such as transcendental subjectivity vs. objectivity.