A Project

As mentioned in notes [], [] and [4], mathematics can intimately express the workings of reality. In the context of physics this feat was probably most obviously uncovered by the mathematics of the infinitesimal and the infinite, i.e. calculus. The notion of integration and its geometrical implementations can be traced back to the Greek (Archimedes). However, in 1615 Kepler published a paper in which he introduced a new method for measuring the volume of wine barrels. This essentially started the mathematical era of the infinitesimal to which many great mathematicians are linked, such as Fermat, Descartes, Cauchy, Johann Bernoulli and Taylor. In 1687 Newton published his famous Principia^B.1 introducing differential calculus and linking it to integration. But most crucially he applied these mathematical formalisms to science, thus starting classical mechanics; [Wa91]. The extent of this physical theory is quite vast, spanning from the description of the motion of macroscopic objects on earth to celestial bodies in the sky. There is another good example of the fruitful implementation of quantitative mathematics to physics coming from Euler, perhaps the most prolific scientist ever.^B.2 His exponential function seems to characterize the dynamics of nature, i.e. the way natures changes, very fundamentally. The exponential function appears in a myriad of physical and chemical manifestations. A very recent example of the profound entwinement of physics and nature is seen in molecular biology, where RNA strands can undergo knotting. The mathematical theory of knots can help explain what kind of enzymatic reaction took place; [Dij01]. But mathematics can not only explain nature, sometimes nature can be used to aid mathematics, as in the case of knot theory, where recently new stimuli have come from particle and statistical physics. Another example of physics aiding mathematics is given in [NS00] where the energy levels of quantum chaotic systems are conjectured to be linked to Riemann's Zeta function and thus ultimately to the distribution of prime numbers. In fact, the whole mathematical formalism of chaos theory and its many physical implementations go back to 1960, when Mandelbrot was working for IBM and was said to be inspired by charts of cotton wool prices and income distributions; [Gle88].

However, there are also some very subtle and unusual links between mathematics and nature. About 900 years ago the Italian mathematician Fibonacci defined a very basic number series: $F(1) = F(2) =1$, $F(n) = F(n-1) + F(n-2)$, $n > 2$. Today the Fibonacci sequence is known to appear in many natural phenomena: sea shell shapes, branching ratios of plants, flower petals, seeds, the pattern on pineapple skins, in broccoli and cauliflower, on pine cones and also in stock market charts; [Kn00]. The Fibonacci numbers even appear in the (fractal) Mandelbrot set. They are also an example of exponential growth. Another rudimentary pattern in nature is given by the golden section $\varphi = (\sqrt{5} -1)/2 \approx 0.618$. As a ratio it is said to reflect aesthetic pleasing in art and architecture. The ratio of Fibonacci numbers converges towards $\varphi$: $F(n) / F(n+1) \to \varphi$ for $n \to \infty$. The so-called law of Benford gives another intriguing example of how reality seems to rest on crude patterns. In 1881 the astronomer Newcomb issued a remark (and an equation) to the American journal of mathematics, referring to a strange phenomenon he had observed in logarithm books. But not until 1938 was any particular interest given to the subject, when the physicist Benford re-discovered the issue. He used numbers originating from statistical studies of geographic, biological, demographic, economical and sociological phenomena to illustrate the law. But not until 1961 could any mathematical explanation for Benford's law be found. The `law' states that in any collection of numbers with statistical origin the probability of any number starting with the digit $D$ is log $_{10} (1 + 1/D)$. Hence $30\%$ of the numbers start with the digit 1 and only $4.5\%$ start with 9. Because many different and unrelated subjects are governed by Benford's law it appears as though propability is violated. Nevertheless, this law has uncovered a very basic and fundamental pattern associated with natural phenomena. Interestingly, it originates in the principles of covariance and invariance: Benford's law is the only distribution of digits which is independent of the units the numbers were measured in and independent of the number system the numbers are expressed in -- log $_{B} (1 + 1/D)$, where $B$ is the base of the number system gives the general probability. In addition Benford's law is scale invariant. Hence this law guarantees that nature can be expressed mathematically in a sensible and unambiguous way. An alternative origin of the law can be seen in the fact that exponential growth is implicitly incorporated in it (however, the question remains why nature chooses exponential changes in its dynamics in the first place). Benford's law also asks the question of what distinguishes natural phenomena related to the world from purely random ones, or equivalently: why does nature manifest a pattern in the numbers describing it? The law can also be seen to arise within the context of probability distributions where Benford's law emerges as a kind of ultimate distribution, i.e. the distribution of distributions. Interestingly, the Fibonacci series (expressed in any number system) also obeys Benford's law, nicely rounding up this notion of an underlying pattern of nature. Reference: [NS99iv].

A conjecture concerning the nature of mathematics and reality could be stated as follows: reality not only operates in accordance with different mathematical branches (the most prominent being geometry), it is also governed by some very basic patterns. Recall the possible status of mathematics within the anthropic principles, stated at the end of the second paragraph in note []. This concludes the intriguing notion of mathematics acting as the blueprint of a reality, which in its complex evolution gives rise to living entities equipped with a consciousness able to tune into this underlying level of reality. Probably the first appearance of the idea of an unphysical realm of thought can be attributed to Plato, the scholar of Socrates, in the fourth century B.C. He divided reality into two realms: the material world and the world of ideas. Plato attributes this world of thought an immaterial, unchanging, absolute and eternal nature. Every manifestation in, and every aspect of the material world is thought to have its origin in the world of ideas. Nearly two millennia later Galileo would remark: `the book of nature is written in the language of mathematics'. It is also interesting to note that schamanism, still found both in North and South America, employs this idea of a spiritual realm. In addition, most religious and mystical beliefs speak of a reality beyond the physical. The $20^{\text{th}}$ century philosopher Popper even envisages a reality of threefold nature. Next to the realms of the material world and the psychological world of abstract thoughts (consciousness) he believed scientific problems and theories to live in an additional world. Hence mathematics has its on reality and existence. The human mind is the link joining the three worlds as it can access reality and the world of theories. The conjecture stated above would attribute Popper's third world the greatest weight, as it is the underlying structure giving rise to a physical world which in turn can manifest life which in turn develops consciousness able to uncover this third world.

Obviously any further development in theoretical physics will be intimately linked to some very deep mathematical aspects (e.g. topology). Thus a basic project would consist in an extensive study of the subjects. Perhaps a small selection would be: [Baa96], [Fu92], [Fra97], [H-et.al.98], [HSW99], [HT94], [JS96], [Lan99], [OR86] and [Sc95]. The topics covered are among others: Chern-Simons-Witten theory (also known as topological QFT or quantum topology), Jones polynomials, Wess-Zumino-Witten theories, quantum groups (e.g. Hopf algebras), twistors and spin networks. It is also very interesting to note that these physical studies are intimately linked to, and interact with modern mathematical research, namely Seiberg-Witten theory, symplectic Floer homologies and 3-manifold invariants; [Ga99]. Conceptually this could be interpreted as a new paradigm shift in science: mathematics not only underlies physics, but at a very high level of abstraction mathematics and physics interact and fuse. In the words of Vafa: `I believe we are now witnessing perhaps an unprecedented depth in this interaction between the two disciplines [physics and mathematics]'; [Va98].

However, even if one day a final, unifying theory does emerge its practical relevance could well be disputed. In having found an ultimate principle and its mathematical expression in accordance with reality one still is not able to address the second biggest challenge of physics: how can complex systems be described? I.e. how does one reach the natural, macroscopic world we perceive (with its highly sophisticated problems) from such an underlying description of reality? The ultimate theory of reality should also allow problems arising in economics and sociology to be addressed, next to natural sciences on a macroscopic level. Interestingly, the above conjecture still holds: in order to understand highly complex systems, our knowledge of mathematics must be deepened. Chaos theory is probably the first mathematical branch to address complex, non-linear phenomena. Perhaps the most promising approach to the problem of complexity is known as information geometry. Here highly sophisticated statistical systems are being analysed in terms of geometry. $^{\text{\scriptsize\cite{proj1}}}$ This allows the possibility of linking an underlying theory of reality (also expressed in terms of geometry) to complex, every-day phenomena. In addition, the ideas of chaos theory and non-linear effects are being studied within the context of theoretical physics, called chaotic field theory. [Cv00] gives an account of how this approach produces a better understanding of perturbation theory and Feynman diagrams.

So, in essence the study of a wide range of mathematical subjects is the best investment for any scientist hoping to glimpse more of the fundamental and universal workings of nature.