"The book of nature is written in the language of mathematics" (Galileo).
"No physical theory that deals only with physics will ever explain physics" (John Wheeler).
"The challenge of physics is to explain how consciousness works" (Roger Penrose).
Mathematical Physics
Physics is geometry. The geometric universe
That there is a close relationship between the physical universe and geometry has always been intuited. But it was in the 20th century when this idea was given shape, mainly by authors such as Einstein, Dirac, Schrödinger and John Wheeler, who claimed that the universe is a geometrical structure, so physics is geometry.
"What we observe as material bodies and forces are nothing but forms and variations in the structure of space" (Erwin Schrödinger).
"There is in the world nothing but curved and empty space. Matter, charge, electromagnetism and other fields are only manifestations of the curvature of space. Physics is geometry" (John Wheeler).
The formal geometrization of physics was initiated by Einstein, who used Riemannian geometry in his theory of general relativity. For Einstein, space and time are bound together (in an entity he called "space-time"), and gravity is a consequence of the curvature of space-time. Riemannian geometry is a local, nonlinear geometry of curved spaces of any number of dimensions, based on the concept of manifold.
A manifold is a geometric object that generalizes the intuitive notion of curve, surface and in general the shape of any geometric object of dimension n. The dimension of a manifold is the number of independent parameters needed to locally locate a point on the manifold. Curves are dimension 1, surfaces are dimension 2, and so on.
A manifold can be visualized as a set of infinitely small elements without curvature. For example, a curve can be thought of as a set of tiny straight segments, and a surface can be thought of as a set of tiny planes.
Kaluza and Klein added to Einstein's equations of general relativity (the geometric Riemann tensor) a fifth geometric dimension that explained electromagnetism.
The modern string theory of quantum mechanics is a geometric theory. The particles are tiny strings (branes or loops) vibrating in the space of 11 or 12 dimensions, 11 dimensions in the M-theory (from "mother", developed by Edward Witten in 1994), and 12 in the F-theory (from "Father", father), by Cumrun Vafa. The M theory contemplates one time dimension and the F theory includes two time dimensions.
It is often claimed that string theory is the best positioned for ToT ("Theory of Everything", a theory that unifies relativity and quantum mechanics. The problem is that the mathematics it uses is extremely complex and a ToT must necessarily be simple.
Physics is symmetry
Symmetry is a key aspect of aesthetics, of harmony, of the fit between the elements of a system. In physics it is identified as the foundation of all laws of nature. In fact, many physical laws can be deduced from symmetry requirements that these laws must obey. Symmetry is nature's way of achieving maximum economy, simplicity.
In special relativity, there is symmetry in the physical laws, since they are the same regardless of the relative motion (at constant velocity) between different observers (inertial systems). In general relativity, the equivalence principle associates symmetry between gravity and acceleration.
The four forces of nature (electromagnetic, gravitational, strong nuclear and weak nuclear) are directly related to the principles of symmetry.
Supersymmetry, abbreviated "SUSY"), a concept arising from string theory, is a more abstract symmetry that relates particle properties to spin values, an intrinsic property (such as mass or electric charge) associated with the spin of a particle about itself.
Emmy Noether's theorem, formulated in 1915, merged symmetry and conservation, as two facets of the same property. It states the following: To every continuous symmetry of physical laws corresponds a conservation law and vice versa. A continuous symmetry is the one that comes from transformations that can vary continuously (e.g., translations and rotations). For example: under translations, momentum is conserved; under the passage of time, energy is conserved; under rotations, angular momentum is conserved.
Symmetry is associated with the mathematical concept of group.
Mathematics as a metaphor for the physical world
Cognitive scientists George Lakoff and Rafel Núñez [2001] argue that mathematics are mental metaphors derived from concepts of the physical world. For example: numbers correspond to sets or collections of objects, geometric figures correspond to objects in space, functions are sets of ordered pairs or curves in the Cartesian plane, etc.
Each mathematical concept corresponds to a cognitive metaphor of the real world.
Mathematics is the result of the human cognitive apparatus and, therefore, must be understood in cognitive terms.
A Platonic type of mathematics, transcendent, independent of human thought is a meaningless or unanswerable question.
Mathematical objects exist only as particular instances of concepts in our brains.
For Saunders MacLane [1986] −creator, together with Samuel Eilenberg, of Category Theory− mathematical concepts are grounded in ordinary human activities, mainly interactions with the physical world.
MENTAL,a Language for Physics
Features
Paradigms.
Whatever the interpretation of physical reality is (mathematical, computational, geometrical or pre-geometrical), MENTAL is a complete language that allows expressing all these paradigms, because it is an operational and descriptive language. Science is restricted by current mathematics, which is of a superficial type (because it is weakly founded) and lacks a language that covers these aspects (operational and descriptive).
Archetypes.
Physics is a manifestation of deep laws. In this sense, MENTAL is a very suitable language for physics, since it is based on archetypes, on universal concepts, thus facilitating the discovery of the deep interrelationship between all phenomena of nature.
Mathematics is not a metaphor for the physical world. Mathematics, the physical world and the mental world share the same archetypes.
Abstract geometry.
MENTAL goes further, in its level of abstraction, than geometric algebra, which many authors consider the language of physics, by identifying physics with geometry. MENTAL is the foundation of abstract geometry, traditional (Euclidean) geometry and other alternative geometries based on the concept of abstract space.
Simplicity.
When mathematicians develop abstract theories, they always try (consciously or unconsciously) to pursue maximum beauty and simplicity. Why? Well, because as we approach the Unified Field, order, simplicity and consciousness increase. As at a deep level there is simplicity, MENTAL is the most appropriate language to express simplicity, because it is a simple language.
Nature uses the simplest, and therefore the most profound and universal, forms and laws of maximum consciousness.
Levels of reality.
The physical world is a manifestation, particularization or projection of the mathematical world, but the physical world is not equivalent to the mathematical world, it is an inferior world. To claim (as Max Tegmark maintains) that the physical world and the mathematical world are the same thing is to confuse the levels of reality.
Symmetry.
Why does symmetry play such an important role in physics? Because symmetry is simplicity and efficiency, and simplicity is consciousness because it unites opposites. MENTAL is a language of primitives with symmetry (they are pairs of opposites or duals).
It is easier to address physical symmetry issues using a language based on conceptual symmetries (dual concepts).
Bibliography
Barrow, John David. ¿Por qué el mundo es matemático? Grijalbo, 1997.
Borovik, Alexandre. Mathematics under the microscope. American Mathematical Society, 2009. Disponible en Internet.
Courant, Richard; Robbins, Herbert. ¿Qué es la matemática?. Aguilar, 1967.
Eckardt, Horst; Felker, Laurence G. Einstein, Carta y Evans. ¿Inicio de una Nueva Era en física? Internet, 2005.
Hamming, Richard. The Unreasonable Effectiveness of Mathematics. The American Mathematical Monthly 87 (2), 1980. Internet.
Hersh, Reuben. What is Mathematics, really?. Oxford University Press (USA), 1999.
Lakoff, George; Núñez, Rafael. Where Mathematics Come From. How the Embodied Mind Brings Mathematics into Being. Basic Books, 2001.
MacLane, Saunders. Mathematics: Form and Function. Springer Verlag, 1986.
Putnam, Hilary. What is Mathematical Truth? Historia Mathematica 2: 529–543, 1975.
Sarukkai, Sundar. Revisiting the ”unreasonable effectiveness” of mathematics. Current Science, 88 (3), 10 Feb. 2005. Disponible online.
Wigner, Eugene P. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics, 13 (1): 1-14, Feb. 1960). John Wiley & Sons. Disponible online.
Wigner, Eugene P. Symmetries and Reflections. MIT Press, Cambridge, 1967.
