"The enormous usefulness of mathematics in the natural sciences is something that borders on the mysterious and has no rational explanation" (Eugene Wigner).
"The most incomprehensible thing about the universe is that it is comprehensible" (Einstein).
Wigner and the "Miracle "of Mathematics
Under the title "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", the physicist (Nobel laureate) Eugene Wigner gave a lecture, in 1959, at New York University. This lecture was published as an article the following year [Wigner, 1960] and had a great impact by raising the question of the interrelationship between mathematics and the natural sciences, physics especially. Wigner's main ideas were as follows:
We do not know the reason why mathematics is so useful in describing physical phenomena; why nature can be described so accurately with mathematics. It is a mystery that has no rational explanation.
Wigner joins the chorus that has for centuries repeated Galileo's famous phrase (among them, Newton and Einstein), "The book of nature is written in the language of mathematics." But he makes two observations:
Only some mathematical concepts are used in the formulation of the laws of nature, implying that mathematics encompasses something more.
Mathematical concepts are neither arbitrarily selected nor accidentally useful. They are useful because they are part of the very language of nature. He cites as examples Newton's second law of motion (F = m*a), the matrix formulation of quantum mechanics, and quantum electrodynamics.
It is surprising that a physicist "stumbles" upon a mathematical concept that best describes a phenomenon, and it turns out that a mathematician had already developed the same concept independently. Wigner cites as examples:
Complex numbers, which are absolutely necessary for the formulation of the laws of quantum mechanics. They are not a mere instrument of calculation.
Hilbert space, a generalization of the concept of Euclidean space that allows spaces of any dimension, even infinite. Hilbert space is also of crucial importance for the mathematical formulation of quantum mechanics.
Riemannian geometry, which Einstein used in his theory of generalized relativity for 4-dimensional space (the 3 dimensions of space and time).
The mathematical structure of a physical theory often points the way to further advances of that theory and even to empirical predictions. This cannot be mere coincidence, so there must be a deep connection between mathematics and physics.
Mathematical concepts are general, since they have application beyond the context in which they were developed, being able to describe a large number of phenomena. And he gives two examples:
The law of gravitation. Initially it was used to model the free fall of bodies on the surface of the earth. But this law was extended to describe the motion of the planets, which has proven to be accurate beyond reason.
Maxwell's equations of electromagnetism. They were established to model the known electromagnetic phenomena in the year 1865, the date of their publication. These same equations also served to describe radio waves, discovered by Heinrich Hertz in 1887.
It is asked whether mathematics can help determine the correct or most appropriate theory to apply to a given physical phenomenon. The problem is that some theories that we know to be false also give good results. He cites Bohr's early atomic model, Ptolemy's epicycles, and the free electron theory as examples.
Wigner concludes that mathematical language is the appropriate language for the formulation of the laws of physics and that it is a gift that we neither understand nor deserve.
Hamming's partial explanations to Wigner's question]
Richard Hamming [1980] −the creator of the error correction code that bears his name− provided four partial explanations to the question posed by Wigner, although he acknowledged that the essential question remains unanswered:
Humans see what they look for. Science is not based on experience alone,
for it also depends on the "glasses" we use, the way we look at phenomena. Many physical laws can be deduced intellectually. Eddington went further by stating that a sufficiently wise mind might be able to deduce all of physics without relying on experience. Hamming gives four examples of non-trivial physical phenomena that arise from the mathematical tools employed rather than from intrinsic properties of physical reality:
Galileo discovered the law of the fall of bodies, not only experimentally, but also by intellectual reflection, by logical reasoning. By combining mathematics and experimentation, Galileo is considered the father of modern science.
The law of universal gravitation, governed by the inverse of the square of the distance is deduced from the law of conservation of energy and the 3 dimensions of space. The exponent (2) itself is more a reflection of Euclidean space than of the properties of the gravitational field.
The uncertainty principle of quantum mechanics follows from the properties of Fourier integrals and assuming time invariance.
Einstein's theories of relativity (special and general) were the result of exploring possible theories with mathematical tools, without any experiment (only considering the well-known result of the Michelson-Morley experiment, the invariance of the speed of light). Einstein believed more in mathematical truth than in physical truth (if observations were inconsistent with his theories, the observations were to blame).
Humans create and select the mathematics that best fits each phenomenon. For example, scalars were not adequate to represent forces, so vectors and tensors had to be invented.
Mathematics covers only a part of human experience. Much of human experience does not fall under mathematics, but under the philosophy of values (ethics, aesthetics, etc.). A broader perspective is needed.
Evolution has supplied the model, so that the humans who have had the best models of reality are the ones who have survived. But evolution may have blocked us in some directions, so that "maybe there are thoughts we can't think."
Physics vs. mathematics
The issue of the relationship between the abstract world of mathematics and physics (which purports to describe the universe) is a recurring theme in the philosophy of mathematics.
The idea in general that the universe is mathematical in some sense goes back to the ancient Greeks, who considered mathematics a higher science.
Pythagoras concluded that number was the essence of reality, and that mathematics was the way to understand the universe. The same mechanisms that explain mathematical reality explain physical reality. Nature can be studied mathematically.
Plato said that mathematical objects existed in a higher dimension (the realm of Ideas or Forms), physical objects being an imperfect projection of that higher, ideal realm.
Kepler, like Pythagoras, believed that the universe could only be understood through mathematics. He discovered his famous three laws of planetary motion, laws that were mathematical and simple.
Galileo, in his work "Dialogue on the Great Systems of the World", sets forth three postulates: 1) There are universal laws, mathematical in character; 2) These laws can be discovered by experiment; 3) These experiments can be replicated.
Descartes believed that physics was a branch of mathematics: "No other principles are required in physics than those used in geometry or abstract mathematics, nor should they be desired, for all natural phenomena are explained by them."
For Berkeley, reality is fundamentally mental. The physical world is a derivative construct, a result of the mental world.
For Kant, the mathematical world is a product of the mental world.
Einstein said that "The most incomprehensible thing about the universe is that it is comprehensible. How can it be that mathematics, being after all a product of human thought that is independent of experience, is so admirably appropriate to the objects of reality?" Einstein often lamented that he did not know more mathematics, for he knew that mathematics opens the door to understanding physical phenomena and to creativity.
Characteristics of mathematics in relation to physics
Theoretical physics.
Physics attempts to understand the universe by constructing a mathematical and conceptual model of reality. Its main core is the so-called "theoretical physics" or "mathematical physics", a bridge discipline between physics and mathematics aimed at mathematically formulating the phenomena of nature. Milestones of this discipline are: the formulation of analytical mechanics (Joseph-Louis de Lagrange and William Rowan Hamilton), an abstract and general formulation of mechanics, as well as the quantum and relativistic revolutions. The great advances in 20th century physics, mainly in quantum theory and relativity, are largely due to theoretical physics.
Transcendence of the physical world.
Mathematics is more universal, more general than physics, it transcends the physical world. "Mathematical language opens a virtual window to spaces beyond our three-dimensional physical world" (Marcus du Sautoy). For example, mathematics allows us to work with more than three dimensions.
Universality.
In general, there is a tendency towards the mathematization of the sciences, especially physics. The reason lies in the fact that mathematics makes it possible to contemplate things from a general or universal point of view, thus favoring the ability to relate everything from a higher point of view.
Predictive ability.
Mathematics has predictive capacity, since it facilitates the way to new discoveries and phenomena not yet observed experimentally. Examples:
The famous Dirac equation (describing elementary particles of spin ½, such as the electron), formulated in 1928, by admitting more than one solution, postulated the existence of antimatter, years before the first such particle (the positron, the antiparticle of the electron) was discovered in 1932. This was considered a great triumph of theoretical physics or mathematical physics. The Dirac equation also predicts monopoles, as yet undiscovered. Dirac said that mathematical entities had to be interpreted in terms of physical entities.
The atomic bomb was a prediction of theoretical physics.
Particles and phenomena (the Higgs particle, micro black holes, etc.) predicted by theoretical models are searched for in particle accelerators.
Symmetry.
Symmetric structures are the most stable and efficient. "Extensive areas of mathematics, physics and chemistry can be explained on the basis of the underlying symmetry of the structures involved." "Symmetry is not only a precursor of meaning and language, but it is also the way that nature follows in order to be efficient and economical" (Marcus du Sautoy).
Simplification.
By using generic and highly abstract mathematical concepts, the description of physical phenomena is simplified. For example, Maxwell's equations of electromagnetism, which in vector notation are four equations, are reduced to one in Clifford algebra. Simplicity should always be sought by using concepts at a high level of abstraction.
Three Worlds
Penrose's "Three Worlds"
For Roger Penrose [2006], reality is a unity structured in three worlds: "We live in a single reality with three dimensions, mathematical, physical and psychic, unified in man."
The physical world.
It has its ontological foundation in the mathematical world. It is an external world, that of sensible and perceptible reality through sensations. The physical world is a manifestation of the mathematical world. There is a profound dependence of the physical world on the mathematical world. The physical explanation of the world is based on mathematics.
The psychic world.
It is a world of internal, personal, intersubjective experiences. Consciousness is a psychic property, but only of some beings in the physical world. There is an interrelation between the psychic world and the mathematical world, which closes the cycle between the three worlds. The three worlds need each other and complement each other.
The Platonic mathematical world.
Penrose is a convinced Platonist. Mathematics inhabits the world of Being, a timeless, harmonic, ideal and perfect world. It is an intelligible world, in a rational and intuitive sense. Mathematical entities possess an existence that can only be discovered by intelligence and intuition. The ontological foundation of the physical world is mathematical. Man is the only being capable of contemplating mathematical realities.
Penrose admits that these worlds may be reflections or manifestations of a higher or deeper reality.
Frege's "Third Realm"
Frege, in "Der Gedanke" (The Idea), Frege asserts that there is a world of non-sentient, independent objects, which he calls the "Third Realm" (Drittes Reich).
According to Frege, there are three worlds or realms:
The external, objective, sensible physical world.
The inner mental world of ideas, mental processes and psychological representations, of a subjective type. The mental world is ontologically superior to the physical world.
The world of ideal concepts and entities of logical type, an objective and timeless world, which is independent of individual subjective minds, which is above the physical and mental worlds and which is independent of the human mind. For example, numbers are logical type entities residing in the Third World, which are independent of all psychological processes and representations.
For Frege, mathematical and logical entities are real, objective entities that dwell in the Third World. It is a world beyond the physical and the mental, an abstract world, which is real, true, eternal, non-sensible and invisible. It is an archetypal world from which the physical and psychic world emerges. It is the world from which all laws emerge. "The laws of arithmetic are not laws of nature, but laws of the laws of nature, that is, fundamental principles about the thinkable." It is a kind of real, objective and eternal Platonic realm, where mathematical entities live that have their own existence, independent of the physical and mental world.
Frege also opposed psychologism, the idea that numbers (or mathematical entities in general) are subjective mental contents. Mathematics is not subordinate to psychology, but above it.
Sense is not something subjective, that is, it does not belong to the mind of the individual, but is an objective entity, ontologically independent and shared by a community of speakers. Meaning is an abstract notion, not a psychological one. It resides in the Third World.
For Frege, abstract formal expressions have meaning, are objective, and belong to the Third World. He rejected the formalism of arithmetic, according to which number and arithmetical expressions are a mere set of meaningless, meaningless symbols. With the idea of the Third World, Frege aligned himself with Platonist realism.
For Frege, logical truth is only predicated of logical relations between mathematical objects in the Platonic logical-mathematical realm. Reference and truth connect language with reality. Simple linguistic expressions refer to simple elements of reality (objects, individuals, etc.) and propositions correspond to facts, which can be true or false. The reference of a proposition is its truth value.
Frege postulated mathematical Platonism by stating in his work "Foundations of Arithmetic" that the objectivity of concepts is dissociated from the cognition of the subject. He considered mathematics as a science of the Platonic domain of mathematical concepts and objects. "Mathematicians, like geographers, cannot create something out of nothing." Mathematics is the language of thought. With the power of thought we grasp or apprehend mathematical objects. "In arithmetic we deal with objects which are not given to us from without, as something foreign, thanks to the mediation of the senses, but which are given directly to reason, which can contemplate them as the deepest part of itself."
Popper's "Three Worlds"
According to Popper, there are 3 worlds:
The external world: the physical entities (the objective and corporeal reality), the physical, chemical and biological world.
The inner world: the non-corporeal of mental entities, the psychological world, including subjective and unconscious experiences.
[The products of the human mind, which are entities that have their own existence. It is the world of culture, including all products of the human intellect (philosophical, scientific, artistic contents, etc.). Scientific theories and logical laws belong to the third world. This world is the most valuable and productive.
These 3 worlds interact with each other. Worlds 2 and 3 are able to interact with world 1. The interaction between world 3 and world 1 is realized through world 2.
According to Popper, the third world is an objective world without a cognizing subject. The proof is that these objects can produce causal effects or manifest themselves in worlds 1 and 2. For example, a sculpture is not only an object of world 1, but is the result of a planned and elaborated project in world 3. And two copies of a book (which are distinct objects because they occupy different spaces) are the same book in world 3. Following this reasoning, we could say that there is only a single DNA in world 3 and multiple manifestations in world 1. This is along the same lines as John Wheeler's proposal that there is only one electron, and that this is the cause of the indistinguishability of electrons (they all have the same charge and the same mass).
Popper drew analogies between cultural and biological evolution, pointing out the similarities between the process of scientific progress and natural selection. In his work "The Logic of Scientific Inquiry" (1934) he proposed a theory of knowledge based on trial and error, i.e., by Darwinian selection.
Popper did not believe in top-down causality, but in bottom-up causality: nature is creative, man being the supreme result of this creativity. Man is an emergent phenomenon, the result of a process of gradual evolution of nature. Mind and consciousness are epiphenomena of the brain.
Popper's conception differs from Penrose's 3 worlds in that for the latter, world 3 is the world of mathematics.
MENTAL, the Third World: A Theory of Everything
Primal archetypes: the key to the answer to Wigner's question
MENTAL, by relying on archetypes of consciousness, provides a simple answer to the question posed by Wigner, unveiling and clarifying the deep relationship between mathematics and physics. Indeed, we can distinguish 3 levels of reality:
Consciousness is the source of the source of all possibilities and of everything that exists.
The possible worlds (manifested and unmanifested). Among the manifested is our universe.
MENTAL is the world that connects the two previous ones, the world of the primary archetypes, the abstract realm of all possibilities. It is the common structure, the Constitution (or Magna Carta) of all possible worlds.
The archetypes of consciousness connect the inner with the outer. It is the third factor, the third world. From consciousness, ontology is the same as epistemology. Meaning emerges from the connection between inner world and outer world through the archetypes of consciousness.
The primitives of MENTAL are primary archetypes, a concept equivalent to Platonic Ideas. Primary archetypes are inexpressible and transcendent, of which we can only see their concrete manifestations. The expressions of MENTAL, which are manifestations of the primary archetypes, can also be considered to reside in a Third Realm of abstract entities, for they are neither physical nor mental. So we can identify the Third World with the archetypal world, a real, objective and profound world that manifests on the physical and mental level.
For Popper there are 3 worlds. But according to the universal principle of downward causality, there is only one deep reality and all the rest are manifestations.
The Third World is real, it exists and is constituted by all the possible expressions that can be formed. It is what we have symbolized by @ (the universal expression or the universe of expressions). When we write an expression, we are accessing something that already exists.
MENTAL is a world in itself to explore. It is autonomous. It underlies everything manifested at the physical and mental level. MENTAL is the third world, the underlying world, a real, objective world, already existing a priori. It is all the possibilities (all the possible expressions), which already exist. The universe is one of the possible manifestations.
The archetypes of MENTAL cannot be expressed (as neither can consciousness), they can only be intuited, because they belong to a deep level. They can only be expressed by means of examples, that is, "materializations" or particularizations at a superficial level of these primary archetypes.
MENTAL goes beyond the mere metaphor of the physical world. MENTAL connects mind and nature. That is why it is a model of mind, nature and possible worlds. MENTAL's set of instructions or primitives are common to mind and nature. Mind and nature share the same primary archetypes.
MENTAL goes beyond the mathematical world. Mathematics and computer science are two of its manifestations.
For Galileo, nature is a book written in the language of mathematics. For Zuse, Fredkin and Wolfram, nature is written in the language of computation, in the language of computers. But both positions are right, for the universe shares the same principles or archetypes as mathematics, computing and the human mind. Truth resides in the deep, and in that place, there are no distinctions or differences or boundaries; it is all the same thing. Mind and nature are, in essence, the same thing. The model of the mind and the model of the universe are analogous (or synchronistic) because both are manifestations of the same primary archetypes. "The objective structure of the universe and the intellectual structure of the human being coincide" (Pope Benedict XVI).
MENTAL is consciousness and creates consciousness. In general, when we use mathematics we are raising our consciousness. MENTAL provides a deep (or higher) point of view, where everything has meaning and significance. Mathematical truth resides in the deep: in the primal archetypes.
So, to the key question (Why is mathematics so useful in describing natural phenomena?), we are now in a position to provide an answer:
Because mind and nature share the same primary archetypes. The universe is a particularization of the Mathematical Realm, which in turn is a manifestation of the primary archetypes. All within a mechanism that goes from the universal to the particular, from the profound to the superficial, from the superior to the inferior.
MENTAL is a "theory of everything":
It is the deepest, at the abstract level, that we can conceive of.
It unites mathematical world, psychic world and physical world.
It is the foundation of consciousness.
It is the common deep structure of all things.
It is the source of all manifestations.
Its primitives are primary archetypes that connect the internal or deep with the external or superficial.
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