"The world is rationally comprehensible because it has structure" (Plato).
"The role of the sciences is to classify rather than to measure" (Aristotle).
"What things are to be reduced to are Platonic ideas" (Kurt Gödel).
Platonism and Aristotelianism
Plato's theory of ideas
Relying on his famous allegory of the cave [Republic VII, 514 d], Plato elaborated his theory of ideas, the core of his philosophy. With it he sought to ground reality and explain unity in diversity. According to Plato, reality has two poles:
Intelligible.
It is the higher world, the world of ideas. These ideas are eternal, perfect, immutable, susceptible of true knowledge (episteme). They represent the true being, They are the models or archetypes of the other pole of reality (the things of the sensible world). They have real and independent existence. They are transcendent. They cannot be the object of sensible knowledge, they are only cognizable by reason. They are a priori knowledge (prior to experience).
Sensible.
It is the lower world, the material world of things. Things are temporal, imperfect, changeable, deceptive, susceptible of opinion (doxa). They are not true being, but the projection or manifestation of intelligible reality (our senses perceive only the shadows or projections on the wall of the cave). But the sensible world cannot be reduced to a mere illusion, for things reflect and participate in ideas. They are a posteriori knowledge, it requires sensory experience, contact with the physical world.
According to the Platonic theory:
The origin of things is in the ideas. His teacher Socrates claimed that knowledge is based on concepts, which represent the reality of things. Plato intuited that there was something superior to the concepts of the mental plane and called it "idea", elevating it to the category of Being. Ideas are abstract entities, models or ideal patterns that constitute a superior and universal ontology.
Ideas have their existence in a realm of their own, in a separate dimension, different from the material and spatio-temporal dimension.
Since the world of ideas is timeless, there are no cause-effect relations either.
Ideas are not creations of the mind, for they exist independently of thought. They are eternal, immutable, incorporeal forms. We can only perceive and know them by pure reason, not by the physical senses.
True reality consists of pure essences of archetypal ideas. The phenomena we perceive are only reflections or manifestations of that higher reality.
Every particular (sensible) thing has its support and foundation in the intelligible, in the universal.
Sensible things participate in ideas. Ideas constitute a higher reality, but they are also present in things, in the lower world.
With understanding and reason knowledge is obtained. With imagination and belief one obtains only opinions. Sensible reality is changeable and does not provide true, full and universal knowledge.
The source of all ideas is God.
Dialectics, the science of ideas, leads us to true knowledge.
"To know is to remember". According to Plato, all human souls once inhabited the higher world of ideas.
Plato has been one of the most influential thinkers in Western culture. Platonism played a crucial role in the development of Christianity, as it was the mainstay of Christian theology. The theologians Clement of Alexandria and St. Augustine were the first exponents of this philosophy. In the 20th century, Heidegger said that "My philosophy consists in rediscovering the Being hidden by Platonic idealism", and Whitehead affirmed that "The history of Western philosophy is nothing but a series of footnotes to Plato's works".
Plato's transcendental mathematics
With the theory of ideas, Plato also wanted to found mathematics, for he was influenced by Pythagorean notions of numerical and geometrical harmony. Actually, the theory of ideas had its origin and inspiration in geometrical forms. In fact, the Greek word "idea" is also translated as "form". For example, a circle is a geometric figure composed of points equidistant from a given one. But no one has ever seen such a figure (beginning because the points have no extension), nor will it ever be seen. The circular objects of the sensible or material world are only approximations to the ideal circle. Plato recommended geometry as the system to follow to intuit or approach this metaphysical realm. Remember that on the frontispiece of his Academy he wrote "Let no one enter here who does not know geometry".
Towards the end of his life, Plato adopted a philosophy closer to the Pythagorean one, interpreting ideas in mathematical terms, which he reflected mainly in the Timaeus. Plato, although not a mathematician, had a considerable influence on the development of mathematics, because of his conviction of the transcendence of this discipline:
Mathematical ideas are the fundamental ideas, the most important of all, since they constitute the essence of reality.
Mathematics is the intellectual aristocracy of knowledge, since its mission is to form the intellect, to found philosophy and all knowledge in general. Mathematics is a universal ontology. The knowledge of mathematics is an essential prerequisite for all wisdom.
Mathematics must be independent of all pragmatism, for its purpose is knowledge itself. The human spirit can explore this territory of ideas without bowing to the sensible dimension of reality. Mathematical applications are at a lower, more superficial level, and they are dealt with by a science of mathematical entities, the "mathematical art".
Mathematics elevates the spirit towards the abstract and towards the truth, to escape from the sensible realm. It lifts the soul from sensible things to intelligible truth, cognizable by the rational way.
Arithmetic and geometry are a propaedeutic (preparatory teaching) for Dialectic, which is a science superior to Mathematics.
Mathematics leads us beyond ordinary existence. And it shows us the underlying structure of all creation. Mathematics is the key to the transcendent world, a world beyond ordinary description.
Mathematics is connected with the divine: "God always does geometry" (Plato). We must try to discover the laws that govern the universe in order to approach the supreme reality of God.
The fundamental objective of mathematics is the contemplation of Being, of the profound.
In the depths of the world of ideas, in its fundamental core, is the mathematical world.
The mathematical world is not a part of the mental world, but the framework, the structure of which the mental world is constituted.
The physical world is a projection or manifestation of the mathematical world. Therefore, there is a correspondence between the laws of nature and those of mathematics. Physical laws are mathematical laws.
Mathematical Platonism
Also called "mathematical realism", it is the application of Platonic idealism to the mathematical world. This term was first coined by Paul Bernays [1935], to refer to the doctrine that mathematical concepts have an objective reality independent of the cognitive subject.
The principles of mathematical realism, as it is conceived today, are as follows:
Existence. Mathematical objects exist, they are real.
Abstraction. Mathematical objects are abstract. They have metaphysical existence, they do not belong to the physical world, they are immaterial. Therefore, they have no spatio-temporal attributes.
Independence. Mathematical objects exist by themselves, and are independent of the physical world and of human beings.
Eternity. Mathematical objects are eternal or timeless. They have existed since the beginning of time and before humans first perceived them. They have, therefore, no causal relationships.
Inalterability. Mathematical objects cannot be modified or destroyed.
Discoverability. Mathematical truths are not invented, they are discovered. They are not constructions of the mind. We discover mathematical entities that already exist.
Apriorism. Mathematical truth is a priori knowledge, prior to all experience.
Intuition. Intuition is the faculty that allows us to know mathematical reality.
Perfection and order. The mathematical world is a perfect and ordered world.
Some examples of Platonic mathematical entities are:
The natural numbers and their structure are abstract entities that have real existence, independent of the human mind. For example, the numbers 16 and 25, perfect squares, have properties as real as the physical properties of light and gravity.
A triangle is a real entity and not a creation of the human mind.
The Pythagorean theorem is eternally true, regardless of place, time or the person using it.
The so-called "last theorem of Fermat", which states that no positive n-th power of an integer can be the sum of two other positive n-th powers if n is an integer greater than 2. This theorem is an objective statement. Femat wrote in 1637, in the margin of a copy of Diophantus' Arithmetic: "I have found a wonderful demonstration, but it does not fit in this narrow margin". Finally, Andrew Wiles proved it in 1995.
Mandelbrot's (fractal) set is a discovery, not an invention of the human mind. Although this geometrical figure needs the help of Computer Science for its visualization and exploration.
The group of symmetries called M (from "monster"), which has really a huge number of symmetries and is representable in a space of at least 196,883 dimensions, is a discovery.
Aristotle's philosophy
Aristotle's view presents differences from that of his teacher. He criticized his theory of ideas for its transcendent and mystical character, posing his philosophy on a more earthly and pragmatic level.
The essence of things (universals) is to be found in things themselves and not separated from them.
Knowledge is acquired in two ways:
In a direct, practical way, through the senses, by making abstractions to grasp the common features or characteristics of the objects of experience. "There is nothing in the intellect that has not first been in the senses."
Indirectly, by deducing new knowledge by applying the rules of logic. He based his formal logic (the logic of syllogisms) on the simplest type of sentence: subject and predicate (substance and attribute, respectively).
Like Plato, he considers abstract and universal knowledge superior to any other, but disagrees with Plato regarding the method of attaining it. For Plato, knowledge is of the up-down type (from the world of ideas to the sensible world). For Aristotle, knowledge is achieved bottom-up (from the physical world to the abstract world). "What we perceive by the senses is necessarily particular, while science consists in recognition of the universal." For Aristotle universals are real, but they are attached to things.
Substance is what is fixed and immutable in reality. Accident is that which is subject to change and mutation. Substance, in turn, is made up of matter and form. It is his hylemorphic theory (from hylé, matter and morphé, form), which states that everything is matter and form.
For Plato, mathematics is discovered. For Aristotle, mathematics is invented, created through abstraction.
God is not the source of ideas, but the highest of ideas.
Plato did not specify which were the ideas or forms that rule in the higher world. Aristotle, on the other hand, drew up a list of 10 abstract categories that constituted the essence of reality, the first of these being precisely substance. "The role of the sciences is to classify rather than to measure" (Aristotle).
The positions of Plato and Aristotle are nicely reflected in Raphael's painting "The School of Athens", which shows Plato and Aristotle in the center, defending their respective theories. Plato raises the index finger of his right hand to the sky, defending the theory of ideas (the ideal, the sublime, the superior, intuition). Aristotle, with the palm of the right hand downward, defending the theory of the forms (reason, logic, the formal, the facts, the practical).
Plato and Aristotle in "The School of Athens" (Raphael)
Platonic mathematicians
Platonism has been −and still is−a very important and influential philosophy, as evidenced by the fact that many important mathematicians have declared themselves Platonists. This is what is called "mathematical Platonism" or "mathematical realism".
Throughout history, many mathematicians have been realists and referred to their own work as "discoveries." When a mathematician feels that he is discovering objective truths, and not simply constructing systems, he is more or less consciously committed to Platonism. Today, Platonism remains one of the most lively and influential philosophies of mathematics. Prominent Platonist mathematicians are:
Pythagoras. He can be considered a Platonist because he believed that the integers constituted the essence of reality, a noumenal type of reality, the form (or ideal) existing behind the reality of the physical (phenomenal) world. For the Pythagoreans, mathematics was more than a science, it was a unified and profound way of looking at the world. They recognized 10 essential principles of all things, which were associated with each of the first 10 numbers. Of these 10 numbers, the number one (or monad) was considered the supreme principle, as it is present in all numbers. The Pythagoreans accepted only the integers and rational numbers as transcendent reality, a conviction that was profoundly altered when they discovered the irrational numbers.
Descartes. His position is one of radical rationalism. Ideas are innate and possessed by every being endowed with reason. Reason is the same for all. Truths are innate, ways of thinking the physical world. Mathematical truths are based on ideas. Mathematical objects are ideas, ways of thinking, they are not sensible objects. The eternity of mathematical truths is due to the immutability of the innate ideas with which we conceptualize objects. Descartes' position can be considered a quasi-platonism.
Leibniz. He follows Descartes and refines the rationalist view of mathematics. The senses connect us with particular or singular entities. But both are not sufficient to establish universal truths. The universal resides in simple principles and basic building blocks. The supreme simplicity resides in his concept of monad: a simple metaphysical substance (metaphysical atom), without parts, without extension, indivisible, eternal and autarchic. Monads are the foundation of everything and are like souls reflecting the entire universe.
Cantor. He brought a modern view of mathematical Platonism based on the universal concept of the set, and discovered the transfinite numbers. Cantor believed in the existence of the Platonic world of ideas, where numbers, sets, infinity had real and independent existence, and where infinite sets (such as natural numbers, real numbers and transfinite numbers) also existed as totalities. For Cantor, mathematics is not only useful for science, but is an inexhaustible source of intuition and insight.
The appearance of paradoxes in Cantor's set theory led to its axiomatization, the best known being the so-called ZFC (the Zermelo-Fraenkel theory with the axiom of choice), which restricted operations to define new sets from given sets. The introduction of axiomatics was a departure from Cantor's initial platonism.
Frege. He postulated mathematical Platonism by asserting 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 discover, not invent. Mathematics is the language of thought. Mathematical objects are grasped with the power of thought. Mathematical objects are not grasped with the ordinary senses, but are given directly to reason, which can contemplate them as the deepest part of itself.
Frege distinguishes 3 realms: that of sensible objects, that of mental representations, and that of mathematical objects. This third realm is an objectively real space, which exists independently of man and is where truth makes sense.
Russell. He was a Platonist during the time he was under the influence of Frege. His Platonist culmination is found in his work "Introduction to Mathematical Philosophy".
Hardy. He expressed his Platonic convictions in his classic "Apology of a Mathematician:" "I believe that mathematical reality exists outside of us, that our task is to discover or observe it, and that the theorems we prove and grandly describe as our 'creations' are simply the notes of our observations."
Gödel. He was influenced early on by the Platonic conception of mathematics (when he was a student at the University of Vienna) by the philosophy professor Heinrich Gomperz, to such an extent that he decided to study mathematics instead of theoretical physics, which was his initial intention. After the publication of his famous Incompleteness Theorem (1931), the question of whether mathematics is only a product of the human mind (and therefore subjective in nature) or whether, on the contrary, it constitutes an objective reality is formally raised. Gödel saw in this issue three alternatives:
Aristotelianism. Mathematical objects can be located in nature.
Psychologism. Mathematical objects are created and reside in the human mind.
Platonism. Mathematical objects reside in the higher world of ideas.
Aristotelianism rejects it because of the difficulty of the task of recognizing mathematical concepts as abstract qualities of sensible things. Psychologism also rejects it, because it would imply denying objective and shared mathematical knowledge. He leans toward platonism because the conclusion of his theorem is that mere syntax, mere external form is insufficient to grasp all mathematical truths. So for Gödel his platonism was well founded:
Based on his theorem, he distinguished between objective and subjective mathematics. The objective ones encompass all properties. The subjective ones are those that mathematicians can elaborate, which are the closest possible to the objective truths, since they cannot get to know them in their entirety.
The Platonic world describes an objective reality, not accessible by the ordinary senses, but only accessible by the human mind and existing independently of it. Mathematical objects are as real as physical objects. Mathematical intuitions are as real as sensory perceptions.
Mathematical concepts are objective and pre-existing. The task of mathematicians is to discover and describe them, to bring them to light.
Like Frege, he spoke of a conceptual space: "Concepts are there, but not in a definite place..... They form the conceptual space."
The mathematical world is as real as the material one, but the senses are different. Mathematical concepts are perceived through intuition.
The human mind cannot be reduced to a mere mechanism.
A priori concepts must be established to ground mathematics. In fact, he drew up a list of 18 categories or fundamental concepts.
Paul Erdös. He was a mathematical realist, as much as Gödel. They argued that the mathematical universe can be perceived analogously to how our senses perceive the physical world.
René Thom. He expressed his belief that mathematics deal with archetypal entities. "Mathematical forms have in fact an existence that is independent of the mind that examines them."
Penrose. An avowed Platonist, he identifies Platonism with objectivity:
Mathematical objects have objective existence. We contact them and discover them through intellect and intuition (insight). He gives as examples the Mandelbrot set, complex numbers and Fermat's last theorem. "To say that a mathematical statement has Platonic existence is simply to say that it is true in an objective sense."
The world of mathematical objects is a more perfect world than the material one, but it is as real as the latter. "The deeper we probe the secrets of nature, the more deeply we are drawn into the Platonic world of mathematical ideas as we seek knowledge."
Complex numbers have a deep and timeless reality. Mandelbrot's set is not an invention, but a discovery, something as real as Mount Everest.
Willard Van Orman Quine and Hilary Putnam declared themselves mathematical realists. The so-called "Quine-Putnam indispensability argument" of mathematical entities is based on the following logic:
We have good reason to believe that our best scientific theories are true.
We must recognize the existence of all entities that are indispensable for our best scientific theories to be true.
Abstract mathematical entities are indispensable to our best scientific theories. Therefore:
We must recognize the existence of abstract mathematical entities.
It has been objected to this argument that point 3 does not explain why mathematics is indispensable to science, since there can be scientific theories without mathematics. In fact, Hartry Field published in 1980 "Science without Numbers. A defense of nominalism" in which he rejected Quine-Putnam's indispensability argument [Field, 1980]. Field considered mathematics "a useful fiction". He said that an expression like 2+2 = 4 was as false as Sherlock Holmes. His doctrine is called "mathematical fictionalism".
Nevertheless, the indispensability argument is considered the foundation of contemporary mathematical empiricism.
Henri Poincaré. There is a mathematical reality, but we also create tools to explore that reality.
Alain Connes. He declares himself a mathematical Platonist. He believes in the existence of a pure and immutable mathematical realm, independent of the human mind.
According to Jean Dieudonné, the active mathematician is a Platonist on working days and a formalist on Sundays. "In foundational matters, we believe in the reality of mathematics, but of course, when philosophers start attacking us with their paradoxes, we run to hide behind formalism and say: 'Mathematics is nothing more than a combination of symbols lacking meaning' ... Finally we are left alone and so we can return to our mathematics, working as we have always done, that is, with something that is real" [Dieudonné, 1970]. In his work "In honor of the human spirit" [1989], he examines the evolution of mathematics in an independent and objective way. This work was the answer he gave to someone who asked him the reason why he devoted himself to mathematics.
For Martin Gardner, mathematics is discovered. He gives the example of Mandelbrot's set, which he considers to be "out there" in the same way that a jungle is subject to exploration.
Mathematical anti-platonism
The schools that oppose the Platonic view of mathematics are those of the materialistic, physicist, scientistic, positivist, experimental and practical types. We highlight the following:
Constructivism.
It considers mathematical objects to be a creation, construction, or invention of the human mind. Mathematics adds nothing new to experience. Mathematical objects are entities of reason and exist only in the thinking of the mathematician. Mathematics are "tools of thought" created by the mathematician.
According to Allan Calder [1979], the acceptability criteria of constructive mathematics are more rigorous than non-constructive mathematics, because the analysis is more rigorous and the theorems are more sound.
Conceptualism.
Universals have no independent existence and are a creation of the cognitive subject.
Nominalism.
They see in abstract or universal objects simply a lexicographical convention with no reality beyond language. These objects have no objective existence.
Intuitionism. Its main inspirer was Brower. It is a particularization of constructivism:
Every mathematical object is a creation of the human mind.
The demonstration of the existence of a mathematical object is not enough. The existence of an object is equivalent to its possibility of construction.
Mathematics must be constructed on intuitively clear principles.
Denies the principle of the excluded third, the law of double negation and the abstraction of infinity.
Attempts to unite theory and practice.
The axiomatic formalism. Its main inspirer was Hilbert. The axiomatic method is an imposition, a more or less subjective mental creation. It is not a discovery, so it goes against Platonism.
Positivism.
Positivism, which was born with Auguste Comte and John Stuart Mill, gives primacy to the objective, the material, the experimental, the external, the practical. It is the opposite of idealism, which gives primacy to the subjective, spiritual, internal and theoretical world:
The only true knowledge is scientific, experimentally verifiable.
All philosophical and scientific activities must always be carried out within the framework of experimentally verifiable facts.
The scientific method must be unique and the same for all sciences.
Reason must be the only means to obtain truth.
Principles or theories that have not been experimentally verified should not be considered.
Rejects intuition and a priori concepts.
Truth is the correspondence between statements and facts.
Mathematical objects are a creation of the mind. Mathematics is an empirical, experimental science.
According to Gödel, positivism is the stance of the left, along with materialism and skepticism. And the idealistic view is the stance of the right, along with metaphysics and theology.
Logical positivism −also called neopositivism or logical empiricism, and whose greatest exponent was the Vienna Circle− was a scientific-philosophical movement that went beyond classical positivism. Its most outstanding thesis is that a statement is cognitively significant only if it is a sentence of empirical science, logic or mathematics.
One of the strongest criticisms against Platonism is that of Banacerraf in his now classic article entitled "Mathematical Truth", where he uses the following reasoning:
Human beings exist entirely in spatio-temporal form.
If there are abstract mathematical objects, they exist outside space-time.
Therefore, human beings cannot have knowledge of them.
If mathematical platonism is correct, human beings could not have mathematical knowledge.
Human beings have mathematical knowledge.
Therefore mathematical platonism is not correct.
Types of mathematical platonism
As opposed to the "standard" mathematical Platonism, there are also various versions or visions, among them the following:
Ontological Platonism.
It refers only to mathematical entities: they exist and are independent of human beings.
Logical Platonism (truth-value realism).
Every well-formed mathematical statement is objectively true or false, independent of human beings.
Natural Platonism.
The laws of mathematics are laws of nature.
Structural Platonism.
Mathematical entities are descriptions of abstract patterns or structures.
Weak Platonism.
Pure Platonism regards the mathematical world as wholly objective. Platonic existence is equivalent to the objectively true. However, one sometimes speaks of "weak" Platonism when a certain subjectivity of perception of the objective mathematical world is considered, i.e., when it is a matter of opinion. An example of this type is the famous "axiom of choice" of set theory, which has generated so much controversy in the foundations of mathematics, mainly because it involves the subject of infinity.
Strong, full, complete or absolute Platonism.
Every imaginable mathematical object actually exists. It is a world of unbounded freedom and creativity. Therefore, to the 9 principles of mathematical Platonism we should add the principles of:
Freedom. The mathematical world is a world of freedom.
Imagination. The mathematical world is open to the imagination, transcending the limits of the physical world.
Creativity. The mathematical world is a world of maximum creativity, as a consequence of freedom and the power of imagination.
Degrees of Platonism
Paul Bernays defined what he called the "degree of platonism" of a mathematical system, based on the class of totalities admitted into the system. He defined three levels:
Degree one is the system that accepts the natural numbers as a single complete entity and to the properties the excluded-third principle can be applied (every property is either true or false, without any intermediate element or "third party"). For example, every real number is zero or non-zero.
Degree two is the system that admits totalities such as the set of all points on the real line or the set of all subsets of the natural numbers.
The third degree is the system that admits Cantor's transfinite numbers.
Mathematical Platonism and imagination
According to Platonism, mathematical entities exist and are objective, in the sense that any mathematician can access and explore that common territory. However, there are mathematical entities that are not expressible in formal language. For example:
All irrational numbers exist in the Platonic world, but most of them are inexpressible in a formal language. Only a minimal part are expressible, which are those that can be described by a finite pattern. Therefore, the inexpressible numbers are imaginary and to access them we need our imaginative faculty.
The real line as a totality is only perceived with the imagination. It is an inexpressible abstract entity.
Gödel's incompleteness theorem also tells us that there are expressions inaccessible by means of the axioms. These expressions are also imaginary.
The "axiom of choice" of set theory. Formulated by Ernst Zermelo in 1904 it states that, "Given a collection of sets (finite or infinite), each with at least one element, another set of higher order can be created by taking one element from each set." This axiom, evident for finite sets, for infinite sets is platonic at the imaginative level (it is possible to imagine choosing one element from each set) but it is only expressible when two generic patterns are established: one for defining the sets themselves, and another for the criterion for selecting the element of each set.
Therefore, we can conclude that the Platonic world consists of two levels: the expressible and the inexpressible, both objective but with different modes of access: the imaginative way and formal linguistics.
The inexpressible −the mystical as Wittgenstein said− is imaginary, since it falls beyond the world of the mind. We can only access it through the imagination, in a subjective way. The objective is what is expressible, what can be shared externally.
From this point of view, the so-called "imaginary numbers" (based on the imaginary unit, based on the expression i2 = −1) are not really imaginary, since they are expressible.
MENTAL, a Platonic and Aristotelian Language
In MENTAL the mathematical conceptions of Plato and Aristotle are harmonized:
The Platonic perspective
There is a correspondence or analogy between MENTAL and Platonic philosophy. In both cases, we have two worlds or levels of reality:
The universal semantic primitives. They correspond to the higher world of Platonic ideas, as Gödel suggested that things should be reduced. They are Platonic ideas for the following reasons:
They are transcendental and universal concepts, present in all manifestations.
They are innate and a priori.
They are eternal. They are immutable. They are the true reality. For according to Platonism, "reality does not change, what changes is not real."
They are abstract archetypes. Platonic ideas today we can identify with archetypes. Archetypes act as intermediaries between the soul plane and the higher mental plane (the intuitive aspect). Intuitions are messages from the soul that reach the higher mind.
They are known or accessed through intuition.
They are unmanifest. They are only manifested in particular expressions.
MENTAL expressions are particularizations of the primitives. They would correspond to the platonic lower world.
They are sensible elements, although abstract. They are not imperfect mathematical objects (like the drawing of the circle) but exact expressions of the universal language.
They are manifest expressions of the unmanifest universal language.
They participate in the higher world through the primitives. In the particular expressions are reflected the universal primitives.
Comparing the Platonic vision with MENTAL, we can make the following observations:
Plato said that ideas are projected into phenomena, but he did not explain how this mechanism works. With MENTAL it is seen that in all phenomena (the expressions) are "made" of ideas (the abstract archetypes).
With MENTAL we have universal, abstract and innate archetypal patterns that make us recognize them in all things. These archetypes act on all levels, including the physical, which explains that the laws and structure of nature is mathematical. There is mind-nature unity.
Plato does not clarify how the Dialectic of ideas is realized or established.
In MENTAL there is a vertical Dialectic between the aspects of the universal-particular consciousness, and also a horizontal Dialectic corresponding to the combinatorics of the primitives. In the vertical Dialectic resides the key to consciousness, the connection that is established between the universal and the particular through the mind.
When Plato speaks of reason he is evidently referring to the mind, as intermediary between the higher world of ideas and the lower world of sensible things. In the case of MENTAL, the two poles are intuition and reason, the two aspects or modes of the mind.
The Aristotelian perspective
MENTAL can also be considered Aristotelian:
Because the primitives, in addition to Platonic ideas, can be considered generalizations or abstractions of the sensory. In this sense they can be considered a posteriori. These abstractions are philosophical categories, in the Aristotelian style, that is, classifications of reality.
Because they are universals that are present in all things.
Because it includes a linguistics, a formalization, as Aristotle did with logic.
Because of its practical character, which complements the theoretical.
The union of the two modes of consciousness
MENTAL unites the two poles or aspects of consciousness, represented by Platonic and Aristotelian philosophies, in a balance between idealism and pragmatism, between the subjective and the objective, between the a priori and the a posteriori, between theory and practice, between syntax and semantics:
MENTAL is a discovery and an invention:
To discover is to bring to light something that was hidden or unknown. Discovery is associated with the profound, the universal and the inner. In this sense, MENTAL is a discovery.
To invent is to establish new and creative relationships with previously known elements. Invention is associated with the superficial, the particular and external. In this sense, MENTAL, in its practical aspect, is an invention because it allows to combine primitives to create all kinds of expressions. Syntax, as a more superficial aspect, is also an invention.
There are two connected or integrated levels. The result of this integration is consciousness:
An expressible, linguistic, rational, conscious, objective level, associated with the consciousness of the left brain mode. The representation is external, digital, superficial.
A non-expressible, alinguistic, imaginative, intuitive, subconscious, subjective level, associated with the consciousness of the right brain mode. The representation is internal, analogical, deep.
Union of abstraction and archetypes.
Intuition (the up-down movement of consciousness, from the soul) and abstraction (the down-up movement, from the physical world to the mind) are concepts that meet at a common point, which are the universal semantic primitives. MENTAL, honoring its name, belongs to the mental realm. The primitives are archetypes of the consciousness that connects the higher unmanifest world and the lower manifest world (the mental and the physical world).
Absolute mathematical Platonism.
Every expression of MENTAL exists, for it is a combination of primitives. The limits of the cognizable are the limits of language. And it has the highest degree of Platonism (according to Bernays' classification), for it is capable of expressing or describing mathematical totalities of every order.
With abstract archetypes, because of their universality, we can construct all kinds of mathematical entities. These archetypes are innate and common to all mankind.
Every expression exists as long as it has been formed by combinatorics of primitives. It is a world of full freedom, creativity and imagination: absolute Platonism.
Addenda
The theory of ideas in Plato's "The Dialogues"
In the Dialogues, Plato used the question-and-answer method of his teacher, Socrates, to reach the truth. The theory of ideas is treated, from different aspects, in several of Plato's works:
The Republic. It expounds the theory of knowledge, including the famous allegory of the cave.
Timaeus. Deals with cosmology and the sciences of nature, influenced by Pythagorean mathematics. Mentions what we now know as Platonic solids. It deals with the theme of true knowledge, which concerns reason and never changes (that of ideas), as opposed to changing opinions that depend only on sensations, on experience.
Phaedrus. Discusses the theory of ideas and the nature of the soul.
Theaetetus. Deals with the different theories of knowledge. In it Plato denies that knowledge can be identified with perception. He distinguishes between belief and knowledge. It is considered the first study on the philosophy of science.
Sophist. It is a later reflection on ideas and forms.
Bibliography on Gödel and mathematical Platonism
In 1944 he published "Russell's Mathematical Logic", in which he expounds a realistic or Platonic philosophy of logic. It is available in [Mosterín, 1989].
In 1947, he publishes "What is Cantor's continuum problem?". In it he states that the world of abstract entities is a necessary and true world, accessible by pure reason. It is also available in [Mosterín, 1989].
In 1951 he gives a lecture entitled "Some basic theorems on the foundations of mathematics and their philosophical implications", where he makes explicit his mathematical realism. It is available in [Rodríguez Consuegra, 1994].
In 1953 he agreed to contribute an essay to a volume on Carnap's philosophy (by the Vienna Circle). He worked on it for several years and made several versions. In 1959 he decided not to publish it. It was finally published in 1994 under the title "Is mathematics the syntax of language?". In this essay he denies that mathematics is the syntax of language, as the logical positivists of the Vienna Circle maintained; mathematics is a science of real objects. It is available in [Rodríguez Consuegra, 1994].
In 1958, he publishes "On an extension of finitary mathematics that has not yet been used", in which he points out the need to specify the concepts of "intuitive evidence and "abstract evidence". It is available in [Mosterín, 1989].
Bibliography
Balaguer, Mark. Platonism and Anti-Platonism in Mathematics. Oxford University Press, 2001.
Bernays, Paul. Platonism in Mathematics. 1935. Disponible en Internet.
Barrow, John D. ¿Por qué el mundo es matématico? Grijalbo, 1997.
Barrow, John D. PI in the Sky: Counting, Thinking, and Being. Back Bay Books, 1992.
Calder, Allan. Matemática constructivista. Investigación y Ciencia, Diciembre 1979.
Cañón Loyes, Camino. La matemática. Creación y descubrimiento. Universidad Pontificia de Comillas de Madrid, 1993.
