"It is common to speak of 'the language of mathematics.' But is mathematics truly a language?"
(F. David Peat)
"In order to interpret mathematical notation it is
you need to know what kind of grammar it uses"
(Stephen Wolfram)
"The essence of mathematics lies
in its complete freedom" (Cantor)
The Nature of Mathematics
Historically, there has been a long controversy over the question of the nature of mathematics. Surprisingly, today we still do not know exactly what it is. It is often claimed that it is better not to define mathematics because, if this were done, it would mean limiting it, circumscribing its domain.
The question "What is mathematics?" has had different answers throughout history, among them the following:
The science of consciousness.
The science of mind and thought.
The science of reasoning.
A form of abstract knowledge or the abstract structure of reality.
The common essence of all things.
The science of the generic and the universal.
The language of nature.
The science of order and harmony.
The science of measure.
The structure of a higher, perfect and ideal world, of which the material world is an imperfect reflection.
A form of logic or a branch of logic.
The science of formal systems.
The formal language of science.
A universal language for studying problems.
A theory of universals.
A set of domains, more or less related, such as arithmetic, geometry, algebra, topology, etc.
A system of symbols and their manipulation.
The science of patterns or shapes and their relationships.
The science of signifying forms.
The logical syntax of language.
The logic of imagination.
The science of structural archetypes.
The formalized philosophy.
The science of infinity.
The study of mental objects with reproducible properties.
The science of what is clear by itself.
The most certain of all sciences.
The supreme form of pure thought.
The foundation of all knowledge.
A symbolic language in which the mind assigns meaning to symbols.
The aesthetics of reason.
The relation of relations.
The science of transcendent knowledge. This would link with metaphysics, with philosophy and even with spirituality.
Among all these answers, the first answer is the highest of all: mathematics as the science of consciousness. From it, all other conceptions must be consequences.
Weak foundation
Without a knowledge of the true nature or essence of mathematics we can hardly ground it. The foundations of mathematics (known in abbreviated form as FOM, "Foundations of Mathematics") is the study of its most basic or fundamental concepts and the relationships (structural and functional) between them. It may seem surprising, but mathematics, a key science that underpins our society, lacks solid foundations or it is not known what exactly its foundations consist of.
FOM is a subject of enormous importance from a theoretical and practical as well as a philosophical and psychological point of view. Today there is a renewed interest in this subject because it is recognized that a true structure of mathematics as a system and as a language is lacking.
One often speaks of the philosophy of mathematics, of the philosophy derived from mathematics. But the point of view must be the opposite: mathematics must be grounded in philosophy. Mathematics cannot ground itself. Its foundations have to be deeper: they have to be philosophical.
Fragmentation
Mathematics has become a tower of Babel, as it has become progressively fragmented into more or less scattered branches or domains:
There are too many mathematical branches, each with its corresponding theories and too many concepts (sets, vectors, matrices, tensors, functions, categories, structures, etc.) without a common conceptual foundation.
Different notations, terminologies and definitions proliferate.
There is redundancy. Many of the discoveries made in one domain are sometimes a replication of results already known from another domain, but with a different formalization.
As a consequence of all this, there is little or no communication between the different areas, creating in fact conceptual and linguistic barriers between them.
The fragmentation of mathematics began in the early 18th century and acquired a pathological character in the 20th century, with a division into increasingly specialized domains. Many of these domains disappeared, others flourished. The result has been the extinction of the species of generalist or universalist mathematicians, whose last exponents were Henri Poincaré and John Von Neumann.
The AMS (American Mathematical Society) lists 50 major mathematical specialties, ranging from algebraic topology to Zermelo-Fraenkel set theory. These specialties are further divided into more than 300 subspecialties.
It is perhaps because of so much fragmentation that they are called "mathematics," plural, for there are many and diverse. Mathematics urgently needs a unified language, with a common conceptual basis, connecting the different domains.
Paradigms
A paradigm is "a view of the world", a form of abstraction of reality. Throughout history there have been different mathematical paradigms, which have been based on only one concept (or conceptual paradigm). The concepts that have been used have been number, set, structure, function and category.
But what we need is a change of paradigm, but not just another one, but a new paradigm of a global and unifying type. A paradigm that must be based on a few primitive concepts and that will allow the rest of the paradigms to be founded and expressed. These primitive concepts should be understood in as simple terms as possible. From these primitive concepts it should be possible to define new (derived) concepts in an environment that facilitates expressiveness and creativity. This unifying approach is precisely the one being followed by physics, which seeks a "theory of everything," although in this case "everything" refers only to the physical world.
Restricted concepts
Mathematical concepts, in general, are usually restricted. But fundamental mathematical concepts should be universal or have the maximum possible generality, depth and abstraction. But they should fulfill three conditions, which are related to each other:
✲
Genericity.
Any fundamental or primary mathematical concept must be generic. Secondary concepts must be derivable from primary concepts. For example:
The concept of imaginary number. If imaginary numbers exist, genericity demands that all kinds of imaginary expressions must exist: imaginary sets, imaginary functions, imaginary categories, etc.
At the level of the mathematical domain, if there is imaginary arithmetic, there must also be imaginary algebra, imaginary logic, imaginary geometry, and so on. And if there exists a fuzzy (or fuzzy) logic, there must also exist a fuzzy arithmetic, a fuzzy algebra, etc.
The concept of vector, defined as an oriented line segment. Its generalization is the multivector: oriented segments of two or more dimensions. This concept, in this case, does exist and was conceived by Grassmann.
Parameterization. The only mathematical object that can be parameterized is a function. But this concept should be applicable to any mathematical object: a set, a sequence, a number, and so on.
If there are classes or categories of sets or functions, there should also be classes or categories for all kinds of mathematical objects, without exception. For example, categories of functions of functions, categories of logical expressions, categories of numbers, etc.
✲
Orthogonality.
Orthogonality refers to the free combinatorics of concepts, entities, objects or mathematical expressions, without restrictions of any kind. For example:
A function should be able to return as a result any mathematical object: a function, a higher-order function, a numerical range, a set, a sequence of sets, etc.
Concepts such as matrices and vectors are composed of "numbers". This is restrictive. They should be able to be composed of any kind of expressions and not just numbers. For example, matrices of vectors, vectors of matrices, vectors of functions, etc.
Logical operations are not generic as they apply only on logical expressions, i.e., those that evaluate to V (true) or F (false).
Arithmetic operations are not generic either since they apply only on numbers or on variables. They cannot be applied over functions, sets, logical values, etc.
Predicates (or attributes) should be able to be applied to any mathematical object (a function, a set, etc.).
The condition is applied to specify elements of a set that have a certain property. There is no standard notation. It is usually represented as "|" (vertical bar) or ";" (semicolon). Neither of these symbols corresponds to an operator with defined semantics. For example, {n | n∈N ∧ n>5} (specifies the set of natural numbers greater than 5). But the condition must be applicable to any type of mathematical object (if the condition is met, that object is selected).
✲
Reflexivity.
When a concept is used, there must always be higher-order concepts, i.e., there must be reflection or conceptual recursion: every concept must also be applicable to itself. Thus arise concepts of concepts (concepts of order 2), concepts of concepts of concepts (concepts of order 3), and so on. In reality it is an auto-orthogonality, that is, a conceptual combinatoriality with the concept itself. And this mechanism should also be reflected in the different mathematical domains, there should be a higher order arithmetic, a higher order logic, a higher order geometry, etc. For example:
Higher-order imaginary expressions: numbers of numbers, imaginary numbers of imaginary numbers, imaginary functions of imaginary functions, functions of functions, vectors of vectors, matrices of matrices, multivectors of multivectors, predicates of predicates, etc.
There are the arithmetic operators addition, product (sum of sums) and exponentiation (product of products), but there is no standard notation for higher-order operations (exponentiation of exponentiations, etc.).
If the concept of infinitesimal exists, there must exist infinitesimals of higher order. And the same with infinite numbers, although this aspect Cantor tried to cover with his transfinite numbers.
Every parameterized expression must be able to be, in turn, parameterized (parameterized expressions of higher order).
The mathematical language
It is common to speak of "mathematical language", but mathematics lacks a formal language. For it to have one, it would need:
A lexical semantics. A set of semantic primitives, primary or fundamental concepts.
A structural semantics. A set of rules on how to combine these fundamental concepts to create expressions and derived concepts.
A syntax or formal notation.
Traditional languages are defined by a formal (or syntactic) grammar. But mathematics is not a language anymore, it is not a concrete language, it is a universal language, and it cannot be limited by a formal grammar. Therefore:
Mathematics (like the mind) must be based on degrees of freedom, which are the fundamental concepts and their combinatorial rules.
Mathematical language must be based on a semantic grammar, a grammar that relates the fundamental concepts without restrictions. The universal principle of economy requires that the lexical semantics be equal to the structural semantics, i.e., that the combinatorics between primitives be realized by the primitives themselves.
It must also have a syntax that is a reflection of the semantics, in such a way that given the syntax the semantics is deduced, and that the semantics has an unambiguous syntactic reflection. Semantics and syntax must be biunivocal, two sides of the same coin.
Lack of humanism
Mathematics has become a complex discipline, difficult to understand, with concepts that are generally not very intuitive and with a somewhat esoteric notation. In short, a not very humanistic discipline, out of the common culture, only for "initiates".
The essayist and poet Hans Magnus Enzensberger [2001] asks: "How is it possible that mathematics has remained a kind of blind spot in our culture, as if it were a foreign territory?".
This situation is paradoxical, because mathematics, if it is the foundation of the formal sciences, should have very simple foundations and be integrated into human culture.
Historically, mathematics has been considered an absolute, perfect, precise, objective and universal discipline, independent of cultural or epistemological context. However, all these characteristics have been declining, especially with the emergence of the humanistic mathematics movement. Humanistic mathematics is a philosophy that attempts to seek the human side of mathematical thinking in two main aspects: 1) Mathematics grounded in humanism; 2) Humanistic teaching/learning of mathematics.
The characteristics of humanistic mathematics are:
Mathematics as a science of mind and consciousness. Mathematics is an intellectual discipline, an activity of the human mind and consciousness.
Mathematics as a paradigm, that is, as a way of conceiving reality, but also as a way of contemplating and approaching subjects from different perspectives and new forms of consciousness.
Mathematics as epistemology, that is, as a system of concepts inherent to the human being.
Mathematics as a philosophy that helps us to understand this world and other possible worlds.
Intuition as a key element for the understanding and elaboration of concepts, which must be supported by metaphors and analogies. Intuition must always be above formalization. The internal must prevail over the external. The external must reflect the internal.
An end to dogmatic and authoritarian mathematics, of absolute and indisputable truths. There should be no dogmas established a priori. Everything is questionable and revisable, especially its foundations.
The paradigm of absolute truth has been challenged by many mathematicians, including Philip J. Davis, Reuben Hersh, Imre Lakatos, Philip Kitcher, Paul Ernest, and Tom Tymoczko.
Mathematics should be open, flexible and relative, with modifiable axioms, encouraging abstraction, creativity, discovery and invention. Mathematical knowledge, as in any other science, must be open and under continuous revision.
Unification versus fragmentation. Unification of "pure" and applied mathematics, between theory and practice, between mathematics as a system and mathematics as a means or tool.
Integration with culture. Mathematics is not an isolated discipline, but has interdisciplinary connections. Mathematics is only one part of the collective knowledge of human beings and makes sense only in a cultural context.
Mathematics as a language. Constructivism as a way of doing mathematics.
Mathematics as the search for simplicity, beauty and harmony.
Search for new ways of teaching and learning, for new educational paradigms. Discovery and freedom must be favored over dirigisme. Education must be progressive, evolutionary, spiraling and utilizing both modes of consciousness (rational and intuitive).
Eliminate boundaries between specialists and ordinary people. Make mathematical knowledge accessible to all.
To deepen the true nature of mathematics and its universalistic character.
Redefining the role of logic in mathematics, because of the limitations revealed by Gödel in 1931 with his famous incompleteness theorem of formal axiomatic systems.
Consider the impact of the computer as a new paradigm of mathematics and logic. Consider also the impact of cognitive science.
Respect for contradictions and paradoxes, as they are a source of creativity.
Include non-linear thinking to address complex issues, such as chaos theory.
Handling degrees of truth, approximate information and fuzzy concepts.
Addendum
Some proposals on the "new mathematics"
Ludwig Von Bertalanffy, the creator of the General Theory of Systems, stated that it was necessary to search for a new mathematics of a gestalt type, in which the notion of relationship, rather than quantity, would be fundamental.
René Thom advocated a new mathematics integrating qualitative and quantitative aspects to be applied to "significant forms".
Edward MacNeal [1994] has coined the term "Mathsemantics" to refer to a revolutionary way of looking at mathematics: as a language of the mind that integrates numbers (mathematics) and meaning (semantics), sciences that were born independently, to offer a new way of looking at the world.
George Spencer-Brown, with his "Laws of Form," advocates a mathematics of consciousness.
Stephen Wolfram advocates "a new kind of science," a new universal scientific paradigm for modeling systems and for understanding the universe itself. The idea is to base mathematics on the concept of computation with cellular automata that use simple local transformation rules that, applied recursively, produce results (patterns) of great complexity. According to Wolfram, the universe is a computer.
History of humanistic mathematics
Humanism dates back to the ancient Greeks, Chinese Confucianism and the Renaissance. It is a philosophy that places man, with his possibilities and his limitations, as the foundation and center of all things, so that:
All human beings have the right to free and independent thought.
Dogmas, ideologies and traditions must be weighed and questioned by each particular individual.
All human beings have the right to give meaning and value to whatever they desire. The meaning of life is to live a meaningful life.
The important thing is the love of life, here and now.
Humanistic mathematics is an idea that dates back to Plato and has been advocated by different mathematicians throughout history by highlighting the role of intuition, imagination and creativity in mathematical ideas.
Humanism as a formal school in philosophy of mathematics was created by Reuben Hersh in 1979 [Hersch, 1999].
The modern revitalization of this subject is mainly due to Alvin White [1993]. In 1986 he organized a conference, attended by mathematicians, philosophers, and educators, to discuss the relationship between mathematics and humanism and to try to discover what was wrong with mathematics education. The enthusiastic response led to the creation of the "Humanistic Mathematics Newsletter" in 1987. In 1992 it became the "Humanistic Mathematics Network Journal". It is currently published only on the Internet.
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