"Insofar as the propositions of mathematics refer to reality, they are not true; and insofar as they are true, they do not refer to reality." (Einstein)
The Duality Formal Sciences - Empirical Sciences
The term "science" comes from the Latin scienctia, "knowledge". There are two types of sciences (or knowledge): formal and empirical. Formal sciences are exclusively internal, mental and rational. The empirical (or factual) sciences are based on the external, but also apply rational methods. The empirical sciences (according to Carnap's classification) are divided into natural sciences and social sciences. Formal and empirical sciences are dual, with dual properties.
There is no general consensus on a precise definition of formal sciences, but they are considered to have the following properties:
They are based on rational, systematic and coherent knowledge. They study the truths of reason, necessary and universal, that is, valid in all possible worlds.
They work exclusively with rational ideas created by the mind. They study the inner world, its structure, its expressive and creative possibilities, in a descending process: from the general principles (or degrees of freedom) of the mind to the concrete rational ideas.
They do not deal with physical-natural reality, but only with the abstract world in its formal aspects, with pure, ideal, abstract forms or structures, independently of their contents. Abstract forms and their relations are codified by means of a symbolic language. Abstract forms are timeless and unspatial.
They are grounded in mathematics, as formal logic is considered to be part of mathematics. Mathematics itself is considered a formal science, the pure formal science and mother of all formal sciences.
They are deductive sciences. They use a deductive, top-down method. They usually take the form of formal axiomatic systems, consisting of general axioms (independent of each other), definitions and rules of inference to produce theorems, new truths or conclusions of a particular type.
In formal axiomatic systems, truth is based on consistency. A statement is true if it is deducible from axioms and rules of inference. If it is not deducible, the statement is false.
Formal entities admit many interpretations. In formal axiomatic systems, the interpretations are called "models".
They are analytic sciences. Their propositions are analytical. They do not add new knowledge. The predicate is included in the subject or they are based on the principle of identity (according to Kant's definition).
They are a priori sciences, prior to experience. They do not have and do not require empirical knowledge. Their validity is independent of all empirical knowledge. Therefore, they are not falsifiable.
They are auxiliary tools of the empirical sciences, providing them with a formal foundation. The theories of empirical sciences or models of reality are elaborated by means of formal systems.
They function like any scientific discipline, but in specific domains and with their own themes and methods.
They are precise, unlike the sciences of the empirical world.
"One of the reasons why mathematics enjoys special esteem, above the other sciences, is that its laws are absolutely certain and indisputable, whereas the other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts" (Einstein).
In contrast, the empirical sciences:
They study the objective external world.
They elaborate theories or models of reality with the help of formal sciences.
They are a posteriori. They require contact with the sensible world.
They use the inductive, ascending, synthetic and progressive method: from particular facts to laws, patterns or general rules.
They are synthetic sciences. Their statements are synthetic. They add new knowledge.
They are predictive. They predict the behavior of a system according to the circumstances existing at each moment.
They are falsifiable. A new fact that does not conform to a theory of reality can invalidate it.
They are progressive. They attempt to approximate the truth by successive theories.
"The aim of science is not to establish immutable truths and eternal dogmas; its aim is to approach the truth by successive approximations, without proclaiming that at any of the intermediate stages the truth has been completely attained" (Bertrand Russell).
The various formal sciences
The formal sciences are more important than the empirical sciences because the formal sciences study systems and possible worlds, one of which is the physical world.
The twentieth century has witnessed a veritable explosion of new formal sciences, all of them grounded in mathematics. Among the new formal sciences are: computational theory, information theory, systems theory, decision theory, operations research, theoretical linguistics, game theory, numerical analysis, systems engineering, control theory, network theory, cellular automata, artificial life, etc.
In addition to these sciences, which have a specific scope, sciences with generic designations have also emerged under the names of "complexity sciences" and "artificial sciences".
Complexity sciences deal with the study of systems with many components, with a high degree of dependence among them, where there is no centralized control, where the components obey simple operational rules, but exhibit a complex collective behavior not predictable (or difficult to foresee) by analyzing the individual components.
The sciences of artificial is one of Herbert A. Simon's [2006] main contributions to the philosophy and methodology of science. These sciences share the characteristic that they are goal-oriented, which can be achieved by various methods or designs that require rationality. For example, economics, pharmacology, documentation, etc.
The various formal sciences, although grounded in mathematics, are disconnected. There is no discipline (or generic science) that integrates them because it is unknown what is the common characteristic of all of them. It has been suggested that:
It is applied mathematics. But this denomination is too generic.
It is mathematical engineering. Nor does it capture the common essence of these sciences.
It is general systems theory. But the term "system" is too generic, for almost anything is a system (a set of interrelated components).
They are general sciences. Nor does this designation clarify their true nature.
The philosophy of formal sciences
Formal systems and formal sciences have attracted the attention of philosophers, ever since ancient Greece. The questions of interest to philosophers are:
The true nature of formal systems, their grounding, their deep meaning and their kind of truth.
Their relation to philosophical categories.
Its possible application to philosophy. Can philosophy be formalized?
The answer to the latter question was the so-called "analytic philosophy," a philosophy that attempts to apply formalization to the problems of philosophy in order to clarify or better understand them. Analytic philosophy, one of the great philosophical currents, arose as a reaction to idealism. Analytic philosophy does not study facts in competition with the empirical sciences but is interested only in the essential concepts expressed through language and their logical relations, i.e., the logical structure of sentences.
This analysis led to three visions or strategies: 1) the analysis of the concepts of natural or ordinary language; 2) the search for the more or less explicit or hidden logical structures in ordinary language, independently of its contents; 3) the creation of a perfect or ideal language.
Four figures stand out mainly in this field:
Frege.
Analytic philosophy was born with Frege, who is considered the father of modern formal logic or mathematical logic. Frege attempted to connect at the formal level philosophy and logic, a path that was followed by Russell and Wittgenstein.
Moore.
He focused on the analysis of ordinary language by attempting to extract the fundamental concepts involved in philosophical reflections.
Russell.
Logic is the underlying structure of language in which the structure of the world is reflected. There are some logical atoms (or primitives of logical character) with which language is constructed, reflecting the structure of reality. This is the isomorphist thesis.
Wittgenstein.
Philosophy is not a science because science discovers and philosophy does not make discoveries but only tries to clarify or improve the understanding of things already known and which were confused. It does so by trying to go beyond the superficial, searching for the essence common to everything, which resides in language.
Logic is also not a science because the propositions of logic are tautologies.
With respect to mathematics, Wittgenstein's thought is difficult to characterize and classify. Nevertheless, we can describe his main ideas:
Mathematics needs no foundation because it needs no explanation. Mathematics, like philosophy, must be merely descriptive, using a formal language. This formal language can illuminate or clarify philosophical problems. Philosophical problems arise when there is no formal language.
Mathematical expressions such as "3 times 3 is 9" are not really genuine propositions, despite having grammatical structure. They are rules expressing identity of descriptions. Rules are formal and are not arbitrary, but necessary because they are based on empirically verifiable regularities.
Mathematical objects (numbers, sets, functions, etc.) exist, but they belong to an abstract realm, not spatio-temporal and independent of the mind. We can know these objects through intuition.
Mathematical propositions do not refer to physical reality. Therefore, it makes no sense to talk about their truth or falsity, but only whether they make sense or not.
Mathematics is not transcendent. It is tied to language and human practices. Language only serves to describe mathematics and philosophy, not to ground them. Mathematics is divorced from the language it uses.
We must distinguish between pure and applied mathematics. They are different things.
On whether or not philosophy is a formal science, the answer is that philosophy must underlie all formal sciences, including mathematics.
The analysis-synthesis question
For Kant, an analytic judgment is one in which the predicate is included in the subject, i.e., it does not add new knowledge, e.g., "A bachelor is an unmarried man." In a synthetic judgment, the predicate is not included in the subject, so it adds knowledge, e.g., "All roses are red" or "Some bachelors are doctors."
Since Kant, the analytic-synthetic debate has interested philosophers, who have tried to clarify and generalize this distinction for all kinds of statements. It is generally admitted:
An analytic statement is one whose truth value can be determined by virtue of the meaning of its component terms and their relations.
A synthetic statement is one whose truth value requires some kind of empirical ascertainment.
Willard van Orman Quine, in "Two Dogmas of Empiricism" [1951], an article considered one of the most important of the 20th century:
It rejects the traditional distinction between analytic and synthetic statements. All statements are synthetic to a greater or lesser degree. Formal sciences such as mathematics and logic are not analytic, but synthetic, although to a lesser degree.
There is no qualitative distinction between formal and empirical sciences. They differ only in degrees of abstraction.
There is linguistic holism. It is necessary to consider the whole field of a science and not isolated sentences, since all scientific statements are interconnected, from empirical statements to the most general principles. Language is a structural whole that responds to experience. The relationship between the statements of a theory and experience is holistic and not reductionist.
Mathematical empiricism
Mathematical empiricism was born with John Stuart Mill (19th century), for whom mathematical knowledge comes from the physical world, although he recognized that mathematical knowledge is "the most general" of all.
Imre Lakatos coined the term "mathematical quasi-empiricism" in his article "A Renaissance of Empiricism in the Recent Philosophy of Mathematics". He meant that mathematical knowledge is not radically different from the rest of scientific knowledge because mathematical knowledge is also grounded, albeit partially or not entirely, in empiricism:
Mathematical theories and empirical theories have in common that both are falsifiable. Fallibility is inherent in mathematical knowledge. Mathematics has made mistakes throughout history, in the same way as the experimental sciences.
In mathematics there are rival theories, as in the empirical sciences, starting with the theories of the foundation of mathematics itself: the formalist, logicist and intuitionist theories.
In mathematics there are axioms, postulates and demonstrations, but also empiricism. The distinction between a priori and a posteriori is relative.
Mathematical demonstrations are not necessarily eternal or necessarily true.
The procedure of mathematics is not axiomatic, but is based on a succession of theories or proofs and errors leading to fallible conclusions. We cannot know if we have arrived at the final truth, but we do progressively improve our knowledge. Every mathematical theory is a conjecture.
The attempt to provide firm foundations for mathematics leads to a regress to infinity because we will never reach the ultimate truth.
There are informal mathematical demonstrations that do not need a perfectly formalized theory to be considered valid demonstrations. There are also formal mathematical demonstrations that are falsifiable by informal demonstrations.
Mathematical theorems have a historical dimension. They are processes that unfold over time. Understanding a theorem must go through knowing the history of its proof.
Mathematics has evolved differently according to its cultural context.
All mathematical theories share a common firm (non-falsifiable) core and a (falsifiable) periphery constituted by auxiliary hypotheses susceptible of being modified or replaced by new ones.
In 1975, Hilary Putnam -one year after Lakatos' death- published "What is Mathematical Truth?" where he stated:
Mathematical knowledge resembles empirical knowledge in the sense that the criterion of truth for both is the success of ideas in practice.
In both cases, knowledge is not absolute, but susceptible to improvement.
Mathematics is not purely logical but quasi-empirical. Mathematics uses logical proofs and quasi-empirical methods. For example, Fermat's last theorem (there does not exist a n>2 such that an + bn = cn with a, b, c and n natural numbers) had been tested for many values of n before being proved by Andrew Wiles in 1995.
Formal language and science
Every science, in order to be rigorous, needs a formal and precise language to represent knowledge, a language as an intermediary between epistemology and ontology, between the internal mental world and the external physical world. There are authors who affirm that science itself is a language.
Languages for the representation of scientific knowledge can be natural or artificial. Natural languages suffer from lack of precision, are centrifugal, superficial and tend to diversification. Artificial languages are centripetal, deep and tend to unification, to universalization. Among the latter are mathematics (including logic) and computer languages (programming, knowledge representation, etc.). The so-called "mathematical language" (which is of a descriptive type) is not such because it does not have a perfectly defined lexical semantics and structural semantics. Computer languages (which are of the operational type) have contributed greatly to the formalization of linguistics.
It is clear that a formal language conditions the development of science and, therefore, of society. The most paradigmatic example was the invention of the decimal positional numbering system, together with the introduction of zero. A universal formal language, common to all sciences, would have an enormous impact on the evolution of science and society.
For Aristotle, scientific language must be perfect to reflect the truth of the facts accurately and clearly. He called scientific language "apophantic."
MENTAL as the Foundation of the Formal and Empirical Sciences
MENTAL is the common essence and foundation of all formal sciences, including mathematics itself. Logic is a dimension or degree of freedom of MENTAL. It also helps formalize the empirical sciences. MENTAL is the universal foundation of the sciences.
Rather than classifying the sciences into formal and empirical, it is better to classify them into internal (or mental) sciences and external (or physical) sciences. Both are linked or connected by a universal formal language.
MENTAL has a deep foundation: the philosophical categories. It is the explicit foundation of the formal sciences and the implicit foundation of the empirical sciences. In MENTAL, philosophy and formalization are two aspects of the same thing. With MENTAL philosophy becomes analytic and at the same time synthetic.
MENTAL, as a universal language, goes beyond the limitations of formal axiomatic systems (based on first-order logic), for it allows to establish operational (functions, procedures, rules) and descriptive forms.
The mathematics with which formal systems are elaborated has expressive difficulties. It is incomplete, it is limited. The limitation comes from the lack of universality of the mathematical concepts used. For example, it is difficult or impossible to formalize generalized quantification, higher order quantification, generalized modal operators, etc.
There are certain domains that cannot be formalized or are very difficult to formalize precisely because of these expressive limitations of mathematical language. This is solved by MENTAL, which sets the limits of what is expressible and formalizable.
In MENTAL, space and time are abstract and united in expressions, they are two aspects of the same thing. Every expression is a form that occupies a space (sequence or set), and the form of that space determines how the expression is evaluated (serially or in parallel, respectively).
MENTAL primitives are always present in all sciences because they are primary archetypes. Formalization of empirical sciences is always possible, although to a certain degree. Every empirical science has to be elaborated with conceptual, mathematical and logical rigor, and rigor implies formalization.
In the empirical sciences, whether or not formal language is used to represent knowledge, primary archetypes are being used more or less consciously at the internal (mental) or linguistic level. MENTAL harmonizes formal science and empirical science because both types of sciences are a manifestation of the same primary archetypes. Therefore, the boundary between formal science and empirical science is blurred.
Leibniz distinguished between truths of reason and truths of fact. But those of fact are supported by those of reason.
MENTAL, as the foundation of the formal sciences, goes beyond the strictly rational, for it allows imaginary expressions.
The complexity of the "complexity sciences" is not such, complexity is only simplicity applied recursively. "Complexity correctly viewed, is only a mask for simplicity" (Herbert Simon). MENTAL offers a universal paradigm that unites simplicity and complexity.
MENTAL is a universal formal language capable of representing descriptive and operational knowledge. Following the principle of economy, it is based on primary archetypes that are common to the inner and outer world, it is devoid of any concrete content, it is pure form, abstract, symbolic. The expressions of language are the valid combinations of these symbols.
The absolute truth resides in the primary archetypes, in the deep.
We must distinguish between analytic/synthetic processes and analytic/synthetic expressions.
The analytic process is descending: from the general to the particular (or less general). The opposite process is symptomatic.
An analytic expression is an expression that represents itself. A synthetic expression is one that represents several expressions, analytic or less synthetic.
There is a hierarchy of synthetic expressions. For example, the expression ⟨x+y⟩ is more synthetic than ⟨x+3⟩ because it includes: ⟨x+y⟩ ⊃ ⟨x+3⟩.
Primitives constitute the universal synthesis. Primal archetypes are not falsifiable because they are a priori. According to Kant, synthetic a priori judgments are at the heart of science.
Every expression (analytic or synthetic) admits infinite interpretations based on the interpretation of the contents. On the other hand, the structural interpretation is unique.
Paradoxically, the ultimate analysis, which leads to the philosophical categories, also leads to the universal synthesis. It is the supreme union of opposites, the universal consciousness. MENTAL is both a reductionist and holistic language.
The primitives of MENTAL are not falsifiable. They can be made to correspond to the "hard core" (non-falsifiable) that Lakatos spoke of.
Mathematics, and science in general, try to capture the common essence of everything, moving towards unification, generalizing concepts and searching for general abstract principles or universals. For example, non-Euclidean geometries were born to generalize Euclidean geometry. And polyvalent logics were born to generalize bivalent logic.
In short, MENTAL is the foundation of the formal sciences, and from this basis the essential unity of all sciences (formal and empirical) is contemplated.
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