Human Evolution

IV Session

Formal Sciences

A multidisciplinary anthropic focus

Creating formal sciences

Interdisciplinarity leads us to formal activity of the mind.The human brain is gifted with functional skills to:

• perceive and analyze a world of structures• abstract certain structural features of reality• conceive and imagine new forms and structures

Why is the human mind capable of creating formal sciences?Could mental forms, formalization, formal sciences, exist without brains? (Hofstadter)

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Structures and rationalityOur human evolutive sensitive experience is structured by the perception of: Differentiated objects as a unity (elements

of a set). Structures, operations and relations

between them Consequential, logical, unity

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Structures and rationality

The world is experienced as a variety of structures logically involved in other structures of superior level. We do not know neither the final micro-physic nor the final macro-physic structures, but we know a multiplicity of consequential intermediate connections.

Counting and measurements

The first formal abstraction was probably that of numbers and geometric figures. Human mind was capable: To perceive differentiated entities: two fish, three stones…measure changes in time: days, seasons, years… compare size, shape and relative position of geometrical figures.To formally represent these entities: by abstract numerical structures concerning numbers and operations on them and abstract geometrical structures with metrical properties.

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Mayan numerals

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Counting and measurements

Classical arithmetic and geometry

Formal Sciences seemed to be firmly established on the simple foundation of numbers and geometry:The formal language of mathematics allowed counting and space-time measurements.Developments in formal logic and set theory led to questions about mathematical certainty:•What is the cause of mathematical certainty?•Why does mathematical reason work like it actually does?•What is the ontological status of formal entities?

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Formal theories

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Foundation of mathematics

The relationship between mathematics and logic.

Philosophers of mathematics began to divide into various schools of thought.

Logic apriorism, formalism and intuitionism emerged partly in response to the search for the causes of mathematical certainty.

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Apriorism

Platonism suggests that mathematical entities exist independently of the human mind.

E. Kant believes that the objectivity of mathematics is based in space and time as an a priori forms of sensibility.

Different forms of apriorism remain present among today mathematicians.

For Roger Penrose some mathematical assertions belong to an unchanging world of essences.

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Logicism

Mathematics can be known a priori because it is part of logic. Logic is the proper foundation of mathematics. Logicism becomes strong with the formalization of logic.

G. Frege constructed a formal logical system that made it possible to represent the logical inferences as formal operations.

This program was continued by Russell and Whitehead.

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Formalism

Formal logic is a part of formal mathematics.

Can mathematics rationally justify itself as a purely formal science?

The meta-mathematical Hilbert program intends to justify mathematics as a pure formal science.

Meta-mathematics uses mathematics as a formal language to speak about mathematics as a formal object.

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Intuitionism/constructivism

Gödel caused a crisis in the Hilbert’s programme proving that, if the formal system of arithmetic is consistent, then it is incomplete.

Intuitionism rejected the meta-mathematical formal foundation of mathematics. Only the mathematical entities which can be explicitly constructed are admitted.

Intuitionist logic does not contain the law of excluded middle. Constructivism regards the sets with infinite elements.

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Objective reality

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Natural aposteriorism

What is the origin of human reason’s formal capacities?Aposteriorism responds to the evolutive adaptation of consciousness to an objective and structural reality.Neurological anthropology, evolutive epistemology, and authors like J. Piaget and X. Zubiri do support this point of view. It is congruent with the paradigm of evolution in modern Science.

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Natural aposteriorism

Aposteriorism states the:structural construction of an objective physical world.emergent properties of mind to adapt behaviourally to this structural world.emergent structural representation of reality and capacity to abstract specific structural features.emergent skills to imagine created structural forms in order to open new possibilities for knowledge and technology

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Computationalism

Computing is formal mathematics applied to the development of algorithms.

From ancient times algorithmic processes have been used in algebra and formal logic.

The old mechanist ideal consisted of obtaining a mechanical artifice by which it would be possible to execute all the deductions.

In 1936 Turing specified the informal idea of an algorithm through what we call the Turing machine.

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Computationalism

By means of his machine, Turing showed that there is no general solution to the Decidability’s Problem: Given a formal statement in a formal system, there is not always a general algorithm which decides if the statement is valid or not.

The incompleteness of arithmetic caused disappointment against the mechanicist ideal, but the negative solution to the problem of decidability by the Turing machine, in some way, involves a deepening in this disappointment.

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Liberation and extension of FFSS

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Liberation and extension of FFSS

The applicability of computing has highlighted applied dimensions of languages and formal models.

This has led to the use of a plurality of logics, suited to several finalities.

The existence of a plurality of logics opens up new perspectives for scientific language.

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Anthropic perspectives

Reality: Formal languages allow a high degree of objectivity. Is this objectivity total? Are formal sciences really objective and independent of the subject that formulates them?The pluralism of formal systems and their undecidability leads us to ask about the rationality of inevitable non formal decisions. Are formal sciences a mask disfiguring the real world? How does mathematics access the real world?

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Anthropic perspectives

Technology: We can say that we control a scientific theory when we have expressed it in formal language. Formal language, as it is objective, permits the technological implementation of scientific theories.

Do formal sciences qualify natural sciences for a new design of theoretical frameworks of new instrumental machinery to formalize and control reality?

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Anthropic perspectives

Metaphysics:Do formal sciences formalize open or closed systems?Will they establish insurmountable limits to human reason?How far will they be able to formalize open or closed natural systems?How does philosophy of formal sciences behave in face of metaphysics?

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Expressing reality

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Knowing reality

New forms to know reality: Classical formal sciences intended to be

mechanicist. Do we have new formalisms for new

holistic ontologies? Formal sciences are analytical, not holistic. They can analytically interpret holistic properties.

The human mind: Do we have new classical-quantum forms to formally describe the functioning of our human mind?

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Creating technology

New forms for manipulating technologically reality:Classical technology intends to be mechanistic. Technology is based in our formal control of reality, and classically the ideal of reality control was mechanist. Quantum technology: quantum properties such as superposition and entanglement can be used to represent and structure formal data. Classical-quantum technology for information processing: new forms to implement new technologies of the mind?

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Speculating in metaphysics

The main problem of metaphysics: Universe’s sufficiency/insufficiency

Physical knowledge is constructed by applying formal models. Do formal sciences offer to physics a closed and self consistent formal system to organize natural knowledge?

Philosophy should think about internal possibilities of formal sciences

Gödel’s theorem shows that if formal systems are consistent, they are incomplete and therefore they are intrinsically open.

Consistency is an unrenounceable value of formal systems

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Speculating in metaphysics

Do formal sciences qualify human reason for an open or a closed metaphysics?

Will formal sciences empower human reason for an absolute consistent dominium over natural reality?

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