Human Evolution. IV Session Formal Sciences. A multidisciplinary anthropic focus. C reating 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
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A multidisciplinary anthropic focus
Interdisciplinarity leads us to formal activity of the mind.
The human brain is gifted with functional skills to:
Why is the human mind capable of creating formal sciences?
Could mental forms, formalization, formal sciences, exist without brains? (Hofstadter)
Our human evolutive sensitive experience is structured by the perception of:
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.
The first formal abstraction was probably that of numbers and geometric figures. Human mind was capable:
Formal Sciences seemed to be firmly established on the simple foundation of numbers and geometry:
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.
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.
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.
What is the origin of human reason’s formal capacities?
Aposteriorism states the:
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.
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.
New forms to know reality:
New forms for manipulating technologically reality:
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
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?