Formalism is one of the approaches to the philosophy of mathematics , trying to reduce the problem of the foundations of mathematics to the study of formal systems . Along with logicism and intuitionism, in the 20th century it was considered one of the directions of fundamentalism in the philosophy of mathematics.
Content
History
Formalism arose at the beginning of the 20th century at the Hilbert School of Mathematics as part of an attempt to bring together rigorous justifications for various areas of mathematics. Developed by employees (students) of Hilbert Ackerman , P. Bernays , von Neumann .
Unlike logicism, formalism did not claim to build a uniform theory for mathematics, like the theory of sets or the theory of types . In contrast to intuitionism, formalism did not refuse to construct theories with “doubtful” grounds from the point of view of intuition, if only the rules of theorem derivation were strictly justified in them. Formalists believed that mathematics should study as many formal systems as possible.
Criticism
Formal axiomatic theories built on the basis of classical logic , it makes sense to consider only in the absence of contradictions in them, since otherwise any proposition of the theory is “proven”. If it is possible to prove a logical lie in such a formal system, then it is contradictory and “rejected”, which invalidates any theorems proved within the framework of this system. Of course, mathematicians were worried about the question of whether it is possible to somehow prove the consistency of the theory. To the chagrin of formalists, it was shown that the question of the inconsistency of theory does not have an adequate solution within any of the formal systems used in mathematics .
Nothing prevents the study of one formal theory with the help of another; this approach is called metamathematical . However, he forces us to use the most reliable foundations for constructing metatheories, with which formalists considered, again, classical logic and formal arithmetic .
Current status
Since the beginning of the 1990s, interest in formalism (in a more applied sense) has increased again in connection with the tasks of automatically proving theorems (see, for example, the ).
Links
- Formalism / N. N. Nepeyvoda // New Philosophical Encyclopedia : in 4 volumes / before. scientific ed. Council V. S. Styopin . - 2nd ed., Rev. and add. - M .: Thought , 2010 .-- 2816 p.