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Published online by Cambridge University Press: 14 March 2022
After several decades during which formalization has flourished it now becomes possible to detect its shortcomings. A definition of formalization is given at the outset. It is next shown that the main justification of formalization as making explicit the form of a proof has serious difficulties. An important shortcoming is found in the fact that many validation procedures in logic and mathematics are not adequately represented deductively. Several such procedures relating to the validation of logical and mathematical sentences are examined. It is concluded that formalization is materially inadequate.
This paper was contributed to the 1964 International Congress for Logic, Methodology and Philosophy of Science.
1 The following is a commonly held definition:
By formalization is understood the transformation of a logical or mathematical field to a form characterized by the existence of
a) A class of signs. Its elements are physical objects called primitive signs.
b) A class of certain finite linear sequences of primitive signs. Its elements are called well-formed-formuli.
c) A class, subclass of the class of well-formed-formuli. Its elements are called axioms.
d) A class of rules of inference for constructing well-formed-formuli given other well-formed-formuli.
e) A class of certain finite sequences of well-formed-formuli. Its elements are called proofs. All the above classes are non-empty and whether an entity is an element of them or not can be decided effectively. Effectiveness can be defined precisely by several methods such as that employing the concept of general recursiveness. The class of rules of inference is effective in the special sense that whether a proof is constructed in accordance with them or not can be decided effectively. Finally, discussion about the whole logical or mathematical field is carried in metalanguage that is elementary to the point of instructing us merely about the possible linear arrangements of observable physical signs and combinations of them.
