The justification of formalisation

Conclusion

Our discussion may have appeared rather abstract and academic, and it is perhaps not useless to draw some conclusions that clarify the main theses of the paper.

• 1) It is necessary and possible to order the class of theories in such a way that theories appearing earlier in this order have lower degree of formality and theories appearing higher in the order have higher degree of formality.

• 2) However, as far as we can see, such order cannot be unique: the concept of formality defines itself by means of many independent characteristics that can be independently approximated.

• 3) It is not possible to define for these various orders of “degree of formality” a natural minimum and a natural maximum. For any system having any arbitrary degree fo formality, we can find a system of a higher and a system of a lower degree of formality.

• 4) It is necessary and possible to characterize in themselves the operators that applied to given systems, increase or decrease their degree of formality; this task has not however been undertaken until now. Our second paragraph has made a very modest start in this direction. In particular it should be analysed if the application of Boolean or relational operations applied to formal systems taken as wholes would have a systematically increasing or decreasing effect upon the degree of formality of the system (but let us not be too conservative).

• 5) It is possible to describe more exactly than has been done until now the various reasons that drive towards formalisation. A formal theory of these reasons can and should be developed. If this attempt is made, it appears: a) that ontological, linguistical or praxeological presuppositions are implied by the option towards formalisation and b) that the option toward formalisation is motivated by a variety of aims that, by their very diversity, explain the multidimensionality of the concept of “degree of formality”.

• 6) It is possible to establish the syntactic properties of the pragmatic metalanguage in which the formalisation operator and the reasons for formalisation can be defined. In this same metalanguage it can be asked how a systematic method can be developed allowing to determine the degree of formality adequate for the solution of a given problem.

• 7) It is only in this pragmatic metalanguage that two of the foremost controversies of present times can be resolved: a) in mathematics, the Bourbaki school defends a high degree of formality in the presentation and research of creative mathematics, while others insist on the dangers of formalisation and the necessity to hold mathematics close to its applications; b) in philosophy logical empiricism defends a high degree of formality in the formulation of philosophical hypotheses while analytical philosophy wants a lower degree of formality.

Our conclusions are of such a nature that we can both explain the continuation of the struggle and prepare its final solution.