Інтенсіонально-прагматичний підхід до аналізу d-пропозицій у формальних системах: метаматематичний аспект

Abstract

У статті пропонується вирішення проблеми верифікації математичних D-пропозицій (пропозицій, істинність яких ґрунтується на їх доведеності в межах однієї або декількох формальних систем) шляхом звернення до засновків метаматематичного характеру, які виключають контекст партикуляризованої формальної системи. До таких засновків можна віднести прагматичні аспекти існування формальної системи, наявність інтенсіоналів трьох класів в рамках кожної конкретно взятої системі, або наявність вирішальних аргументів «А-типу» в рамках кожного конкретного доказу. Дані аргументи, на думку автора, слід відокремлювати від сингулярних Р-пропозицій, які виступають антецедентами доказу істинності / хибності конкретно взятої D-пропозиції, але не здатні виступати в якості повноцінних епістемічних гарантій її істинності за відсутності вирішальних аргументів, які імплікують істинність / хибність D-пропозицій.

The paper is dedicated to the problem of analysis of D-propositions in mathematics (from the position of metamathematics). D-propositions, as for Mouloud and Lorenzen, are understood as sets of mathematical proposition which represent the result of conjectures-proofs/refutations process or logical deduction (which, in their turn, consist of singular propositions, represented by P-propositions). The author tries to indicate the insufficiency of the analysis of the course of the proof and the proof system itself in a particular formal system; the algorithms, provided by the formal system; the operational scope of the formal system for the verification of particular exemplified D-proposition. Instead of narrowing the analysis of D-propositions within the formal system itself, it is proposed, following Wittgenstein, to come out of the limits of a singular formal system and to turn to metamathematics, which is seen as the grounding for the construction of each particular formal system within the scope of mathematical knowledge. While analyzing the problem from the metamathematical point of view, it is also proposed to include to the field of analysis the pragmatic aspect. In case for metamathematics, the concept ‘pragmatic’ means: the efficiency of formal system and the judgments derived from it for the salvation of the particular mathematical (as well as trans-mathematical) tasks; the author’s intentions concerning the creation of the formal system and his expectations of its scope and limits. The second metamathematical aspect, which is proposed to be included into the analysis, is the intensional aspect, by which the philosophers shouldn’t see only the meanings of concepts or mathematical propositions. The analysis of intension should also include the research of quality and character of relations between the three classes, which, to author’s mind, are inevitably exist in every formal system: the primordial set of axioms and intensional-extensional meanings of the terms and concepts introduced as the basic elements of formal system set of proofs, grounded on the axioms, from which the particular propositions are derived (the amount of proposition may be potentially infinite or, vice versa, strictly quantified due to the conditions and rules, set by the first class); the singular D-propositions themselves, which are derived from the operations within the scopes of second class (including the false D-propositions, the possibility of falsity of which is an important element of a primordial set of algorithms and axioms, enabling the possibility of falsity of the proposition of a particular type, but not a falsity «in general»). Background of the problem is analyzed within the context of Frege – Russell – Peano interpretation of mathematical knowledge and the theories ‘hard core’ and their criticism by late Ludwig Wittgenstein, particularly, taking into consideration his understanding of the essence of ‘proof’ in mathematics and the concept of ‘mathematical language-game’. The author also appeals to David Liggins’ concepts of ‘causal relation’ and ‘conceptual relation’ which represent the development of Carnap’s intension / extension dichotomy, however, grounded on metaphysics similar to the correspondence theory of truth of Tarsky or Churchland. The foundations of the method of intensional-pragmatical analysis of formal system are, eventually, introduced in 6 paragraphs. To author, the proposed demands to the formal system evaluation, as well as the procedures of analysis themselves, are seen only as a beginning of the new metamathematical research program, which should be complemented by the new procedures and demands, reconceptualized and, surely, argued, as well as moved beyond the D-propositions of the mathematics to other areas of language.