Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.pl Data: 27/02/2019 11:52:41 Annales UMCS Annales UMCS Informatica AI 3 (2005) 119-130 Informatica Lublin-Polonia Sectio AI http://www.annales.umcs.lublin.pl/ Church’s thesis as an empirical hypothesis Adam Olszewski Pontifical Academy of Theology, Kanonicza 25, 31-002 Kraków, Poland The studies of Church’s thesis (further denoted as CT) are slowly gaining their intensity. Since different formulations of the thesis are being proposed, it is necessary to state how it should be understood. The following is the formulation given by Church: [CT] The concept of the effectively calculable function1 is identical with the concept of the recursive function. The conviction that the above formulation is correct, finds its confirmation in Church’s own words:2 We now define the notion, […] of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers […]. Frege thought that the material equivalence of concepts is the approximation of their identity. For any concepts F, G: Concept FUMCS is identical with concept G iff ∀x (Fx ≡ Gx) 3. The symbol Fx should be read as “object x falls under the concept F”. The contemporary understanding of the right side of the above equation was dominated by the first order logic and set theory. 1Unless stated otherwise, the term function in this work always means function defined on the set natural numbers, that is, on natural numbers including zero. I would rather use the term concept than the term notion. Instead of if and only if will be used iff. 2This fragment comes form the seventh paragraph of the work ‘Unsolvable Problem of Elementary Number Theory’. It is similar in the first paragraph although the formulation of this definition occurs in the third paragraph:’ a positive integer function will be named effectively calculable if it is λ-definable in the sense of the second paragraph below’. 3Frege was not supposed to distinguish the signs ≡ and = (identity and equivalence). In his work of 1879 he used the sign ≡, while in the writings of 1893 and 1903 he used =. See: Jan H. Alnes, ‘Sense and Basic Law V in Frege’s Logicism’, Nordic Journal of Philosophical Logic, 4(1999), pg. 1-30, in particular s. 4. Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.pl Data: 27/02/2019 11:52:41 120 Adam Olszewski This leads to regard (after Frege) the proposition expressing the material equivalence of concepts as identical with the proposition stating the equality of their extensions (sets).4 In the present work, the term hypothesis is understood as a proposition valued absolutely as true or false. However, the current state of investigations precludes precise determination of such values. Since the times of Popper, falsifiability is the criterion determining whether a theory or a hypothesis is scientific. Here, falsifiability will be understood broader as a possibility to prove the falsity of propositions. Although this is a necessary feature of empiricity, it is not sufficient for mathematical hypotheses fall under such category as well (e.g., continuum hypothesis). A hypothesis will be named empirical when it possesses empirical content. Empirical content of a hypothesis (theory) is a set of base propositions (observational) that can be deduced from the hypothesis by the use of logical methods. An observational proposition is a proposition whose logical value can be determined based on sensorial cognition (confrontation with reality). Such propositions contain observational terms. Their truth is established by contingent properties of the Universe, the world in which we live. From the epistemological point of view, one can say that these are a posteriori propositions. 1. A sample of the views on the empirical character of CT Of the “fathers of calculability”, Emil Post supported the understanding of CT as an empirical hypothesis. In his three page article “Finite Combinatory Processes. Formulation I.” he gives his analysis of the process of calculation. Although independent, this method is almost identical with the Turing’s analysis. Post’s articleUMCS reached the editorial office of Journal of Symbolic Logic on October 7, 1936 and, although it was published earlier, it is posterior to the famous work of the English logician. In the last chapter Post writes: The writer expects the present formulation to turn out to be logically equivalent to recursiveness in the sense of the Gödel-Church development.[…] Its purpose, however, is not only to present a system of a certain logical potency but also, in its restricted field, of psychological fidelity. In the latter sense wider and wider formulations are contemplated. On the other hand, our aim will be to show that all such are logically reducible to formulation I. We offer this conclusion at the present moment as a working hypothesis. And to our mind such is Church’s identification of effective calculability with recursiveness. 4 In Frege’s system, the transition from the identity of the extension of notions to the identity of notions was possible based on the so called Basic Law V. This law was contradictory within Frege’s system of the second order logic. Remarks on that matter are given in S. Shapiro, ‘Foundations without Foundationalism’, Oxford 2002, pg. 16-17. Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.pl Data: 27/02/2019 11:52:41 Church’s thesis as an empirical hypothesis 121 Here footnote number eight appears: […] Actually the work already done by Church and others carries this identification considerably beyond the working hypothesis stage. But to mask this identification under a definition hides the fact that a fundamental discovery in the limitations of the mathematicizing power of Homo Sapiens has been made and blinds us to the need of its continual verification. And further on in the main text: Out of this hypothesis […] flows a Gödel-Church development. The success of the above program would, for us, change this hypothesis not so much to a definition or to an axiom but to a natural law. The excerpts given above indicate that Post maintained the following: – CT is neither a definition nor an axiom.5 – CT is a working hypothesis that demands continuing verification. – CT was elevated by Church and others to the level of a valid law of nature. – CT limits the (psychological) mathematical possibilities of man. This understanding did not accord with that of Church. For him, the thesis was a definition. Therefore, in his review of the cited paper of Post, Church criticizes the understanding of CT as a working hypothesis that requires constant verification. He thinks that the notion of the effective calculability does not have precise sense and as a result CT does not have precise meaning as a working hypothesis. Calculability with the use of a machine (limited by the condition of finiteness) is supposed to be an adequate representation of an intuitive notion6. Then, according to Church, the need to assume such a hypothesis disappears. However, he did not relate to Post’s view expressed in the quote cited. This is one of the few Church’s comments on CT aside from the article in which it was formulated. It seems that later on he modified his standpoint in that matter. In 1940, he wrote the followingUMCS7: Now a formal definition of effective calculability, for functions of positive integers, has been proposed by the author, […] and the adequacy of this definition to represent the empirical notion (A.O) of an effective calculation finds strong support in the recent result of Turing. It is not clear what kind of definition CT is for him. In Reference 168 of his Introduction to mathematical Logic he distinguishes four types of definitions within logic: definitions that are abbreviations (a kind of synthetic definitions?), 5It is not clear how Post understood definitions. It seems that he understood them as what is sometimes called synthetic definition. Post’s graduate student, Martin Davies, thought like his professor: ‘[…] how can we ever excluded the possibility of our being presented, some day (perhaps by some extraterrestrial visitors), with a (perhaps extremely complex) device or “oracle” that “computes” a noncomputable (in the sense of Turing; A.O.) function?’, M. Davis, ‘Computability and unsolvability’, Dover, New York 1982, pg. 11. 6This review was published in: Journal of Symbolic Logic, 2 (1937) pg. 43. 7See: Alonzo Church, ‘The Concept of Random Sequence’, Bulletin of the American Mathematical Society, 46(1940), pg. 133. Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.pl Data: 27/02/2019 11:52:41 122 Adam Olszewski definitions that explicate the notation of the language (similar to real definitions)8, semantic rules (metalanguage) that establish the interpretation of the system’s language9, definitions that extend the language of a formal system (they are a part of object language and must fulfil the conditions of correctness formulated by Leśniewski). It seems that CT can fall only under the second category of definitions due to the fact that Church attempted to provide justification of his thesis. If this is true, the opposition of Church towards Post’s approach to CT seems strange because these two positions with regard to the character of CT could be reconciled. Hao Wang was another logician with strongly philosophical inclinations who pointed out the empirical character of CT. In the book, A survey of Mathematical Logic, he writes10: It does seem that in the concept of effectiveness, there is a core in mechanical terms, and at the same time, there is an idealization which brings to infinity. Something which a physical object can do reliably and systematically would seem to be effective, no matter whether we understand the process or not.