Stanislaw Krajewski

Context is essential in virtually all human activities. Yet some logical notions seem to be context-free. For example, the nature of the universal quantifier, the very meaning of “all”, seems to be independent of the context. At the same time, there are many quantifier expressions, and some are context-independent, while others are not. Similarly, purely logical consequence seems to be context-independent. Yet often we encounter strong inferences, good enough for practical purposes, but not valid. The two types of examples suggest a general problem: how to characterise the context-free logical concepts in their natural environment, that is, in the field of their context-dependent associates. A general Thesis on Quantifiers is formulated: among all quantifiers, the context-free ones are just those definable by the universal quantifier. The issue of inferences is treated following the approach introduced by Richard L. Epstein: valid ones are an extreme case, the result of the disappearance of context-dependence. This idea can be applied to an analysis of a form of abduction, called “reductive inference” in Polish literature on logic. A tentative Thesis on Inferences identifies the validity of a strong inference that is context-independent.

Modern mathematics has been shaped by the process of secularization of science. Yet even some present-day mathematicians use religious terms behind the (mathematical) scenes. How essential this is remains debatable. Before modernity, everything, including mathematics, was perceived from a religious perspective. Deeper connections existed for Pythagoreans, and nowadays there is a revival of Pythagoreanism. Mathematics was used by medieval theologians, and even the founders of modern science – for instance, Newton, Leibniz, Boole, Cantor – referred to religion. From the present perspective, those theological aspects seem mostly irrelevant. The concept of actual infinity had a fundamentally religious meaning before the nineteenth century. Now, only a few see mathematical infinity as a notion common to mathematics and theology.Religious origins of some mathematical concepts can be detected, for example, Indian origins of zero, possible Christian sources of the initiation of the algebraic approach, the creative function of naming applied to infinite entities. On the other hand, mathematical illustrations of theological concepts are possible, even though genuine mathematical models do not seem to be essential or testable. Generally, religious needs and motivations played a role in the development of mathematics, and mathematical metaphors helped theologians. To perceive this connection as relevant now requires sensitivity to the realm of the religious.

Sometimes an oscillation takes place between two incompatible approaches to an experienced situation: from one to another, then back and then again, and again. The oscillation is not an additional ingredient but an essential aspect of the situation. Both approaches are needed: by assuming one of them the participant is led to understand the need of the other. A process of this kind occurs in interfaith dialogue: we oscillate between considering the other religion from an objective standpoint and perceiving it from the perspective of my own religion. Some kind of oscillation is present in other situations. Several examples are given to show that the process can be seen as familiar even though it has hardly been identified as a separate phenomenon to be analyzed. According to the present author, it is related to but possibly distinct from complementarity as it is known in physics. A preliminary attempt is made to formulate the logic of the oscillation process, which can be called sequential paraclassical logic.

Examples of possible theological influences upon the development of mathematics are indicated. The best known connection can be found in the realm of infinite sets treated by us as known or graspable, which constitutes a divine-like approach. Also the move to treat infinite processes as if they were one finished object that can be identified with its limits is routine in mathematicians, but refers to seemingly super-human power. For centuries this was seen as wrong and even today some philosophers, for example Brian Rotman, talk critically about “theological mathematics”. Theological metaphors, like “God’s view”, are used even by contemporary mathematicians. While rarely appearing in official texts they are rather easily invoked in “the kitchen of mathematics”. There exist theories developing without the assumption of actual infinity the tools of classical mathematics needed for applications (For instance, Mycielski’s approach). Conclusion: mathematics could have developed in another way. Finally, several specific examples of historical situations are mentioned where, according to some authors, direct theological input into mathematics appeared: the possibility of the ritual genesis of arithmetic and geometry, the importance of the Indian religious background for the emergence of zero, the genesis of the theories of Cantor and Brouwer, the role of Name-worshipping for the research of the Moscow school of topology. Neither these examples nor the previous illustrations of theological metaphors provide a certain proof that religion or theology was directly influencing the development of mathematical ideas. They do suggest, however, common points and connections that merit further exploration.

Examples of possible theological influences upon the development of mathematics are indicated. The best known connection can be found in the realm of infinite sets treated by us as known or graspable, which constitutes a divine-like approach. Also the move to treat infinite processes as if they were one finished object that can be identified with its limits is routine in mathematicians, but refers to seemingly super-human power. For centuries this was seen as wrong and even today some philosophers, for example Brian Rotman, talk critically about “theological mathematics”. Theological metaphors, like “God’s view”, are used even by contemporary mathematicians. While rarely appearing in official texts they are rather easily invoked in “the kitchen of mathematics”. There exist theories developing without the assumption of actual infinity the tools of classical mathematics needed for applications. Conclusion: mathematics could have developed in another way. Finally, several specific examples of historical situations are mentioned where, according to some authors, direct theological input into mathematics appeared: the possibility of the ritual genesis of arithmetic and geometry, the importance of the Indian religious background for the emergence of zero, the genesis of the theories of Cantor and Brouwer, the role of Name-worshipping for the research of the Moscow school of topology. Neither these examples nor the previous illustrations of theological metaphors provide a certain proof that religion or theology was directly influencing the development of mathematical ideas. They do suggest, however, common points and connections that merit further exploration.

The Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the “maverick” trend in the philosophy of mathematics. Several points made by its main representatives are mentioned – from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the new approach questions Hilbert’s Thesis, according to which a correct mathematical proof is in principle reducible to a formal proof, based on explicit axioms and logic.

In courses of logic for general students the general and existential quantifiers are the only ones distinguished from among all possible quantifier expressions of the natural language. One can argue that other quantifiers deserve mention, even though there are good reason for emphasizing the familiar ones: namely, they are the simplest, the universal quantifier is a counterpart of the operation of generalizing, the number of nested quantifiers is a good measure of logical complexity, and the expressive power of the general quantifier and its dual is considerable.Yet, even in the teaching about these two simplest quantifiers it has not been resolved how to indicate the realm to which a given quantifier refers. The methods range from the Fregean assumption that they refer to the totality of objects in the world to the restricted quantifiers to many sorted logic. It turns out that these approaches are not fully equivalent, because the sorts are usually assumed to be nonempty, which results in a problem similar to the well-known issue with non-emptiness of names in syllogistics.Logicians have studied various generalized quantifiers. It is, however, unclear how to treat the quantifier “many” and similar heavily context-dependent ones. They are not invariant under isomorphisms so no purely logical or mathematical treatment seems applicable. How else can one characterize the context-independent quantifiers among all possible quantifiers corresponding to quantifier expressions in natural language? The following thesis on quantifiers is proposed: Context-independence = definability in terms of the universal quantifier.This thesis provides an additional reason for distinguishing the universal quantifier from among all other quantifiers: it suffices for defining all context-independent ones.

The alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel’s incompleteness theorem is revisited. Its history is reviewed. The proof, also known as the Lucas argument and the Penrose argument, is refuted. It is claimed, following Gödel himself and other leading logicians, that antimechanism is not implied by Gödel’s theorems alone. The present paper sets out this refutation in its strongest form, demonstrating general theorems implying the inconsistency of Lucas’s arithmetic and the semantic inadequacy of Penrose’s arithmetic. On the other hand, the limitations to our capacity for mechanizing or programming the mind are also indicated, together with two other corollaries of Gödel’s theorems: that we cannot prove that we are consistent, and that we cannot fully describe our notion of a natural number.

In the paper a new model of God, or rather of the relation man-God, is presented. It uses the model of the projective plane. The resulting picture illustrates Martin Buber’s conception, and in fact his statements inspired the construction presented here. Further, it is shown how to apply this model to visualization in the course of the Jewish prayer involving the verse “Hear, oh Israel…”. Having indicated the merits of the model, the author critically analyses its adequacy, and, more generally, the suitability of mathematical models in theology.

The professional mathematician is a Platonist with regard to the existence of mathematical entities, but, if pressed to tell what kind of existence they have, he hides behind a formalist approach. In order to take both attitudes into account in a possibly serious way, the concept of suprasubjective existence is proposed. It involves intersubjective existence, plus a stress on objectivity devoid of actual objects. The idea is illustrated, following William Byers, by the phenomenon of the rainbow: it is not an object but can be said to possess a subjective objectivity.

Contacts of the two logicians are listed, and all Gödel's written mentions of Tarski's work are quoted. Why did Gödel almost never mention Tarski's definition of truth in his notes and papers? This puzzle of Gödel's silence, proposed by Feferman, is not merely biographical or psychological but has interesting connections to Gödel's philosophical views.No satisfactory answer is given by the three “standard” explanations: no need to repeat the work already done; Tarski's achievement was obvious to Gödel; Gödel's exceptional caution. In fact, , Tarski had done the work, but Gödel almost never mentioned the achievement; , the obviousness is no explanation for the omission of Tarski's work in contexts in which an application of the definition of satisfaction was useful, and even necessary; , the point was not just caution: if Gödel had felt the need to mention the program of scientific semantics he could easily have done that in his manuscripts, or in conversations.Three ideas, detectable in Gödel's approach, can help us understand Gödel's silence: the idea of truth as the intuitive provability in the most general sense; defining it set-theoretically would contribute nothing. the idea of truth as an inexhaustible idea in the sense of Kant; “truth in general” is a category that must be applicable to all kinds of sentential expressions; also, while for Gödel language was secondary, Tarski's definition is focused on language. the idea of logic as the universal language, in Hintikka's sense, as opposed to the perception of logic as a reinterpretable calculus; hence the thesis that semantics is inexpressible. Gödel always remains a Platonist who asks a natural question: what does really happen in the realm of abstract objects?

Philosophy should seriously take into account the presence of computers. Computer enthusiasts point towards a new Pythagoreanism, a far reaching generalization of logical or mathematical views of the world. Most of us try to retain a belief in the permanence of human superiority over robots. To justify this superiority, Gödel’s theorem has been invoked, but it can be demonstrated that this is not sufficient. Other attempts are based on the scope and fullness of our perception and feelings. Yet the fact is that more and more can be computer simulated. In order to secure human superiority over robots, reference to the realm of human relations and attitudes seems more promising. Insights provided by philosophy of dialogue can help. They suggest an ultimate extension of the Turing test. In addition, it seems that in order to justify the belief in human superiority one must rely on the individual experiences that indicate a realm that is not merely subjective. It makes sense to call it religious.

God’s name “Elohim,” common in the Hebrew Bible and Jewish tradition, is always used with verbs in the singular even though it is in the plural form. It is shown here that the ungrammatical usage can be seen as the best solution to a natural problem. Namely, tradition assumes that it should be impossible to talk about a general category of gods within which the one God could be located. The best and perhaps the only way to prevent the implicit pluralization of the unique God is to put his name in plural even though it is intended to be used as if it were singular. One cannot form the plural form of the name that is already grammatically plural! Surprisingly, this explanation seems to have been considered by neither classical nor modern commentators.

It is shown that there exists no grammatical translation into classical (propositional) logic of the modal logics, nor of intuitionistic logic and of the relatedness and dependence logics, as defined in Richard L. Epstein's bookThe Semantic foundations of logic. In the book the result is proved for translations without parameters.Classical propositional logicPC can be translated into other logics. Usually the grammatical structure of propositions is preserved, in the sense of the following definition