Stanislaw Krajewski

Andrzej Grzegorczyk zmarł 20 marca 2014 r. Miał 91 lat, ale nie spodziewaliśmy się jego odejścia, bo prawie do ostatnich chwil pracował twórczo, a jego siły intelektualne wydawały się nienaruszone. We wspomnieniach podkreślano, że był ostatnim dobrze znanym na świecie przedstawicielem polskiej logiki ubiegłego stulecia. To prawda, ale jeszcze ważniejsze wydaje się to, że był bardzo nietuzinkową postacią, człowiekiem aktywnym naukowo i społecznie, a zarazem duchem osobnym, chadzającym własnymi ścieżkami.

This paper introduces the special issue on the Concept of God of the Journal of Applied Logics (College Publications). The issue contains the following articles: Logic and the Concept of God, by Stanisław Krajewski and Ricardo Silvestre; Mathematical Models in Theology. A Buber-inspired Model of God and its Application to “Shema Israel”, by Stanisław Krajewski; Gödel’s God-like Essence, by Talia Leven; A Logical Solution to the Paradox of the Stone, by Héctor Hernández Ortiz and Victor Cantero; No New Solutions to the Logical Problem of the Trinity, by Beau Branson; What Means ‘Tri-’ in ‘Trinity’ ? An Eastern Patristic Approach to the ‘Quasi-Ordinals’, by Basil Lourié; The Éminence Grise of Christology: Porphyry’s Logical Teaching as a Cornerstone of Argumentation in Christological Debates of the Fifth and Sixth Centruies, by Anna Zhyrkova; The Problem of Universals in Late Patristic Theology, by Dirk Krasmüller; Intuitionist Reasoning in the Tri-unitrian Theology of Nicolas of Cues, by Antonino Drago.

Mathematicians use theological metaphors when they talk in the kitchen of mathematics. How essential is this talk? Have theological considerations and religious concepts influenced mathematics? Can mathematical models illuminate theology? Some authors have given positive answers to these questions, but they do not seem final. It is unclear how religious views influenced the work of those mathematicians who were also theologians. Religious background of some mathematical concepts could have been inessential. Mathematical models in theology have no predictive value. It is, however, important to continue the recently initiated search for the mutual influences of mathematics and theology., 2016.)

The penitential prayer of the Polish Catholic Bishops on May 27, 2001 in Warsaw was a memorable and important event. In an unequivocal way they accepted moral responsibility for the massacre of the Jewish community of Jedwabne on July 10, 1941, in which Poles and Catholics took part. This was a symbolic gesture of great significance in the current debate about the massacre, one which has changed the Polish perception of World War II: although the Germans conceived and executed the plan of killing all Jews, in some cases local Poles were actively involved. The question may be asked of how to measure the sincerity and the impact of such symbolic gestures. One approach is to look at practical consequences. In this case, two tests seem in order. One is the existence in the church in which the prayer took place of a bookshop with anti-Semitic books on sale. The prayer service wrought no change here at all. Another test would be the attitude of the local church in Jedwabne and its parishioners towards those who attempted to help Jews in 1941 and later, as well as towards those who spoke the painful truth after the War. Unfortunately, they continue to be marginalized by the hostile majority.

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.