The General Character of Mathematical Logic, Part 2
Sku: 10200A0E050
Archival Number: CD/mp3 102
Author: Lonergan, B.
Language(s): English,
Decade: 1950

CD/mp3 102, part 2 of lecture 1 on mathematical logic 1957. Corresponds to CWL 18, pp. 21-37. Sponsor: William F. and Susan Doran. The movement toward the technical arises with an increasing accumulation of insights leading to a detailed plan of operation with both its organizational and its mechanical aspects. In such an arrangement only a fragmentary understanding of the total process is found in any individual. The process is throughout intelligible, controlled by intelligence, and yet there is no individual that does the understanding of everything. Symbolism is a kind of technique. People can take the square root even if they know nothing about the technique. There is an economy of intelligence and of reason and an objectification of mind involved in technique. Traditional logic claimed to be engaged in investigating the immutable laws of the mind, whereas symbolic logic is concerned with what a machine can do. Symbolism is also a model, something we see or imagine and understand. The sensible signs are sensible data in which a form can be grasped. 'Model' is not the same as object. The object is what is conceived, thought about, considered, as a result of insight into the model. There are explicit and virtual elements in the model. Symbolism is both technique and model, and one must know the game, as it were, to know what moves to make. Isomorphism is the similarity of relational structure. It may have all sorts of different meanings, but in general it enables us to see that symbolic logic is possible. There is an isomorphism, for example, of geometry, algebra, physics, in that the same relational structure can be found in all three. The symbols are representing logical relations, and when they represent deductions in general, one is in symbolic logic. Thus mathematical logic is the investigation of the field of logical relations through the development of suitable symbolic techniques. It has developed along different lines: Russell-Whitehead, Hilbert, the Polish school, and others. The classical propositional calculus is one system, another is based on and, and another on if then. In general, they are all equivalent, though Lewis's modal system adds strict implication in the ordinary sense of if-then, and there is an intuitionist school that omits the principle of excluded middle, and three-valued logics such as those that consider the contingent future. The questions concerned implicit definition and further epistemological and philosophic difficulties that arise because the people involved have very little knowledge of the philosophical tradition.


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