The Truth of a Mathematical-Logical System, part 1
The Truth of a Mathematical-Logical System, part 1
Sku: 10500A0E050
Archival Number: CD/mp3 105
Author: Lonergan, B.
Language(s): English
Decade: 1950
Description:
CD/mp3 105, part 1 of lecture 3 on mathematical logic 1957. Corresponds to CWL 18, pp. 68-78 and 333-36. What kind of truth do mathematical-logical systems have? That is the question for this lecture. And first it must be asked, The truth of what? To ask about the truth of a system is to ask about the truth of all the propositions that constitute the system, whether or not they ever were or will be written down. That is not the truth of expressions but of judgments. The truth of a mathematical logical system is the truth of a virtual totality of propositions. The truth we are asking about is the truth of something universal and indeterminate and to be determined. The abstraction of a mathematical-logical system is of form from matter, not of universal from particular: a structure of relations. So we are asking about the truth of a relational structure into which any particular that one wishes may be put. What is meant by 'truth'? The notion of the grasp of the virtually unconditioned states what one gets hold of when one grasps that the evidence is sufficient for a judgment. Lonergan exemplifies this in deductive inference and in concrete judgments of fact. But the key distinction for the truth of a mathematical-logical system is the distinction between an analytic proposition and an analytic principle. In an analytic proposition the definition of terms provides the fulfillment of the conditions for a virtually unconditioned. The analytic principle adds on concrete judgments of fact. The terms in the sense in which they are defined are found in the existential order. Lonergan distinguishes three levels of analytic principles: absolute (as in metaphysics), provisional (as in the sciences), and serial (as in mathematics). Every mathematical-logical system that is respectable possesses the truth of the analytic proposition. But Lonergan raises the question, in what sense do they also possess the truth of analytic principles? In the next section (second half of the lecture) he will claim that various types of mathematical-logical systems contain fragments of factual truth. The questions had to do mainly with the distinction of analytic propositions and analytic principles.