Log24

Friday, January 5, 2018

Types of Ambiguity

Filed under: General,Geometry — Tags: , — m759 @ 2:56 AM

From "The Principle of Sufficient Reason," by George David Birkhoff
in "Three Public Lectures on Scientific Subjects,"
delivered at the Rice Institute, March 6, 7, and 8, 1940 —

From the same lecture —

 Up to the present point my aim has been to consider a variety of applications of the Principle of Sufficient Reason, without attempting any precise formulation of the Principle itself. With these applications in mind I will venture to formulate the Principle and a related Heuristic Conjecture in quasi-mathematical form as follows: PRINCIPLE OF SUFFICIENT REASON. If there appears in any theory T a set of ambiguously  determined ( i e . symmetrically entering) variables, then these variables can themselves be determined only to the extent allowed by the corresponding group G. Consequently any problem concerning these variables which has a uniquely determined solution, must itself be formulated so as to be unchanged by the operations of the group G ( i e . must involve the variables symmetrically). HEURISTIC CONJECTURE. The final form of any scientific theory T is: (1) based on a few simple postulates; and (2) contains an extensive ambiguity, associated symmetry, and underlying group G, in such wise that, if the language and laws of the theory of groups be taken for granted, the whole theory T appears as nearly self-evident in virtue of the above Principle. The Principle of Sufficient Reason and the Heuristic Conjecture, as just formulated, have the advantage of not involving excessively subjective ideas, while at the same time retaining the essential kernel of the matter. In my opinion it is essentially this principle and this conjecture which are destined always to operate as the basic criteria for the scientist in extending our knowledge and understanding of the world. It is also my belief that, in so far as there is anything definite in the realm of Metaphysics, it will consist in further applications of the same general type. This general conclusion may be given the following suggestive symbolic form: While the skillful metaphysical use of the Principle must always be regarded as of dubious logical status, nevertheless I believe it will remain the most important weapon of the philosopher.

Related remarks by a founding member of the Metaphysical Club: