The Development of Self-Reference: Löb’s Theorem

Abstract

Social processes operate on the sciences and social history must be understood in order to understand the history of science. This simple truth has transcended mere axiomatic truth; it has transcended mere fashion; today, the social treatment is de rigueur in the history of science and the history of computing. In the history of mathematics, however, the social emphasis is not so firmly entrenched. Historians who discuss the history of mathematics certainly steep their discussions in social contexts, but the history of mathematics is quite often written by mathematicians or, at least, mathematical educators, and mathematicians generally do not care about the overall societal background. Modern mathematics has largely divorced itself from the sciences and has become something of an intellectual sport, a game of ideas. Now, mathematicians are somewhat social and love anecdotes about fellow mathematicians, but this is as far as their interest in social history goes: Where the sciences inject history into their textbooks and discuss the growth of ideas, the history in most calculus texts limits itself to paragraph-length biographies, and one of the most successful textbooks on the history of mathematics is highly anecdotal, its author having additionally published no fewer than three books consisting of nothing but anecdotes. When the mathematician wants to discuss history (i.e., history of mathematics) and he wants to discuss it seriously and will not settle for mere anecdotes, it is not the social forces that created a need for a given type of mathematics that he writes about—mathematics has the odd habit of developing long before it is useful (i.e., before it is needed by society at large); it is not a simple chronology that he lists—despite all one reads about how the mathematician is like a fish out of water when he is not doing mathematics, he is not as stupid as all that; it is the development of the subject that interests him: What was the key idea? How did it come about? etc.KeywordsModal LogicProof TheoryDerivability ConditionCompleteness TheoremModal LanguageThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.