Academic literature on the topic 'Refreshing semantics'

Abstract Modern logic provides a theoretical analysis of mathematics which cuts across its traditional subdivisions into algebra, geometry, analysis, etc. Actually, there is no single logical theory but a variety of such whose aim is to model the reasoning of, say, an idealized platonistic (set-theoretical) mathematician or an idealized constructivist mathematician at work in one of these areas of mathematics. What is common to the various theories is the use of formal systems to describe that activity, namely, as being one of drawing consequences within a specified class of symbolic expressions called formulas from certain formulas called axioms by use of certain specified rules of inference. Some logicians call themselves formalists and think that there is nothing more to mathematics than what is pictured as the production of derivations within formal systems. But those who take seriously the platonistic, constructivist, or other such views (for example, finitist, predicativist, etc.) also concern themselves with the meaning of what is expressed in formal languages. There is then the question as to which choices of axioms and rules are legitimate and-in case the systems are incomplete-how they might be legitimately extended. This leads one into controversial areas of the foundations of mathematics. I stress here the syntactic (formal) aspect as opposed to the semantic (and/or foundational) aspect of the logical description of mathematical activity. This is more neutral territory, but one within which there are a number of notions and results concerning the logical structure of mathematical proofs that are relevant to the remarks by Y. Manin in “Digression: proof” (Manin 1977, pp. 48-51). It should be added that I found those remarks both refreshing and stimulating.