A variant of Automath

We describe and document a variant of Automath which we have developed in the course of thinking about some ideas in the philosophy of mathematics. The Automath implementation may be viewed as a tool for building implementations of parts of the the mathematical world under the philosophical view. It is an important aspect of the philosophical view that completely general mathematical objects can be accessed by a computer program working in a finitary way (as a computer program must), and that, as we shall see, the mathematical world accessed can be the world of classical, impredicative, nonconstructive mathematics with high orders of infinity. And yet, at the same time, the underlying view is that we only deal with infinities (however large) as potential rather than actual. The moral is that classical mathematics is no less Aristotelean than constructive or impredicative mathematics, properly viewed. An obvious eccentricity of this implementation is that the usfer never writes any expression with bound variables. This is not to say (as I have carelessly said at times) that this implementation contains no λ-abstractions: it does, but they are generated internally (and are displayed to the user). The system could be extended with the ability to directly enter λ-abstractions in some contexts, but a philosophical point about the nature of abstraction is made by avoiding this. The description of the system which follows may appear quite mysterious when read by itself: I hope that reading it along with the examples of system output given in the last section will clarify matters.