Transformations of Discourse

It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth;  for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based.  The same thing cannot, however, be said about mathematics;  for here we have the new method of thought, pure intellect, the very well‑spring of the times, the fons et origo of an unfathomable transformation.

— Robert Musil • The Man Without Qualities

Here we take up the general study of logical transformations, or maps relating one universe of discourse to another.  In many ways, and especially as applied to the subject of intelligent dynamic systems, the argument will develop the antithesis of the statement just quoted.  Along the way, if incidental to my ends, I hope the present essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.

The goal is to answer a single question:  What is a propositional tangent functor?  In other words, the aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.

As a first step we examine the types of transformations we already know as extensions and projections and we use their special cases to illustrate several styles of logical and visual representation which figure in the sequel.

Resources

• Differential Logic and Dynamic Systems
• Differential Logic • Transformations of Discourse

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