Why Mathematical Reasoning Cannot Be a Simple Matter of Definitions and Formal Rules

by Gene Callahan

The point I wish to make here has been made before, notably, by Lewis Carroll in his essay “What the Tortoise Said to Achilles“, as well as by Wittgenstein, in his work on what it means to “follow a rule,” and by Gödel in his famous paper on undecideability. But, as I recently encountered a very, very bright young philosopher who seemed unaware of the import of such arguments, it is, perhaps, a point worth making once again.

The contention at hand is that, contrary to those who hold that mathematical knowledge offers us an example of objective truths that are of a non-physical nature, mathematical truth is “simply” a matter of positing some arbitrary set of definitions and rules for drawing conclusions from them. Continue reading

What Is a Model?

by Gene Callahan

I’ve been pondering this point a bit lately, and this seems like a good place to share my musings and get some feedback. The main questions I’ve been pondering are things like, ‘How is a model “accurate”?’ ‘What makes something a model of one thing and not another?’ ‘How do we know how to “use” the model in some activity?’

Let’s consider a blueprint for a house. It consists of some blue lines on white paper. You give it to me, ignorant of building practice, and tell me ‘Build this right here’, and indicate a piece of ground. I see there is a scale conversion on the blueprint, say, 1 inch = 3 feet. I figure out the requisite enlargement of the figure — then I go and paint a white rectangle on the ground of that size, and proceed to paint blue lines on it. Continue reading