Practical reason and mathematical argument

Practical reason and mathematical analysis

That mathematics employs a species of reasoning that is akin to practical reasoning is an ancient observation. When Aristotle wants a model for deliberation in ethics it is to mathematics that he turns. Consider his outline of the nature of practical deliberation in the Nicomachean Ethics:

[W]e first lay down the end, and then examine the ways and means to achieve it. If it appears that any of several [possible] means will reach it, we consider which of them will reach it most easily and most

How to reason theoretically and practically at the same time

The use of forms of reasoning PR1 and PR2 is not peculiar to Greek mathematics. It is to be found implicitly in modern mathematical texts. A good example is to be found in Hamilton's development of quaternion theory in the last century.⁴ The problem Hamilton has to solve is the extension of the use of imaginary numbers to represent directed line segments from two to three dimensions. He attempts to solve the problem

Mathematics, morals and science

In mathematics the distinction between theoretical and practical reasoning, between reasoning in which the conclusions are assertives and reasoning in which they are directives, is not the deep dichotomy it is taken to be. A piece of reasoning can end with either an indicative or imperative without much hinging on the difference—hence the different presentations Hamilton offers for his derivation of the equations for quaternions. Related contrasts between `discovery' and `invention', `finding'

Unlinked references

Hamilton (1967); this ref does not appear in the text.

Acknowledgements

I am grateful to John Benson, Jimmy Lenman and the journal referees for comments on an earlier draft of this paper. The paper was completed during a fellowship funded by the Leverhulme Trust.