Invention or Discovery?

A current | LinkedIn discussion asks "Did man invent Maths or was it there all along waiting to be discovered?". So I thought I'd take the return of that old question as a prompt to review and express some thoughts on the nature of the subject, of "truth" in general, (and of us).

In fact Elias Gourtsoyannis' description of the views Lakoff and Nunez (which I guess I should make a point of reading!) sounds very like my own feelings on this issue - which also echo my (possibly not entirely faithful) reading of Alexander Pope's "The proper study of mankind is man".

As I see it, the rules of logic and axioms of set theory (from which all else in conventional mathematics can be constructed) are really as much expressions of how our minds work as of any external "reality" with which we have to deal.

And, as Colin says, when we find interesting new new tautologies to add to the body of mathematics (which consists of nothing else after all), the question of whether we are inventing or discovering is really just about the attitude with which we do so.

Of course, the reference to set theory as a foundation is merely an example. But most of the body of any mathematical system consists of tautological consequences of some small set of propositions which are taken as axioms.

I think that the part of mathematical activity which builds an edifice of theory on top of its axiomatic foundation is (contra the building metaphor) actually more a voyage of discovery than an act of invention. After all, once the axioms and rules are set, we have no choice to invent anything but what is already there.

But a claim for invention can surely be made about the axioms, and I believe, also about the rules of logic.

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