It was a week of mathematics here at The Big Questions. I am still reeling from the momentous events that inspired Monday’s post; we now know that the Internet has changed mathematics forever. On Friday, we celebrated the momentous achievenments of the new Fields Medalists.
In between, we began what will be an occasional series on the foundations of arithmetic. In Part I, we distinguished truth from provability. In Part II, we distinguished theories (that is, systems of axioms) from models (that is, the mathematical structures that the theories are intended to describe). A theory is a map; a model is the territory. In Part III we talked about consistency and stressed that it applies only to theories, not to models. A purported map of Nebraska can be inconsistent; Nebraska itself can’t be.
It turns out (a little surprisingly) that any consistent map must describe multiple territories. (That is, any consistent set of axioms must describe many mathematical structures — or in other words, any consistent theory must have many models.) (This assumes the map has enough detail to let us talk about addition and multiplication.) These territories—i.e. these mathematical structures, all look very different, even though they all conform to the map. Conclusion: No map can fully describe the territory. No set of axioms can fully describe the natural numbers.
I’ll continue this series sporadically, and eventually we’ll get into some controversial philosophical questions. So far we haven’t.
Speaking of controversy, I’ve increased the default font size for this blog. Tell me if you like it.