Posts tagged with toposes

We often speak of an object being composed of various other objects. We say that the deck is composed of the cards, that a road is [composed of asphalt or concrete], that a house is composed of its walls, ceilings, floors, doors, etc.

Suppose we have some material objects. Here is a philosophical question: what conditions must obtain for those objects to compose something?

If something is made of atomless gunk then it divides forever into smaller and smaller parts—it is infinitely divisible. However, a line segment is infinitely divisible, and yet has atomic parts: the points. A hunk of gunk does not even have atomic parts ‘at infinity’; all parts of such an object have proper parts.

The definition of toposes has surprisingly powerful consequences. (For example, toposes have all finite colimits.)

Probably the best analogy elsewhere in which a couple of mild-sounding hypotheses pick out a very narrow and interesting class of examples is the way in which the Cauchy-Riemann equations select the analytic functions from all smooth functions of a complex variable.

Michael Barr & Charles Wells, Toposes, Triples, and Theories (p 64)

The ontological commitment of a sentence is whatever must be among the values of bound variables for the sentence to come out as true.

Willard van Orman Quine

I bring up this quotation not to agree or disagree with Quine, but just to point out the connection between everyday language and mathematics. The connection is well-known in some circles — like,  anyone for whom the phrase “possible world semantics” rings fifty different bells.

If you’re not schooled in such stuff, just notice this: Quine is talking about regular declarative sentences in natural languages, yet using the word “variables”.

One reasonable conclusion to draw is that, through the machinations of Analytical-Philosophy-Of-Language, the reach of mathematics can extend very, very far. How useful is stuff, anyhow? What if mathematics appeared in every declarative sentence you wrote or uttered?

(Source: supervenes)

"Have you ever done acid, kid? This book is like Acid."
—John L. Rhodes, speaking of the book Topos Theory by Peter Johnstone