University of Oregon

Department of Mathematics

Email: dspivak

Meeting place: 103 Peterson Hall.

What is information? What aspects of the world can be fully communicated? It seems that mathematics comes close to being "pure information." Of course, people often define a set to be "a collection" (which is nothing more than a synonym for "a set") -- this is circular. Thus we realize that even sets, the most basic objects of mathematical thought, are not

Thus, whatever we define information to be, mathematics should be "at home" there. This suggests perhaps that we

The goal of this seminar is to define a reasonable notion of "information," using mathematics as a foundation and language for the definition. In particular, I will discuss my research in this direction. I began this research with the belief that the most appropriate vantage point from which to view information is category theory. My studies have led me more specifically to topos theory.

While I don't yet have a finished definition, I do have a prototype: an

We each have our own olog (which is the situation as we see it, and the language we use to describe it). The name "olog" is short for "ontology log", where ontology is a subject studied both in computer science and philosophy referring to "the way things are." The word olog is supposed to be both reminiscent of "blog" and of the suffix "-ology", meaning the study of something. Ologs are created by studying the way things are and logging it.

The relation of ologs to blogs (weblogs) is that if everyone "ologs" the situation as they see it, and we attempt to connect our ologs together into one big human olog (in the appropriate topos-theoretic way), then we'll have a better understanding of each other's viewpoint on the world. Information will flow between humans more readily and accurately. I imagine a "Facebook" type system where users olog their ideas and lives, and connect them with their friends. Doing so would force us to understand each other. Moreover, computers would be able to "understand" us quite well.

Your job in this course is to understand these ideas, to improve upon them, and to challenge them.

2. Lawvere. Taking categories seriously A great 25 page article, including how to think of dynamical systems from a categorical perspective.

3. A short paper on topoi. It is good, but I think that there is at least one medium-sized error; ask me about it some time.

4. Stanford Encyclopedia entry on Information. I personally found sections 2.2 and 3.2.2 particularly relevant.

5. For topoi: Moerdijk and Mac Lane.

6. Andreas Blass, Topoi and Computation.

7. Doring and Isham, `What is a thing?': Topos Theory in the Foundations of Physics. Well-written and understandable (at least the first few sections are). This brings home the point that toposes are appropriate for modeling the "real world."

8. Bell, The Development of Categorical Logic. In particular, this discusses the lambda calculus, as well as proving the theorem "every topos is linguistic."

See me.

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This work by David I. Spivak is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.