Visiting Assistant Professor
University of Oregon
Department of Mathematics
Office: 317 Fenton Hall
Winter 2010 -- Informatics Seminar
Meeting time: Tuesday and Thursday at 4pm
Meeting place: 103 Peterson Hall.
What is the mathematics of information?
1928: "the fundamental idea [of Spivak's Informatics Seminar] is
none other than to describe the activity of our understanding, to make a
protocol of the rules according to which our thinking actually proceeds."
Purpose of this seminar:
The world is too complex to understand, but aspects of it can be
understood. Some things which are understood can be expressed in words,
not only for the purpose of mumbling to ourselves but for communicating to
others. Let's call anything of this form, anything that can be understood
and communicated in words to others, "information."
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
fully defined in mathematics, so math is not really "pure"; it
relies on agreement. But perhaps that's fair. Regardless, math certainly
comes as close as anything else to being pure information.
Thus, whatever we define information to be, mathematics should be "at
home" there. This suggests perhaps that we use mathematics to
information; otherwise, math will always be a foreigner within our
information in terms of mathematics may be seen as circular in that
math is seen both as a basic kind of information and as a language
in which to
define information. The implications of this circularity require
may begin to consider information in mathematical terms before having to
take on that analysis.
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
ontology log (or olog) is a topos sketch. In other words,
it's a graph together with a set of diagrams that are said to commute, a
set of diagrams which are said to be limits, and an object which
represents those limits which enforce monomorphism. The topos sketch
may also be enriched in the monoidal category of sets of strings. Less
formally, an olog is a system of boxes and arrows, each labeled by english
text, with some amount of the additional structure guaranteed by the
axioms of a topos.
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
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.
1. Lawvere and Schanuel. Conceptual
a first introduction to categories. An excellent book. Amazon.com link
2. Lawvere. Taking
categories seriously A great 25 page article, including how to think
of dynamical systems from a categorical perspective.
A short paper on topoi. It is good, but I think that there is at
least one medium-sized error; ask me about it
Stanford Encyclopedia entry on Information. I personally
found sections 2.2 and 3.2.2 particularly relevant.
5. For topoi: Moerdijk and Mac Lane. Sheaves in geometry and Logic: a
first introduction to topos theory.
6. Andreas Blass, Topoi and
7. Doring and Isham, `What is a
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
8. Bell, The
Development of Categorical Logic. In particular, this discusses the
lambda calculus, as well as proving the theorem "every topos is
An appropriate amount of work for this class:
Ideas to consider:
1. Global truth versus local truth. Can something be true only
locally? For example, should a pencil be considered "for all time and
space, a pencil" or just "for the last 5 days, a pencil".
2. Context. Related to (1), information is always contextual. In
other words, we are always looking at information from the viewpoint of
a certain audience (e.g. just myself). Given a sensible
"change-of-audience" morphism, should we
get a change-of-information functor, that takes information from one
perspective to information from another? Please consider
slice-toposes and change-of-base
functors in a topos and see how well they stack up to this idea.
3. Measurements. Suppose we say something has weight 5 lbs. What
does that mean? It means that when we put it on a good scale, the
scale returns the number 5. (What does it mean that the scale is
good?) Does weight, as a function from physical objects to numbers,
exist as a stand-alone concept, or should it be considered only as
a function from
pairs (physical object, scale) to numbers?
4. Objects are like sets? The objects in our ologs are labeled as
though they are sets (as one
interprets a box labeled "a person" as the set of all persons). Are
they? Toposes are categories in which
the objects feel like sets, but do not have to be such. Perhaps
we should think of objects as moduli spaces instead. See "moduli
5. Parts of speech. Objects in an olog should be labeled with noun
should be labeled with verb phrases. Adjectives are used in noun
suggest subobject-hood. As such, they should suggest a name for the
characteristic morphism corresponding to that subobject.
Adverbs correspond to subobjects of exponential objects.
6. Two types of info? The
Stanford Encyclopedia article made a
distinction between information that is
instructional (e.g. a computer program) and information that is factual (e.g. like
a database). Is that distinction important? How do ologs handle these two types
This work by David I. Spivak is licensed under a Creative Commons
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