Department of Mathematics
"Mathematics is the pursuit of that upon which
we can surely agree."
Essential to success in anything is the capacity to duplicate success, to
make it happen again. However, success in a given situation depends on
the precise nature of the situation, and this situation will never be
precisely repeated. Hence we must find "invariants" of the situation,
i.e. ways to describe it, which may be recognized in other situations and
hence clue us in to the possibility of a similar success.
An individual creating such a description for himself may use his own
jargon, so long as he will understand it in the future. (Within the
scientific model, we would note that a neuronal firing pattern in his
brain may serve as "jargon" in this sense -- an identifiable pattern used
to indicate something.) He may also jot down some notes, or
write himself a manual. Regardless of its form, in order to
be effective, this description must be informative to the person at a
later time, so that he may
act in accord with it.
This may be seen as a more primative form of communication -- the person
in the present communicates to the person he imagines himself to be in the
future. Certainly not all actions by the person in the present would
serve as working communications. Again, to be useful, a communication
must be informative, whatever that may mean. Creating an informative
communication, whether between two people or between a person and his
later self, relies on correctly determining a set of similarities between
two communicating entities. These similarities, agreements, and
conventions serve as the basic materials used for building or structuring
What is the structure of informative
The issue of duplicating success in tandem with another person is similar,
but even more difficult. As the observation of others is less direct than
the observation of the self, it is harder to know what actions we can take
to be informative to the other, and hence build a working relationship.
Still, a primary goal in any agreeable relationship is to create working
communication. This requires finding a set of similarities with which to
build the communication. The purpose of culture is to have such a set of
similarities "ready-made." Here, I intend the word culture to include
groups like native english speakers, academics, members of a tribe, or
even a group consisting of just one individual -- a culture of one.
When people are instilled with a set of shared experiences
and understandings, communication becomes possible.
To study effective communication, it is useful to ask: "what are the types
of things upon which we can come to agreement, and how can we communicate
them?" It seems that mathematics is our best attempt to create
unambiguous agreement. It also may be the case that mathematics is the
appropriate language and environment in which to create a working
definition of communication itself and to explore its possibilities. In
regard, we ask "what is the mathematical structure of communication and
"Information is the new physics."
In the modern world, information is increasingly becoming a commodity -- it's
value is becoming increasingly apparent. Whereas the "physical world" (the
world as studied by physics) is hypothesized to be lacking in deliberacy, our
human world is just the opposite: our concerns are intent, purpose, and
meaning. In order for me to bring an intention to fruition, I must know
exactly what I mean, what it is that I want to occur. This object of my
intention is really nothing more than structured meaning -- anything I want
is simply a concept I have, but a very precise one. In order to manifest it,
I must know exactly what it is.
We are often intending and meaning things, and hoping to inform others of
our intentions so that they may help. The desire to achieve success more
frequently -- not specific success but just success generally in life --
this itself is an intention, albeit of a "higher" order. At this level our
goal becomes to know what success itself is. Success begins with creating
a desired outcome, a set of observations that will be the case in order to
declare success. The desired outcome is not yet present -- it is imagined
-- but it has some resemblance to the current situation. How are these
similarities and differences understood? The issue rests on knowing what
information is. Once we know what information is, we will be more
effective at using it to describe situations, and hence achieve our
purposes more easily.
We must search for a model of information which is both flexible and
rigorous and which is able to clearly and effectively describe all
imaginable types of information.
Using mathematics to model the physical universe has proven quite
successful (for both math and physics). To model the world of
communication and information we begin by finding better ways to classify,
organize, and process information. To understand what information is, we
must first understand what it does, how it works. For this, we should
again turn to mathematics.
Category theory is the mathematics of structure.
I think that category theory has a lot to offer in this regard. In order
to find somewhere to start, I've been
working on aspects of computer science which are relevant to the goal of understanding
information. The idea is that
we already do have ways of classifying, organizing, and processing information;
by modeling them mathematically, we can understand their conceptual underpinnings, which
in turn can help us create new and better systems in the
My work on this subject.
To see some of my work on the subject of information and communication (and a
mathematical formulation thereof), see my page on Categorical
Informatics, or in particular some pure math papers
and applied math
which discuss various aspects of this work.
on the importance of the theory of knowledge.
review in which the author explains why Shannon's theory of information
is inadequate to explain the real mystery.
Hilbert "This formula game is carried out according
to certain definite rules, in which the technique of our thinking is
expressed. [...] The fundamental idea of my proof theory 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" (Hilbert,
Weiner wrote a book called "The mathematics of self-organizing
systems." See also the wikipedia article on
Teleology. The idea
that some things are done with intent and purpose.
This work by David I. Spivak is licensed under a Creative Commons
Attribution-Share Alike 3.0 Unported License.