David Spivak

Research Scientist
Department of Mathematics

Office: 4-182
Email: dspivak--math/mit/edu

"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 a communication.

What is the structure of informative communication?

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 this regard, we ask "what is the mathematical structure of communication and information?"

"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 future.

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 papers, which discuss various aspects of this work.

Related ideas.

Albert Einstein on the importance of the theory of knowledge.

Book review in which the author explains why Shannon's theory of information is inadequate to explain the real mystery.

David 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, 1928.)

Norbert Weiner wrote a book called "The mathematics of self-organizing systems." See also the wikipedia article on this.

Teleology. The idea that some things are done with intent and purpose.

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