Recently there has been some controversy over David Mumford’s Nature magazine invited obituary of Alexander Grothendieck being initially rejected on submission (see here and here). At issue was the attempt to explain the mathematical idea of schemes (one of Alexander Grothendieck’s most important contributions) to a non-mathematician audience. Professor Mumford is a mathematician of great stature and his explanation is better than anything I could even attempt. However, in addition to the issues he raises I don’t think he was sensitive enough to what a non-mathematician considers motivation.
I’ll take a quick stab at explaining a very tiny bit of the motivation of schemes. I not sure the kind of chain of analogies argument I am attempting would work in an obituary (or in a short length), so I certainly don’t presume to advise professor Mumford on his obituary of a great mathematician (and person). Continue reading Let’s try to motivate schemes
von Neumann and Morgenstern’s “Theory of Games and Economic Behavior” is the famous basis for game theory. One of the central accomplishments is the rigorous proof that comparative “preference methods” over fairly complicated “event spaces” are no more expressive than numeric (real number valued) utilities. That is: for a very wide class of event spaces and comparison functions “>” there is a utility function u() such that:
a > b (“>” representing the arbitrary comparison or preference for the event space) if and only if u(a) > u(b) (this time “>” representing the standard order on the reals).
However, an active reading of sections 1 through 3 and even the 2nd edition’s axiomatic appendix shows that the concept of “events” (what preferences and utilities are defined over) is deliberately left undefined. There is math and objects and spaces, but not all of them are explicitly defined in term of known structures (are they points in R^n, sets, multi-sets, sums over sets or what?). The word “event” is used early in the book and not in the index. Axiomatic treatments often rely on intentionally leaving ground-concepts undefined, but we are going to work a concrete example through von Neumann and Morgenstern to try and illustrate a bit more of the required intuition and deep nature of their formal notions of events and utility. I also will illustrate how, at least in discussion, von Neuman and Morgenstern may have held on to a naive “single outcome” intuition of events and a naive “direct dollars” intuition of utility despite erecting a theory carefully designed to support much more structure. This is possible because they never have to calculate in the general event space: they prove access to the preference allows them to construct the utility funciton u() and then work over the real numbers. Sections 1 through 3 are designed to eliminate the need for a theory of preference or utility and allow von Neuman and Morgenstern to work with real numbers (while achieving full generality). They never need to make the translations explicit, because soon after showing the translations are possible they assume they have already been applied. Continue reading Working an example of von Neumann and Morgenstern utility
Having a bit of history as both a user of machine learning and a researcher in the field I feel I have developed a useful perspective on the various trends, flavors and nuances in machine learning and artificial intelligence. I thought I would take a moment to outline a bit of it here and demonstrate how what we call artificial intelligence is becoming more statistical in nature. Continue reading A Personal Perspective on Machine Learning
This article is a quick appreciation of some of the statistical, analytic and philosphic techniques of Deming, Wald and Boyd. Many of these techniques have become pillars of modern industry through the sciences of statistics and operations research.
Continue reading Deming, Wald and Boyd: cutting through the fog of analytics
What is “genetic art?” My answer to this is http://www.geneticart.org (redirects to http://www.mzlabs.com), but this requires some explanation. Continue reading What is “Genetic Art?”
Hello World: An Instance Of Rhetoric in Computer Science
John Mount: email@example.com
February 19, 2008
Computer scientists have usually dodged questions of intent, purpose or meaning. While there are theories that assign deep mathematical meaning to computer programs we computer scientists usually avoid discussion of meaning and talk more about utility and benefit. Discussions of the rhetorical meaning of programs is even less common. However, there is a famous computer program that has a clean an important rhetorical point. This program is called “hello world” and its entire action is to write out the phrase “hello world.” The action is simple but the “hello world” program actually has a fairly significant purpose and meaning.
I would like to briefly trace the known history of “hello world” and show how the rhetorical message it presents differs from the rhetoric embodied in earlier programs. In this sense we can trace a change in the message computer scientists felt they needed to communicate (most likely due to changes in the outside world).
Continue reading Hello World: An Instance Of Rhetoric in Computer Science