NMSU seminar in constructive mathematics

Las Cruces 1971-1989

I was on sabbatical leave during Spring 1970, so I missed the arrival of constructive mathematics in Las Cruces. Upon returning, I heard about a seminar given by a visitor named Stanley Tennenbaum, and I noticed that everyone seemed to own a copy of a book by Errett Bishop. When Mark Mandelkern tried to interest me in the subject, I remarked that it seemed like recursive function theory. He assured me that this was not at all the case, but I saw no reason to believe him. Since then I have often encountered this erroneous identification of constructive mathematics with recursive function theory, and I have had as little success in correcting this impression as Mark.

Although it should be commonplace for mathematicians to get together and talk mathematics, my experience has been that this is the exception rather than the rule. So when Mark suggested that a few of us meet during the Spring semester to discuss Bishop's book, I was more than willing to participate. This was the start of the longest lasting, most active, working seminar that I have ever been involved in. Those early days were particularly exciting as the five of us, two abelian group theorists, two general topologists, and an applied mathematician, tried to understand and participate in a revolution in mathematical thought.

I remember the moment when I ceased being a curious observer and adopted Bishop's program as my own. I don't recall what was being discussed, but I related it to a puzzle that had been bothering me for some time. The theory of primary abelian groups relies heavily on ordinals; and since a complete theory for the countable case was developed in the 30's, big ordinals are essential for new results. Now perhaps big ordinals are taken with a grain of salt by logicians, but abelian group theorists, at least in those days, considered them to be very real. I had been disturbed by my inability to get a clear picture of big ordinals, yet that had not prevented me from proving theorems about them. It suddenly occurred to me that this was the poverty of meaning that Bishop was talking about in his criticism of classical mathematics.

The first research effort of our constructive seminar was to prove a constructive Jordan curve theorem. This theorem made a lot more sense to me when I thought of it from a constructive point of view. The standard proofs are usually an annoying mixture of constructive and idealistic techniques---if you are going to do something by magic, you should do it all by magic. More importantly, I found the constructive interpretation of the theorem much more appealing: how to join two points by a polygonal path bounded away from the curve if you have certain kinds of data about the curve at your disposal. Since the Jordan curve theorem, the constructive group in Las Cruces has carried out Bishop's program in such areas as dimension theory, Alexander duality, valuation theory, finite dimensional algebras, and abelian groups.

Participants: Gordon Berg, Douglas Bridges, Allan Calder, Y. K. Chan, Henry Cheng, Bill Julian, Mark Mandelkern, Ray Mines, Fred Richman.
Last modified May 23, 1997