John Baez

Public Lecture at Categories in Algebra, Geometry and Mathematical Physics

July 14, 2005

The Mysteries of Counting:
Euler Characteristic versus Homotopy Cardinality

We all know what it means for a set to have 6 elements, but what sort of thing has -1 elements, or 5/2? Believe it or not, these questions have nice answers. The Euler characteristic of a space is a generalization of cardinality that admits negative integer values, while the homotopy cardinality is a generalization that admits positive real values. These concepts shed new light on basic mathematics. For example, the space of finite sets turns out to have homotopy cardinality e, and this explains the key properties of the exponential function. Euler characteristic and homotopy cardinality share many properties, but it's hard to tell if they are the same, because there are very few spaces for which both are well-defined. However, in many cases where one is well-defined, the other may be computed by dubious manipulations involving divergent series - and the two then agree! The challenge of unifying them remains open.

The Mysteries of Counting - transparencies in PDF format.

For more information, try these papers:

Also try these expository treatments in This Week's Finds in Mathematical Physics:

For much more, try these course notes:

Finally, here's an earlier talk on this subject:

The imaginary expression √-a and the negative expression -b resemble each other in that each one, when they seem the solution of a problem, they indicate that there is some inconsistency or nonsense. - Augustus De Morgan

Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature. - Hermann Weyl

© 2005 John Baez