On-line papers

Here is the directory containing my papers listed below together, in some cases, with their plain TeX source files.

Reflections on retirement

A slightly mangled version of this brief note appeared with a different title in the Spring 2013 issue of Fine Letters, published by the Princeton mathematics department. Several people did not get the point about the anecdote of Stanley Jordan's comment. If more undergraduates greeted new ideas with such wonder and eagerness, faculty would be clamoring to teach undergraduates.

Review of stochastic mechanics

Slides for a talk at the 2011 conference on Emergent Quantum Mechanics. The paper is here.

Il potere del paradosso

A power point presentation in connection with the exhibit From one to infinity. At the heart of mathematics, with a video in Italian at the Rimini Meeting, August 2010, together with notes (also in Italian). And here is a video of a panel discussion I was in.

Warning signs of a possible collapse of contemporary mathematics

A paper given at a meeting of mathematicians, philosophers, scientists, and theologians in San Marino, August 2006. A very heavily edited version appears as Chapter 3 of the conference proceedings: Infinity: New Research Frontiers, Michael Heller and W. Hugh Woodin, eds., Cambridge University Press, 2011.

A talk for a general audience

An untitled talk given in January, 2009, at the Crossroads Cultural Center.

Review of 18 Unconventional Essays on the Nature of Mathematics

A review of the book edited by Reuben Hersh, which appeared in the November 2007 issue of the American Mathematical Monthly.

Hilbert's Mistake

Here are the slides for a talk given at the Second New York Graduate Student Logic Conference, March 18, 2007.

L'infinito nell'aritmetica

A portion of a talk, in pdf format with links, given in Italian at the Rimini Meeting 2006.

The mystery of stochastic mechanics

These are notes, in a large font for transparencies, of a talk pointing out two defects in stochastic mechanics. The mystery is how a theory with such a natural and appealing derivation can be so right and yet so wrong. I do not understand this. For more on stochastic mechanics, see Quantum Fluctuations and the references therein; also Dynamical Theories of Brownian Motion.


A talk, Completed versus Incomplete Infinity in Arithmetic, addressed to non-mathematicians, for the conference on Infinity in Science, Philosophy, and Theology, Pontifical Lateran University, Vatican City, November 9-11, 2005 -- also in Italian.

Book Review

A review of Dennis Hesseling's "Gnomes in the Fog: The Reception of Brouwer's Intuitionism in the 1920s", in the Bulletin of the American Mathematical Society. The expository paper mentioned in the review is Understanding Intuitionism.

Syntax and Semantics

This is a paper that I will give at the International Symposium "Foundations and the Ontological Quest" to be held at the Pontifical Lateran University in Rome, January 7--10. An almost identical talk was given at the international conference Il problema dei fondamenti oggi held in Mendrisio, Switzerland, on November 16, 2001.

Mathematics and Faith

I was asked to give a paper (or pdf format) on this topic at the Jubilee for Men and Women from the World of Learning, held at the Vatican, 23-24 May 2000.

Mathematics and the Mind

Text of a talk (or pdf format) at the Tokyo conference Toward a Science of Consciousness -- Fundamental Approaches, Tokyo, May 25-28, 1999.

Understanding Intuitionism

This is a paper (or pdf format) presented at the Rencontre du Reseau Georges Reeb, March 24-28, 1997.


In November 1995, Ennio De Giorgi and I gave lectures at the University of Rome "La Sapienza" and then debated the foundations of mathematics in Forlì. Here are the revised texts of my lecture Confessions of an apostate mathematician, in English (or pdf format) and Italian (or pdf format) (with many thanks to Professor Gianfranco Basti for correcting the Italian).


A paper published in 1977: Internal set theory.

Ezekiel's Wheel

This is a poem I wrote many years ago for Nancy Wong Nelson (1930-1988). ps or html.

Ramified recursion and intuitionism

Presented to the Colloque Trajectorien, Strasbourg/Obernai, June 12-16, 1995. It is an introduction to recent work of Bellantoni, Cook, and Leivant on ramified recursion with some remarks on a possible logic of the feasible. This paper (or pdf format) is thinking out loud, not a finished work.


A non-technical talk on Space-Time-Chance. Also available in Romanian, translated by Alexandra Seremina and courtesy of azoft. A Polish translation is here, courtesy of Olga Babenko. A Danish translation is here, courtesy of Einar Solbakken. A Czech translation is here, courtesy of Olga Babenko.

To my home page.