# GÖDEL AND THE NATURE OF MATHEMATICAL TRUTH II

*I doubt that pure philosophical discourse can get us anywhere. Maybe phenomenological narrative backed by psychological and anthropological investigations can shed some light on the nature of Mathematical Truth.*

As to Beauty in mathematics and the sciences, here speaks Sophocles' eyewitness in Antigone:

*"..... Why should I make it soft for you with tales to prove myself a liar? Truth is Right."*

**Einstein & Gödel,** Princeton, 1950s (Photo by Oskar Morgenstern, IAS Archives)

*A true Realist, a true Platonist will not stoop to choose between Beauty and Truth, he will have the tenacity to stick it through until Truth is caught shining in her own Beauty. Sure there are messy proofs, we have to bushwhack trough a wilderness of ad hoc arguments, tours de force, combinatorial jungles, false starts and the temptations of definitions ever so slightly off target. Eventually, maybe not in our own lifetime, a good proof, a clear and beautiful proof will be honed out.*

**VDS Self-Portrait**

*That, I think, is the belief of the true Platonist. What Gödel and Einstein were doing when walking together over the Institute's grounds may have been just that; bush whacking, comparing mental notes and encouraging each other not to give up while getting all scratched and discouraged. Yet finding solace in speaking to each other in their mother tongue about their deepest concerns, and the state of the cosmos, the world, the weather and their households to*

**Introduction**

Verena Huber-Dyson, a Swiss national born in Naples in 1923, was educated in Athens before returning to Zurich to study mathematics (with minors in physics and philosophy), obtaining her PhD under Andreas Speiser in 1947. She moved to the United States in 1948, and happened to be in exactly the right field, the right place, and the right time to witness her two particular areas of interest — group theory and formal logic — have an unexpected impact (via particle physics and digital computing) on the real world.

She considers herself an Intuitionist, and this prompts the question she is asking herself:

"When I think with what a sure touch Bernoulli, Euler and their contemporaries summed infinite series without having a precise definition of convergence, which only came over a century later with Weierstrass and Cauchy, I am starting to wonder whether we are not witnessing a typical evolutionary phenomenon here.

"I don't think any contemporary analyst (Walter Hayman, Wolfgang Fuchs, Lars Ahlfors etc) would nowadays have that skill although they have other, more precise reasons for seeing that a series converges and more sophisticated and powerful methods for summing them.

"I am thinking both of the way our appendix has become obsolete, and of how some aborigines in central Australia are still able to hear what is happening at distances farther than we can perceive noises.

"So I wonder whether it might not be possible that mathematical intuition is regressing (atrophying by disuse) just as the discipline is evolving and so much can be accomplished without that extra fine sense — a phenomenon now due to the IT escalation, but already started with the surge of precision via formalization and mathematical logic."

Some of what follows is a challenge for the layperson...a rewarding challenge. "People are not prepared to roll up their sleeves and do some hard thinking and figuring," Huber-Dyson says, "and read a book with a a pad of paper and a few sharp pencils on the side, or their laptop in action. Reminds me of some of my colleagues, who said 'Oh it's brilliant, but soooo dense'. But to make up for that I had a few students who thrived on my dense cooking."

-**JB**

VERENA HUBER-DYSON is emeritus professor of the Philosophy department of the University of Calgary, Alberta Canada, where she taught graduate courses on the Foundations of Mathematics, the Philosophy and Methodology of the sciences.

Before the Vietnam war she was an associate professor in the Mathematics department of the University of Illinois. She taught in the Mathematics department at the University of California in Berkeley. She is the author or a monograph, *Gödel's theorems: a workbook on formalization, *which is based on her experience of teaching graduate courses and seminars on mathematical logic, formalization and its limitations to mathematics, philosophy and interdisciplinary students at the Universities of Calgary, Zürich and Monash.

She lives in Berkeley, California.

VERENA HUBER-DYSON's *Edge* Bio Page