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


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."


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.




A recent, important turn in my life occurred when I realized that something that I have long been stating in footnotes should be put on the marquee. I have engaged myself, without realizing it, in undertaking a theory of roughness. Think of color, pitch, heaviness, and hotness. Each is the topic of a branch of physics. Chemistry is filled with acids, sugars, and alcohols; all are concepts derived from sensory perceptions. Roughness is just as important as all those other raw sensations, but was not studied for its own sake.

Introduction by John Brockman

During the 1980s Benoit Mandelbrot accepted my invitation to give a talk before The Reality Club. The evening was the toughest ticket in the 10 year history of live Reality Club events during that decade: it seemed like every artist in New York had heard about it and wanted to attend. It was an exciting, magical evening. I've stayed in touch with Mandelbrot and shared an occasional meal with him every few years, always interested in what he has to say. Recently, we got together prior to his 80th birthday.

Mandelbrot is best known as the founder of fractal geometry which impacts mathematics, diverse sciences, and arts, and is best appreciated as being the first broad attempt to investigate quantitatively the ubiquitous notion of roughness.

And he continues to push the envelope with his theory of roughness. "There is a joke that your hammer will always find nails to hit," he says. "I find that perfectly acceptable. The hammer I crafted is the first effective tool for all kinds of roughness and nobody will deny that there is at least some roughness everywhere."

"My book, The Fractal Geometry of Nature," he says, reproduced Hokusai's print of the Great Wave, the famous picture with Mt. Fuji in the background, and also mentioned other unrecognized examples of fractality in art and engineering. Initially, I viewed them as amusing but not essential. But I soon changed my mind.

"Innumerable readers made me aware of something strange. They made me look around and recognize fractals in the works of artists since time immemorial. I now collect such works. An extraordinary amount of arrogance is present in any claim of having been the first in "inventing" something. It's an arrogance that some enjoy, and others do not. Now I reach beyond arrogance when I proclaim that fractals had been pictured forever but their true role had remained unrecognized and waited for me to be uncovered."


BENOIT MANDELBROT is Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) at the IBM T.J. Watson Research Center. His books include The Fractal Geometry of Nature; Fractals and Scaling in Finance); and (with Richard L. Hudson) The (mis)Behavior of Markets.

Benoit Mandelbrot's Edge Bio Page 

Smolin Vs. Susskind: The Anthropic Principle



Leonard Susskind          Lee Smolin      


Recently, I received a copy of an email sent by Leonard Susskind to a group of physicists which included an attached file entitled "Answer to Smolin". This was the opening salvo of an intense email exchange between Susskind and Smolin concerning Smolin's argument that "the Anthropic Principle (AP) cannot yield any falsifiable predictions, and therefore cannot be a part of science".

After reading several postings by each of the physicists, I asked each if (a) they would consider posting the comments on Edge, and (b) if they would write a new, and final "letter".

Both agreed, but only after a negotiation: (1) No more than 1 letter each; (2) Neither sees the other's letter in advance; (3) No changes after the fact. A physics shoot-out.

While this is a conversation written by physicists for physicists, it should nonetheless be of interest for Edge readers as it's in the context of previous Edge features with the authors, it's instructive as to how science is done, and it's a debate that clarifies, not detracts. And finally it's a good example of what Edge is all about, where contributors share the boundaries of their knowledge and experience with each other and respond to challenges, comments, criticisms, and insights. The constant shifting of metaphors, the intensity with which we advance our ideas to each other — this is what intellectuals do. Edge draws attention to the larger context of intellectual life.

Below are the original email pieces, followed by the final letters presented side-by-side.


LEE SMOLIN, a theoretical physicist, is concerned with quantum gravity, "the name we give to the theory that unifies all the physics now under construction." More specifically, he is a co-inventor of an approach called loop quantum gravity. In 2001, he became a founding member and research physicist of the Perimeter Institute for Theoretical Physics, in Waterloo, Canada. He is the author of The Life of The Cosmos and Three Roads to Quantum Gravity. Lee Smolin's Edge Bio Page

LEONARD SUSSKIND, the discoverer of string theory, is the Felix Bloch Professor in theoretical physics at Stanford University. His contributions to physics include the discovery of string theory, the string theory of black hole entropy, the principle of "black hole complementarity," the holographic principle, the matrix description of M-theory, the introduction of holographic entropy bounds in cosmology, the idea of an anthropic string theory "landscape." Leonard Susskind's Edge Bio Page




As we spent more time climbing to the cliffs and hanging out on and around them, they rewarded us more and more. They taught us this: most of the amazingness of the Clock we can borrow from the amazingness of the mountain. The more we highlight and blend in with the most spectacular features of the mountain, the more memorable a Clock visit will be for the time pilgrims. It's a Mountain Clock....

Only the Salon Knows the Answer
But who asks the questions? Even scientists of the Third Culture look for natural laws.

By Jordan Mejias
New York, 19 January 2004

The most recent edition of this parlor game, partly earnest but also beset with irony and serious jokes, takes the natural law as its theme. What law, Brockman asks the great minds, could be filtered out of their empirical research and would be worthy of carrying their names? If Kepler and Newton could have their laws, why shouldn't J. Craig Venter be worthy of one today? He, with no less ambition than his agent, names five laws, the third of which states, "We have the tools for the first time in the history of humanity to answer virtually any question about biology and our own evolution." [more]

[Original German text]



What's new at Long Now? It's been nearly 10 years since Stewart Brand picked up on Danny Hillis' idea to build a 10,000 year clock. Brand co-founded The Long Now Foundation to help Danny build the clock, wrote a book, The Clock of the Long Now. These were initial steps in beginning to address the bigger issue involved: how to get people to think in a longer term, how to stretch out their sense of time.

In an feature published in Edge #46 in 1998, his opening comments began as follows:

Three years we've been working on building a ten-thousand-year clock, and as of this year, '98, we're building a prototype eight feet tall, probably about the size of two refrigerators back to back, and we've got an invitation to debut it at the World Economic Forum in Davos next January, '99 - perfect place to get world leaders and corporate leaders and so on thinking in ten thousand year terms. Danny's clock is I believe the world's first Year 10,000-compliant computer. There's all this ruckus about—correct ruckus—about the year 2000 problem in computers—that they can't handle 2-digit year dates. Well, 4-digit year dates also become a problem after the year 10000—five-digits in your year dates. In the Long Now Foundation we refer to this year as 01998, and 01999, and 02000 and so on. So the clock is ready for the year 10000 and carries on in a completely accurate time-telling fashion until the year 12000.

The prototype was built and it toured the world. Hillis was one of the first recipients of the $1,000,000 Dan David Prize in 2001 for the category of "present" - Technology. The Long Now Foundation board raised money from software and dot.com entrepreneurs and purchased land on Mt. Washington, Nevada for a permanent home for the clock. This summer, an exploratory expedition was carried out on the mountain to check out locations and "to contemplate deeply whether it is the right place for a 10,000-year Clock." The following is an abridged version of Brand's report to the Long Now board on the summer's activities.


Stewart Brand's Edge Bio Page




"What we've discovered in the last several years is that string theory has an incredible diversity—a tremendous number of solutions—and allows different kinds of environments. A lot of the practitioners of this kind of mathematical theory have been in a state of denial about it. They didn't want to recognize it. They want to believe the universe is an elegant universe—and it's not so elegant. It's different over here. It's that over here. It's a Rube Goldberg machine over here. And this has created a sort of sense of denial about the facts about the theory.



What we've discovered in the last several years is that string theory has an incredible diversity—a tremendous number of solutions—and allows different kinds of environments. A lot of the practitioners of this kind of mathematical theory have been in a state of denial about it. They didn't want to recognize it. They want to believe the universe is an elegant universe—and it's not so elegant. It's different over here. It's that over here. It's a Rube Goldberg machine over here. And this has created a sort of sense of denial about the facts about the theory. The theory is going to win, and physicists who are trying to deny what's going on are going to lose.

LEONARD SUSSKIND, the discoverer of string theory, is the Felix Bloch Professor in theoretical physics at Stanford University. His contributions to physics include the discovery of string theory, the string theory of black hole entropy, the principle of "black hole complementarity," the holographic principle, the matrix description of M-theory, the introduction of holographic entropy bounds in cosmology, the idea of an anthropic string theory "landscape." Leonard Susskind's Edge Bio Page

The Reality Club: Paul Steinhardt, Lee Smolin, Kevin Kelly, Alexander Vilenkin, Lenny Susskind, Steve Giddings, Lee Smolin, Gino Segre, Lenny Susskind, Gerard 't Hooft, Lenny Susskind, Maria Spiropulu

by John Brockman

For some people, the universe is eternal. For me, it's breaking news.

Recently I sat down to talk with Lenny Susskind, the discoverer of string theory. After he left, I realized I had become so caught up in his story-telling that I forgot to ask him "what's new in the universe?" So I sent him an email. Here's his response...


The beginning of the 21st century is a watershed in modern science, a time that will forever change our understanding of the universe. Something is happening which is far more than the discovery of new facts or new equations. This is one of those rare moments when our entire outlook, our framework for thinking, and the whole epistemology of physics and cosmology are suddenly undergoing real upheaval. The narrow 20th-century view of a unique universe, about ten billion years old and ten billion light years across with a unique set of physical laws, is giving way to something far bigger and pregnant with new possibilities. 

Gradually physicists and cosmologists are coming to see our ten billion light years as an infinitesimal pocket of a stupendous megaverse. At the same time theoretical physicists are proposing theories which demote our ordinary laws of nature to a tiny corner of a gigantic landscape of mathematical possibilities.

This landscape of possibilities is a mathematical space representing all of the possible environments that theory allows. Each possible environment has its own laws of physics, elementary particles and constants of nature. Some environments are similar to our own corner of the landscape but slightly different. They may have electrons, quarks and all the usual particles, but gravity might be a billion times stronger. Others have gravity like ours but electrons that are heavier than atomic nuclei. Others may resemble our world except for a violent repulsive force (called the cosmological constant) that tears apart atoms, molecules and even galaxies. Not even the dimensionality of space is sacred. Regions of the landscape describe worlds of 5,6…11 dimensions. The old 20th century question, "What can you find in the universe?" is giving way to "What can you not find?"

The diversity of the landscape is paralleled by a corresponding diversity in ordinary space. Our best theory of cosmology called inflationary cosmology is leading us, sometimes unwillingly, to a concept of a megaverse, filled with what Alan Guth, the father of inflation, calls "pocket universes." Some pockets are small and never get big. Others are big like ours but totally empty. And each lies in its own little valley of the landscape.

Man’s place in the universe is also being reexamined and challenged. A megaverse that diverse is unlikely to be able to support intelligent life in any but a tiny fraction of its expanse. Many of the questions that we are used to asking such as 'Why is a certain constant of nature one number instead of another?' will have very different answers than what physicists had hoped for. No unique value will be picked out by mathematical consistency, because the landscape permits an enormous variety of possible values. Instead the answer will be "Somewhere in the megaverse the constant is this number, and somewhere else it is that. And we live in one tiny pocket where the value of the constant is consistent with our kind of life. That’s it! There is no other answer to that question."

The kind of answer that this or that is true because if it were not true there would be nobody to ask the question is called the anthropic principle. Most physicists hate the anthropic principle. It is said to represent surrender, a giving up of the noble quest for answers. But because of unprecedented new developments in physics, astronomy and cosmology these same physicists are being forced to reevaluate their prejudices about anthropic reasoning. There are four principal developments driving this sea change. Two come from theoretical physics, and two are experimental or observational.

On the theoretical side, an outgrowth of inflationary theory called eternal inflation is demanding that the world be a megaverse full of pocket universes that have bubbled up out of inflating space like bubbles in an uncorked bottle of Champagne. At the same time string theory, our best hope for a unified theory, is producing a landscape of enormous proportions. The best estimates of theorists are that 10500 distinct kinds of environments are possible.

Very recent astronomical discoveries exactly parallel the theoretical advances. The newest astronomical data about the size and shape of the universe convincingly confirm that inflation is the right theory of the early universe. There is very little doubt that our universe is embedded in a vastly bigger megaverse.

But the biggest news is that in our pocket the notorious cosmological constant is not quite zero, as it was thought to be. This is a cataclysm and the only way that we know how to make any sense of it is through the reviled and despised anthropic principle.

I don’t know what strange and unimaginable twists our view of the universe will undergo while exploring the vastness of the landscape. But I would bet that at the turn of the 22nd century, philosophers and physicists will look back nostalgically at the present and recall a golden age in which the narrow provincial 20th century concept of the universe gave way to a bigger better megaverse, populating a landscape of mind-boggling proportions.


Below is a wide ranging discussion with Lenny. "To this day," he says, "the only real physics problem that has been solved by string theory is the problem of black holes. It led to some extremely revolutionary and strange ideas."

"Up to now string theory has had nothing to say about cosmology. Nobody has understood the relationship between string theory and the Big Bang, inflation, and other aspects of cosmology. I frequently go to conferences that often have string theorists and cosmologists, and usually the string theory talks consist of apologizing for the fact that they haven't got anything interesting to tell the cosmologists. This is going to change very rapidly now because people have recognized the enormous diversity of the theory."

Read on...





September 15, 1929 – May 24, 2019

Uncharacteristically, I discussed my application to Yale with my father, who asked, "What were you thinking of putting down?" I said, "Whatever would be appropriate for archaeology or linguistics, or both, because those are the things I'm most enthusiastic about. I'm also interested in natural history and exploration."

He said, "You'll starve!"

After all, this was 1944 and his experiences with the Depression were still quite fresh in his mind; we were still living in genteel poverty. He could have quit his job as the vault custodian in a bank and taken a position during the war that would have utilized his talents — his skill in mathematics, for example — but he didn't want to take the risk of changing jobs. He felt that after the war he would regret it, so he stayed where he was. This meant that we really didn't have any spare money at all.

I asked him, "What would you suggest?"

He mentioned engineering, to which I replied, "I'd rather starve. If I designed anything it would fall apart." And sure enough when I took an aptitude test a year later I was advised to take up nearly anything but engineering.

Then my father suggested, "Why don't we compromise — on physics?"

Edge is pleased to bring you a conversation (and video) with Murray Gell-Mann conducted in SantaFe over the Christmas holiday in 2003 — in which he conveyed "something about his life and his attitude toward the world and toward physics."

— JB

MURRAY GELL-MANN (September 15, 1929 – May 24, 2019) was a theoretical physicist and, until his death, Robert Andrews Millikan Professor Emeritus of Theoretical Physics at the California Institute of Technology; winner of the 1969 Nobel Prize in physics; a cofounder of the Santa Fe Institute, where he is a Distinguished Fellow; a former director of the J.D. and C.T. MacArthur Foundation; one of the Global Five Hundred honored by the U.N. Environment Program; a former Citizen Regent of the Smithsonian Institution; a former member of the President's Committee of Advisors on Science and Technology; and the author of The Quark and the Jaguar: Adventures in the Simple and the Complex.

Murray Gell-Mann's Edge Bio Page




"I'm interested in bending the edges of the spectrum to make the abstract and the concrete hit one another more directly."




Peter Galison, Professor of the History of Science and of Physics at Harvard, asks how Poincaré and Einstein "could have radically reformulated our ideas of time and space by looking at the way that philosophically abstract concerns, physics concerns, and ... technological problems of keeping trains from bashing into each other and coordinating mapmaking across the empires might fit into a single story."

Regarding Einstein’s and Poincaré’s account of simultaneity, he wonders: "Is it really physics, or fundamentally technology, or does it come down to philosophy?" He calls it "an extraordinary moment when philosophy, physics and technology cross, precisely because of the intersection of three very powerful streams of action and reasoning at the turn of the century."

This moment resonates with many recent discussions on Edge, and to what Galison terms "the collection of sciences that have grown up around computation. Here, ideas about the mind, about how computers function, and about science, codes, and mathematical physics all come together. Von Neumann thinks about the mind and its organs (memory, input-output, processing) as a way of designing a programmed computer. The programmed computer then becomes a model for the mind. The ideas of information, which are encoded into the development of computation, also become ways to understand language and communication more generally, and again feed back into devices. Information, entropy, and computation become metaphors for us at a much broader level."

Convergences such as Einstein’s and Poincaré’s account of simultaneity or new the sciences of computation are "opalescent moments" that "point to science in times and places where we’re starting to think with and through machines at radically different scales—Where we are flipping back and forth between abstraction and concreteness so intensively that they illuminate each other in fundamentally novel ways, in ways not captured by models of simple evaporation or condensation. When we see such opalescence, we should dig into them, and deeply, for they are transformative moments of our cultures."


PETER GALISON is the Mallinckrodt Professor of the History of Science and of Physics at Harvard University and the author of How Experiments End; Image and Logic; and Einstein's Clocks and Poincaré's Maps: Empires of Time.

Two collections of interpretive essays have been published about Image and Logic, which also won the Pfizer Prize for Best Book in the History of Science. For his work on these books he was awarded a MacArthur Fellowship in 1997, and a Max Planck Prize by the Max Planck Gesellschaft and Humboldt Stiftung in 1999.

Peter Galison's Edge Bio page

Peter Galison presents his new book: Einstein's Clocks and Poincaré's Maps: Empires of Time:

True time would never be revealed by mere clocks—of this Newton was sure. Even a master clockmaker's finest work would offer only pale reflections of the absolute time that belonged not to our human world, but to the "sensorium of God." Tides, planets, moons—everything changed, Newton believed, against the universal background of a single, constantly flowing river of time. In Einstein's electro-technical world, there was no place for such a "universally audible tick-tock" that we can call time, no way to define time meaningfully except in reference to a definite system of linked clocks...Two events simultaneous for a clock-observer at rest are not simultaneous for one in motion. With that shock, the foundation of Newtonian physics cracked; Einstein knew it. Late in life, he interrupted his autobiographical notes to apostrophize Sir Isaac as if the intervening centuries had vanished; reflecting on the absolutes of space and time that his theory of relativity had shattered, Einstein wrote: "Newton, forgive me; you found the only way which, in your age, was just about possible for a man of highest thought and creative power."...At the heart of this radical upheaval in time lay an extraordinary yet easily stated idea that has remained dead-center in physics, philosophy, and technology ever since: To talk about simultaneity, you have to synchronize clocks with a flash from one clock to another, adjusting for the time that the flash takes to arrive. What could be simpler? Yet with this definition of time, the last piece of the relativity puzzle fell into place, changing physics forever.


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