Rebels With a Cause

I’m long fascinated by the varying lives and minds of mathematicians. Thusly, one of my favorite sections of Thomas Lin’s (editor) fabulous volume, “The Prime Number Conspiracy,” is Part 4 “How Do the Best Mathematical Minds Work?,” offering fascinating profiles of several individual prominent mathematicians. One of them, Freeman Dyson held a well-known disdain for PhD. degrees, and I’ll just pass along these lines (that just maybe will inspire some):

“…I’m very proud of not having a Ph.D. I think the Ph.D. system is an abomination… It’s good for a very small number of people who are going to spend their lives being professors. But it has become now a kind of union card that you have to have in order to have a job, whether it’s being a professor or other things, and it’s quite inappropriate for that… The Ph.D takes far too long and discourages women from becoming scientists, which I consider a great tragedy. So I have opposed it all my life without any success at all…

…So I’m very proud that I don’t have a Ph.D. and I raised six children and none of them has a Ph.D., so that’s my contribution.”

[Dyson is still going strong at 95, contemplating unsolved math problems, and 5 of his 6 children, by the way, are women.]

...and then, on a separate note, a bit ago I came across this wonderful Paul Lockhart quote (from "A Mathematician's Lament") in a piece by Sunil Singh:

Of Birds and Frogs

A well-known passage from Freeman Dyson today:

"Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking... Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time. I happen to be a frog, but many of my best friends are birds."

"mathematics is permanent"

This  week's Sunday reflection from Freeman Dyson:

"...it’s the beauty of mathematics, as opposed to physics, that it’s forever. I published my selected papers recently in one volume, and I found out that when you publish your selected papers most of the physics is ephemeral, that you don’t want to publish stuff that was written 10 or 20 years earlier, but the mathematics is permanent. So essentially everything I’ve ever published in mathematics is there, whereas only about a quarter of what I published on physics was worth preserving."

Expectations and Discoveries... and a Passing

This Sunday's reflection from Freeman Dyson (in "Dreams of Earth and Sky"):

"A typical proton-proton collision in the LHC will produce a large spray of secondary particles, and the collisions are occurring at a rate of millions per second. The machine must automatically discard the vast majority of the collisions, so that the small minority that might be scientifically important can be precisely recorded and analyzed, The criteria for discarding events must be written into the software program that controls the handling of information. The software program tells the detectors which collisions to ignore. There is a danger that the LHC can discover only things that the programmers of the software expected. The most important discoveries may be things that nobody expected. The most important discoveries may be missed."

                                                                      __  __  __

....on a side note, Oliver Sacks died this morning; I don't consider that a sad note -- his life and contributions were so glorious, and his approach to death so thoughtful and intrepid, that death is more like a punctuation mark at the end of another inspiring sentence from his rich, productive life and writings.
There will no doubt be profuse tributes/obits in coming days. This from the NY Times:

http://tinyurl.com/pxg3ak7

 

Listening To Feynman...

Tomorrow is Richard Feynman's birthday... in his honor, for this week's 'Sunday Reflection,' a blurb from Freeman Dyson's "The Scientist as Rebel":

"Before I met Feynman, I had published a number of mathematical papers, full of clever tricks but totally lacking in importance. When I met Feynman, I knew at once that I had entered another world. He was not interested in publishing pretty papers. He was struggling, more intensely than I had ever seen anyone struggle, to understand the workings of nature by rebuilding physics from the bottom up. I was lucky to meet him near the end of his eight-year struggle. The new physics that he had imagined as a student of John Wheeler seven years earlier was finally coalescing into a coherent vision of nature, the vision that he called 'the space-time approach.' The vision was in 1947 still unfinished, full of loose ends and inconsistencies, but I saw at once that it had to be right. I seized every opportunity to listen to Feynman talk, to learn to swim in the deluge of his ideas. He loved to talk, and he welcomed me as a listener. So we became friends for life.
"For a year I watched as Feynman perfected his way of describing nature with pictures and diagrams, until he had tied down the loose ends and removed the inconsistencies. Then he began to calculate the numbers, using his diagrams as a guide. With astonishing speed he was able to calculate physical quantities that could be compared directly with experiment. The experiments agreed with his numbers."

...and from Feynman himself:

"We have found it of paramount importance that in order to progress we must recognize our ignorance and leave room for doubt. Scientific knowledge is a body of statements of varying degrees of certainty — some most unsure, some nearly sure, but none absolutely certain."....

"It is in the admission of ignorance and the admission of uncertainty that there is a hope for the continuous motion of human beings in some direction that doesn't get confined, permanently blocked, as it has so many times before in various periods in the history of man."....

"I can live with doubt, and uncertainty, and not knowing. I think it's much more interesting to live not knowing than to have answers which might be wrong. I have approximate answers, and possible beliefs, and different degrees of certainty about different things, but I'm not absolutely sure of anything. There are many things I don't know anything about, such as whether it means anything to ask 'Why are we here?' I might think about it a little bit, and if I can't figure it out then I go on to something else. But I don't have to know an answer. I don't feel frightened by not knowing things, by being lost in the mysterious universe without having any purpose — which is the way it really is, as far as I can tell. Possibly. It doesn't frighten me."

Couple of Physicists' Views

Sunday thoughts on science...:

"Science in its everyday practice is much closer to art than to philosophy. When I look at Gödel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as a Chartres cathedral. The proof destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas."
-- Freeman Dyson in "Nature's Imagination"

...and from another physicist, a related view:

"…to sum up, science is not about data; it's not about the empirical content, about our vision of the world. It's about overcoming our own ideas and continually going beyond common sense. Science is a continual challenging of common sense, and the core of science is not certainty, it's continual uncertainty -- I would even say, the joy of being aware that in everything we think, there are probably still an enormous amount of prejudices and mistakes, and trying to learn to look a little bit beyond, knowing that there's always a larger point of view to be expected in the future."
-- physicist Carlo Rovelli in "The Universe" edited by John Brockman

or, just perhaps, Richard Feynman summed it up succinctly ;-):

"Physics is like sex: Sure, it may give some practical results but that’s not why we do it.”


[…If you have a favorite math-related passage that might make a nice Sunday morning reflection here let me know (SheckyR@gmail.com). If I use one submitted by a reader, I'll cite the contributor.]

Prime Synchrony

Sunday reflection today, courtesy of Freeman Dyson and Hugh Montgomery (this reflection actually ties nicely into last week's Sunday offering as well)...:

"[Freeman] Dyson helped bring together the continuous and the discrete understandings of subatomic behavior. Similarly, by fusing his love of number theory with his expertise in creating the mathematical tools of physics, he would make the initial observation that would reinforce the connections between the discrete world of the integers and the continuous world of analysis, and thus galvanize research on the Riemann hypothesis.
"As Dyson recalls it, he and [Hugh] Montgomery [number theorist] had crossed paths from time to time at the [Princeton] Institute [for Advanced Study] nursery when picking up and dropping off their children. Nevertheless, they had not been formally introduced. In spite of Dyson's fame, Montgomery hadn't seen any purpose in meeting him. 'What will we talk about?' is what Montgomery purportedly said when brought to tea. Nevertheless, Montgomery relented and upon being introduced, the amiable physicist asked the young number theorist about his work. Montgomery began to explain his recent results on the pair correlation, and Dyson stopped him short -- 'Did you get this?' he asked, writing down a particular mathematical formula. Montgomery almost fell over in surprise: Dyson had written down the sinc-infused pair correlation function.
"Dyson had the right answer, but until that moment he had associated this formula with understanding a phenomenon that seemed completely unrelated to the primes and the Riemann hypothesis. In a flash he had drawn the analogy between the sinc-described structured repulsion of the zeta zeros and a similar tension seemingly exhibited by the different levels of energy displayed by atomic nuclei. Whereas Montgomery had traveled a number theorist's road to a 'prime picture' of the pair correlation, Dyson had arrived at this formula through the study of these energy levels in the mathematics of matrices. This connection is the source of most of the current excitement surrounding the Riemann hypothesis..."

-- from "Stalking the Riemann Hypothesis" by Dan Rockmore

Sunday Thoughts…

From Freeman Dyson in  "The Scientist As Rebel":

"Gödel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.
"It is a curious paradox that several of the greatest and most creative spirits in science, after achieving important discoveries by following their unfettered imaginations, were in their later years obsessed with reductionist philosophy and as a result became sterile. Hilbert was a prime example of this paradox. Einstein was another…

"Science in its everyday practice is much closer to art than to philosophy. When I look at Gödel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral… The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Gödel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalization into unexplored territory."

Freethinker Dyson...

via Wikipedia

Fascinating article and interview with mathematician/thinker/puzzlemeister/physicist (and "rebel") Freeman Dyson over at Quanta Magazine:

https://www.simonsfoundation.org/quanta/20140326-a-rebel-without-a-ph-d/

Somehow I never realized that he DOESN'T have a PhD. and at 90-years-old sounds as feisty, insightful, brilliant as ever! I'd quote some lines from the article but it's SO good (especially the interview portion) it's impossible to pick, so go enjoy the whole thing (also includes a 5-minute video).

A couple years back I did a short post on "Dyson numbers" which included a famous anecdote about Freeman here:

http://math-frolic.blogspot.com/2012/10/next-up-dyson-numbers.html

I think some evening I'd like to have Dyson and Raymond Smullyan (94 years-old) over for pizza (maybe throw in 76-yr.-old whippersnapper John Horton Conway, as well) and see if, bouncing ideas off one another, they couldn't prove the Riemann Hypothesis before David Letterman comes on ;-)

[side-note: tomorrow I should have some weekly potpourri picks posted at MathTango.]

A Little Bedtime Reading…

…okay, NOT really!

This is probably the most heavy, dense mathematical piece (on the Riemann Hypothesis, quasicrystals, Freeman Dyson, and more) I've ever linked to, and I don't comprehend it… but, all things Riemannian fascinate me, and even without following the mathematics here I think this very long piece from John Baez communicates the beauty and profundity of methodical, thoughtful, mathematical analysis. Simply put, it is, I think, a beautiful piece of mathematical exposition (not at all unusual for Baez)….

http://golem.ph.utexas.edu/category/2013/06/quasicrystals_and_the_riemann.html#more

It starts off thusly:

"Freeman Dyson is a famous physicist who has also dabbled in number theory quite productively.  If some random dude said the Riemann Hypothesis was connected to quasicrystals, I’d probably dismiss him as a crank.  But when Dyson says this, it’s a lot more interesting.  So I’ve been trying to understand his remarks on this.  And it’s been productive, in that I’ve learned some interesting things, and I now feel closer to seeing why the Riemann Hypothesis is a natural and important conjecture."

 ...On a side-note, I also find it interesting that Freeman Dyson advanced this original quasicrystal notion, which has generated a lot of interest (like so many things Dyson has advanced over the years), at the ripe young age of 85! Who says mathematics must be a young man's game!

Riemann, Dyson, Montgomery, Quasi-crystals, Matrix Theory...

....and, quantum physics.

Below a longish and deep read from the Institute For Advanced Study on the mysterious linkage between the Riemann zeta function and physics (a lot to chew on or just try to comprehend):

https://www.ias.edu/about/publications/ias-letter/articles/2013-spring/primes-random-matrices


Just a couple of passages from the long piece:

 “ 'The Riemann zeta function remains one of the mysteries of modern mathematics. It is a function that we understand a lot about except for the most important question,' says Peter Sarnak, Professor in the School of Mathematics. 'It connects the theory of prime numbers, or encodes deep information about the theory of prime numbers, with the zeros. It controls the prime numbers in a way that nothing else we know does. While understanding prime numbers is an important problem, it is the generalizations of the Riemann zeta function and the objects associated with these that make it more significant.' ”....

"Quasi-crystals were discovered in 1984 and exist in spaces of one, two, or three dimensions. Dyson suggests mathematicians obtain a complete enumeration and classification of all one-dimensional quasi-crystals, the most prevalent type, with the aim of identifying one with a spectrum that corresponds to the Riemann zeta function and one that corresponds to the L-functions that resemble the Riemann zeta function. If it can be proved that a one-dimensional quasi-crystal has properties that identify it with the zeros of the Riemann zeta function, then the Riemann Hypothesis will have been proved."

I haven't had occasion to read the 2nd and 3rd volumes of Matthew Watkin's trilogy on prime numbers, but I can't help but think that the above article may overlap with or relate to some of the notions reached by Watkins who also writes of the 'vibrational' nature of the number system.