Friday, November 30, 2018

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Wednesday, November 28, 2018

What A Life!

Many readers here will have enjoyed various videos done by Stanford mathematician Tadashi Tokieda for the Numberphile site or elsewhere, but probably, like me, didn’t know much about his background. Erica Klarreich has now given us a wonderful profile of the man behind the mellifluous voice and sparkling “toy” videos (and his "unusual path into mathematics"). I encourage everyone to read one of the most fascinating brief portraits I’ve ever encountered:

Here’s a summary of some of the high points, but do read the entire interview to fill in all the details:

1)  He doesn’t find puzzles, games, and the like particularly interesting, because they are man-made with artificial man-made rules and set-ups. The “toys” he finds intriguing are simply a part of nature or life that become interesting when closely examined.

2)  When young he thought he would become a painter/artist — something he was very good at. And even now acknowledges that “In some sense, drawing and pictures are still what I care about most.”

3)  Later, living in Japan, he had “a real epiphany” about language, eventually leading to a decision to learn various languages and become a philologist.

4)  By chance, needing something to read on a train trip, he picked up a biography of Russian physicist Lev Landau. It opened his eyes to science, math, and specifically calculus, which he then decided he must learn.

5)  Landau suggested learning math, not with classes or lectures, but by finding “a book with the largest number of solved exercises and go through them all.” The book Tokieda found was in Russian which he didn’t know, but as a philologist was willing to learn.

6)  Eventually (within months actually) he says he found he “was fairly good at this kind of silly manipulative exercise,” and then proceeded to enroll at Oxford in a two-year undergraduate program in mathematics. He didn’t know English, by the way, but what the heck, just another language to quickly learn!

7)  Soon he realized mathematics was what he wanted to do for a living, and was off to Princeton for a Ph.D. program.
…And, as they say, the rest is history. ;)
(He's currently a professor of mathematics at Stanford.)

I’ll stop here, but the last several paragraphs of the Quanta piece are also great reading (Tokieda’s take on math and his own videos... including making children happy), so be sure to read the whole piece.

…If I had read of Tokieda’s life, as Klarreich reports it here, in a novel, I would’ve thought, ‘what a pile of non-believable fiction this is… there could never be such a character in real life.’ …And yet apparently there is!

One last note: Quanta is recently out with two compendiums of their many superb articles on math, and on the sciences, particularly physics... this Klarreich profile is one more of many they've now done on individual scientists/mathematicians. I suspect that somewhere in the future there may be a collection of these profiles available as well.
[Addendum: I've now noticed that several of these profiles are already included in the math volume, The Prime Number Conspiracy.]

Sunday, November 25, 2018

A Li'l Physics, and more, With Eric Weinstein

Polymath (and member of the so-called “Intellectual Dark Web”) Eric Weinstein is a bit of an odd-duck — I often agree with his take on various matters and usually find him interesting (whether in agreement or not), but also occasionally find him infuriating. Anyway, he recently did a long (almost 4 hr.!) stint with Joe Rogan where he attempts at one point (beginning ~41:20 mark) to give Joe and listening audience a primer on fundamental physics that lasts about an hour:

A couple other shorter segments I enjoyed:

A bit of discussion of the ‘craft’ of comedy versus that of physics (starting ~1:36:10):

Followed by a short lesson from Eric on harmonica music (starting ~1:40:20):

There are many other interesting bits in the whole podcast (which touches on a LOT of diverse topics), though I thought the first half more interesting overall than the 2nd half.

Eric, an economist by trade and mathematical physicist by academic degree (who, in his spare time, is working on his own 'theory of everything'), has appeared in many other podcasts/videos as well.

[ the way, he should NOT be confused with "Eric Weisstein" who runs "Wolfram MathWorld" -- 2 completely different people.]

Friday, November 23, 2018

Chi-i-i-i-i-i-ll Friday *

[*  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Monday, November 19, 2018

Book Mentions....

First off, John Golden recently tweeted out this video on math and analogies from Kalid  Azad over at “Better Explained,” that I thought worth passing along (once again it touches on the interplay of math and language as I was musing about in my recent post, and as Jim Propp also broaches in his latest offering:

(via: )
Moving on, it’s been another great year in mathy popular books, with a mini-flurry of volumes showing up in the final three months of the year.  Of the books I’ve already read, I recently tweeted out my 5 faves, one of which I’ll soon cite as my 2018 ’book-of-the-year,’ but for now will hold ya all in suspense ;)

When Einstein Met Gödel -- Jim Holt
Math With Bad Drawings -- Ben Orlin
Exact Thinking In Demented Times -- Karl Sigmund
Closing the Gap -- Vicky Neale
Hello World -- Hannah Fry

Meanwhile, Thomas Lin (as editor) is newly-out with “The Prime Number Conspiracy,” a collection of pieces from that outstanding stable of writers at Quanta Magazine.  I haven’t seen it yet, but no doubt it would easily break into my select group above if I had.

And since citing the above, Mircea Pitici’s latest “Best Writing On Mathematics” for 2018 has appeared; delighted to see it materialize in my mailbox... knowing that Mircea has had some topsy-turvy changes in his life this year, I wasn’t sure he’d have the time/inclination to do another edition — but he has and it contains his typical variety of diverse selections (something for everyone). I received the volume just a couple of days after my Nov. 11 posting and so was heartened to see several picks in it dealing with creativity, paradox/puzzles, and beauty in math, as well as Wigner’s “unreasonable effectiveness” notion (all things I’d been thinking about lately).

From Brian Kernighan (and Princeton University Press) comes Millions, Billions, Zillions: Defending Yourself in a World of Too Many Numbers a small, stocking-stuffer-sized book aimed at bestowing basic numeracy to readers.

And am currently reading/enjoying “The Model Thinker” by Scott Page (from Basic Books) — a book for which I'd seen no prior buzz or publicity before its arrival. It’s another in the string of volumes covering big data, modeling, algorithms, probabilities and the like, for a mass audience, but with more textbook-like intros to an array of data/statistical subjects than most previous volumes have offered. Lots of interesting topics. If you’ve been looking for a ‘meatier,’ more mathematical, take in this genre this may be it.

Enough said for now… in another week or two I’ll have my final book wrap-up of the year (including a few additional book mentions) for anyone with math bibliophile friends-or-family on their Holiday list… or, for themselves!

Friday, November 16, 2018

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Sunday, November 11, 2018

Sunday Night Ramble…

"If my mental processes are determined wholly by the motion of atoms in my brain, I have no reason to believe that my beliefs are true... and hence I have no reason for supposing my brain to be composed of atoms."--- J.B.S. Haldane, "Possible Worlds" (1927)
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.— Eugene V. Wigner
(longish ramble ahead....)
Last Friday was one of those oddly serendipitous days in some ways — and that’s despite the fact that I was stewing over missing Ben Orlin’s night-before presentation at our local University. ARRRRRRRGH!! — had planned for weeks to attend, but for a whole series of reasons didn’t make it. Luckily, someone had posted his hour-talk from a previous stop (which I assume was the same as here), so I went online later and viewed that.

A couple days prior, a mathematician/blogger had sent along something to read for any comments, and parts of it reminded me of a favorite quote I’ve used here previously from provocateur David Berlinski (for those who’ve seen me employ it multiple previous times I beg forgiveness, and indeed apologies for all the long quotes coming below):
"Like any other mathematician, Euclid took a good deal for granted that he never noticed.  In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means. Euclid expressed himself in Greek; I am writing in English. Neither Euclid's Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn. These examples suggest what is at work in any attempt to describe once and for all the beliefs 'on which all men base their proofs.' It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact."-- D. Berlinski (from "The King of Infinite Space")
The interplay of language, meaning, abstraction, perception… and, mathematics/science is an ongoing interest of mine. Another quote I’ve used elsewhere is from venerable Bertrand Russell toward the end of a frustrating career (1957) trying to formalize all of mathematics:
"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal."  

Again, I think this reflects on the complex interplay of language and human thought — hmmm, all of mathematics as tautologies?… maybe all of meaning is tautological, just substituting one set of human scratchings or sounds for another… is all of knowledge just one gigantic Thesaurus? ;)) No, I don’t believe that, but a lot of “knowledge” does seem illusory or mirage-like and certainly changing. Often meanings, metaphors, analogies and the like simply refer back on themselves within an enclosed bubble.

Later on Friday afternoon I stumbled across this new essay at the Scientific American site, “Proofs and Guarantees” (actually reprinted from “The Mathematical Intelligencer”), which seemed to hit some of the same notes, questioning assumptions. It ends accepting the “fallibility of the initial axioms or other first principles” of math, and thus acknowledging an evolutionary nature to mathematics, as something that is not necessarily fixed over time (just as science is not static and self-corrects over time):

In arguments between math Platonists and non-Platonists it’s often contended that even (philosophical) non-Platonists ARE Platonists-at-heart when it comes to their livelihood… i.e., a pure mathematician must presume math exists out there to be discovered, in order to carry on their daily work. That may overstate the case, but on-the-other-hand one of my favorite volumes is by non-Platonist, retired mathematician William Byers, “How Mathematicians Think.” The subheading to the book title is: “Using Ambiguity, Contradiction, and Paradox to Create Mathematics” and that succinctly sums up what he argues in the volume, that mathematics is “created” out of the very things that most people presume run counter to it.
Interestingly, a second major influence in Byers’ life (besides mathematics) is Zen Buddhism, known for its mystical focus on contradictions. Here is one passage where he touches upon it:
The second strand in my life was and is a strenuous practice of Zen Buddhism. Zen helped me confront aspects of my life that went beyond the logical and the mathematical. Zen has the reputation for being antilogical, but that is not my experience. My experience is that Zen is not confined to logic; it does not see logic as having the final word. Zen demonstrates that there is a way to work with situations of conflict, situations that are problematic from a normal, rational point of view. The rational, for Zen, is just another point of view. Paradox, in Zen, is used constructively as a way to direct the mind to subverbal levels out of which acts of creativity arise.
Later in the volume he writes:
…every human being lives in a bubble. This bubble contains all their perceptions and cognitions. What exists outside the bubble is not knowable. Radical constructivists 'do not make claims about what exists in itself, that is, without an observer or experiencer.' This is a point that I also made earlier when I claimed that there exists no mathematical knowledge that is completely objective. Mathematical knowledge and truth must be considered as a package with both objective and subjective aspects. The belief in ‘objective mathematical knowledge,’ that is, knowledge that is independent of the beings who know it, is itself a belief and therefore nonobjective. There is no knowledge that is independent of knowing. There is no absolute, objective truth.”
And finally, here’s Byers, at length, in another book, “The Blind Spot,” hitting the same theme:
"It is certainly conceivable that the clarity we perceive in the world is something we bring to the world, not something that is there independent of us. The clarity of the natural world is a metaphysical belief that we unconsciously impose on the situation. We consider it to be obvious that the natural world is something exterior of us and independent of our thoughts and sense impressions; we believe in a mind-independent reality. Paradoxically, we do not recognize that the belief in a mind-independent reality is itself mind-dependent. Logically, we cannot work our way free of the bubble we live in, which consists of all of our sense impression and thoughts. The pristine world of clarity, the natural world independent of the observer, is merely a hypothesis that cannot, in principle, ever be verified. To say that the natural world is ambiguous is to highlight this assumption. It is to emphasize that the feeling that there is a natural world 'out there' that is the same for all people at all times, is an assumption that is not self-evident. This is not to embrace a kind of solipsism and to deny the reality of the world. It is to emphasize that the natural world is intimately intertwined with the world of the mind. In consequence, the natural world is a flow just like the inner world. By stabilizing the inner world through language, logic, mathematics, and science, we simultaneously stabilize the outer world. The result of all this is the recognition that the clarity we assume to be a basic feature of the natural world merely masks a deeper ambiguity. One of the functions of mathematics and science is precisely to deny this ambiguity. This is really the motivation behind the science of certainty." 
Anyway, finally coming back around to Dr. Orlin’s talk, which is all about the relationship between mathematics and science, Ben concludes that they share a symbiotic relationship — two quite DIFFERENT activities feeding off one another (as opposed to the more common conception of math being foundational to science). I couldn’t help but think that perhaps that viewpoint might be broadened out to describe the relationship/interplay between language, thought, and math… entities that are separate but very much feed off one another (though many mathematical aspects of language may not even yet be understood/appreciated).
Give Ben's entertaining, thoughtful talk a watch if you’ve not seen it:

David Chalmers famously talks about consciousness as the “hard problem” of philosophy and psychology, left untouched by resolving the other “easy” or soft problems. How do subjectively-felt  experiences arise out of the conglomeration of matter that is our physical brain? Or, in Thomas Nagel’s famous take, what does it feel like to be a bat?
In recent years there has been a lot discussion and competing theories over “consciousness.” Certainly some of Doug Hofstadter’s past writings touch on these matters, as does Joselle sometimes over at her Mathrising blog… and many many more [including, if you haven't seen it, John Horgan's latest 'free' volume on consciousness, where he speaks to several major thinkers on the topic, HERE].
But then another favorite quote of mine (from Emerson Pugh) is, “If the human brain were so simple that we could understand it, we would be so simple that we couldn’t,” implying that  we will never be able to turn the brain on itself to reveal its own deepest secrets. That’s a sort of “Mysterian” viewpoint (and I’m in the mysterian camp with Colin McGinn, Martin Gardner, Roger Penrose, and others, but plenty of folks oppose it, believing the brain can be fully understood, even duplicated).
Anyway, similarly, I think mathematics has a ‘hard’ problem (philosophically-speaking). It is the one made famous by physicist Eugene Wigner (quoted above). How does one account for the exquisite fit of abstract mathematics with the physical world as we interpret it? How indeed! Max Tegmark’s tempting answer is that fundamentally, mathematics is all there is… mathematics IS the core foundational structure/component of the Universe, or of reality; an intriguing notion, but difficult to flesh out, and not one I see a lot of others gravitating toward. Even if you're a full-out Platonist (like Gödel) and believe mathematics exists in the world, independent of humans, the question remains where did it come from and how are we humans able to access it so successfully? Or has some alien civilization, a million years more advanced than us, recognized mathematics as a truly tautological illusion, and moved on to something else more fundamental by now?
Finally, there's been a lot of emphasis in recent years on "beauty" in mathematics, but now even that view is being drawn into question, with a lot of buzz in particular around Sabine Hossenfelder's recent volume, "Lost In Math" (on modern physics), where she argues the myopic focus on beauty simply leads us astray. Is nothing sacred anymore!... first taking away certainty and truth, and now even our wistful love of beauty. ;)
More and more, I find pieces I'm reading connect somehow back to these interests in language, cognition, consciousness, recursion... maybe some day our grandchildren's grandchildren... or, Ben Orlin... will actually make sense of it all! 

Well, I've let the atoms in my brain (assuming they exist) bounce around a bit too much tonight... time to put them to bed, and perhaps go flying -- something I do quite splendidly in my dreams, but, frustratingly, can't seem to do upon awakening (...though, as Chuang Tzu might ponder, perhaps my flying is real and it's this blog-writing that is just a dream....)

==> Enough of my stream-of-consciousness, IF you want to hear from a real mathematician, I listened to Numberphile's maiden podcast earlier today with Grant Sanderson (of 3Blue1Brown), and it's quite good:

Friday, November 9, 2018

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]

Sunday, November 4, 2018

More Collaboration and Less Collusion!


January marks the 10-year anniversary of the collaborative “Polymath Project” begun by Fields Medalist Tim Gowers. Let’s celebrate early!
Here is the main page for the Project:
…and here is the Tim Gowers posting that originally proposed such a crowdsourcing effort be attempted:

I find it interesting how little widespread digital collaboration has made a dent in other sciences, but been productive in mathematics — one suspects there’s some sort of whole sociological dissertation that could be written here! (although perhaps the pen-and-paper, and computer-assisted nature of mathematics makes it more suited to such endeavors than other more applied sciences)
Anyway, Vicky Neale in her fabulous volume “Closing the Gap,” writes a lot about Polymath’s contributions to the ongoing work on the Twin Primes conjecture. She includes many wonderful quotes from Gowers along the way. Here he is when first proposing a digital collaboration on significant unsolved math problems:
“The ideal solution would be a solution of the problem with no single individual having to think all that hard. The hard thought would be done by a sort of super-mathematician whose brain is distributed amongst bits of the brais of lots of interlinked people. So try to resist the temptation to go away and think about something and come back with carefully polished thoughts: just give quick reactions to what you read and hope that the conversation will develop in good directions.”
Neale then writes, “Gowers was very keen to encourage participants to share their immediate thoughts, rather than working on ideas independently in private, with an emphasis on expressing their immediate thoughts as clearly as possible so that others could build on them.” 
And then she quotes Gowers again:
“When you do research, you are more likely to succeed if you try out lots of stupid ideas. Similarly stupid comments are welcome here. (In the sense to which I am using ’stupid,’ it means something completely different from ‘unintelligent’. It just means not fully thought through.)”
Later on he writes, “…even ‘frivolous’ observations can (and should) be posted on this thread, if there is even a small chance that some other participant may be able to find it helpful for solving the problem.
“Similarly, ‘failed’ attempts at a solution are also worth posting: another participant may be able to salvage the argument, or else the failure can be used as a data point to eliminate some approaches to the problem, and to isolate more promising ones.”

Gowers also foresaw the problem of assigning 'credit’ for such widespread collaborations, and so addressed that ahead of time as well:
“Suppose the experiment actually results in something publishable. Even if only a very small number of people contribute the lion’s share of the ideas, the paper will still be submitted under a collective pseudonym with a link to the entire online discussion.”
Later in the book Neale references how Terry Tao deferred to recent-grad James Maynard to insure he would get the most credit for advances to the Twin Prime conjecture that both had made independently — Tao, absolutely established in his field with no need for acclaim, knowing that credit to Maynard might boost his career greatly — I don’t know that this type of gesture/collegiality would occur very often in any field outside of mathematics!
At this point the Polymath Project has met much success (with fits and starts) and is working on its 16th project. Congratulations to all participants involved, and to Tim Gowers' foresight. As the post-title above suggests, here's to more collaboration in life, and less collusion! ;)

For entertainment purposes only, I’ll end with a completely different take on collaboration in math, in the form of a wonderful short piece of film fiction (on the Goldbach conjecture) that I haven’t re-posted for awhile — for any who have never seen it, “The Calculus of Love”:

Friday, November 2, 2018

Chi-i-i-i-i-i-ll Friday *

[ *  "Chill Friday" is Math-Frolic's meditative musical diversion, heading into each weekend]