Pickover on Platonism…

Some extended discourse via Cliff Pickover today from his volume, "A Passion For Mathematics" (one of my favorite Pickover offerings):

"I think that mathematics is a process of discovery. Mathematicians are like archaeologists. The physicist Roger Penrose felt the same way about fractal geometry. In his book The Emperor's New Mind, he says that fractals (for example, intricate patterns such as the Julia set or the Mandelbrot set) are out there waiting to be found:

'It would seem that the Mandelbrot set is not just part of our minds, but it has a reality of its own… The computer is being used essentially the same way that an experimental physicist uses a piece of experimental apparatus to explore the structure of the physical world. The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there.'

I think we are uncovering truths and ideas independently of the computer or mathematical tools we've invented. Penrose went a step further about fractals in The Emperor's New Mind: 'When one sees a mathematical truth, one's consciousness breaks through into this world of ideas… One may take the view that in such cases the mathematicians have stumbled upon works of God.'

Anthony Tromba, the coauthor of Vector Calculus, said in a July 2003 University of California press release, 'When you discover mathematical structures that you believe correspond to the world around you, you feel you are seeing something mystical, something profound. You are communicating with the universe, seeing beautiful and deep structures and patterns that no one without your training can see. The mathematics  is there, it's leading you, and you are discovering it.'

"Other mathematicians disagree with my philosophy and believe that mathematics is a marvelous invention of the human mind. One reviewer of my book The Zen of Magic Squares used poetry as an analogy when 'objecting' to my philosophy. He wrote,

'Did Shakespeare 'discover' his sonnets? Surely all finite sequences of English words 'exist,' and Shakespeare simply chose a few that he liked. I think most people would find the argument incorrect and hold Shakespeare created his sonnets. In the same way, mathematicians create their concepts, theorems, and proofs. Just as not all grammatical sentences are theorems. But theorems are human creations no less than sonnets.'

Similarly, the molecular neurobiologist Jean-Pierre Changeux believes that mathematics is invented: 'For me [mathematical axioms] are expressions of cognitive facilities, which themselves are a function of certain facilities connected with human language.'"


[…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.]

Introduction to Incompleteness

"Incompleteness is one of the most beautiful and profound proofs that I’ve ever seen. If you’re at all interested in mathematics, it’s something that’s worth taking the effort to understand."  -- Mark Chu-Carroll 

Mark Chu-Carroll (of "Good Math, Bad Math") is in the process of re-posting his own splendid discussion/explanation of Gödelian Incompleteness this week. If it's a subject that interests you, or you've always wanted a detailed introduction, his first three four posts (with more to come) are here:

http://www.goodmath.org/blog/2014/08/25/godel-reposts/

http://www.goodmath.org/blog/2014/08/26/godel-numbering/

http://www.goodmath.org/blog/2014/08/27/godel-part-3-arithmetic-and-logic/

[just added] http://www.goodmath.org/blog/2014/08/28/gdel-part-3-meta-logic-with-arithmetic/

Ahhh, Mickey Mantle...

Just for fun today, contemplating the wonderful number "7" with these two appreciations:





(image via SGT141/WikimediaCommons)

Sunday Morning With Paul

"Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don't know of any human activity as demanding of one's imagination, intuition, and ingenuity. But I do it anyway. I do it because I love it and because I can't help it. Once you've been to the jungle, you can never really leave. It haunts your waking dreams….

"The solution to a math problem is not a number; it's an argument, a proof. We're trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying."

-- Paul Lockhart from "Measurement"


[…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.]

Short Film About Prime Numbers

 Kinda coool! (h/t to Gary Davis for pointing to this):



You can read more about it here (with several more links):

http://fadesingh.tumblr.com/post/91837545374/prime-numbers-magic-cicadas-and-mexican-bus-drivers

Epiphanies...

Love this newly-posted (by MAA) video of James Tanton answering the question, "What was the hardest thing you learned when studying math?" Especially timely to me since it ties in beautifully with the last two 'Sunday Reflections' I've posted here:

http://math-frolic.blogspot.com/2014/08/prime-synchrony.html
http://math-frolic.blogspot.com/2014/08/somethings-going-on-here.html

 

And, for more mathy stuff check out this Friday's link collection over at MathTango.

 

And a Few More Puzzles

 If the prior puzzle was a bit too much for you, a few below that are more manageable...

Been reading "Mathematical Curiosities," new from Alfred Posamentier and Ingmar Lehmann. It is, as the subtitle suggests, "a treasure trove of unexpected entertainments" -- especially entertaining if you have a geometry bias.

In the middle of it come 90 "curious problems with curious solutions." Several of these are classics with which you'll be familiar, and others are a little fresher, all interesting. I'll pass along three to whet your appetite (these are paraphrased from the volume):


#1.  I feel like EVERYone should know this first one, so just passing it along for any readers not already familiar with it:
On a certain pond the water lilies double in number every single day. After the 50th day the pond is completely covered. How many days were required for the pond to be half-covered?

#2. Given the following four numbers:

8,932
12,668
85,423
9,165

What percentage of their sum, is their average?

#3. What time is it now if in 2 hours it will be one-half as long 'til noontime as in 1 hour from now?

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1)  49 days

2)  25%  (if you work this out the 'long' way, you may then see there's an easier, more general solution)
3)   9 am.

This Is a Puzzle You've Hated (or Loved) Before

A puzzle re-run today....
Two years ago I ran the below mind-numbing, self-referential puzzle that became one of the most frequent links back to this blog... I think primarily from computer programmers who enjoyed writing code to solve it. Anyway, if you missed it first go-around, here's another chance (answer posted further down):

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Given the following list of 12 statements which of the statements are true?

1.   This is a numbered list of twelve statements.
2.   Exactly 3 of the last 6 statements are true.
3.   Exactly 2 of the even-numbered statements are true.
4.   If statement 5 is true, then statements 6 and 7 are both true.
5.   The 3 preceding statements are all false.
6.   Exactly 4 of the odd-numbered statements are true.
7.   Either statement 2 or 3 is true, but not both.
8.   If statement 7 is true, then 5 and 6 are both true.
9.   Exactly 3 of the first 6 statements are true.
10.  The next two statements are both true.
11.  Exactly 1 of statements 7, 8 and 9 are true.
12.  Exactly 4 of the preceding statements are true.

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answer:  1, 3, 4, 6, 7, 11 are true

Dr. Yanofsky and Category Theory

Delighted to see Dr. Noson Yanofsky getting some further publicity in this piece from FQXi on category theory:

http://fqxi.org/community/articles/display/192

I interviewed Yanofsky last year after reviewing his popular work, "The Outer Limits of Reason," which I regard as the best, most important book I've read in a very long spell. I'll again reiterate that anyone interested in cross-disciplinary math-science-related fields ought devour this volume!

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

Of Love and Hate… sort of

"...journalism has rules about writing stories that don’t really work for math. When journalists are told to 'put a face on the story,' they end up with all face and no story."

'Mathbabe' (Cathy O'Neil) hits another home run, or should I say home rant, today with this piece on math and the Fields Medal... and "the incredible collaborative effort that is modern mathematics":

http://mathbabe.org/2014/08/15/i-love-math-and-i-hate-the-fields-medal/


...and for more mathy links this morning see the weekly MathTango potpourri:
http://mathtango.blogspot.com/2014/08/friday-bonanza.html

An Even Better Day Will Come…

I wrote a couple days back that I didn't plan to cover the Fields Medals here, since I believed they would receive good and widespread coverage elsewhere… little did I realize what an understatement that would be! Because of the first-ever female winner, Maryam Mirzakhani, the reportage has been even beyond what I anticipated, in both the popular press as well as math sites.
 I hope that everyone is right in thinking that this will be a huge boost for women in math -- that Maryam can be a role-model and inspiration to young female math enthusiasts everywhere. I almost fear that the continual, overriding emphasis on her gender plays into a perception that she has achieved some rarefied, super-human feat, no ordinary female can aspire to… but then, I probably worry too much. Still... better will be the day when there is no special hoopla surrounding a woman winning a major math prize… it will just be a common ordinary happenstance! Until then though, indeed, congratulations to Dr. Mirzahkani and her co-recipients, Manjul Bhargava, Artur Ávila, and Martin Hairer… I just wish I could understand anything that they wrote :-((

Anyway, here is a smidgen of the coverage that is out there (if you've been living under a rock and missed it somehow ;-):

Part of the original press release for both Fields and other prize winners:
http://www.mathunion.org/general/prizes/2014

Quanta Magazine's nice profiles of the winners, starting here:
http://www.simonsfoundation.org/quanta/20140812-a-brazilian-wunderkind-who-calms-chaos/

Lots more roundup of the Fields coverage from The Aperiodical:
http://aperiodical.com/2014/08/2014-fields-medals-awarded-coverage-round-up/

Also, Keith Devlin's quick take on the awards for NPR today (Keith and Maryam are both at Stanford):
http://hereandnow.wbur.org/2014/08/13/math-prize-mirzakhani

Meanwhile, on a side-note, the IMU has also created a 'Women in Mathematics' website:
http://www.mathunion.org/wim/
(not clear to me if this has been around for awhile, or was possibly created in anticipation of the first female Fields Medal winner being announced?)

Carpe Diem... no more

"Oh Captain! My Captain!"

Today, in remembrance of Robin Williams, am just re-running material from a post I did one year ago:

Came across this quirky little posting that linked together math, teaching, and one of my favorite Robin Williams' movies, "Dead Poets Society":

http://casualcalculations.wordpress.com/2013/08/21/dead-mathematicians-society/
(the post is probably even more pertinent today with all the debate over math reform, than it was a year ago)

Watch the scene in the above post and then, if you've seen the movie, re-live the ending, that still tugs at me (not specific to math and perhaps only meaningful if you've seen the film):



...and apparently I'm not the only one moved by the above scene; check out the Twitter feeds started last night for "stands on desk" and "standing on desk":
https://twitter.com/search?f=realtime&q=stands%20on%20desk&src=typd

https://twitter.com/search?f=realtime&q=standing%20on%20desk&src=typd

R.I.P.  Mr. Keating. . . .

ADDENDUM:  [There are lots of wonderful tributes to Williams pouring in today, but the best one I've read thus far comes from Russell Brand in The Guardian:  http://tinyurl.com/ltr2qo9
(VERY worth reading; H/T to N. Ghoussoub for pointing me to it)]

Of Medals and Cocktail Party Chatter

Evelyn Lamb ran a timely piece at her Roots of Unity blog today on the Fields Medal, to be awarded to four people this coming Wednesday (I hadn't previously seen the number of recipients listed this year, so I'll assume she's right; it's always 2-4 individuals):

http://blogs.scientificamerican.com/roots-of-unity/2014/08/11/how-to-talk-about-the-fields-medal-at-your-next-cocktail-party/

She touches on the 'age-ist' nature of the Fields, before discussing some of math's other prestigious prizes. Most interesting part to me was learning of the "Chern Medal" for lifelong achievement in math, which I'd not heard of, and which Evelyn calls "The hipster candidate for the 'Nobel Prize of mathematics,'” first awarded in 2010 (and only every four years thereafter). She urges we keep an eye on it now, well before it becomes "cool." :-)

Anyway, the excitement is building for Wednesday's announcement, so stay tuned.
[...since it'll get widespread coverage on the InterTubes, I'll likely tweet about the Fields awards, but not blog about it here, other than to include as part of the Friday potpourri on MathTango]

'Something's Going On Here'

Today's Sunday reflection, a passage about the role of intuition in mathematics (from "How Mathematicians Think" by William Byers):

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"For the mathematician, the idea is everything.  Profound ideas are hard to come by, and when they surface they are milked for every possible consequence that one can squeeze out of them. Those who describe mathematics as an exercise in pure logic are blind to the living core of mathematics -- the mathematical idea -- that one could call the fundamental principle of mathematics. Everything else, logical structure included, is secondary.
"The mathematical idea is an answer to the question, 'What is going on here?' Now the mathematician can sense the presence of an idea even when the idea has not yet emerged. This happens mainly in a research situation, but it can also happen in a learning environment. It occurs when you are looking at a certain mathematical situation and it occurs to you that 'something is going on here.' The data that you are observing are not random, there is some coherence, some pattern, and some reason for the pattern. Something systematic is going on, but at the time you are not aware of what it might be…
"The feeling that 'something is going on here' can even be brought on by a single fact, a single number. A case in point happened in 1978, when my colleague John McKay noticed that 196884 = 196883+1. What, one might ask, is so important about the fact that some specific integer is one larger than its predecessor? The answer is that these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics. The first arises in the context of the mathematical theory of modular forms. The second arises in the context of the irreducible representations of a finite simple group called the Monster. McKay intuitively realized that the relationship between these two constants could not be a coincidence, and his observation started a line of mathematical inquiry that led to a series of conjectures that go by the name 'monstrous moonshine.' The main conjecture in this theory was finally proved by Fields Medalist winner Richard E. Borcherds. Thus the initial observation plus recognition that such an unusual coincidence must have some deep mathematical significance led to the development of a whole area of significant mathematical research….

"…But still it is possible to say 'we do not really understand what is going on.' Understanding what is going on is an ongoing process -- the very heart of mathematics."

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I think, in essence, this is very much what Eugene Wigner's famous notion of "the unreasonable effectiveness of mathematics" revolves around… how is it that the human brain is capable of such seemingly successful intuitions about the world around us….

[…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 you submit, I'll cite the contributor.]

The Nobel and Noble In Math

Today's NY Times carries an interesting historical piece on the Fields Medal (to be awarded to unannounced winners next Wednesday). The interesting part is the story of Stephen Smale, a University of California at Berkeley winner back in 1966, and the political shenanigans that shadowed him because of his political activism. Politics and mathematics, to the surprise of many, do sometimes intertwine. With many mathematicians today taking stands against NSA work or certain other Government activities… and, with the Fields Medals being given out next week... it's a timely piece:

http://www.nytimes.com/2014/08/10/opinion/sunday/how-math-got-its-nobel-.html?_r=2

"math proofs that look like Nadal and math proofs that look like Federer"

That's just one of the gems from this wonderful new interview (30+ mins.) with Steven Strogatz via Santa Fe Institute:

http://www.santaferadiocafe.org/science/2014/08/06/steven-strogatz/

Funny Stuff...

 Okay, a lot of stuff that gets posted on the Web as math humor, I've either seen a 100 times, or just doesn't strike my funny-bone that hard… but THIS one, on the evolution of "the new, new math," did give me a belly laugh today (read all the way through):

http://www.forabodiesonly.com/mopar/showthread.php?t=176888

Simple Logic...

List:

   1.  This sentence contains eight words.

   2.  This sentence contains five words.

   3.  Exactly one sentence in this list is true.

Fiction and Non-Fiction

Want some novel math....

Not much of a fiction reader myself… but if I were, this site listing over 1100 entries of math-related fiction might interest me (broken down below by several categories, including novels, films, short stories, TV, comics, etc.):

http://kasmana.people.cofc.edu/MATHFICT/browse.php

...in other matters, someone writes me asking for the answer to a Richard Wiseman puzzle I posted over at MathTango on Friday. The puzzle read as follows:

Can you create a 10-digit number, where the first digit is how many zeros in the number, the second digit is how many 1s in the number etc. until the tenth digit which is how many 9s in the number.

Richard has posted the correct answer, with all the reader-comments, indicating the range of approaches people took toward a solution here:

http://richardwiseman.wordpress.com/2014/08/04/answer-to-the-friday-puzzle-269/
If you don't care to check out Richard's post the solution is below:
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answer:  6210001000

Sunday Morning With Max and Martin

"The Mathematical Universe Hypothesis offers a radical solution to this problem: at the bottom level, reality is a mathematical structure, so its parts have no intrinsic properties at all! In other words, the Mathematical Universe Hypothesis implies that we live in a relational reality, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks. The external physical reality is therefore more than the sum of its parts, in the sense that it can have many interesting properties while its parts have no intrinsic properties at all."

-- Max Tegmark from "Our Mathematical Universe"

…and in a footnote on the same page Max pushes the idea further, writing:

"Our brain may provide another example of where properties stem mainly from relations. According to the so-called concept cell hypothesis in neuroscience, particular firing patterns in different groups of neurons correspond to different concepts. The main difference between the concept cells for 'red,' 'fly,' and 'Angelina Jolie,' clearly don't lie in the types of neurons involved, but in their relations (connections) to other neurons."

Martin Gardner almost seemed to pre-sage Tegmark when he wrote the following more than a decade earlier (in his essay, "Is Mathematics 'Out There'?"):
 

"No mathematician, Roger Penrose has observed, probing deeper into the intricate structure of the Mandelbrot set, can imagine he is not exploring a pattern as much 'out there,' independent of his little mind and his culture, as an astronaut exploring the surface of Mars.
"In the light of today's physics the entire universe has dissolved into pure mathematics. The cosmos is made of molecules, in turn made of atoms, in turn made of particles which in turn may be made of superstrings. On the pre-atomic level the basic particles and fields are not made of anything. They can be described only as pure mathematical structures. If a photon or quark or superstring isn't made of mathematics, pray tell me what it is made of?"

Or similarly, this from another Gardner piece, "In Defense of Platonic Realism":

"In a curious way, numbers may be more real than pebbles. Matter first dissolved into molecules, then into atoms, then into particles, which are now dissolving into tiny vibrating strings or maybe into Penrose's twistors. And what are strings and twistors made of? They are not made of anything except numbers. If so, the numbers are as much 'out there' as molecules. They could be the only things out there. As a friend once said, the universe seems to be made of nothing, yet somehow manages to exist. As Ron Graham remarked, mathematical structure may be the fundamental reality."

heehee

Will close out the week (or, start the new month, if you prefer) with a little piece of math humor:



...or, if you want something slightly more substantive, check out the Friday pickings at MathTango, including a brand new piece from Dr. Keith Devlin, and latest puzzle from Richard Wiseman.