Foundational Math... to the Defense

Carnegie Mellon University has been awarded a whopping $7.5 million Dept. of Defense grant "to reshape the foundations of mathematics by developing a new approach that allows for large-scale formalization and computer verification," arriving at a "new, computational foundation for mathematics," based on "homotopy type theory"... a combination of 'homotopy theory' and 'type theory':

https://www.cmu.edu/news/stories/archives/2014/april/april28_awodeygrant.html#.U2An4hZDTn4

The press release ends thusly:

"CMU's Steven Awodey and his research team will use a $7.5 million grant from the Department of Defense to reshape the foundation of mathematics, building on his groundbreaking discovery in 2005 of a connection between abstract, mathematical geometry and computational logic."

I had trouble finding any references or descriptions of 'homotopy type theory' that I felt the average reader (including MYSELF!) could follow well. But this year-old post from Christian Perfect helps some:

http://aperiodical.com/2013/06/homotopy-type-theory-a-new-foundation-for-21st-century-mathematics/  

and Christian's piece has other links including this introduction to the subject:

http://www.carloangiuli.com/blog/homotopy-type-theory-univalent-foundations-of-mathematics/

Homotopy type theory tries to unify the foundations of mathematics that were originally shattered when set theory fell victim to Russell's Paradox. Needless to say, it's a fairly cutting-edge area of mathematical thought that apparently shows a lot of promise.  Indeed, I find it fascinating that the Defense Dept. is pouring money into it... but that's partly my naivete in associating the DOD with more practical 'hardware' sorts of expenditures... they of course back LOTS of more abstract study; it's just not what most people think of front-and-center when you say "Department of Defense!" Anyway, not bad funding if you can get it! ;-)

Let the Fun Begin!

In this Centennial year for Martin Gardner a new math magazine is born…
For those spoiled by Gardner growing up, the semi-annual (FREE!) online magazine entitled simply "Recreational Mathematics," has been launched, as reported by Christian Perfect here:

http://aperiodical.com/2014/04/a-new-recreational-mathematics-magazine-from-the-ludus-association/

It will include, "games and puzzles, problems, mathmagic, mathematics and arts, history of mathematics, math and fun with algorithms, reviews and news," and the editors add that they "will try to bring in focus amazing mathematical ideas. We seek sophistication, imagination and awe."

The first issue (March 2014) containing six varied articles (from different authors) looks good, and is here:

http://rmm.ludus-opuscula.org/PDF_Files/RMM_Number1_March_2014_low.pdf

Of Financial Advisors and Algorithms

If you've ever perused the promotional material of a stock market advisor/prognosticator you may be impressed at how often their charts and models seem indeed to fit the history of market activity (…of course when you give them YOUR money, suddenly their predictions aren't quite as stellar). This article (h/t to Jennifer Ouellette) recounts the ubiquitous idea of "backfitting" or "backtesting" (a "subtle mathematical con" as the author calls it)… let's just say that statistics, hedge funds, and stock modeling aren't always the most forthright mix:

http://tinyurl.com/n3mlxuv

The actual paper on which the article is based is here:
http://www.ams.org/notices/201405/rnoti-p458.pdf

And finally, a related blog, "The Mathematical Investor," is here:  
http://www.financial-math.org/blog/

It's no wonder that "past performance is no guarantee of future results..."

Sunday With Einstein

Another Sunday meditation of sorts today, about the Albert Einstein quote I employ in the right-hand column of this blog...

"... As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. "

I've always loved those thought-provoking lines. Once or twice a year someone emails me with some sort of question or complaint about them. Often readers don't realize they can simply click on the quotation to go to the fuller speech Einstein gave from which the words come.
Anyway, since it's a famous quote, I decided to check around the internet to see what sort of discussion it may have generated elsewhere.
It arises in a lot of places, but one of the best treatments I came across was entitled "Unpacking An Einstein Aphorism" with a lengthy discussion of it (and its possible meanings) here:

http://geopolicraticus.wordpress.com/2011/03/05/unpacking-an-einstein-aphorism/

It's a bit heavy on philosophy and semantics, but introduces many points I hadn't thought about. For one thing, I hadn't even thought about the fact that Einstein originally gave this talk in German, and so what we commonly read is of course an English translation… which immediately raises some uncertainty because of the imprecision of translation, especially of a word like "reality." The piece is a bit of a slog by the end (but no more so than any philosophical piece addressing subtleties and nuances of language), and makes interesting points along the way. If the quote interests you give the above piece by J.N. Nielson a whirl...


[elsewhere... if you know who blogger Mike Lawler is, get to know him better by visiting MathTango now, or if you don't know of Mike, by all means, go there to learn about him.)]

Just A Random Post Today

Fantastic, and mostly very comprehensible, article (long read) on randomness by Scott Aaronson over at American Scientist:

http://www.americanscientist.org/issues/pub/2014/3/the-quest-for-randomness/3

"Randomness" is a topic that potentially cuts across so many different boundaries… mathematics, logic, computer science, philosophy, psychology, physics, engineering… even if you think you're not particularly interested in it, you should be! The article touches on a number of topics, including Kolmogorov complexity, the Berry Paradox, quantum computing, random number generation, and best of all it's just the first of two pieces Aaronson is doing for Am. Scientist.

And over at MathTango right now another Friday list of links to catch up on for the weekend.

Truthiness From Yablo

This isn't everyone's cup-o-tea, but I've mentioned Yablo's Paradox before, and, since I love it, will do so again! As Sam Alexander states, Yablo's Paradox is "a cute version of the Liar’s Paradox" that manages "to achieve paradox without any direct self-reference." The simply-stated paradox involves a countably infinite number of sentences, each of which refer only to sentences that come after it:

• Sentence 1:  Sentence n is false for every n > 1
• Sentence 2:  Sentence n is false for every n > 2
• Sentence 3:  Sentence n is false for every n > 3
• Etc....

Read Alexander's post here:  http://www.xamuel.com/dangerous-graphs/  (he discusses it in terms of "graphs").
And here is (Stephen) Yablo's amazingly short, original (1993) piece introducing the paradox:

http://www.mit.edu/~yablo/pwsr.pdf

Worth noting, that while Yablo claims his paradox involves neither self-reference nor circularity (because all steps along the way reference sentences that are yet to come), others disagree with this contention. If you're logically-inclined, see Graham Priest here:

http://www.accionfilosofica.com/misc/1183297103crs.pdf

and JC Beall here: http://ferenc.andrasek.hu/papersybprx/jcbeal_is_yablo_non_circular.pdf

(This is another good example of how things become muddled when infinity is involved; in the case of Yablo's Paradox, an infinite number of sentences.)

Teaching, Conceptualizing, Common Core

As much as I try to avoid the primary/secondary education debates, I'd feel remiss if I didn't pass along two of the pieces from this morning's Web:

1) California teacher Fawn Nguyen interviewed by Cathy O'Neil on Common Core:
http://mathbabe.org/2014/04/23/interview-with-a-high-school-math-teacher-on-the-common-core/

2) Grant Wiggins on "conceptual understanding" (longread +comments):
http://grantwiggins.wordpress.com/2014/04/23/conceptual-understanding-in-mathematics/

Fascinating (to me) how something (Common Core) intended to bring unification/standards, is instead stirring up vastly more debate and discussion than likely ever anticipated, with resolution seemingly a long way off. One thing I'm not clear on (as a non-educator), maybe someone can quickly clarify, is Common Core being sharply debated in realms OTHER than math (i.e., English/verbal content), or is the debate mostly a math thing?

Sociopaths, PhDs, Mathematicians, and more

Jordan Ellenberg offers an observation on why mathematicians (generally) are NOT sociopaths (unlike, perhaps, investment bankers ;-):

http://quomodocumque.wordpress.com/2014/04/22/why-arent-math-professors-sociopaths/

Meanwhile, possibly on the opposite end of the spectrum, is Jason Padgett's story, which I've reported here previously -- an individual with "acquired" savant syndrome, which appeared later in life following significant brain trauma, as recounted in Padgett's new volume "Struck by Genius: How a Brain Injury Made Me a Mathematical Marvel."  Interesting stuff...
Another review here: http://tinyurl.com/nqk8g8t
Or, you can learn a lot more about Jason by googling him:  http://tinyurl.com/kbozcnm

Family Math

Just filling some time today passing along this li'l algebraic Friday puzzler that Richard Wiseman offered last week (I've re-stated it slightly):

In a certain family with 2 or more children, each daughter has the SAME number of brothers as she has sisters, while each son has TWICE as many sisters as brothers. How many sons and daughters are in this family?

answer below:
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answer:  4 daughters & 3 sons

Thursday Meditation From Mathodist Pastor Keith Devlin ;-)

Just a mathematical meditation for today...:

About 10 days ago in a longish post over at MathTango I included some links to Keith Devlin materials -- one of my favorites was a podcast he did for NPR's "On Being," an always-wonderful radio-hour.  In that conversation he made the following observation, which I fancy, regarding a transition on the road to becoming a mathematician [bold added]:

"...that's when I became a mathematician; that's what I stumbled on at age 15 or 16 when here I was learning all this mathematics because I needed  it. I had a utilitarian view of mathematics. I was learning it because I needed to solve the equations because I was going to be solving them in physics. And then, at the age of about 16 or 17, it all fit because it all came together in my mind. It was no longer this disjointed collection of techniques you could use to solve problems. It all fell into place, into this wonderful landscape. It was as if I'd been stumbling around in a forest, and suddenly I've climbed to the top of a tree and looked out and thought, this is the most beautiful place in the world. You can't tell it when you're down in the trees, which I had been, but the moment you reach an elevation where it all falls into place and you can see the whole topographic display in front of you, then the beauty is incredible. And the moment I discovered it, I said, um, I want to study mathematics. And I've been studying it ever since."

I love this imagery of most of us exploring down amongst the trees and forest floor, versus looking out above the canopies over the whole forest landscape that is mathematics. This is what Ed Frenkel essentially talks about, especially in working on the Langlands Program which brings together disparate aspects of mathematics... the separate sections of the forest linking together when we zoom out far enough.  The same idea comes up in discussions of symmetry, group theory, number theory, and other areas of math, unlike the more disjointed way math must often, perhaps of some necessity, be taught in the classroom. The individual trees and groves can of course be beautiful and stately all by themselves… but, ohhh, to take in that majestic view from above the treetops!!

Numerosities

For the philosophically, or foundationally, inclined, the below post argues for something called "numerosities" that give "part-whole" set relationships priority over "one-to-one" relationships, and in so doing, counter the usual Cantorian orthodoxy, which permits a partial set (say odd numbers), to be deemed equal in size to an entire set (all integers):

http://www.newappsblog.com/2014/03/counting-infinities.html

Excerpt: "The philosophical implications of the theory of numerosities for the philosophy of mathematics are far-reaching... Philosophically, the mere fact that there is a coherent, theoretically robust alternative to Cantorian orthodoxy raises all kinds of questions pertaining to our ability to ascertain what numbers ‘really’ are (that is, if there are such things indeed)."

This Sunday's Math Sermon…

"...every decent maths teacher knows that teaching maths is about understanding how different people think. Some students need to see the bigger picture first – they need context, they need a reason for doing things, they need to know what the end result will be. Others are happy to discover things for themselves and they enjoy the process as much as the outcome. Some are comfortable thinking in an abstract way; others need a more concrete approach. Many students impose their own rules which don’t quite work and we need to unpick what they are doing and figure out what their underlying thinking is.  I like this challenge of figuring out how people think. Studying history gave me a good grounding in understanding that not everyone sees the world in the same way."

The above comes from a li'l musing (dare I say rant?) from a British secondary math teacher who has a history degree where one might expect a math background. I LOVE what she has to say here:

http://mathematicalmagpie.blogspot.co.uk/2014/04/you-mean-you-have-arts-degree-pulls.html

Toward the end she writes, "I’d like to finish by arguing that, rather than being afraid of people who cross the invisible arts/science divide, more of us should come out and celebrate it."

...Amen

And lastly she vows: "All I can say is, if anyone tries to put me in a box, I’ll soon be fighting my way out."

As a bird lover, I even love this blogger's Internet handle: "the mathematical magpie"... go read the whole lovely post.

Stats and Primes...

Today, a couple of statistics-related readings from the latest John Brockman/Edge volume, "What Should We Be Worried About?"

1) Bart Kosko on what he deems five "lamplight probabilities" that help explain the world we observe, but also limit our analysis, and 'especially restrict modern Bayesian inference':

http://edge.org/response-detail/23856

2) And Nassim Taleb talking about one of his favorite subjects, "fat tails," and why "having skin in the game" is a necessary component for honest, accurate measurement of risk:

http://edge.org/response-detail/23839

...In other matters, an update on the twin-prime conjecture, now down to an upper bound of 252 (from the original approach based on Zhang's work):

http://tinyurl.com/m3pt98s

From Seinfeld to Alex Bellos

Apparently George Costanza was onto something…:



"1" is perhaps the loneliest number, and "110," according to Alex Bellos, the least loved, but his research pinpoints "7" as the public's favorite number:

http://tinyurl.com/n24nqjr

ADDENDUM (I'll just pass along a few more points from the article):

Bellos admits it wasn't necessarily a rigorous or fair statistical sampling, but nonetheless over 30,000 respondents took part. A few interesting tidbits:

8 of the top 10 favorite numbers are single digits, the other two slots taken by 11 and 13 (but all numbers from 1 to 100 were chosen at least once). I was surprised that 13 came out as high as sixth place, but perhaps that has something to do with Taylor Swift's choosing it as her well-publicized favorite number.  Numbers ending in either 0 or 5 (except for 5 itself) were particularly avoided as favorites. Bellos notes that 7 is unique in many ways and that probably accounts for its popularity.

This material all comes from Alex's new book, "Alex Through the Looking-Glass," (out in the UK, but not available until June in the US, and then under the title "The Grapes of Math").

Good Listenin' and Readin'

Sol Lederman's latest inspiring interview is with Tim Chartier, newly out with his slim, but rich, book, "Math Bytes" (also famous for his basketball "bracketology" research):

http://wildaboutmath.com/2014/04/06/tim-chartier-inspired-by-math-34/

As usual, Sol covers a lot of interesting ground here, and we learn a lot about Dr. Chartier.

And another new book, "Infinitesimal," by Amir Alexander is reviewed in NY Times and sounds good. A lot of folks who take calculus somewhere along the way, don't ever learn about the checkered history of it, and the original controversy surrounding the concept of an "infinitesimal." So read up:

http://tinyurl.com/mnj9wzy

It's going to be another fertile year for popular math books!

If you're in a hurry for more good and varied readin' then the 109th Carnival of Mathematics is newly-posted and just a click away:

http://tonysmaths.blogspot.co.uk/2014/04/109th-carnival-of-mathematics.html

Four From Friday

Friday brought several interesting, varied reads across my computer screen:

1) Scientific American got us started with a little math history lesson, centered around "infinitesimals" coming in and out of favor, and the re-casting of calculus:

http://tinyurl.com/k5ym5qd

2) On the light side, The Guardian offered an excerpt about our response to numbers, from Alex Bellos' newest book, "Alex Through the Looking Glass":

http://www.theguardian.com/books/2014/apr/04/why-all-love-numbers-mathematics

It starts off telling about a man named Jerry Newport, a retired taxi driver with Asperger's Syndrome who has "an extraordinary talent for mental arithmetic." Also, turns out that Jerry's living room includes "a cockatoo, a dove, three parakeets and two cockatiels"… THIS is a man I can relate to! ;-) -- I've had a similar living room in the past!… though I lack Jerry's number talents. Anyway, many of Jerry's unexplained skills interestingly center around prime numbers. (BTW, I touched on the subject of linkage between Asperger's and math ability a bit ago.)
The rest of Bellos' piece deals with various subjective (and seemingly inexplicable) oddities about integers, and our relationships to them.
An interesting, fun read, touching on language and psychology in addition to math.  If the rest of the book is this entertaining, jolly good!

3) A bit more advanced, Adam Kucharski writes a piece for Nautilus on the startling Weierstrass Function, which pre-saged "fractals" -- a fascinating function that is continuous, yet lacks a derivative at any given point (is "smooth" NOwhere). This was contrary to all prior math thought, and "With one bizarre equation, Weierstrass had demonstrated that physical intuition was not a reliable foundation on which to build mathematical theories":

http://nautil.us/issue/11/light/maths-beautiful-monsters

(this interesting extract comes from a longer article Nautilus subscribers can access)

4) Finally, in a fascinating bit of logical legerdemain, Arkady Bolotin has linked the P vs. NP Millennium Problem to quantum mechanics, and in so doing reached conclusions about both. A longstanding puzzle in quantum theory is how to apply equations that work so well at the quantum level to the world we actually live in and experience. Bolotin argues that while Schrodinger's equation has relatively simple solutions at the atomic-level, at the macro-level it becomes NP-hard (essentially unsolvable). Essentially he's killing two birds with one stone: claiming that P ≠ NP (which is what most assume, but have yet to prove) and that the quantum inscrutability of our world is the result of Schrodinger's equation being essentially unsolvable (within reasonable time) at the macro level:

https://medium.com/the-physics-arxiv-blog/7ef5eea6fd7a

And if you have any energy left after reading all these you can go catch the latest lengthy blather (re: math education) I've put up on MathTango this morning.

Visit MathTango

Nothing new here, but linkfest up at MathTango now, and on Sunday will be a longish post there related to math education.

The Joy of Number Theory ;-)

Number theory is both one of the most interesting and arcane areas of all mathematics… arcane in that oftentimes findings or proofs within the field appear to have no practical application. Such was true of a finding from four decades ago which had no known application until this very week. As reported back in 1975:

"…when the transcendental number e is raised to the power of π times √163, the result is an integer. The Indian mathematician Srinivasa Ramanujan had conjectured that e to the power of π√163 is integral in a note in the Quarterly Journal of Pure and Applied Mathematics (vol. 45, 1913-1914, p. 350). Working by hand, he found the value to be  262,537,412,640,768,743.999,999,999,999,…. The calculations were tedious, and he was unable to verify the next decimal digit. Modern computers extended the 9's much farther; indeed, a French program of 1972 went as far as two million 9's. Unfortunately, no one was able to prove that the sequence of 9's continues forever (which, of course, would make the number integral) or whether the number is irrational or an integral fraction.
"In May 1974 John Brillo of the University of Arizona found an ingenious way of applying Euler's constant to the calculation and managed to prove that the number exactly equals 262,537,412,640,768,744. How the prime number 163 manages to convert the expression to an integer is not yet fully understood."

Only recently was it realized that this arcane mathematical number-theory result, which is now much better understood, could be put to practical use… by bloggers wishing to entertain on April 1st, 2014! ;-))

Yes, the above, for any who don't immediately recognize it, was an April Fool's fabrication from prankster Martin Gardner for his classic April 1975 column in Scientific American. While the reference to Ramanujan is true (except that Ramanujan knew the number involved was transcendental), and the computed number, as given, is accurate as far as it goes, the rest of the passage was a hoax that fooled many at the time ("John Brillo" was a play on the name of another number theorist). The next digit following the string of 9's that Martin listed, is actually a "2".
You can read a bit more about the interesting number from this old journal article (Pi Mu Epsilon Journal, Vol. 5, Fall 1972, No. 7, pgs. 314-15; "What Is the Most Amazing Approximate Integer in the Universe?" by I.J. Good):

http://www.pme-math.org/journal/issues/PMEJ.Vol.5.No.7.pdf

My quotation above comes from chapter 50 of Martin Gardner's "The Colossal Book of Mathematics." Of course, after all these years he still entertains us.

Ohhh, and by the way, your shoelaces are untied!....