Words/Math Puzzle For Today

"Mathematical Mind-reading" problem from NY Times' 'Numberplay':

http://tinyurl.com/2c4qmt2

ALL Data/Knowledge on a Computer

For Real??

It's not April 1st so I guess Stephen Wolfram's stated goal to make ALL the world's data computable is a real one...:

http://tinyurl.com/39stuzc (very long post... transcript actually)

Wolfram lays forth his belief that it may be algorithmically possible to save ALL the world's data/knowledge into a working computer format for practical use:

"The idea is: take all the systematic knowledge—and data—that our civilization has accumulated, and somehow make it computable. Make it so that given any specific question one wants to ask, one can just compute the answer on the basis of that knowledge and data."

Although admitting to skepticism earlier on that this could be achieved, Wolfram (encouraged by the huge success of his Mathematica program) now seems optimistic that it is do-able.

Meanhwile, for the audiophiles, a new (hour-long) math podcast here:

http://pulse-project.org/node/230

Finally, I awoke this morning to find this problem posted on Twitter:

Between them, two numbers use all the digits (1 thru 9). What are they, such that their product is as big as possible?

hmmmm...???

New Hersh Book Forthcoming

Reuben Hersh has a new book, with Vera John-Steiner, upcoming (December?) that looks to be good: "Loving and Hating Mathematics: Challenging the Myths of Mathematical Life."

Hersh is one of the premier popular explicators of mathematics of the 20th century. But interestingly, he and Martin Gardner (probably the more popular writer, but less-schooled in academic mathematics than Hersh) feuded over the years, with their opposing views of math's relationship to 'reality.'
Gardner was the adamant and traditional "Platonist" believing (as most intuitively do) that mathematics is a true reflection of the real world (outside the human mind). Seems obvious to many... but in fact a surprising number of professional mathematicians hold to a different view of mathematics, as just another creation of the human mind (not objectively discovered, but very much influenced and created by human culture, cognition, psychology, etc.). Remove humans (and their minds) from the Universe and there would be no mathematical laws, as we perceive them, operating.

I didn't realize, until reading his brief Wikipedia entry, that Hersh actually originally earned a B.A. degree in English Literature (Harvard), and worked as a machinist and writer for Scientific American, before eventually getting his Ph.D. in mathematics in 1962 (New York University).
Here is what Gardner had to say of Hersh in one of his many online interviews:

"Reuben Hersh is a marvelous example of a person who thinks that mathematics is entirely a human product and has no reality outside of human culture. He has written a whole book about this called "What Is Mathematics Really?" To Reuben Hersh, mathematics is no different from art or fashions in clothes. It’s a cultural phenomenon. The postmodernists in France have essentially this point of view. And it drives me up the wall. I like to say, 'If two dinosaurs met two other dinosaurs in a clearing, there would be four of them even though the animals would be too stupid to know that.' Of course, the argument as to whether the universe exists outside of the human mind goes back to the middle ages."

(I suspect Hersh might object to at least part of this characterization.) The irreconcilable differences between these two pillars of 20th century math reporting/education is fascinating, and both views have very bright, significant supporters on their sides (if anything, the Hersh view may even have made gains in recent years, though the Platonist view still predominates).

This simple 2008 piece on the Web addresses the issue:

http://www.canadafreepress.com/index.php/article/2805

And Gardner himself has a wonderful 2005 essay (actually a book review), "A Defense of Platonic Realism," reprinted as chapter 9 in his volume "The Jinn From Hyperspace," if you have access to that.
Or, an earlier Gardner essay entitled "How Not To Talk About Mathematics" in his "The Night Is Large" book (chapter 24) covers much the same ground as well (with more specific reference to Hersh). It's a fascinating debate that won't end anytime soon, and that non-mathematicians often aren't even aware of.
  

There Was A Young Lady With Curves...?

Not exactly Robert Frost, but just for fun today a page of math limericks:

http://www.trottermath.net/humor/limricks.html

Here's a couple of samples, just to get you in the right mood:

"A graduate student at Trinity
 Computed the square of infinity.
    But it gave him the fidgets
    To put down the digits,
 So he dropped math and took up divinity."

"Archimedes, the well known truth-seeker,
      Jumping out of his bath, cried, "Eureka!"
        He ran half a mile,
        Wearing only a smile,
      And became the very first streaker."

Open Science?

Without even mentioning P vs. NP (though obviously there are implications for that debate), RJ Lipton's latest post interestingly addresses the question of whether the sharing of ideas and partial results in math/science is a valuable thing (before all issues are worked out). Such 'open science' seems to me to be the wave of the future, but there are undoubtedly habits and considerations that will need to be overcome:

http://tinyurl.com/37hcy7z

(...and I love the quote he includes from Howard Aiken: “Don’t worry about people stealing an idea. If it’s original, you will have to ram it down their throats.”) ;-)

Lipton employs several famous math examples for his discussion, and ends with this question for his readers:

"Should researchers be encouraged to share ideas earlier than we do now? What mechanisms are needed to be sure that proper credit is given out? Would you publish a partial result?"

 ADDENDUM: another pertinent post here:

http://tinyurl.com/38d67g9

 

Paul Erdos

Image via Wikipedia

A documentary portrait of the amazing Paul Erdos (1913-1996) via YouTube clips here:

http://tinyurl.com/2f2776n

To this day Erdos is the most published mathematician of all time. He was well-known as a traveling math vagabond who would simply hang out with colleagues he had all over the world, after showing up on their doorstep and announcing, "My brain is open," essentially meaning 'hey, let's do some math!' And yet despite many award-winning accomplishments in several math fields he was initially flummoxed by the well-known 'Monty Hall problem.' A fascinating character, whose likes we may never see again.

Have I Mentioned Any Math Books Before...

Well, here's some mmmmore:

Browsing in a local bookstore I noticed the new edition of David Foster Wallace's 2003 "Everything and More" is now out. It has a newly-written Introduction, but otherwise, so far as I could tell, there were no changes/corrections to the text (???). Wallace died tragically at his own hands a couple years ago. Many mathematicians would not recommend this work, but if you're a lover of math and infinity, combined with wordplay (or what some call Wallace's "verbal pyrotechnics"), I think it a worthwhile, if eclectic, read (my previous mini-review HERE).

Wallace's volume was part of Norton's "Great Discoveries Series." Though not truly a math book, I'll just point out that another upcoming volume in that series will be a new biography of Richard Feynman from physicist Lawrence Krauss... ought to be good, keep an eye out.

Also worth noting that Stephen Wolfram has announced that his massive 2002 tome "A New Kind Of Science" is now available on the iPad:

http://tinyurl.com/33sjswd

Finally, what I did pick out (based solely on Martin Gardner's ringing endorsement on back cover) during my browse through the bookstore, was Brian Hayes' 2008 "Group Theory In the Bedroom, and Other Mathematical Diversions." Gardner wrote, "Every essay in this book is a gem of science writing at its highest level...Its scope is awesome... There isn't a dull page in the book."  Into the reading queue it goes....

Factoid...

The formula n^2 - n + 41 produces prime numbers for all n's from zero to 40. At 41 it fails.

Primes and Quantum Physics

We normally think of mathematics underlying and driving ideas in physics... In this interesting 2006 article (focused around a chance 1972 meeting between Freeman Dyson and number theorist Hugh Montgomery) Marcus du Sautoy writes of how ideas from quantum physics may underlie and help deepen our understanding of prime numbers:

http://seedmagazine.com/content/article/prime_numbers_get_hitched/

Speaking of du Sautoy, you may wish to check out his website for  "The Story of Maths" series he did for the BBC:

http://www.open2.net/storyofmaths/index.html

'nuther Oldie but Goodie

This is an old puzzle that comes in a variety of forms. I've adapted it here from a Martin Gardner version in his "Aha! Gotcha" volume:

6 students make reservations for a dinner at a popular pizza place. But at the last minute a 7th student decides to join them.
When the kids arrive the hostess immediately sees that there are 7 diners for her table set-up of 6. But she is a clever one. Thinking on her feet, she decides to seat the first student in chair #1 and then have his girlfriend (the second student) sit on his lap temporarily. Then the hostess can sit student #3 in the 2nd chair, student #4 in the 3rd chair, student #5 in the 4th chair, and finally student #6 goes into chair #5. Chair #6 is thus still leftover, and so of course the hostess can now move the original girlfriend to that seat. Waaah-laaaaah!!
Hope she gets a big tip... or, maybe not!?

Do you see the flaw in her method?
(For any who don't see through the flaw I'll wait 24 hrs. and explain the simple answer in the comments below... and then you can go "DOH!")

I'm sure the young lass (another child prodigy) reported on in this nice article can spot the catch in the math:

http://tinyurl.com/34arukx
 

Boston Museum

Nice article from Boston Globe on the "Mathematica" exhibit at  their city Museum of Science, still enthralling young and old after all these years:

http://tinyurl.com/329upk5

(I was especially heartened to see the quincunx or Galton Box  mentioned which I previously wrote about as one of my joys as a youngster; although it is only designated in the article as a "probability" demonstration.)

Proofiness, Numerical Mendacity, and Civics

What has Stephen Colbert wrought....

NY Times favorable review (from Steven Strogatz) of Charles Seife's latest book, "Proofiness: The Dark Arts of Mathematical Deception":

http://tinyurl.com/2dphrtd

(promoted as a book about "the art of using pure mathematics for impure ends")

Strogatz concludes thusly, "For the most part, though, he [Seife] is deadly serious. A few other recent books have explored how easily we can be deceived — or deceive ourselves — with numbers. But “Proofiness” reveals the truly corrosive effects on a society awash in numerical mendacity. This is more than a math book; it’s an eye-opening civics lesson."

Have It Your Way...

This ain't your Daddy's Burger King...

Math on the menu (billions of burger choices) at a new NY restaurant:

http://tinyurl.com/22pddta

ohhh, and do you want fries with that. . . .

More Books

How timely!
RJ Lipton, who has been one of the primary communicators/bloggers of the recent P vs. NP happenings, and someone with a long-time particular interest in the issue, has newly published a book on the topic, "The P = NP Question and Gödel’s Lost Letter":

http://tinyurl.com/2v88wwd

(it's a tad expensive, and NOT bedtime reading!)

What ought make the book particularly interesting though is that Lipton is among the minority who still believe that P may indeed EQUAL NP (versus the majority belief that P ≠ NP).

Changing gears from one deep problem to another, Harvard mathematician and Fields Medalist Shing-Tung Yau has a recent well-received book out, "The Shape of Inner Space," which recounts the geometry/mathematics underlying string theory:

http://tinyurl.com/26u4o2q

ADDENDUM: Peter Woit's review of Yau's book is now up here:
 http://www.math.columbia.edu/~woit/wordpress/?p=3165
   

The Special Number 3816547290

Read all about it (find out why it's special):

http://everything2.com/title/3816547290

...and some more on this peculiar number here (from a computational-science slant):

http://scienceblogs.com/builtonfacts/2010/08/a_conspiracy_of_digits.php

...and one more further follow-up here:

http://tinyurl.com/29rvhae

Pi: The Number That Keeps On Giving....

Ever get the feeling that perhaps some people just have too much free time on their hands ;-)....

In news, that we've all been hungry for, it's been announced that the two-quadrillionth digit of pi has been determined... in binary... to be a zero!:

http://www.bbc.co.uk/news/technology-11313194

It's still the case that only a mere 2.7 trillion consecutive digits of pi are known, but powerful cloud computer techniques are now able to find specific digits much farther out. But more importantly... can these techniques help me find my car keys in the morning???

Seriously though, congratulations to those involved!

Just a Few Thangs

RJ Lipton offers a brief update on where things stand with Vinay Deolalikar’s PNP “proof” here:

http://tinyurl.com/2wqandk 

(essentially, it's still being discussed and worked on, and we're a long way from final resolution) 

The 3rd "Mathematics and Multimedia Blog Carnival" is up-and-running here:

http://math4allages.wordpress.com/2010/09/13/blog-carnival-3/

Some unresolved prime number questions here:

http://twitpic.com/2ou1x8

And lastly, another quickie puzzle (that MAA's "MinuteMath" ran recently):

A certain positive integer "x" has the property that x% of x = 4. What is "x"?

answer below:
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.
.
.
.
.
.
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.
.
.
.
.
.
solution x = 20
 

Nerds, Geeks, Dorks, Dweebs

Why didn't they have Venn Diagrams like this back when I was in school:

http://tinyurl.com/ydvzqld

(but it doesn't include 'buffs'...)

The Jordan Curve Theorem

The Jordan Curve Theorem (named for a French mathematician who first proved it) states that any continuous simple closed curve in a plane, separates the plane into two disjoint regions, the inside and the outside.

...Seems intuitively pretty straightforward, or as one of the books I have on my shelf says,

"The theorem seems like a statement of the blindingly obvious. If a curve proceeds continuously without any breaks in it and returns to its starting point without crossing itself, then there will be a region outside the curve and a region inside. The two regions are separate, one is finite and the other is infinite."

While this is indeed clear for the run-of-the-mill closed curves we are accustomed to seeing, there are far more complex topological curves, including for example the Koch Snowflake, that may help one see why the theorem's proof (and it has been proved) is not at all easy (some of the proofs run to 1000's of lines).

A couple of discussions of the theorem here:

http://tinyurl.com/26tfk7v

http://www.math.ohio-state.edu/~fiedorow/math655/Jordan.html

Number Digit Puzzle

Can you find a 10 digit number such that:

• the 1st digit tells how many zeros are in the number, 
• the 2nd digit tells how many 1's are in the number,
• the 3rd digit tells how many 2's are in the number,
• the 4th digit tells how many 3's are in the number, etc. etc. (10th digit tells how many 9's are in no.)

 answer below....
.
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answer: 6210001000

This is called a "self-descriptive number," BTW, and it is the only one in base 10.

Einstein In Context...

A couple of folks have emailed me over time with inquiries about the (somewhat famous) quotation from Albert Einstein used in the right-hand column of this blog. For anyone interested in its fuller context, it comes from this 1921 address of Einstein's to the Prussian Academy of Sciences in Berlin, entitled "Geometry and Experience."  (Einstein was essentially addressing the notion of what would later come to be known as "the unreasonable effectiveness of mathematics" (Wigner) when he made the remark):

http://tinyurl.com/2b4375e

This and That

A mathematics blogger comments on the "hoopla" around Stephen Hawking's new book here:

http://mathrising.com/?p=238

Does learning higher mathematics take away from the ability to think sharply on a problem? The question is raised here (includes a nice 'algebra' problem):

http://tinyurl.com/2v7uz7f

... and Princeton University Press is giving away a copy of the new book, "The Calculus Lifesaver," by Adrian Banner:

http://tinyurl.com/3426tfj

DOH!! Page 29...

Some more video entertainment today. A clip from back in 2008 of someone expressing the joy/passion that is evoked by a claim that the Riemann Hypothesis has been proven:



And now check out this note:

http://arxiv.org/abs/0807.0090
  

A Book, and a Crafty Law Student

An old Dover publication I only recently stumbled upon is one of the best I've seen at succinctly covering many of the most trenchant paradoxes and self-reference issues underlying mathematics:

Bryan Bunch's "Mathematical Fallacies and Paradoxes" (1982) covers a wide selection of, guess what, fallacies and paradoxes; some fairly light, others deeper and heavier. Below I've totally re-adapted one of the many self-reference paradoxes contained in the volume:

To demonstrate what a fine teacher he is, Larry the Lawyer contracts with each of his students such that they need only pay him for his individual instruction IF and ONLY IF they win their first law case. If they lose that first case they pay him no fee.
However, one of his students, Squiggy, upon completing the course, opts simply not to try any cases at all to avoid paying any fee. Perturbed, Larry feels compelled to sue Squiggy for payment (since avoiding trial cases in order to avoid payment was not intended as an option). Once the case comes to court Squiggy represents himself. IF he loses, then by the original contract he does NOT have to pay Larry! IF he wins the suit then the court will have ruled that he does NOT have to pay! Squiggy appears well on his way to being a superb lawyer....

The Bunch book definitely isn't for everyone (not even for all math buffs), but if you have an especial interest in the paradoxes and intrinsic issues that underlie uncertainty in mathematics, as well as in science and knowledge more generally, it's worth a look.

"Nerd Superbowl" & The Halting Problem

Some stuff to chew on for weekend:

Nice wrap-up to the current state of the recent P vs. NP excitement from "Math Trek" here:

http://tinyurl.com/32smtbf

And below, Mark Chu-Carroll takes his readers through the 'halting problem' and its unsolvability in a recent post:

http://scientopia.info/blogs/goodmath/2010/09/08/1069/#more-1069

The Mathematics of Sunscreen

Higher and higher SPFs....

Older NY Times article here:

http://www.nytimes.com/2009/05/14/fashion/14SKIN.html?_r=2

Visualizing Data...

This David McCandless (journalist) 18-minute TEDTalk has been bopping around the Web of late, in which he promotes the utility of 'visualizing' numbers/data/context; probably a tad oversimplified, but still interesting:




David's blog is here: http://www.informationisbeautiful.net/

Factoid...?

Read this on Twitter once:

2^86 is the longest known power that contains no "0" digits.

A Number Savant

Video clip of a savant who's friends are numbers:

Math: A Young Man's Game?

Another interesting post from Alex Bellos, this time on "maths' cult of youth" (the common notion that great mathematicians invariably do their best work in youth, and are past their prime by age 40).

http://alexbellos.com/?p=1354

Starts off focusing on now 15-year-old child prodigy Arran Fernandez, before addressing the issue more broadly.

(...Hopefully, we'll get a few more productive years out of 35-year-old Terence Tao , who some consider the world's greatest living mathematician, before he commences his downward slide. ;-))

My Fab Four

Speaking of Euler, as I did in the prior post....

It's of course impossible to name the four most important or influential or greatest mathematicians of all time. The arguments could go 'round-and-'round forever without resolution, and math is a surprisingly diverse, robust field where different individuals make all kinds of different contributions.

Nonetheless, one of the math T-shirts I offer at my Zazzle store is "The Fab Four" where I've designated the four I would pick out for such an honor if forced to choose. For the combined breadth, depth, and variety of their contributions they are (with their Wikipedia links):

Leonhard Euler
(1707 - 1783)

Carl Friedrich Gauss
(1777 - 1855)

David Hilbert
(1862 - 1943)

G.F. Bernhard Riemann
(1826 - 1866)

I'm leaving out both a lot of hugely important ancient and modern-day mathematicians, but these are just my subjective choices. Who might you choose for a Fab Four? Here's a list of 'greatest mathematicians' as formulated by someone else:

http://fabpedigree.com/james/greatmm.htm

e^i(π) + 1 = 0

Astrophysicist Adam Frank waxes poetic (almost) over Euler's "incomparable and glorious" Identity, involving "five magic numbers," and "beauty" (...with Justin Bieber thrown in for good measure):

http://www.npr.org/blogs/13.7/2010/09/02/129610905/best-equation-ever?ft=1&f=114424647

Frank asks, "Why are so many mathematically inclined folks sent into paroxysms of delight over this string of symbols which seem like gibberish to others," and then he proceeds to answer the question for the reader.

New Carnival of Mathematics

The 69th "Carnival of Mathematics" it is now available (with the usual variety of math-festive offerings) here:

http://jd2718.wordpress.com/2010/09/03/carnival-of-mathematics-69/

Fermat's Last Theorem Used in Proof

For any greater than is irrational...  

A demonstration of Fermat's Last Theorem being used in a very efficient ("charming") proof of the above:

http://tinyurl.com/2bof7l6

(...this is almost too simple)

Who Knew!

I just discovered there is a YouTube channel devoted to number theory. Latest offering here:

http://tinyurl.com/26unyae

...and a few problems (from elsewhere) to play with here:

http://threesixty360.wordpress.com/2010/09/02/its-a-new-newsletter/

(I suspect 4.2.3 has some shortcut solution, though I'm not seeing it?)

Oldie But Goodie

Famous old Sam Loyd 'Chinese Warrior' fun visual puzzle here:

http://alexbellos.com/?page_id=865

Cryptography Basics

Nice basic intro to cryptography (and especially RSA encryption) from Daniel Chiquito over at Equalis site (including a recommendation for Simon Singh's "The Code Book" as a good source of further learning):

http://www.equalis.com/members/blog_view.asp?id=565749&post=108495

Women In Math

Over a year ago on her blog Tanya Khovanova mentioned the numerical outcome of googling "male mathematicians" and "female mathematicians."  So I just did the same and similarly found a heavily-weighted favoritism toward females:

"male mathematician" --- 3420 hits
"female mathematician" --- 8990 hits

At first glance of course this seems peculiar given the preponderance of male mathematicians in society over females, but of course upon a moment's reflection it's clear that someone doing a search would be far more likely to "search" for find "female mathematicians" than "male mathematicans" (which is, at least subconsciously, for a lot of Americans, almost a redundancy!).

At any rate, many are working hard today, to alter this gender-bias and specifically encourage females to pursue mathematics, at least as an interest, if not even a vocation. One organization active in that endeavor is the Association For Women In Mathematics:

http://www.awm-math.org/

Check 'em out, especially if you're interested in math, and you're a possessor of two 'X' chromosomes!

Prime Numbers Always Intrigue

In 1968, a Russian mathematician conjectured that, on average, the number of prime numbers for which the sum of their digits was even was equal to the number of prime numbers for which that sum was odd.

The conjecture has now been proven true, by French mathematicians:

http://www.physorg.com/news192907929.html

Can't honestly say I understand what the significance of the finding is 8-\ but the article concludes thusly:

"The methods employed to arrive at this result, derived from combinatorial mathematics, the analytical theory of numbers and harmonic analysis, are highly groundbreaking and should pave the way to the resolution of other difficult questions concerning the representation of certain sequences of integers.
Quite apart from their theoretical interest, these questions are directly linked to the construction of sequences of pseudo-random numbers and have important applications in digital simulation and cryptography."

Pretty much any proof that involves prime numbers has a lot of potential consequences...