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Happy New Year, everyone!

Sorry, there hasn’t been much activity on this blog over the past few months, but 2011 will be the year when you can expect a lot more stuff to be posted as I find more time to do so. Todd has been busy on other sites/blogs, especially MathOverflow and n-Category Cafe, but he is still an author/admin on this blog.

I got an email from WordPress, summarizing the level of activity on our blog for the year 2010. I thought it wouldn’t be so bad if I shared the information with you all.

# 2010 in review

The stats helper monkeys at WordPress.com mulled over how this blog did in 2010, and here’s a high level summary of its overall blog health:

The *Blog-Health-o-Meter™* reads Wow.

## Crunchy numbers

About 3 million people visit the Taj Mahal every year. This blog was viewed about **33,000** times in 2010. If it were the Taj Mahal, it would take about 4 days for that many people to see it.

In 2010, there were **2** new posts, growing the total archive of this blog to 122 posts.

The busiest day of the year was January 20th with **196** views. The most popular post that day was Continued fraction for e.

## Where did they come from?

The top referring sites in 2010 were **terrytao.wordpress.com**, **mathoverflow.net**, **en.wordpress.com**, **qchu.wordpress.com**, and **gilkalai.wordpress.com**.

Some visitors came searching, mostly for **imaginary numbers**, **platonic solids**, **euler’s formula**, **vertex**, and **challenging integrals**.

## Attractions in 2010

These are the posts and pages that got the most views in 2010.

Continued fraction for e August 2008

18 comments

Platonic Solids and Euler’s Formula for Polyhedra March 2008

10 comments

High IQ and Mathematics December 2007

Problem-Solving Hall of Fame! May 2008

5 comments

Integration Bee, Challenging Integrals January 2008

16 comments

Who doesn’t like self-referential paradoxes? There is something about them that appeals to all and sundry. And, there is also a certain air of mystery associated with them, but when people talk about such paradoxes in a non-technical fashion indiscriminately, especially when dealing with Gödel’s incompleteness theorem, then quite often it gets annoying!

Lawvere in ‘Diagonal Arguments and Cartesian Closed Categories‘ sought, among several things, to demystify the incompleteness theorem. To pique your interest, in a self-commentary on the above paper, he actually has quite a few harsh words, in a manner of speaking.

“The original aim of this article was to demystify the incompleteness theorem of Gödel and the truth-definition theory of Tarski by showing that both are consequences of some very simple algebra in the cartesian-closed setting. It was always hard for many to comprehend how Cantor’s mathematical theorem could be re-christened as a“paradox” by Russell and how Gödel’s theorem could be so often declared to be the most significant result of the 20th century. There was always the suspicion among scientists that such extra-mathematical publicity movements concealed an agenda for re-establishing belief as a substitute for science.”

In the aforesaid paper, Lawvere of course uses the language of category theory – the setting is that of cartesian closed categories – and therefore the technical presentation can easily get out of reach of most people’s heads – including myself. Thankfully, Noson S. Yanofsky has written a nice paper, ‘A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points’, that is a lot more accessible and fun to read as well.Yanofsky employs only the notions of sets and functions, thereby avoiding the language of category theory, to bring out and make accessible as much as possible the content of Lawvere’s paper. Cantor’s theorem, Russell’s Paradox, the non-definability of satisfiability, Tarski’s non-definability of truth and Gödel’s (first) incompleteness theorem are all shown to be paradoxical phenomena that merely result from the existence of a cartesian closed category satisfying certain conditions. The idea is to use a single formalism to describe all these diverse phenomena.

(Dang, I just found that John Baez had already blogged on this before, way back in 2006!)

Sorry, there hasn’t been much activity on this blog lately – which is an understatement, I acknowledge! But, for now, here’s a small article by Terry Tao on CNNOpinion.

Among several things, these days, I have been doing some (serious) reading of literature on psychology, cognitive development, learning, linguistics, philosophy and a few other subjects. Well, the ones I just named happen to be parts of interdisciplinary areas, which are precisely the ones I am interested in. Of course, on many levels those parts also have a lot to do with mathematics, especially mathematical education. Ok, that was just a little background I wanted to provide for the content of today’s post.

I need to do a small (online) experiment in order to test a hypothesis, which will be the subject of my next post. Let me not reveal too much for now. The experiment is in the form of two puzzles that I ask readers (you!) to solve. They are both “multiple choice” puzzles with exactly **two** correct answers to each. Please bear in mind that this is NOT an IQ test. It is also not meant to test how good you are at solving puzzles individually. I am really interested in “aggregate” results. That is, for testing my hypothesis, I am only interested in what the majority thinks are the right answers. What is more, you won’t be graded, and no one (not even me) will ever know if you got the right answers. Please submit answers to both the puzzles.

Lastly, please don’t cheat or try to look for answers offline or online. As I said, this is NOT a test!

Let us now look at the puzzles.

**Puzzle 1:**

There are four cards, labelled either X or Y on one side and either 3 or 7 on the other. They are laid out in a row with their top (visible) sides shown like this: X Y 3 7. A rule states: “If X is on one side then there must be a 3 on the other.” Which two cards do you need to turn over to find out if this rule is true?

1) X

2) Y

3) 3

4) 7

**Puzzle 2:**

**As you walk into a bar, you see a large sign that reads, “To drink alcohol here you must be over 18.” There are four people in the bar. You know the ages of two of them, and can see what the other two are drinking. The situation is: Alisa is drinking beer; Dymphna is drinking Coke; Maureen is 30 years old; Lauren is 16 years old. Which two people would you need to talk to in order to check that the “over 18 rule” for drinking alcohol is being followed?**

1) Alisa

2) Dymphna

3) Maureen

4) Lauren

If you think you have the answers to those puzzles, then please click here **Puzzles** to submit your answers. (I couldn’t use PollDaddy to embed the above puzzles in this post because I am not allowed more than 160 characters in a single question. *What a pain!*) So, please go ahead and click the above link to submit yours answers.

**Note:** I will keep this “poll” open for a week to collect as much data as possible. Thanks!

**Update:** It seems many readers weren’t aware of the short duration of the “poll” and that they would have very much liked to participate. So, I am extending the poll till Jan 7, 2010 for them. (Doing so would also help me in collecting more data.)

The following piece of news, in my humble opinion, deserves more mention in the math blogosphere than it has garnered so far. At the center of the story is an Iranian student of mathematics, Mahmoud Vahidnia, who was invited to a meeting between Iran’s Supreme Leader Ayatollah Ali Khamenei and the country’s scientific elite on Oct 28, 2009. An excerpt from a Guardian report on what transpired during a (perhaps, routine) question and answer session:

“I don’t know why in this country it’s not allowed to make any kind of criticism of you,” he told Iran’s most powerful cleric, who has the final say in all state matters. “In the past three to five years that I have been reading newspapers, I have seen no criticism of you, not even by the assembly of experts [a clerical body with the theoretical power to sack the leader]. I feel that if this doesn’t happen this situation will lead to discord and grudge.”

Vahidnia, who achieved nationwide recognition two years ago by winning Iran’s annual mathematics Olympiad, made his remarks at a meeting between Khamenei and the country’s scientific elite. They came after the supreme leader asked at the end of a question-and-answer session if anyone else wanted to speak. He chose Vahidnia after seeing him being pushed down by officials when he stood to ask a question.

Referring to the post-election crackdown sanctioned by Khamenei, he asked: “Wouldn’t our system have a better chance of preserving itself if we were using more satisfactory methods and limited the use of violence only to essential circumstances?”

I discovered the above piece of news “accidentally” via the Colbert Nation. The Associated Press also did a report. I wonder if our Iranian readers (if there is any!) could furnish more information on this.

There is quite a buzz on the physics (and also math) blogospheres over the release of seven videotaped lectures, which were delivered by Richard P. Feynman as part of Cornell University’s *Messenger Lecture Series* of November 1964. The videos have been released by Microsoft Research with quite a few enhancements, though, I believe, they have been around on YouTube for quite some time.

I watched the first two video lectures, titled ‘*Lecture 1: The Law of Gravitation – An Example of Physical Law*‘ and ‘*Lecture 2: The Relation of Mathematics and Physics*‘. It goes without saying that they are spell-binding and brilliant! Of course, the textbook ‘The Feymnan Lectures on Physics‘ (which was followed later by a problem-solving supplement that I highly recommend) is such a joy to read, but if you wish to learn physics “face to face” from *the* master, then I exhort, nay implore, you to watch those video lectures.

(I came to know about the existence of the videos released by the Microsoft Research group from Terence Tao.)

High-school students and undergraduates are (almost) always taught the following definition of an *equivalence relation*.

A *binary relation* on a set is an *equivalence* iff it satisfies

- the
*reflexive*property: for all in , , - the
*symmetric*property: for all in , if , then , and - the
*transitive*property: for all in , if and , then .

However, there is another formulation of an equivalence relation that one usually doesn’t hear about, as far as I know. And, it is the following one.

A *binary relation* on a set is an *equivalence* iff it satisfies

- the
*reflexive*property: for all in , , and - the
*euclidean*property: for all in , if and , then .

**Exercise**: Show that a binary relation on a set is reflexive, symmetric and transitive iff it is reflexive and euclidean*.*

Welcome to the 54th Carnival of Mathematics, and Happy Fourth of July to our American readers! Indeed, the carnival should have been hosted yesterday, and I apologize for being a day late.

Trivia:Today, we have the 234th Independence Day celebrations in the US, and ours is the 54th carnival. 2+3+4 = 5+4, see? Boy, do I feel so clever!

Ok, let’s begin, now!

We start off with a post, submitted by Shai Deshe, that presents a collection of YouTube videos explaining different kinds of infinities in set theory, causality vs conditionality in probability and some topology. The videos are the kind of ones that “math people” could use to explain a few mathematical concepts to their friends, family members and colleagues who may not be enamored of math very much but may still possess a lingering interest in it.

Experimental philosophy, according to the Experimental Philosophy Society, “*involves the collection of empirical data to shed light on philosophical issues*“. As such, a careful quantitative analyses of results of experiments are used to shed light on many philosophical issues/debates. Anthony Chemero wrote a post titled, ‘What Situationist Experiments Show‘, that links to a paper with the same title that he coauthored with John Campbell and Sarah Meerschaert. In the paper, the authors, through quantitative analyses of actual experimental data, argue that virtue ethics has not lost to the siuationist side, whose critiques of virtue theory are far from convincing.

Next, I would like to bring the readers’ attention to two math blogs that came into existence somewhat recently and which I think have a lot of really good mathematical content. They are* Annoying Precision* and *A Portion of the Book*. In my opinion, their blog posts contain a wealth of mathematical knowledge, especially for undergraduates (and graduate students too!), who, if inclined toward problem-solving, will enjoy the posts even more. Go ahead and dive into them!

At *Annoying Precision*, a project aimed at the “Generally Interested Lay Audience” that Qiaochu Yuan started aims “*to build up to a discussion of the Polya enumeration theorem without assuming any prerequisites other than a passing familiarity with group theory*.” It begins with GILA I: Group Actions and Equivalence Relations, the last post of the series being GILA VI: The cycle index polynomials of the symmetric groups.

Usually, undergrads hardly think integrals have much to do with combinatorics. At *A Portion of the Book*, Masoud Zargar has a very nice post that deals with the intersection of Integrals, Combinatorics and Geometry.

Tom Escent submitted a link to an article titled, “Introduction to Nerds on Wall Street“, which actually provides a very small snapshot of the book named, *Nerds on Wall Street: Math, Machines and Wired Markets* whose author is David J. Leinweber. I haven’t read the book yet, but based on generally good reviews, it seems like it chronicles the contribution of Quant guys to Wall Street over the past several decades. Should be interesting to Math and CS majors, I think.

Let’s have a post on philosophy and logic, shall we? At *Skeptic’s Play*, there is a discussion on Gödel’s modal ontological argument regarding the possibility of existence of God. As someone who has just begun a self-study of modal logic, I will recommend Brian K. Chellas’ excellent introduction to the subject, titled Modal Logic: An Introduction.

Then, there is the *Daily Integral*, a blog dealing with solving elementary integrals and which I think may be particularly useful for high-school students.

Let me close this carnival by asking the reader, “What do you think is the world’s oldest mathematical artifact?” There are several candidates, and according to The Number Warrior, candidate #1 is The Lebombo Bone, found in the Lebombo Mountains of South Africa and Swaziland, that dates back to 35,000 BC!

That’s all for now! Thanks to everyone who made submissons.

I thought I would share with our chess-loving readers the following interesting (and somewhat well-known) mathematical chess paradox , apparently proving that , and the accompanying explanation offered by Prof. Christian Hesse, University of Stuttgart (Germany). It shows a curious connection between the well-known Cassini’s identity (related to Fibonacci numbers) and the chessboard ( being a Fibonacci number!). The connection can be exploited further to come up with similar paradoxes wherein any -square can always be “rerranged” to form a -rectangle such that the difference between their areas is either or . Of course, for the curious reader there are plenty of such dissection problems listed in Prof David Eppstein’s Dissection page.

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