P!=NP?

Keeping track on what's going on:
http://rjlipton.wordpress.com/
http://michaelnielsen.org/polymath1/index.php?title=Deolalikar%27s_P!=NP_paper
http://en.wikipedia.org/wiki/Vinay_Deolalikar

Euler

|V| - |E| + |F| = 2

Pro-Choice vs Pro-Life

Surely, Facebook gives us a choice here:

Facebook: I support the right to choose one element from each set in a collection

Do you believe that an infinite product of nonempty sets should be nonempty? Do you feel that non-measurable subsets of the reals should exist? Or games of perfect information with no winning strategy for either player? Do you believe that every set should have a well-ordering, or that any poset in which every chain has an upper bound is entitled to a maximal element? Do nonzero rings have the right to a maximal ideal? Is every vector space entitled to a basis? Should fields have algebraic closures? Should products of compact topological spaces be compact, and countable unions of countable sets be countable? Do you want to be able to cut a sphere up into a finite number of pieces and reassemble them, with only rigid motions, into a sphere twice as large?

It's all possible if you're pro-axiom-of-choice!

versus

Facebook: The Axiom of Life (aka Negation of Axiom of Choice)

Do you have nightmares of being split apart, reassembled, and finding two of yourself? Do you wonder just how large is a non-measurable set, roughly speaking? Then this is the group for you, advocating the negation of AC (or at least replacement by the axiom of dependent choice if you actually need to prove a theorem for some reason).

Memorable Quotes by Erik Demaine

Memorable quotes from Lecture 2 of SMA 5503, attributed to Erik D Demaine.

• I have a feeling, by the end of the lecture, this blackboard will be totally stink.
  
• This is, again, divine inspiration. And, if you have a good connection to some divinity, you are all set. But it is a little bit harder for the rest of us mere mortals.
  
• It usually just works. That's the great thing about it. It provides you intuition for free. It tells you what the answer is pretty much.
• There are three kinds of mathematicians, those who can add and those who cannot, and I am the latter kind so I need your help.
• And, if we believe in fate and we see this three number recurrence, we know that we have the right answer.

Good Math, Bad Math

A week ago, I chanced upon a good blog - Good Math, Bad Math. Basically, its a blog that discusses a wide range of science, from computer science to mathematics and physics, but with an emphasis on mathematics. The blog is written by a really cool guy - Mark Chu-Carroll, a PhD Computer Scientist working at Google. In many ways, the blog offers a platform for intellectual discourse - criticising and highlighting bad/flawed science and mathematics. This is a reminder that the very notion of science, no matter how grand it may seem, is susceptible to those who either manipulate it for their own means (e.g. General Relativity is not about all things being relative [not absolute]... just as relativistic speed do not refer to speed of light being undefined.), or confuse it unintentionally.

Currently, I am reading Mark's recent articles on Graph Theory.

The Mathematical Century

Note: I shall devote this entire post to Mathematics.

I have been reading a book - "The Mathematical Century: The 30 Greatest Problems of the Last 100 Years", written by Italian mathematician Piergiorgio Odifreddi. The one I have been reading is a translated version (from Italian to English). The Mathematical Century provides interesting insight into the development and history of Mathematics in the past decade. By looking at some of the most famous problems in mathematics (eg Fermat's last theorem, Abel's Impossibility Theorem), Piergiorgio Odifreddi traces the development of mathematics since ancient times, from the days of the Geeks, to Pythagoreas and then to modern day mathematics, the progress of which has been largely influenced by Hilbert's problems and perhaps the unification of which by the Bourbaki group.

The book is written in prose-style, free of technical details (this refers to equations, symbols, but does not refer to mathematical terms). (However, some of the ideas presented are rather abstract.) Odifreddi starts by discussing the four main philosophical foundations of mathematics in the 1900s, mainly - Sets (ZFC, etc), Structures, Categories and Functions.

While I admit that I could only understand less than 40% of what was written in the book, the book provided insightful breadth, (rather than depth). Indeed, modern day mathematics is no longer limited to abstract algebra or topology. With the birth of the PC, computer-assisted proofs have become acceptable (eg the Four Colour Theorem) and the P vs NP complexity problem is indisputably one of the most open problem in computer science and mathematics.

Overall, Odifreddi offers a sagacious bird's eye view on mathematics from the perspective of mathematical problems.


Anyone looking forward to appreciating/understanding mathematics should also read A Mathematician's Apology (PDF), an essay written by G. H. Hardy in 1940. It is an essay that is not only reflective, but personal as well. In it, Hardy made one of his most famous observations:

No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game.

While one may agree or disagree with the above statement, the essay certainly proffers intriguing views.

Eulid Infinitude of Primes

I was browsing through sci.math on Google Groups when I came across the newest post... from Archimedes Plutonium... (yes! THE Archimedes Plutonium [his website])

And the topic of discussion is... proving that there are infinite primes ("Which is more flawed and erroneous, MathWorld's Euclid Infinitude of Primes or Wikipedia's")

Well... some people do have all the time in the whole universe...

Axiom of choice

I find the quotes in Wikipedia's article on axiom of choice very amusing.

The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma? — Jerry Bona

This is a joke that although the axiom of choice, the well-ordering principle, and Zorn's lemma are mathematically equivalent, most mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn's lemma to be too complex for any intuition.

Notice the use of the words intuitive, counterintuitive and too complex for any intuition. Even more fascinating, I also read that there is something called intuitionistic logic. Certainly, all these terms are not intuitive.

MathML in Firefox

I just found out that to view Mathematical Markup Language (MathML) properly in Firefox, you need to download the fonts.

As for IE 6, external plugins are needed to view MathML properly.

And if you want to check if your browser can support MathML, go to this test page.
For list of supported browsers, see this.

For not-so-apparent reasons, IE always seems to have problems following W3C's recommendations. I wonder if IE 7 will support XHTML 2.0 and MathML (unlikely, Microsoft will most likely leave it to external plugins to "demonstrate" the "flexibility" of IE).

I just looked at the IE blog.


If I ever get to install IE 7... the first thing I will do is to find a way to fix that MSN Search to Google Search.