Why math-fans really love set theory

Turns out, math fans dig set theory for almost exactly the same reason that some Christian fundamentalists absolutely hate it — all that messy uncertainty, which is either an affront to the idea of intelligent design or really, really sexy and fascinating, depending on your outlook.

At Nautilus, which is currently hosting an entire issue on topic of uncertainty, math professor Ayalur Krishnan writes about an idea in set theory that he calls "The Deepest Uncertainty". This is the Continuum Hypothesis — an idea that, paradoxically, can be proven to be unprovable and proven to be something you can't disprove. (And, with that, I've just typed the word "proven" so many times that it has lost all meaning in my brain.)

The uncertainty surrounding the Continuum Hypothesis is unique and important because it is nested deep within the structure of mathematics itself. This raises profound issues concerning the philosophy of science and the axiomatic method. Mathematics has been shown to be “unreasonably effective” in describing the universe. So it is natural to wonder whether the uncertainties inherent to mathematics translate into inherent uncertainties about the way the universe functions. Is there a fundamental capriciousness to the basic laws of the universe? Is it possible that there are different universes where mathematical facts are rendered differently? Until the Continuum Hypothesis is resolved, one might be tempted to conclude that there are.

Read the full story, which explains what set theory and the Continuum Hypothesis actually are. I could that here, but then this link would end up being as long as the story it's trying to link you to. Ahhhh, set theory.

Fabergé Fractals


Here's a mesmerizing gallery of "Fabrege Fractals" created by Tom Beddard, whose site also features a 2011 video of Fabrege-inspired fractal landscapes that must be seen to be believed. They're all made with Fractal Lab, a WebGL-based renderer Beddard created.

Fabergé Fractals by Tom Beddard, using his WebGL-based fractal engine, Fractal Lab. (via Colossal)

Unknown mathematician makes historical breakthrough in prime theory

Yitang Zhang is a largely unknown mathematician who has struggled to find an academic job after he got his PhD, working at a Subway sandwich shop before getting a gig as a lecturer at the University of New Hampshire. He's just had a paper accepted for publication in Annals of Mathematics, which appears to make a breakthrough towards proving one of mathematics' oldest, most difficult, and most significant conjectures, concerning "twin" prime numbers. According to the Simons Science News article, Zhang is shy, but is a very good, clear writer and lecturer.

For hundreds of years, mathematicians have speculated that there are infinitely many twin prime pairs. In 1849, French mathematician Alphonse de Polignac extended this conjecture to the idea that there should be infinitely many prime pairs for any possible finite gap, not just 2.

Since that time, the intrinsic appeal of these conjectures has given them the status of a mathematical holy grail, even though they have no known applications. But despite many efforts at proving them, mathematicians weren’t able to rule out the possibility that the gaps between primes grow and grow, eventually exceeding any particular bound.

Now Zhang has broken through this barrier. His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N. No matter how far you go into the deserts of the truly gargantuan prime numbers — no matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million.

The result is “astounding,” said Daniel Goldston, a number theorist at San Jose State University. “It’s one of those problems you weren’t sure people would ever be able to solve.”

Unknown Mathematician Proves Elusive Property of Prime Numbers [Erica Klarreich/Wired/Simons Science News]

(Photo: University of New Hampshire)

Life of astronaut Sally Ride honored in Kennedy Center tribute


American astronaut Sally Ride monitors control panels from the pilot's chair on the flight deck in 1983. Photo by Apic/Getty Images, via PBS NewsHour.

Tonight, PBS NewsHour science correspondent Miles O'Brien will serve as master of ceremonies in a Kennedy Center gala honoring the life and legacy of astronaut Sally Ride. The tribute will highlight her impact on the space program and her lifelong commitment to promoting youth science literacy.

Her Sally Ride Science organization reached out to girls, encouraging them to pursue careers in the Science, Technology, Engineering and Math (STEM) fields, where a gender gap persists.

At the PBS NewsHour website, read the column Miles wrote immediately following Ride's death in July 2012, 17 months after she was diagnosed with pancreatic cancer.

Death, be not infrequent

The oldest person in the world died this year. But don't worry if you missed the event. The oldest person in the world will likely die next year, as well. In fact, according to mathematician Marc van Leeuwen, an "oldest person in the world" will die roughly every .65 years.

Looking for mathematical perfection in all the wrong places

The Golden Ratio — that geometric expression of the Fibonacci sequence of numbers (1, 1, 2, 3, 5, etc.) — has influenced the way master painters created art and can be spotted occurring naturally in the seed arrangement on the face of a sunflower. But its serendipitous appearances aren't nearly as frequent as pop culture would have you believe, writes Samuel Arbesman at The Nautilus. In fact, one of the most common examples of mathematical perfection — the chambered nautilus shell — actually isn't. Even math can become part of the myths we tell ourselves as we try to create meaning in the universe.

Image: Golden Ratio, a Creative Commons Attribution (2.0) image from ernestduffoo's photostream

The mathematics of tabloid news

Leila Schneps and Coralie Colmez have an interesting piece at The New York Times about DNA evidence in murder trials, the mathematics of probability, and the highly publicized case of Amanda Knox. What good is remembering the math you learned in junior high? If you're a judge, it could be the difference between a guilty verdict and an acquittal.

Weird probabilities of non-transitive "Grime Dice"

Michael de Podesta has been doing the math on "Grime Dice" -- six sided cubes whose sides average out to 3.5, but whose face values are all radically different:

The interesting thing about these is that the odds of one die beating another are simple to calculate, but shift radically once you start rolling dice in pairs. It's a beautiful piece of counterintuitive probability math:

The amazing property of these dice is discernible when you use them competitively – i.e. you roll one dice against another. If you roll each of them against a normal dice then as you might expect, each dice will win as often as it will lose. But if you roll them against each other something amazing happens.

  • Dice A will systematically beat Dice B
  • Dice B will systematically beat Dice C

and amazingly

  • Dice C will systematically beat Dice A

So the fact that Dice A beats Dice B, and Dice B beats Dice C does not ensure that Dice A will beat Dice C. Wow!

And how about this: If you ‘double up’ and roll 2 Dice  A‘s against 2 Dice B‘s – the odds change around and now the B‘s will beat the A‘s ! Is that really possible? Well yes, and just to convince myself I wrote a Spreadsheet (.xlsx file) and generated the tables at the bottom of the article. If you download it you can change the numbers to try out other combinations.

Amazing Dice: Rediscovering surprise (via Hacker News)

Celebrate "Pi Day" by throwing hot dogs down a hallway

No, that's not a euphemism for anything. Buffon's Needle is an 18th-century experiment in probability mathematics and geometry that can be used as a way to calculate pi through random sampling. This WikiHow posting explains how you can recreate Buffon's Needle at home, by playing with your food.

Calculus-performing mechanical calculator

A clip from the Discovery Channel's Dirty Jobs program on tanneries demonstrates the workings of a calculus-performing mechanical calculator that measures the surface-area of irregularly shaped hides with a fascinating and clever set of gears, calipers and ratchets.

Dirty Jobs - Tannery Mechanical Surface Integrator (Thanks, Dad!)

Game theory and bad behavior on Wall Street

An opinion piece by Chris Arnade on the asymmetry in pay (money for profits, flat for losses), which he describes "the engine behind many of Wall Street’s mistakes" That asymmetry "rewards short-term gains without regard to long-term consequences," Chris writes in a new guest blog at Scientific American. "The results? The over-reliance on excessive leverage, banks that are loaded with opaque financial products, and trading models that are flawed." [Scientific American Blog Network]

The world's largest prime number — visualized

Philip Bump took the recently discovered 17-million-digit prime number and, six digits at a time, converted it into RGB colors. This is the result.

Neil deGrasse Tyson on pi and other constants

Both the Bible and the Indiana State Legislature have tried to redefine pi to equal something much more simple than 3.14159265358979323846264338327950 ...

86.54% liked this

Science blogger Matt Springer analyzes the surprisingly fascinating math behind Reddit upvotes.

Probability theory for programmers


Jeremy Kun, a mathematics PhD student at the University of Illinois in Chicago, has posted a wonderful primer on probability theory for programmers on his blog. It's a subject vital to machine learning and data-mining, and it's at the heart of much of the stuff going on with Big Data. His primer is lucid and easy to follow, even for math ignoramuses like me.

For instance, suppose our probability space is \Omega = \left \{ 1, 2, 3, 4, 5, 6 \right \} and f is defined by setting f(x) = 1/6 for all x \in \Omega (here the “experiment” is rolling a single die). Then we are likely interested in more exquisite kinds of outcomes; instead of asking the probability that the outcome is 4, we might ask what is the probability that the outcome is even? This event would be the subset \left \{ 2, 4, 6 \right \}, and if any of these are the outcome of the experiment, the event is said to occur. In this case we would expect the probability of the die roll being even to be 1/2 (but we have not yet formalized why this is the case).

As a quick exercise, the reader should formulate a two-dice experiment in terms of sets. What would the probability space consist of as a set? What would the probability mass function look like? What are some interesting events one might consider (if playing a game of craps)?

Probability Theory — A Primer

(Image: Dice, a Creative Commons Attribution (2.0) image from artbystevejohnson's photostream)