Fermat's Dilemma

Is Algebra Necessary? Andrew Hacker, a political scientist writing in the New York Times, believes that too many students in high school and college are subjected to the ordeal of learning algebra. He spends much of the article claiming that algebra is both useless and harmful to most of the students forced to take this horrible subject. He goes on to point out that taking an algebra class often doesn't even lead to an appreciation of the cultural significance of mathematics as a whole:

I WANT to end on a positive note. Mathematics, both pure and applied, is integral to our civilization, whether the realm is aesthetic or electronic. But for most adults, it is more feared or revered than understood. It’s clear that requiring algebra for everyone has not increased our appreciation of a calling someone once called “the poetry of the universe.” (How many college graduates remember what Fermat’s dilemma was all about?)

He's absolutely correct here, college graduates don't remember anything at all about Fermat's dilemma - that's because it was never mentioned! He probably meant to say Fermat's Last Theorem, a widely known mathematical curiosity. The end of algebra by Alexandra Petri in the Washington Post is a clever and funny response to Hacker's screed.

Unintuitive Mathematics

Some mathematical methods can seem rather unintuitive in practice and worse, sometimes yield startling and uninintuive results - for example Goedel's incompleteness theorem and Cantorian set theory.  Terence Tao continues his discussion of mathematical techniques in The “no self-defeating object” argument, and the vagueness paradox.

Problem Solving Strategies in Real Analysis

Fields medallist Terence Tao let's us in on his problem solving strategies - many of which are specific to the subject he was teaching, Real Analysis, but some of them are more generic. 

Counterfactual Reasoning in Mathematics

In mathematics, we often assume something (not P), show that it leads to a logical contradiction, and then conclude that P is in fact true.  Mathematician Terence Tao has an interesting discussion of proof by contradiction and related techniques in The “no self-defeating object” argument, revisited.

Universality

A first draft of a non-technical article on universality by Fields medallist Terence Tao. Talks about universality: when important attributes of a large-scale system can be independent of how the system works at smaller scales.

The Mafia Game

The mafia game (or werewolf or assassin game) is a party game in which the organiser divides the players into two groups: the citizens and the mafia. The mafia are told who the other mafia are, but the citizens don't know. The game alternates between the "day" when all the players decide who to "lynch" and the "night" when just the mafia decide who to eliminate. The game ends when either all the citizens or all the mafia are eliminated.
The preprint A mathematical model of the Mafia game provides an analysis of strategy.

Probability Puzzles

There's an intersting discussion of some confusing Probability Puzzles on John Baez's new blog.

“I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?”

The "born on a Tuesday" information can't possibly make a difference, can it? So the answer should be same same as:

“I have two children. One is a boy. What is the probability I have two boys?”

Shoudn't it?

There's a thorough analysis at Some Thoughts on Tuesday's Child by Greg Egan.

More about the mathematician Perelman

He Conquered the Conjecture by John Allen Paulos in the New York Review of Books is a review of Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen a biography of the Russian mathematician Grigoriy Perelman who recently solved a long-standing problem in topology, the Poincaré conjecture.

First Clay Mathematics Prize Awarded

The First Clay Mathematic Prize was awarded to Grigoriy Perelman for solving the Poincare Conjecture.

Group Extensions

Some notes on group extensions by Fields Medallist Terance Tao, contains a "high-concept" overview of methods for extending mathematical spaces, a typical and powerful technique.

Logicomix

Logicomix is a graphic novel which features a fictionalized life story of the Philosopher/Logician Bertrand Russell. I thought the story was quite charming and did manage to give some nice hints about the controversies in the foundations of mathematics, including Russells' Paradox, one of my favorites. I've never tried to read Principia Mathematica (Russell and Whitehead's magnum opus) myself, though I once had a meeting at a professor's office who showed me a copy - he was a big fan of the work - it looked pretty hairy!

Zipf's Law

On the Universality of Zipf's Law

Zipf's law is the most common statistical distribution displaying scaling behavior. Cities, populations and firms are just a few of the examples of this seemingly universal law. Although many different models have been proposed, no general theoretical explanation has been shown to exist for its universality. Here we show that Zipf's law is, in fact, an inevitable outcome of a very general class of stochastic systems. Borrowing concepts from Algorithmic Information Theory, our derivation is based on the properties of the symbolic sequence obtained through successive observations over a system with an unbounded number of possible states. Specifically, we assume that the complexity of the description of the system provided by the sequence of observations is the one expected for a system evolving to a stable state between order and disorder. This result is obtained from a small set of mild, physically relevant assumptions. The general nature of our derivation and its model-free basis would explain the ubiquity of such a law in real systems.

Grigori Perelman Biography

I just read "Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century" by Masha Gessen a biography of Grigori Perelman, the eccentric Russian mathematician. Interesting depictions of the former Soviet system for training mathematicians and the international mathematics community.

Mandelbulbs

Mandelbulbs

Monty Hall and John Von Neumann walk into a bar ...

I recently read The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser .
Here's my version of the Monty Hall Problem.
The Basic Situation is as follows. There are two individuals: Monty Hall and Alice. There is a car; there are three closed curtains; and the car is behind one of the curtains, which hides it completely. There is nothing behind the other two curtains. The action proceeds as follows:
1. Monty hides the car behind one of the curtains (H); Alice has no idea which one.
2. Alice chooses a curtain (C), but it is left closed. Alice doesn't know yet whether she chose the car or not.
3. Monty opens one of the curtains (S) showing Alice what's behind it.
3a. Monty is not permitted to open curtain Alice's curtain C; S cannot equal C.
4. Alice finally chooses another curtain (F) which can be different than C or the same. Alice gets what's behind the curtain she finally chose. If F=H, Alice wins the car, otherwise Alice gets nothing.

We can nail this down so that it is an exercise in pure logic and probability for Alice by further specifying what Monty does at steps 1 and 3. Here are the Additional Stipulations.
1a. Monty chooses where to hide the car (H) by randomly picking one of the three curtains: each of the three curtains is equally likely.
3b. Monty will only show Alice an empty curtain in step 3. Monty never opens the curtain with the car. S cannot equal H.

Given these Additional Stipulations. the consequences of the Alice's choice in step 4 are completely unambiguous - however the results surprise many people. Alice's two main strategies are Stay (F=C) and Switch (F≠C≠S) Many people guess that Stay and Switch are equivalent and that Alice wins 1/2 the time either way. Surprisingly Stay only wins 1/3 of the time while Switch wins 2/3's of the time.

Suppose Alice elects to follow the following strategy: always choose curtain #1 in step 2 and always Stay with curtain #1 in step 4.
In step 1. Monty hides the car behind curtain #1 1/3 of the time.
In step 2. Alice always chooses curtain #1.
In step 3. Monty will open either curtain #2 or curtain #3. Note that this does not change the actual location of the car, it's still behind curtain #1.
In step 4. Alice will always choose curtain #1 again. Alice wins the car.

In step 1. Monty hides the car behind curtain #2 or curtain #3 2/3's of the time.
In step 2. Alice always chooses curtain #1, which is empty.
In step 3. Monty will open either curtain #2 or curtain #3. Note that this does not change the location of the car, curtain #1 is still empty.
In step 4. Alice will always choose curtain #1 again, which is empty. Alice gets nothing.

So we see that by following the strategy of always chosing curtain #1 both times, Alice only wins the car 1/3 of the time.

Suppose on the other hand Alice uses another strategy: in step 2. she always chooses curtain #1; in step 4. shes always Switches to the only other curtain which is still closed.

In step 1. Monty hides the car behind curtain #1 1/3 of the time.
In step 2. Alice always chooses curtain #1.
In step 3. Monty will always open curtain #2 or curtain #3. Note that this does not change the actual location of the car, it's still behind curtain #1.
In step 4. because Alice always switches she will choose either curtain #2 or curtain #3. But the car is still behind curtain #1 and Alice gets nothing.

In step 1. Monty hides the car behind curtain #2 1/3 of the time.
In step 2. Alice always chooses curtain #1, which is empty.
In step 3. Monty must open curtain #3 - because it is the only door which is empty and is not Alice's. That of course does not change the location of the car - which is still behind curtain #2.
In step 4. Alice always switches to curtain #2 - because it is still closed. Alice wins.

In step 1. Monty hides the car behind curtain #3 1/3 of the time.
In step 2. Alice always chooses curtain #1, which is empty.
In step 3. Monty must open curtain #2 - because it is the only door which is empty and is not Alice's. That of course does not change the location of the car - which is still behind curtain #3.
In step 4. Alice always switches to curtain #3 - because it is still closed. Alice wins.

So Alice always wins if Monty hid the car behind curtain #2 or curtain #3 and Alice always loses if Monty hid the car behind curtain #1. Perhaps suprisingly Alice wins 2/3's the time when she always switches.


There's another formulation of the problem which only uses the facts in the Basic Situation. The Additional Stipulations are not included. Surprisingly Alice can guarentee the same favorable outcome in the Basic Situation that was achievable with the Additional Stipulations! Monty is permitted to choose where to hide the car (H) in step 1. any way he likes. In step 3. he is permitted to choose which curtain to open (S) by any method, as long as he doesn't open curtain C (still forbidden by 3a). It doesn't matter how Monty makes his choices (as long as he obeys 3a), Alice can still win the car surprisingly often.

Here's a paper (in pdf) which explains the Game Theory approach to the Monty Hall Problem: Probabilistic and Game Theoretic solutions to The Three Doors Problem

On proof and progress in mathematics

On proof and progress in mathematics by Fields medallist William P. Thurston discusses the following question: "What do Mathematicians Accomplish?".

The Monty Hall Problem: A New Book

Two Doors and a Goat is a review in Science Magazine of The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser by Jason Rosenhouse.

Supersymmetry and Division Algebras

There's a surprising connection between supersymmetry , a hot topic in theoretical physics, and division algebras: the real, complex, quaternion and octonion number systems. Supersymmetry only works in 3,4,6 and 10 space time dimensions. That's because there's a cute little trick which allows us to represent an n+2 dimensional spacetime vector using one number from a division algebra and two additional real numbers. There are only four division algebras - which just happen to have 1,2,4 and 8 dimensions. John Baez Explains It All To You here.

Statistics - Levy Laws and 1/f noise

A unified and universal explanation for Lévy laws and 1/f noises

Lévy laws and 1/f noises are shown to emerge uniquely and universally from a general model of systems which superimpose the transmissions of many independent stochastic signals. The signals are considered to follow, statistically, a common—yet arbitrary—generic signal pattern which may be either stationary or dissipative. Each signal is considered to have its own random transmission amplitude and frequency. We characterize the amplitude-frequency randomizations which render the system output's stationary law and power-spectrum universal—i.e., independent of the underlying generic signal pattern. The classes of universal stationary laws and power spectra are shown to coincide, respectively, with the classes of Lévy laws and 1/f noises—thus providing a unified and universal explanation for the ubiquity of these classes of “anomalous statistics” in various fields of science and engineering.

Musical Numerology

Why the Kirnberger Kernel Is So Small discusses some interesting numerical coincidences in the most commonly used musical scale.