Pi v. Tau

Summary
Pi is considered one of the most important mathematical constants. However, there is a growing movement that suggests that a different constant may be easier to use and easier to teach.

We've already computed trillions of digits of pi. Don't make me restart.
(Photo: Chris Blakeley on Flickr)

Commentary
The value of pi is a constant that relates the circumference of a circle (length around the circle) to its diameter (length through the center). In 2000, Bob Palis wrote a short article called Pi is Wrong! where he outlined arguments for a different circle constant: one that relates the circumference to the radius (length from the center to any point on the circle). In June 2010, Michael Hartl published Palis' arguments as The Tau Manifesto where he suggested that the new circle constant be represented by the Greek letter tau.

The main arguments are as follows:

1. While pi appears in many equations, it most frequently appears as 2pi. All instances of 2pi can be replaced by tau.
   
   
2. Measuring angles in radians is much more straightforward because there are tau radians in a circle (rather than 2pi radians).
   
   
3. The relationship between the trigonometric functions and the unit circle is easier to grasp.
   
   
4. Euler's identity ends up sounding even more powerful: e^it = 1 (A rotation by one turn in the complex plane is 1.)
   
   
5. The area of a circle is in quadratic form similar to many other physical phenomena where two values are proportional to each other. Examples:
   • Falling in a uniform gravitational field (velocity is proportional to time)
   • Potential energy in a linear spring (force is proportional to distance)
   • Energy of motion (force is proportional to acceleration)
   So now we can add: Area of a circle (area is proportional to radius).
   

The arguments are persuasive and merit thought, especially for the pedagogical benefits tau provides. I suspect it will be some time before anyone adopts this constant as a matter of course, but I have no problem writing "tau = 2pi" and moving on from there.

See Also

• Pi is Wrong! by Bob Palis for the original paper.
• The Tau Manifesto by Michael Hartl for why tau ought to be the new circle constant.
• Turn at Wikipedia for a historical discussion of using a turn as a unit of rotation.

Disease Screening & Base Rate Fallacy

Definition
The base rate fallacy refers to the neglect of prior probability of the evidence that supports the conditional probability of a hypothesis.
(Based on Wikipedia)

(Photo: mayaevening at Flickr)

Commentary
A recent example is the controversy about breast cancer screenings.

Imagine that about 1% of women (1 in 100) have breast cancer. You have a diagnostic test that correctly detects cancer 85% of the time (i.e. if the test is given to 100 women with cancer, it catches 85, but misses 15 of them).

Also, the test sometimes incorrectly detects cancer (when none is present) about 10% of the time (i.e. if the test is given to 100 women without cancer, it accidentally tells 10 of them they have cancer, but correctly tells the other 90 they don't have cancer).

Now the tricky bit: Imagine we give the test to 1,000 women in the population. If the test says a women has cancer, what is the probability she actually has cancer?

This question is hard for many people (including doctors!) because it's hard to make the trade-offs in our head about whether or not the test is accurate for this particular woman. Here's how you would do the calculation correctly:

1. Based on the rate of cancer in the population (1%), how many of the 1,000 women tested do we expect to have cancer?
   Answer: About 10.
   
   
2. Of those 10 who have cancer, about 9 will be told they have cancer, and 1 will missed. (Recall, the test only catches 85% of cancers.)
   
   
3. Now of the remaining 990 women who don't have cancer, about 99 of them will be told they have cancer (10% false-alarm) while the rest (891) will be correctly told they don't have cancer.
   
   
4. So how many women are told they have cancer?
   Answer: 9 + 99 = 108.
   
   
5. How many of those women actually have cancer?
   Answer: Just the 9.
   
   
6. So if you're told you have cancer, what's the chance you actually have cancer?
   Answer: 9 / 108 = 8.3%

Pretty strange, right? What about the people who are told they don't have cancer? What's the probability you actually do have cancer?

1. How many women are told they don't have cancer?
   Answer: 1 + 891 = 892.
   
   
2. How many of those actually have cancer?
   Answer: Just the 1.
   
   
3. So if you're told you don't have cancer, what's the chance that you actually do have cancer?
   Answer: 1 / 892 = 0.1%

That means that it's pretty unlikely for you to have cancer if the test says you don't.

The reason this occurs is because the number of women who have breast cancer to begin with is not that high (10 of 1,000). Therefore, the mistakes the test makes start to matter when applied to the entire population.

Naturally, this has policy implications: if you test more and more people, a large percentage of people will be told they have cancer when they don't-- leading to more invasive testing that has other real side-effects. The trick is either to try to test a high-risk subpopulation (where the prevalence rate is higher) or to improve the test by reducing its false-positive rate.

See Also

• Screening for disease and dishonesty at Understanding Uncertainty for an excellent interactive representation of base rates.
• Screening for Breast Cancer at the U.S. Preventive Services Task Force for a summary of its recommendations.
• The base rate fallacy reconsidered for a counter-argument about base rate neglect.

Review: The Autonomy of Mathematical Knowledge

(Image: Amazon)
Review
Curtis Franks' (full disclosure: he's is a friend of mine) PhD-turned-book Autonomy of Mathematics: Hilbert's Program Revisited is an exciting new look at an overlooked aspect of early twentieth century mathematics. Franks' writing is crisp and engaging, as he paints the picture of a man and his philosophy that so many have spurned.

The face of a brilliant mathematician or of a sun-hat enthusiast.
(Photo: Wikimedia)

Commentary
In the 1920's, Hilbert launched a program that was ostensibly aimed at solving the foundational crisis of mathematics-- the issues of paradoxes (e.g., Russell's paradox). The traditional understanding is that Hilbert's program failed because Gödel's incompleteness theorems threw a monkey wrench into any sufficiently sophisticated system that tried to prove itself.

Franks' thesis is that this is a narrow understanding of Hilbert's goals. While Gödel's results did complicate certain endeavors, Franks' suggests that Hilbert was really trying to take back mathematics. That is, certain other endeavors were trying to resolve the foundational crisis by rooting mathematics in some other discipline (e.g., philosophy). Hilbert's goal was to keep mathematics strictly within the realm of mathematics-- a unique feature of the discipline.

The Autonomy of Mathematical Knowledge is admittedly not for everyone (perhaps not even for me)-- about 20% of the book involves theorems I faithfully assume describe what the surrounding text tells me they do. Yet, about 80% of the book is eminently accessible-- the historical context, the epistemic issues, and the attempt to reconstruct a neglected approach combine for a great read.

See Also

• Russell's paradox for our discussion of that issue.
• Computational Theory for Lawyers for our discussion of the issues of intensionality (discussed in the book; not to be confused with intentionality).

fMRI & False Positives

This post is based on a submission by reader Professor Daniel Bitran. Please submit suggestions for posts to metaist.blog@gmail.com.

Definitions
fMRI is a way of measuring blood-flow in the brain or spinal cord and, by extension, neural activity in those areas.

A false positive is the sort of mistake your smoke detector makes when it goes off, but there's no smoke.

Summary
Despite the widespread use of fMRI, a few false positives may result in inaccurate results.

The fMRI is showing neural activity, but the salmon is definitely dead.
(Image: Courtesy of Prefrontal.org)

Commentary
The image above is striking because the false positives seem to show neural activity in a dead salmon's brain. According to researchers at UCLA Santa Barbra, these errors are due to a problem of multiple comparisons.

Imagine we're playing One of These Things is not Like the Others with several tin cans of Atlantic salmons. At first, it's hard to tell which one is not like the others -- there's a bunch of canned salmon. They have similar color, weight, shape, etc. But as we add different ways of comparing the cans (or more cans to compare), we increase the probability that there will be some way in which one of them differs from the rest -- particularly because of small differences, say manufacturing defects.

[Note: Corrections appreciated.] Now imagine we're collecting data for an fMRI. Each each point (called a voxel) is measured several times with certain extreme values discarded. Now we want to figure out which of the voxels is not like the others -- that's because that's where we expect to see differences in blood flow. However, by comparing voxels we're actually comparing multiple measurements of each voxel to multiple measurements of other voxels. This is like adding more ways of comparing the cans. Moreover, because there is a small bit of noise, the measurements for each voxel can be slightly different each time. This is why we occasionally find some difference between neighboring voxels that isn't really there -- it's a false positive. Luckily there are ways of correcting for this sort of error, but unfortunately, it is not applied as frequently as it should.

Meta
What are other examples of widespread errors of multiple comparisons or false positives?

Acknowledgements
Thanks to Craig Bennett of Prefrontal.org for providing a high resolution version of the Atlantic salmon fMRI.

See Also

• Human Brain Mapping 2009 – Presentations at Prefrontal.org for the conference posters & slides.
• Multiple Comparisons with Repeated Measures by David Howell for a lengthy discussion on the subject.
• The Importance of Numeracy

Zeno's Paradox

Summary
Zeno's Paradox involves a race between a tortoise and Achilles that suggests that motion is an illusion.

This is not a picture of Zeno of Elea.
(Photo: Wikimedia)

Commentary
I remember when I was first introduced to this paradox in 8th grade; it was utterly puzzling (until I heard a solution). A simplified version goes something like this:

Achilles and the tortoise are having a 1000 paces race, but the tortoise has a head start of 800 paces (Achilles is much faster than the tortoise). As the tortoise inches forward, Achilles makes a plan.

First, he'll get to the halfway point (let's call it Bob) between himself and the tortoise. Of course, to get to Bob, Achilles realizes, he has to get to the halfway point between himself and Bob (let's call it Jane). And before he can consider anything else, he must first get to the halfway point between himself and Jane (called Sam). [...]

As Achilles continues to think through his plan, he realizes he can never even catch up to the tortoise, let alone win the race.

When viewed abstractly, the problem seems to show that nothing can ever pass anything else-- that is the motion is an illusion. Until the early 20th century, there wasn't really a way to handle this paradox appropriately. However, with the advent of infinite series, we can say that the reason Achilles does pass the tortoise is because if you add up all the little pieces (1/2 + 1/4 + 1/8 + 1/16 ...) you get 1 (which represents the total distance between Achilles and the tortoise).

Fun fact: Zeno's paradoxes are considered some of the earliest examples of reductio ad absurdum, also known as proof by contradiction.

Russell's Paradox

Summary
Russell's paradox describes a fundemental conundrum with set theory that is sometimes illustrated by way of a story about a barber.

(Photo: Wikimedia)

Paradox
This presentation is a modified version of Russell's original presentation.

There once was a platoon of men who were very punctilious about following orders. One day, the commander decided that the men needed to look less disheveled. One of the men, a barber named Bob, proposed that he shave everyone everyday. Some of the men complained and asked if they could shave themselves instead. As a compromise, the commander ordered Bob to shave all and only the men who do not shave themselves.

The paradox is this: who shaves Bob? If he shaves himself, then he ends up being prohibited from shaving himself. In which case he must shave himself. Continue ad infinitum.

Commentary
The significance of this paradox was that it undermined the existence of certain types of sets, potentially undermining all of set theory. There were some interesting responses, but ultimately Gödel showed the inescapable nature of such paradoxes for most logical systems.

The Importance of Numeracy

Definition
Numeracy is a portmanteau of "numerical literacy" and refers to the understanding of mathematical concepts such as orders of magnitude, probability, and statistics.

(Photo: Wikimedia)

Conjecture
Certain concepts in math are necessary for critical thinking. Many people, however, struggle to learn these concepts well thereby reducing the effectiveness of their policy choices.

Commentary
In a country of over $12 trillion dollars deficit (and counting), about 80% of people cannot concieve of the magnitude of a trillion. Very small quantities are equally confusing, because people are unaccustomed to seeing them in their everyday lives. However, thanks to wonderful videos and interactive comparisons you can get a better sense of how orders of magnitude work.

Of course, that's only a very small part of overall numeracy. Probability and statistics, it can be argued, play an important role in everyday discourse, especially as we continue to be bombarded by facts and figures of every sort. When I was younger, someone told me a joke:

Two boys are walking home from school and one asks the other,
"What's the chance that I'll see a man riding a dinosaur in the street?"
His friend thinks for a moment, and responds,
"Fifty percent. Either you will or you won't."

At the time I recall laughing very hard. As I got older, this joke became less funny for I encountered more and more people who surprised me with their ignorance and actually maintain variations of this sort of thinking.

I currently do not have any solutions to this problem, yet I do not believe it unsolvable. Part of the issue may lie with trying to solve the wrong problem. But that will have to be another post.

More on this later.

Computational Theory for Lawyers

Definition
Computational Theory is a branch of mathematics that morphed into computer science. Among its goals is to look at a problem and ask: Is there a series of steps that will solve this problem? Those steps are also known as an algorithm, and finding a fast & efficient algorithm for something people care about, such as searching for a piece of information, is often a great way to start a business.

(Photo: Wikimedia)

Commentary
Groklaw has an interesting (and very long) discussion about computational theory and how it applies to patent law. From the article:

For example consider I write a program. Some outside party has written a similar program. This party sees my program and thinks I infringe on his proprietary rights. There are two scenarios depending on whether he makes his claim according to copyright law or patent law.

For claims of copyright infringement the text of the source code matters. If I wrote my program independently and I can prove the texts are different, I don't infringe on his rights. This is an intensional point of view.

For claims of patent infringement then differences in the text of the source code won't matter. It won't even matter if the code is written in a different programming language. If my program uses the same method that is covered by the patent according to whatever legal test of "same method" is applicable, I will infringe. This is an extensional perspective.

The notion of an intensional and extensional perspectives is actually an amazing insight into the world of mathematics that I hope to discuss more as I make my way through the excellent book Autonomy of Mathematical Knowledge by Curtis Franks. (Disclosure: the author is a friend of mine.)

Meta
The basic question is how can we say that two items or methods "are the same?" In what way are they the same? Computational theory suggests that two methods are the same if they are reducible to each other. That is, if under the same circumstances they both produce the same result, the methods are considered the same.

More on this later.

See Also (Updated 2010-02-24)

• Review: The Autonomy of Mathematical Knowledge for my review of that book.

Pareto Efficiency

Definition
If a change from X to Y results in at least one agent better off without making any agents worse off, we call Y a Pareto improvement relative to X. If no more Pareto improvements are possible, the situation is said to be Pareto efficient or Pareto optimal.

(Image: Wikipedia)

Meta
When thinking about a proposed Pareto improvement, consider the resistance to change on the part of the people who will implement your proposal. You may realize that your proposal is no longer a Pareto improvement because of the non-monetary costs involved.