The grand unified theory of 19th century math

Posted on 11 November 2010 by John

The heart of 19th century math was the study of special functions arising from mathematical physics.

It is well known that the central problem of the whole of modern mathematics is the study of the transcendental functions defined by differential equations.

The above quote was the judgment of Felix Klein (of Klein bottle fame) in 1893. The differential equations he had in mind were the second order differential equations of mathematical physics.

Special functions were the core of 19th century math, and hypergeometric series were the grand unifying theory of special functions. (Not every special function is hypergeometric, but quite a few are.) And yet they’re hardly taught any more. I never heard of hypergeometric series in college, even though I studied differential equations and applied math. Later I encountered hypergeometric functions first in combinatorics and only later in differential equations.

It’s odd that what was “the central problem of the whole of modern mathematics” could become almost a lost art a century later. How could this be? I believe part of the explanation is that special functions, and hypergeometric function in particular, fall between two stools: too advanced for undergraduate programs but not a hot enough of a research area for graduate programs.

Earliest personal account of slavery

Posted on 31 August 2010 by John

According to William R. Cook, there is only one ancient account of slavery written by a slave that still survives: a letter written by Saint Patrick. We have many ancient documents that were written by slaves, but not documents about their experience of being a slave.

Patrick was born in Britain. He was kidnapped at age 16 and became a slave in Ireland. He served as a slave for six years before escaping and returning to Britain. Later he returned to Ireland as a missionary. Although there are many legends surrounding Patrick, historians generally agree that his autobiographical letter, now known as the Confession of St. Patrick, is authentic.

I was surprised to hear that there are no other extant autobiographies of slaves since there were many literate slaves in antiquity. Obviously slaves were not given the liberty to write about whatever they pleased, and slave owners would be unlikely to request candid biographies of their chattel. Still, I imagine some slaves wrote autobiographies, perhaps secretly. But it makes sense that such documents would not likely be preserved.

The lack of first-hand accounts of slavery may contribute to our rosy mental image of classical history. When we think of ancient Greece, we think of Plato and Aristotle, not the anonymous slaves who made up perhaps 40% of the population of classical Athens.

[Update December 2014: The information above comes from a Teaching Company course by William Cook. The original link is dead, and I don’t remember now which of his courses it was from.]

Computing before Fortran

Posted on 25 May 2010 by John

In the beginning was Fortran. Or maybe not.

It’s easy to imagine that no one wrote any large programs before there were compilers, but that’s not true. The SAGE system, for example, involved 500,000 lines of assembly code and is regarded as one of the most successful large computer systems ever developed. Work on SAGE began before the first Fortran compiler was delivered in 1957.

The Whirlwind computer that ran SAGE had a monitor, keyboard, printer, and modem. It also had a light gun, a precursor to the mouse. It’s surprising that all this came before Fortran.

Army surplus flamethrowers

Posted on 21 May 2010 by John

It’s hard to imagine the amount of equipment the US Army decommissioned after World War II: parachutes, diesel engines, Jeeps, etc. Apparently even flamethrowers were up for grabs.

An enterprising merchant in Quakertown, Pennsylvania ran a newspaper ad for military flamethrowers, pitching the weapon as a handy household gadget that “destroys weeds, tree stumps, splits rocks, disinfects, irrigates. 100 practical uses. $22 for 4 gal. tank, torch, hose.”

Source: Bright Boys: The Making of Information Technology, page 145.

Deconstructing Thomas Edison

Posted on 4 May 2010 by John

I’m reading Remarkable Engineers to write a review for a website. The prose is pretty bland, though it got spicier in the chapter on Thomas Edison. It seems the author felt he needed to take Edison down a notch.

The career of Thomas Edison was not that of a great man of science, or even that of an inventive genius … His only major scientific discovery was the fact that a vacuum lamp could act as a rectifier, passing only negative electric currents. … He was said to have invented the business of invention.

So Edison was an engineer rather than a scientist. This criticism seems odd in a book devoted to remarkable engineers.

Surely Edison was an inventive genius; he held over a thousand patents, more than anyone has ever held. That is not to say anyone believes he came up with over a thousand unprecedented ideas completely by himself. He built on the work of others. He coordinated the work of his employees. He took ideas that were not being used and commercialized them. Perhaps he was more of an entrepreneurial genius than a scientific genius, but he was a genius nonetheless.

Technology history quiz

Posted on 14 January 2010 by John

I was skimming through Big Ideas: 100 Modern Inventions the other day and was surprised at the dates for many of the inventions. I thought it would be fun to pick a few of these and make them into a quiz, so here goes.

Match the following technologies with the year of their invention.

First the inventions:

The computer mouse
Radio frequency identification (RFID)
Pull-top cans
Bar codes
Touch tone phones
Cell phones
Car airbags
Automated teller machines (ATM)
Magnetic resonance imaging (MRI)
Latex paint

Now the years:

1948
1952
1953
1963
1968
1969
1973
1977

Two of the years are used twice. Quiz answers here.

All examples taken from Big Ideas: 100 Modern Inventions That Have Transformed Our World

How the central limit theorem began

Posted on 5 January 2010 by John

The Central Limit Theorem says that if you average enough independent copies of a random variable, the result has a nearly normal (Gaussian) distribution. Of course that’s a very rough statement of the theorem. What are the precise requirements of the theorem? That question took two centuries to resolve. You can see the final answer here.

The first version of the Central Limit Theorem appeared in 1733, but necessary and sufficient conditions weren’t known until 1935. I won’t recap the entire history here. I just want to comment briefly on how the Central Limit Theorem began and how different the historical order of events was from the typical order of presentation.

A typical probability course might proceed as follows.

Define the normal distribution.
State and prove a special case of the Central Limit Theorem.
Present the normal approximation to the binomial as a corollary.

This is the opposite of the historical order of events.

Abraham de Moivre discovered he could approximate binomial distribution probabilities using the integral of exp(-x²) and proved an early version of the Central Limit Theorem in 1733. At the time, there was no name given to his integral. Only later did anyone think of exp(-x²) as the density of a probability distribution. De Moivre certainly didn’t use the term “Gaussian” since Gauss was born 44 years after de Moivre’s initial discovery. De Moivre also didn’t call his result the “Central Limit Theorem.” George Pólya gave the theorem that name in 1920 as it was approaching its final form.

For more details, see The Life and Times of the Central Limit Theorem.

Related links:

Cinco de Mayo and the world’s largest cake

Posted on 5 May 2009 by John

Today is Cinco de Mayo, the holiday that celebrates the Mexican army’s defeat of French forces at the Battle of Puebla on May 5, 1862.

Cinco de Mayo is unusual in that it is a Mexican holiday more popular in the United States than in Mexico. According to Wikipedia,

While Cinco de Mayo has limited or no significance nationwide in Mexico, the date is observed in the United States and other locations around the world as a celebration of Mexican heritage and pride.

Cinco de Mayo is a bigger holiday in Texas than Texas Independence Day. (Readers unfamiliar with Texas history may be surprised to learn that Texas was once a sovereign nation. The Republic of Texas existed for nearly a decade between gaining independence from Mexico in 1836 and joining the United States in 1845.)

Texas Independence Day, March 2, usually goes virtually unnoticed. However in 1986, the sesquicentennial, there was a big celebration in Austin. Activities included baking the world’s largest cake. The left-overs were distributed to the dorms at the University of Texas and so I had some of the cake. Quite a bit, actually. You might think that a cake baked for the purpose of setting a world record would be barely edible, but it was actually pretty good lemon cake.

Innovation I

Posted on 25 March 2008 by John

Innovation is not the same as invention. According to Peter Denning,

An innovation is a transformation of practice in a community. It is not the same as the invention of a new idea or object. The real work of innovation is in the transformation of practice. … Many innovations were preceded or enabled by inventions; but many innovations occurred without a significant invention.

Michael Schrage makes a similar point.

I want to see the biographies and the sociologies of the great customers and clients of innovation. Forget for a while about the Samuel Morses, Thomas Edisons, the Robert Fultons and James Watts of industrial revolution fame. Don’t look to them to figure out what innovation is, because innovation is not what innovators do but what customers adopt.

Innovation in the sense of Denning and Schrage is harder than invention. Most inventions don’t lead to innovations.

The simplest view of the history of invention is that Morse invented the telegraph, Fulton the steamboat, etc. A sophomoric view is that men like Morse and Fulton don’t deserve so much credit because they only improved on and popularized the inventions of others. A more mature view is that Morse and Fulton do indeed deserve the credit they receive. All inventors build on the work of predecessors, and popularizing an invention (i.e. encouraging innovation) requires persistent hard work and creativity.

Four characterizations of the normal distribution

Posted on 13 March 2008 by John

The normal distribution pops up everywhere in statistics. Contrary to popular belief, the name does not come from “normal” as in “conventional.” Instead the term comes from a detail in a proof by Gauss discussed below where he showed that two things were perpendicular in a sense.

(The word “normal” originally meant “at a right angle,” going back to the Latin word normalis for a carpenter’s square. Later the word took on the metaphorical meaning of something in line with custom. Mathematicians sometimes use “normal” in the original sense of being orthogonal.)

The mistaken etymology persists because the normal distribution is conventional. Statisticians often assume anything random has a normal distribution by default. While this assumption is not always justified, it often works remarkably well. This post gives four lines of reasoning that lead naturally to the normal distribution.

1) The earliest characterization of the normal distribution is the central limit theorem, going back to Abraham de Moivre. Roughly speaking, this theorem says that if you average enough distributions together, even if they’re not normal, in the limit their average is normal. But this justification for assuming normal distributions everywhere has a couple problems. First, the convergence in the central limit theorem may be slow, depending on what is being averaged. Second, if you relax the hypotheses on the central limit theorem, other stable distributions with thicker tails also satisfy a sort of central limit theorem. The characterizations given below are more satisfying because they do not rely on limit theorems.

2) The astronomer William Herschel discovered the simplest characterization of the normal. He wanted to characterize the errors in astronomical measurements. He assumed (1) the distribution of errors in the x and y directions must be independent, and (2) the distribution of errors must be independent of angle when expressed in polar coordinates. These are very natural assumptions for an astronomer, and the only solution is a product of the same normal distribution in x and y. James Clerk Maxwell came up with an analogous derivation in three dimensions when modeling gas dynamics.

3) Carl Friedrich Gauss came up with the characterization of the normal distribution that caused it to be called the “Gaussian” distribution. There are two strategies for estimating the mean of a random variable from a sample: the arithmetic mean of the samples, and the maximum likelihood value. Only for the normal distribution do these coincide.

4) The final characterization listed here is in terms of entropy. For a specified mean and variance, the probability density with the greatest entropy (least information) is the normal distribution. I don’t know who discovered this result, but I read it in C. R. Rao‘s book. Perhaps it’s his result. If anyone knows, please let me know and I’ll update this post. For advocates of maximum entropy this is the most important characterization of the normal distribution.

Related post: How the Central Limit Theorem began

History