Richard Weaver argues in Visions of Order that our privacy and dignity depend on our being rooted in space. He predicted that as people become less attached to a geographical place, privacy and dignity erode.
There is something protective about “place”; it means isolation, privacy, and finally identity. … we must again become sensitive enough to realize that “place” means privacy and dignity …
When Weaver wrote those words in 1964, he was concerned about physical mobility. Imagine what he would have thought of online life.
I find it interesting that Weaver links privacy and dignity. There is a great deal of talk about loss of privacy online, but not much about loss of dignity. The loss of dignity is just as real, and more under our control. We may lose privacy through a third party mishandling data, but our loss of dignity we often bring on ourselves.
Related post: Emily Dickinson versus Paris Hilton
Here are a couple details of UNIX history I ran across this week.
Why AT&T first licensed UNIX to universities:
At this time , AT&T held a government-sanctioned monopoly on the US telephone system. The terms of AT&T’s agreement with the US government prevented it from selling software, which meant that it could not sell UNIX as a product. Instead … AT&T licensed UNIX for use in universities for a nominal distribution fee.
And why later they turned it into a commercial product:
… US antitrust legislation forced the breakup of AT&T (… the break-up became effective in 1982) with the consequence that, since it no longer held a monopoly on the telephone system, the company was permitted to market UNIX.
Source: The Linux Programming Interface
For daily tips on using Unix, follow @UnixToolTip on Twitter.
Donald Knuth recommends using the symbol ⊥ between two numbers to indicate that they are relatively prime. For example:
The symbol is denoted
perp in TeX because it is used in geometry to denote perpendicular lines. It corresponds to Unicode character U+27C2.
I mentioned this on TeXtip yesterday and someone asked for the reason for the symbol. I’ll quote Knuth’s original announcement and explain why I believe he chose that symbol.
When gcd(m, n) = 1, the integers m and n have no prime factors in common and we way that they’re relatively prime.
This concept is so important in practice, we ought to have a special notation for it; but alas, number theorists haven’t agreed on a very good one yet. Therefore we cry: Hear us, O Mathematicians of the World! Let us not wait any longer! We can make many formulas clearer by adopting a new notation now! Let us agree to write ‘m ⊥ n ’, and to say “m is prime to n,” if m and n are relatively prime.
This comes from Concrete Mathematics. In the second edition, the text is on page 115. The book explains why some notation is needed, but it does not explain why this particular symbol.
[Correction: The book does explain the motivation for the symbol. The justification is in a marginal note and I simply overlooked it. The note says “Like perpendicular lines don’t have a common direction, perpendicular numbers don’t have common factors.”]
Here’s what I believe is the reason for the symbol.
Suppose m and n are two positive integers with no factors in common. Now pick numbers at random between 1 and mn. The probability of being divisible by m and n is 1/mn, the product of the probabilities of being divisible by m and n. This says that being divisible by m and being divisible by n are independent events. Also, independent events are analogous to perpendicular lines. The analogy is made precise in this post where I show the connection between correlation and the law of cosines.
So in summary, the ideas of being relatively prime, independent, and perpendicular are all related, and so it makes sense to use a common symbol to denote each.
From Kevin Kelly’s book What Technology Wants:
Our body size is, weirdly, almost exactly in the middle of the size of the universe. The smallest things we know about are approximately 30 orders of magnitude smaller than we are, and the largest structures in the universe are about 30 orders of magnitude bigger.
In C. S. Lewis’ book The Magician’s Nephew, the horse Strawberry becomes Fledge, the father of winged horses. It didn’t occur to me until today why Lewis chose that name. I just thought it was an odd, arbitrary choice.
This morning I saw something that referred to a bird as unfledged which made me suspect the base “fledge” had something to do with flight, which it does. I knew the word fledgling — a bird just beginning to fly — but I had not made the connection between fledglings and Fledge.
If you’d like to read another etymology post, see Cats, Calendars, and Connections.
I ran into a lot good links this week related to programming, so this is a specialized weekend miscellany.
How to become a Python guru
Unicode in Python, and how to prevent it
Modern Perl (free PDF book)
Languages in general
The three programming languages you need to know
How do I make this hard to misuse?
How long computer operations take
It’s usually a compliment to call someone a “wizard.” For example, the stereotypical Unix wizard is a man with a long gray beard who can solve any problem in minutes by typing furiously at a command prompt.
Here’s a different take on wizards. Think about wizards, say, in the Harry Potter novels. Wizards learn to say certain spells in certain situations. There’s never any explanation of why these spells work. They just do. Unless, of course, they don’t. Wizards are powerful, but they can be incompetent.
You might use wizard to describe someone who lacks curiosity about what they’re doing. They don’t know why their actions work, or sometimes even whether they work. They’ve learned a Pavlovian response to problems: when you see this, do this.
Wizards can be valuable. Sometimes you just need a problem solved and you don’t care why the solution works. Someone who doesn’t understand what they’re doing but can fix your problem quickly may be better than someone who knows what they’re doing but works too slowly. But if your problem doesn’t quite fit the intended situation for a spell, the wizard is either powerless or harmful.
Wizards can’t learn a better way of doing anything because “better” makes no sense. When you see problem A, carry out procedure B. That’s just what you do. How can you address problem A better when “solving A” means “do B“? Professional development for a wizard consists of learning more spells for more situations, not learning a better spell or learning why spells work.
Wizards may be able to solve your problem for you, but they can’t teach you how to solve your own problems.
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.
Consulting in differential equations
Derek Sivers tells how a mentor was able to teach him a semester’s worth of music theory in three hours. His mentor also prepared him to place out of four more classes in four sessions. He gives the details in his blog post There’s no speed limit. It’s an inspiring story.
However, Sivers didn’t go through his entire education this way. He finished his degree in 2.5 years, but at the rate he started he could have finished in under a semester. Obviously he wasn’t able to blow through everything as fast as music theory.
Some classes compress better than others. Theoretical classes condense better than others. A highly motivated student could learn a semester of music theory or physics in a short amount of time. But it would take longer to learn a semester of French or biology no matter how motivated you are because these courses can’t be summarized by a small number of general principles. And while Sivers learned basic music theory in three hours, he says it took him 15 years to learn how to sing.
Did Sivers’ mentor expose him to everything students taking music theory classes are exposed to? Probably not. But apparently Sivers did learn the most important material, both in the opinion of his mentor and in the opinion of the people who created the placement exams. His mentor not only taught him a lot of ideas in a short amount of time, he also told him when it was time to move on to something else.
It’s hard to say when you’ve learned something. Any subject can be explored in infinite detail. But there comes a point when you’ve learned a subject well enough. Maybe you’ve learned it to your personal satisfaction or you’ve learned it well enough for an exam. Maybe you’ve reached diminishing return on your efforts or you’ve learned as much as you need to for now.
One way to greatly speed up learning is to realize when you’ve learned enough. A mentor can say something like “You don’t know everything, but you’ve learned about as much as you’re going to until you get more experience.”
Occasionally I’ll go from feeling I don’t understand something to feeling I do understand it in a moment, and not because I’ve learned anything new. I just realize that maybe I do understand it after all. It’s a feeling like eating a meal quickly and stopping before you feel full. A few minutes later you feel full, not because you’ve eaten any more, but only because your body realizes you’re full.
From Economics in One Lesson:
… the whole of economics can be reduced to a single lesson, and that lesson can be reduced to a single sentence. The art of economics consists in looking not merely at the immediate but at the longer effects of any act or policy; it consists in tracing the consequences of that policy not merely for one group but for all groups.
Related post: One thing to remember in economics
I’ve never kept many icons on my desktop, and tonight I decided to reduce the number to zero. Deleting the recycle bin icon took a little research.
Windows Vista will let you simply delete the recycle bin but other versions of Windows will not.
On Windows 7 you can right-click on the desktop, select Personalize -> Change desktop icons, and uncheck the box for the recycle bin.
On Windows XP, you can edit the registry as described here. The registry changes will take effect next time you log in. [Update: unfortunately the link is no longer available.]
If you don’t want to edit your XP registry, you can right-click on the desktop, select the Arrange Icons By menu, and uncheck the Show Desktop Icons menu. However, this will hide all icons, not just the recycle bin, and will not let you see anything you drag to the desktop until you re-check Show Desktop Icons.
If you miss the recycle bin icon, it’s still in the file explorer on the left side.
Related post: Using Windows without a mouse
Yesterday I wrote about even perfect numbers. What about odd perfect numbers? Well, there may not be any.
I couldn’t care less about perfect numbers, even or odd. But I find the history and the mathematics surrounding the study of perfect numbers interesting.
As soon as you define perfect numbers and start looking for examples, you soon realize that all your examples are even. So people have wondered about the existence of odd perfect numbers for at least 2300 years.
No one has proved that odd perfect numbers do or do not exist. But people have proved properties that odd perfect number must have, if there are any. So far, although the requirements for odd perfect numbers have become more demanding, they are not contradictory and it remains logically possible that such numbers exist. However, most experts believe odd perfect numbers probably don’t exist. (Either odd perfect numbers exist or they don’t. How can one say they “probably” don’t exist? See an explanation here.)
Wikipedia lists properties that odd perfect numbers must have. For example, an odd perfect number must have at least 300 digits. It’s interesting to think how someone determined that. In principle, you could just start at 1 and test odd numbers to see whether they’re perfect. But in practice, you just won’t get very far.
A year is about 10^7.5 seconds (see here). If you had started testing a billion (10^9) numbers a second since the time of Euclid (roughly 10^3.5 years ago) you could have tested about 10^20 numbers by now. Clearly whoever came up with the requirement N > 10^300 didn’t simply use brute force. There may have been some computer calculation involved, but if so it had a sophisticated starting point.
Related: Applied number theory
I just got a review copy of Maths 1001 by Richard Elwes. As the title may suggest, the book is a collection 1001 little math articles. (Or “maths articles” as the author would say since he’s English.) Most of the articles are elementary though some are an introduction to advanced topics. Here’s something I learned from an article that was somewhere in the middle, the connection between perfect numbers and Mersenne primes.
Euclid (fl. 300 BC) proved that if M is a Mersenne prime then M(M+1)/2 is perfect. (A number is “perfect” if it equals the sum of its divisors less than itself. For example, 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. A Mersenne prime is a prime of the form 2n – 1.) Euclid didn’t use the term “Mersenne prime” because Mersenne would come along nearly two millennia later, but that’s how we’d state Euclid’s result in modern terminology.
The converse of Euclid’s result is also true. If N is an even perfect number, then N = M(M+1)/2 where M is a Mersenne prime. Ibn Al-Haytham conjectured this result in the 10th century but it was first proved by Leonard Euler in the 18th century. (What about odd perfect numbers? See the next post.)
I’ve enjoyed reading Maths 1001. I’ll flip through a few pages thinking the material is all familiar but then something like the story above will stand out.
Update: Richard Elwes informs me that his book is published under the title Mathematics 1001 in the US. My review copy was a British edition.
Related: Applied number theory
In 1916, Marshall Mabey was working on a subway tunnel under New York’s East River. Compressed air was pumped into the tunnel to keep the soft earth between the river and the tunnel from caving in. A crack formed in the tunnel ceiling and Mabey was blown through the crack, through the river, and 25 feet into the air. He fell back into the river and was rescued. He survived unscathed and said he planned to go right back to work. The original New York Times account of the blow out is available here.
Marshall Mabey’s story is amazing. But I also found his wife’s reaction remarkable even though I imagine it was unremarkable at the time.
Of course I know that Marshall is in danger every time he goes to work but all work is dangerous and my husband is as careful as he can be. His job is a good one and I am glad he has it.