Mathematics
jrosenhouse | November 28, 2007I have in front of me an anthology of bridge (as in the card game) essays entitled For Experts Only, edited by Pamela and Matthew Granovetter. Essay number six was written by Phil Martin, and is entitled “The Monty Hall Trap.” Sounds interesting, but I am most definitely not an expert at bridge. In fact, I know nothing about the game beyond the basic rules. So I was hoping there was someone out there who could explain to me what Martin has in mind. Below the fold I have transcribed a lengthy excerpt from the article, starting at the beginning. All italics are in the original.
“Behind…
pontiff | November 15, 2007Uncertain Chad asks "What's your favorite dubious proof technique?" I just don't have one dubious proof technique: I have an entire book of dubious proof techniques! Seriously, I have a book where I write them all down.
But if I had to choose a dubious proof technique that was my favorite, it would have to be "proof by logical exhaustion." Now you might think that this means that I logically list all possibilities to prove something, a technique which is perfectly valid and not very dubious at all. No, no. "Proof by logical exhaustion" is where you put forth a chain of logical reasoning…
jrosenhouse | November 15, 2007In the Monty Hall problem, you are confronted with three identical doors, one of which conceals a car while the other two conceal goats. You choose a door at random, number one say, but do not open it. Monty now opens a door he knows to conceal a goat. He then gives you the option of sticking or switching. What should you do to maximize your chances of winning the car.
As we are all by now aware, the correct answer is that you double your chances of winning by switching doors. Most people find this counterintuitive on the grounds that after Monty opens a door, only two, equally likely…
jrosenhouse | November 7, 2007You might have noticed that blogging has been a bit erratic lately, and I have fallen off my usual pace of updating every weekday. There's a reason for that! Regular readers of this blog are aware that I have a small obsession with the Monty Hall problem. I managed to convince Oxford University Press that a book on the subject would be a good idea, and now I'm supposed to submit a manuscript to them by New Year's. And since there is a limit to how many hours a day I can spend typing away at a computer, well, you get the idea.
I've also learned something else lately. Writing a book is…
mikethemadbiologist | October 18, 2007Over at Karmatics, Rob Brown thinks the counter-intuitiveness of natural selection is a big reason why people find evolution difficult to comprehend. In that way, natural selection is similar to prediction markets, where people bet on the chances of future events, such as the outcomes of sports events or political elections:
Prediction markets turn out to be remarkably accurate, typically more accurate than any individual expert can predict, as non-intuitive as it may seem. Like Wikipedia, prediction markets also tap into the power of selection, but the most dramatic similarity they share…
jstemwedel | October 6, 2007Yesterday, one of the elder Free-Ride offspring's teammates brought a Rubik's Cube to soccer practice. While this youngster fiddled with the cube during a water break, I mentioned that I knew how to solve it. I was asked to transmit this knowledge, and I promised to write it up and send it to the player at this morning's soccer match.
And I thought, "You know, there are probably others who might like this information." So I made a quick detour to the scanner, and am sharing the very same information with you all.
I'm pretty sure that revealing this knowledge won't get me drummed out of the…
jrosenhouse | September 4, 2007As a companion piece to yesterday's post have a look at this essay in the religious periodical First Things, written by Amanda Shaw. The purpose is to draw a parallel between imaginary numbers and belief in God. You see, for centuries mathematicians scoffed at the idea of imaginary numbers, but a few brave folks were able to look beyond the stiflng orthodoxy of their times and now imaginary numbers are commonly accepted.
See where this is going?
Shaw presents a passable history of imaginary numbers. But if her intention is to develop a parallel between belief in imaginary numbers and…
jrosenhouse | September 3, 2007I think my calculus students would probably not think so. But as John Allen Paulos reports, not everyone agrees:
Consider first a Baptist school in Texas whose description of a geometry course begins:
Students will examine the nature of God as they progress in their understanding of mathematics. Students will understand the absolute consistency of mathematical principles and know that God was the inventor of that consistency. They will see God's nature revealed in the order and precision they review foundational concepts while being able to demonstrate geometric thinking and spatial…
jrosenhouse | August 1, 2007Sorry for the sporadic blogging. For the past week I've been working on the Progressive Monty Hall problem, and it has proven to be considerably more complicated than I at first realized. I had expected to polish it off with a few hours work. Instead I have thought about little else for the past several days, and have grown very unhappy with my inability to prove certain statements that are obviously true. Adding to my frustration is the nagging feeling that I am overlooking something simple and conclusive. So, I figured, why not turn it over to my readers?
I blogged about this…
jstemwedel | July 27, 2007Chad and Rob have already noted this piece of news about soon-to-be-published research indicating that the order in which high school students are taught physics, chemistry, and biology makes very little difference to their performance in science classes at the college level, while a rigorous math curriculum in high school gives their college science performance a significant boost.
I have a few things to say about this.
Good math instruction is good for students.
As Chad points out, it helps you build problem solving skills and think systematically. To the extent that these skills are…
jrosenhouse | July 5, 2007On Sunday I will be flying across the ocean to participate in the 2007 British Combinatorics Conference, at the University of Reading. If you peruse the book of abstracts, you will see that I will be delievering an edge-of-your-seat barn-burner of a talk entitled, “Decomposition Theorems for Cayley Graphs of the Modular Group Over a Finite Field.” Don't you wish you were going? Though the barn will already have burnt down by the time he speaks, my friend and collaborator Dominic Lanphier will be discussing related work in a rhetorical tour-de-force entitled, “Isoperimetric Sets of Cayley…
jrosenhouse | April 9, 2007First, the good news: I will be attending the spring section meeting of the Mathematical Association of America this Friday and Saturday. The meeting is being held at Roanoke College, and as you can see here, I'm giving one of the Big Shot talks. Lucky me! I'll be talking about the Monty Hall problem. In particular, I will show how you can teach an entire course in probability centered around nothing more than variations on the basic scenario. If you live anywhere near Roanoke, stop by for a visit!
Now, sadly, the bad news. The preparations for the conference, coupled with all of my…
jrosenhouse | April 2, 2007Paul Cohen, one of the giants of twentieth century mathematics, has died of lung disease at 72.
Cohen's major claim to fame was his resolution of the Continuum Hypothesis. Here are the basic ideas:
Suppose you have two finite sets and you want to show they contain the same number of elements. You might try to do that by counting the number of elements in each set. But another method would be to pair up elements of one set with elements of the other. If the sets run out at the same time, then you know they have the same number of elements.
As an example, suppose you have a group of…
evolvingthoughts | March 30, 2007
Here is a list of Basic Concept posts in Mathematics.
Recently Added: Fractals by Karmen at Chaotic Utopia; Innumeracy by Mark Chu-Carroll at Good Math, Bad Math
Statistics
Normal Distribution by Mark Chu-Carroll at Good Math, Bad Math
Mean, Median and Mode by Mark Chu-Carroll at Good Math, Bad Math
Standard Deviation by Mark Chu-Carroll at Good Math, Bad Math
Margin of Error by Mark Chu-Carroll at Good Math, Bad Math
Correlation (and Causation, and Random Variables) by Mark Chu-Carroll at Good Math, Bad Math
Percentage and percentage points by Kristjan Wager at Pro-Science
Statistics…
cevans | March 24, 2007
Here's a beautifully esoteric piece of math news: a team of mathematicians has meticulously explored and completely mapped a hitherto-unknown 248-dimensional structure, called E8. The E8 is an example of a Lie Group, which represent the best developed theory of continuous symmetry of mathematical objects and structures. Lie Groups underlie any symmetrical object.
From the Atlas Team's website:
"Lie groups come in families. The classical groups rise like gentle rolling hills towards the horizon. Jutting out of this mathematical landscape are the jagged peaks of the exceptional groups and,…
jrosenhouse | March 23, 2007In the mood for a good brain workout? Well search no farther! The fourth installment of the Carnival of Mathematics has arrived! Just look beneath the fold for some first rate math blogging:
Over at Universe of Discourse, Mark Dominus gets us started with some wise words about Bernoulli Processes. Also have a look at this post, in which continued fractions are employed in the service of the age old question: How old are you, really.
Then move over to Fightin' the Resistance of Matter for a heavy-duty post on homological algebra, from sirix. Brought back some bad memories from my…
jrosenhouse | March 13, 2007I'm a little late to the party, but do go have a look at the third installment of The Carnival of Mathematics over at Michi's Blog. Lots of good procrastination material!
The fourth installment is set to go up on March 23. And I will be the host! If you write any sort of math-related blog entry, let me know about it by Wednesday the 21st. Go here if you want to submit an essay, and many thanks to Alon Levy for making the submission process so simple. No more than two contirbutions per author plese. I've already received a fair number of submissions, and the deadline is still a ways away.
jrosenhouse | February 10, 2007Alon Levy of Abstract Nonsense has posted the inaugural edition of the Carnival of Mathematics. If you're looking for some great math blogging, I recommend having a look.
jrosenhouse | February 6, 2007Having just spent three hours explaining the value of trigonometric substitutions and partial fraction expansions to not very enthusiastic calculus students, I'm not really in the mood for a lengthy post today. So how about yet another variation on the Monty Hall problem.
In this version the contestant is shown 10 identical doors. One contains a valuable prize, the other nine contain goats. The contestant chooses one door. The host then opens a door he knows to be empty and gives the contestant the choice of switching to one of the remaining eight unopened doors. After the contestant…
jrosenhouse | January 23, 2007I was really impressed by this post from Polymathematics. He discusses a proof of Morley's Theorem, which is a result from Euclidean geometry. Start with any triangle. Trisect each of the three angles. Then the points of intersection of pairs of adjacent trisectors from the vertices of an equilateral triangle. Take one look at the pictures Polymathematics provides and you'll see what I mean. The details of the proof are ingenious, and not too hard to follow. Highly recommended.