hudebnik: (Default)
President Obama took years after entering the White House to publish a paper in a refereed academic journal.

President Trump seems to be on his way after only two weeks, with an illuminating example of the Alternative Facts approach to mathematical proof.
hudebnik: (devil duck)
I was sitting in the front row at some kind of math colloquium, where a woman was describing something called "abstract pairs" of matrices. (Even in the dream, I thought this was a lousy name, as there was nothing abstract about them.) The defining property had to do with the sums of various 2x2 sub-matrices, and she demonstrated how, if this property held, any 2x2 submatrix that appeared in one quadrant of the matrices must also appear in the other three quadrants, or something like that. I asked a question to clarify: "so, if I understand this, abstract pairs are really rare, but if you've got one, you only need to know what's in one quadrant and that tells you everything?" The speaker replied "No, not quite," but before she could explain why not, a woman in the back row had a question of her own, which she introduced by saying "I was the one who replied to your paper in the Bratislava Proceedings with 'I love you!'" And I don't remember what her question was. And then I woke up.

Why I'm having dreams about a math colloquium in the first place, much less one about this particular class of matrices, I don't know....
hudebnik: (teacher-mode)
Two weeks ago, Howard Dean wrote an op-ed in the Times favoring Instant Runoff Voting, and a number of readers weighed in. (I didn't get a letter in within the Times's 7-day window.) Two supported Dean's call for "ranked voting, aka instant runoff voting"; two supported "approval voting, a simpler system that voids [various weird properties of IRV]"; one supported voting under the current system for your favorite candidate, even if that candidate has no chance of winning; one supported proportional representation [although how that applies to a directly-elected President is unclear]; one pointed out that the system isn't going to be changed by vote of people who were elected under the current system; and one makes the peculiar argument that IRV fails "even when the sequential elimination of weaker candidates whittles the number down to two. The reason is that voters who supported weaker candidates can have all their preferred candidates eliminated, so in the end these voters are not counted in the contest between the final two."

I find this last argument peculiar because it's the opposite of an argument I was going to make. Voters whose first choice got knocked out early do have their votes counted in the contest between the final two. The problem is voters whose first choice is one of the final two: nobody ever even looks at their preferences other than the top. Consider an election among Alice Awesome, Peter Prettygood, Oliver OK, and Dr. Evil. If Awesome and Evil are the final two, and Alice is your first choice, it doesn't matter whether you put Dr. Evil in second, third, or fourth place because nobody will ever look at that part of your ballot. IRV pays attention to whom you like, not whom you dislike.

Related to this is another feature of IRV: it tends to favor divisive, polarizing candidates over broadly-acceptable, unifying candidates. Suppose, for example, in the above election Peter Prettygood was the second choice of every single voter in the country, although they divided bitterly between Alice Awesome and Dr. Evil for first choice. The argument could be made that Peter Prettygood is the best candidate to lead the country, because everybody finds him acceptable -- yet Peter Prettygood is the first one knocked out of the race because he's not anybody's first choice.

None of the readers seemed to question the equation of "ranked voting" with "instant runoff voting". In fact, "ranked voting" is about how you cast your vote, while "instant runoff" is one of several possible ways to count ranked votes, the other two leading ones being Borda and Condorcet. Condorcet says "for each pair (A,B) of candidates, how many people prefer A over B?" and declares the election for whichever candidate is preferred over the largest number of other candidates. Borda says "your first-choice candidate gets 4 points, second choice 3 points, third choice 2 points, and fourth choice 1 point" and declares the election for whichever candidate gets the most points. Both of these systems pay attention to all of your ballot, not just your first choice, and both of them would be likely to elect Peter Prettygood in the above scenario.

Of course, both Borda and Condorcet have better mathematical properties, e.g. "if your preferences are the exact opposite of mine, then your vote and mine exactly cancel one another out", which isn't true of IRV or single-vote plurality. But that's perhaps of more interest to mathematicians than the general public.
hudebnik: (devil duck)
[ profile] shalmestere brought home an advance reader's copy of The Math Myth (to be published in March 2016), thinking I might be interested. The main point of the book is that American schools, from middle school through college, require students to take a lot of math courses that they will never use, that do not make them better thinkers, that do not increase their earning ability or their competence as citizens of a democracy, but that do serve to filter out and discourage a lot of kids from reaching their potential in non-mathematical fields.

Which is almost certainly true. I've actually had occasion to use a little bit of calculus on the job (in doing analysis of algorithms), and in answering idle-curiosity physics problems, and I've used trigonometry to design tents, and I've used both trigonometry and linear algebra to write graphics programs, but 99.99% of the U.S. population will never need to do any of those things, either on the job or in private life. Most people need to be able to do arithmetic (with the aid of a calculator, but they need enough of a feel for numbers that they can tell whether the answers are at all plausible), and read a graph, and it would be nice if they knew that correlation isn't causation, and what statistical significance means. As a logician, I'd like it if ordinary citizens knew that "not all cats are grey" is equivalent to "at least one cat is not grey", and that "if that's a duck, then I'm Henry Ford" is not equivalent to "if that's not a duck, then I'm not Henry Ford".

Anyway, the author documents vast numbers of students whose only academic problem is an inability to pass such irrelevant math classes (middle school, high school, or college), but who are denied the opportunity to study Shakespeare or Swahili or spot-welding. He points out that most of the doomsaying about an imminent shortage of STEM-qualified workers comes from employers who have a strong vested interest in creating an overabundance of such workers. And he likes to illustrate things with sample test questions.

Here's a question he likes:

A rectangular-shaped fuel tank measures 27-1/2 inches in length, 3/4 of a foot in width, and 8-1/4 inches in depth. How many gallons will the tank contain? (231 cubic inches = 1 gallon)

(a) 7.366 gallons
(b) 8.839 gallons
(c) 170,156 gallons

He likes this because it tests "did you read the question carefully?" -- specifically, did you convert 3/4 of a foot into 9 inches -- and do you know what needs to be multiplied and what divided? He says if you failed to notice the "feet", you would get the incorrect answer (a) (in fact, he's misplaced a decimal point: you would get .7366 gallons, which you should also be able to rule out through common sense because a tank that big has got to hold more than a gallon!)

Here's a question he doesn't like:

Two charges (+q and -q) each with mass 9.11 x 1031 kg, are place 0.5 m apart and the gravitational force (Fg) and electric force (Fe) are measured. If the ratio Fg/Fe is 1.12 x 10-77, what is the new ratio if the distance between the charges is halved?

(a) 2.24 x 10-77
(b) 1.12 x 10-77
(c) 5.6 x 10-78
(d) 2.8 x 10-78

I have to confess I do like this question, because it doesn't require doing any arithmetic at all, only remembering that both gravitational and electrical forces follow an inverse-square law. Although if you have two masses of almost 1032 kg half a meter apart from one another, they're both black holes and you have bigger things to worry about than measuring the forces between them.

However, I don't see a lot of benefit in asking this question on an MCAT (which is where it allegedly came from). Yes, identifying the relevant and irrelevant features of a problem is important to a doctor, but physics isn't.

For a certain board game, two dice are thrown to determine the number of spaces to move. One player throws the two dice and the same number comes up on each of the dice. What is the probability that the sum of the two numbers is 9?
(a) 0
(b) 1/6
(c) 2/9
(d) 1/2
(e) 1/3

Again, he doesn't like this question, and I do: it requires no arithmetic, no probability, no combinatorics, only the ability to see past the irrelevant stuff to what matters.

Is this what they call a "trick" question? One that requires common-sense reasoning, not just the application of a memorized procedure? If so, I'm all for them.

Anyway, I've only read a quarter of the book; we'll see what else he has to say.
hudebnik: (teacher-mode)
When I was about about four years old and my mother was teaching me math, she mentioned at one point that numbers could be divided into "even" and "odd", and that even numbers were those you could get by doubling another number. I thought it was really cool that there could be different kinds of numbers that had their own names, and I resolved to make up my own: "even even numbers", which were even more even than regular even numbers. They were the numbers you could get by nothing but doubling over and over again: 2, 4, 8, 16, etc.

Decades later, I read in Isidore's Etymologies:

Numbers are divided into even and odd numbers. Even numbers are subdivided into these categories: evenly even, evenly odd, and oddly even.... An evenly even number is one that is divided equally into even numbers until it reaches the indivisible unity, as, for example, 64 has 32 at its midpoint; 32 has 16, 16 has 8, 8 has 4, 4 has 2, 2, has 1, which is an indivisible singularity. An evenly odd number is one that can undergo a division into equal parts, but then its parts cannot immediately be evenly dissected, like 6, 10, 38, 50. As soon as you divide this kind of number, you run into a number that you cannot cut evenly. An oddly even number is one whose parts can be divided equally, but the division does not go to the point of one, like 24.

(From The Etymologies of Isidore of Seville, transl. by Stephen A. Barney, W. J. Lewis, J. A. Beach, and Oliver Berghof, ISBN 978-0-521-83749-1, Cambridge University Press 2006.)

OK, so Isidore beat me out by 1350 years....


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