There’s always room at the Hilbert Hotel

I stumbled upon this series on mind-blowing math facts from a couple of years ago.  It’s by Cornell mathematician Steven Strogatz and treats things like the weirdness of pi, the quirks of probability, Zeno’s paradox, and some of the fun things you can do with calculus.

(Homeschoolers and other educators, take note:  Recovering mathematics and its different applications is urgently needed today and is the missing piece of a truly classical education.  We are doing things with the language part, the trivium, but we now must bring back the mathematics part, the quadrivium, which is far more than just Saxon math.  What Strogatz does here is show that math is far more than memorizing tables and working out problems, showing that it is wonderful, mysterious, philosophical, and imaginative, something that students need to realize.)

Anyway, here he treats the mathematics of infinity, along with the paradox that some sets are more infinite than others:

Some of its [infinity's] strangest aspects first came to light in the late 1800s, with Georg Cantor’s groundbreaking work on “set theory.” Cantor was particularly interested in infinite sets of numbers and points, like the set {1, 2, 3, 4,…} of “natural numbers” and the set of points on a line. He defined a rigorous way to compare different infinite sets and discovered, shockingly, that some infinities are bigger than others.

At the time, Cantor’s theory provoked not just resistance, but outrage. Henri Poincaré, one of the leading mathematicians of the day, called it a “disease.” But another giant of the era, David Hilbert, saw it as a lasting contribution and later proclaimed, “No one shall expel us from the Paradise that Cantor has created.”

My goal here is to give you a glimpse of this paradise. But rather than working directly with sets of numbers or points, let me follow an approach introduced by Hilbert himself. He vividly conveyed the strangeness and wonder of Cantor’s theory by telling a parable about a grand hotel, now known as the Hilbert Hotel.

It’s always booked solid, yet there’s always a vacancy.

For the Hilbert Hotel doesn’t merely have hundreds of rooms — it has an infinite number of them. Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on. That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).

Now suppose infinitely many new guests arrive, sweaty and short-tempered. No problem. The unflappable manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on. This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests.

Later that night, an endless convoy of buses rumbles up to reception. There are infinitely many buses, and worse still, each one is loaded with an infinity of crabby people demanding that the hotel live up to its motto, “There’s always room at the Hilbert Hotel.”

The manager has faced this challenge before and takes it in stride.

First he does the doubling trick. That reassigns the current guests to the even-numbered rooms and clears out all the odd-numbered ones — a good start, because he now has an infinite number of rooms available.

But is that enough? Are there really enough odd-numbered rooms to accommodate the teeming horde of new guests? It seems unlikely, since there are something like “infinity squared” people clamoring for these rooms. (Why infinity squared? Because there were an infinite number of people on each of an infinite number of buses, and that amounts to infinity times infinity, whatever that means.)

This is where the logic of infinity gets very weird.

via The Hilbert Hotel – NYTimes.com.

Does it ever.   Including a set of guests that there is no room for after all.  Read the whole post and the whole series (which is reportedly coming out as a book).

If they can’t pass the test, get rid of the test

For all that I love my native Oklahoma, education is not one of its strong points.  Harold Cole, writing in the Daily Oklahoman, gives  an example of the mindset that keeps holding it back:

A group of school superintendents recently expressed concern that about 6,000 high school seniors won’t graduate this year because of mandatory end-of-instruction tests. While others attempt to ascertain why this problem exists in order to propose preventive measures, Rep. Jerry McPeak, D-Warner, already knows what to do — simply pass legislation eliminating the tests. According to McPeak, “Every youngster who lives up to the contract the state has set up for them — which is complete this amount of coursework and you can graduate” — should receive their high school diplomas.

McPeak’s message seems to be, after serving time in classrooms, give students diplomas whether they learned anything or not. Never mind that giving out undeserved diplomas sets up students to fail in colleges and deceives prospective employers and military branches that require applicants to have legitimate high school diplomas that signify basic competency in math, logic and communication skills.

Rather than making conditions worse, legislators should do something to transform a public education system whose students struggle to pass end-of-instruction tests and, in comprehension of science and math, lag significantly behind students in other industrialized countries.

Correcting these deficiencies requires implementing mandatory coordinated science and math curricula targeted to grades 1-6. Quizzical young minds must be rewarded by instruction that expands understanding of surroundings. Such structured learning provides students with foundations and confidence to excel in future courses.

Science and math courses in grades 7-12 should be reviewed to ensure coverage of core subject matter. Weekly detailed course objectives and outlines should be made available to students, parents and others interested in improving student learning.

And most importantly, administrators and teachers must leave comfort zones and exert tough love by requiring students to earn grades by learning subject matter as indicated by performances on well-written quizzes, periodic exams and mandatory comprehensive final exams. Initially, enforcing this policy will cause consternation among students and teachers since traditions of allowing students to pass science and math courses without learning subject matter will end, and teachers’ abilities to educate will be spotlighted.

via Status quo in Oklahoma education not good enough | NewsOK.com.

I’ve heard school compared to prison, but this takes it to a new level.  If you do the time, they have to let you out!

Battle of the eighth graders

Webmonk alerted me to a post on Freakonomics about a test for eighth graders from 1895.

The urban legend site Snopes labeled this as “False.” But the only false part of it seems to be the claim that it shows a decline in educational levels from then to now. The Snopes writer says that any test will look hard if you haven’t studied for it.

But he doesn’t dispute that this is an actual test from 1895. In fact, here it is from the library that holds the original document.

What this does show is what eighth graders studied and were expected to learn in 1895.

Take a look at the math section and compare it to this eighth grade math test from today. What can you conclude from the comparison about what was taught in the respective classes?

Finally, speaking of eighth graders, consider this.

HT: Webmonk


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