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).

Contraception is not health care

The great Anthony Esolen reminds us, in the midst of the Obamacare insurance mandate, that contraception is NOT, strictly speaking,  a medical issue:

The use of estrogen as contraception is not medical at all. Quite the contrary. A couple who use estrogen to prevent the conception of a child do not ingest the drug to enhance the performance of their reproductive organs, or to heal any debility therein. Their worry is rather that those organs are functioning in a healthy and natural way, and they wish they weren’t. They want to obtain not ability but debility. They want not to repair but to thwart.

Here it is usually argued that the drug is medical because it prevents a disease. But that is to invert the meaning of words. When the reproductive organs are used in a reproductive act, the conception of a child is the healthy and natural result. That is a plain biological fact. If John and Mary are using their organs in that way, and they cannot conceive a child, then this calls for a remedy; that is the province of medicine. It is also the province of medicine to shield us against casual exposure to communicable diseases—exposure that we cannot prevent, and that subjects us to debility or death. Childbearing and malaria are not the same sorts of thing.

via A Tale of Two Sex Hormones « Public Discourse.

The use of artificial estrogen to prevent conception is, in fact, he argues, parallel to the use of artificial testosterone–a.k.a. steroids–by baseball players.  (You’ve really got to read how he ties baseball into all of this!)

HT:  Mark Misulia

The moon as our 51st state

It isn’t just that Newt Gingrich wants us to go back to the moon or that he wants to set up a colony there.  He is thinking that when the population of the lunar colony reaches 13,000 the moon could apply for statehood.  Yes the notion is crazy, absurd, and inappropriate.  But imagine!  The moon as the USA’s 51st state!  Think how indignant the rest of the world would be, looking up in the night sky and seeing America.

First Read – Gingrich promises US moon colony by 2020.

Put a bird on it, but not a real bird

More Portlandia. . . .

This is more than a satire of artsiness.  It cuts to the human condition:   how we idealize nature while also loathing and fearing actual nature.

HT:  Joanna

A new kind of environmentalism

Classic environmentalism wants to restore things to their pristine condition, untouched by man.  But a new kind of environmentalism thinks that man should actively take the lead in steering “spaceship earth.”

More and more environmentalists and scientists talk about the planet as a complex system, one that human beings must aggressively monitor, manage and sometimes reengineer. Kind of like a spaceship.

This is a sharp departure from traditional “green” philosophy. The more orthodox way of viewing nature is as something that must be protected from human beings — not managed by them. And many environmentalists have reservations about possible unintended consequences of well-meaning efforts. No one wants a world that requires constant intervention to fix problems caused by previous interventions.

At the same time, “we’re in a position where we have to take a more interventionist role and a more managerial role,” says Emma Marris, author of “Rambunctious Garden: Saving Nature in a Post-Wild World.” “The easy answer used to be to turn back time and make it look like it used to. Before was always better. Before is no longer an option.”

Although Marris is speaking about restoration ecology — how to manage forests and other natural systems — this interventionist approach can be applied to the planet more broadly. In his book “The God Species: Saving the Planet in the Age of Humans,” environmental activist Mark Lynas writes, “Nature no longer runs the Earth. We do. It is our choice what happens from here.”

via Spaceship Earth: A new view of environmentalism – The Washington Post.

Read the whole article for examples of this “ecopragmatism,” which depends on technology to give us a better environment.

How is this different from not being an environmentalist?  Doesn’t this describe the “dominion” over nature that the Bible describes and that human beings have been carrying out for millennia?  It seems different mainly in its utopian trust in human capabilities, which nature has been humbling for a long time.

China to put a man on the moon

China, the new America:

China has declared its intention to land an astronaut on the moon, in the first official confirmation of its aim to go where Americans last set foot nearly 40 years ago.

While Chinese scientists have previously discussed the possibility of a manned lunar mission, a government white paper published on Thursday is the first public government document to enshrine it as a policy goal.

China will “conduct studies on the preliminary plan for a human lunar landing”, the white paper said.

Although a manned moon mission is still some time off – Chinese experts say after 2020 – the statement highlights Beijing’s soaring ambitions just five months after the US retired its space shuttle programme . “Chinese people are the same as people around the world,” Zhang Wei, an official with China’s National Space Administration, said at a briefing. “When looking up at the starry sky, we are full of longing and yearning for the vast universe.”

According to the white paper, which serves as a blueprint for the next five years, China will develop new satellites, accelerate efforts to build a space station and strengthen its research in space. Laying the foundation for a mission to the moon, the government also plans to launch unmanned lunar probes and make “new technological breakthroughs” in human space flights by 2016.

via China push to put astronaut on the moon – FT.com.

Remember when we used to have grand ambitions like that, thinking we could do anything and then doing it?  Our last manned moon landing was in 1972.  Back then we were in a competition with the Soviets in a “space race.”  As the new and improved version of communism that China has devised outperforms us economically, I doubt that we will even care if China takes up where we left off in outer space.  For better or worse, we don’t have the same energy and optimism that we used to have.  Evidently, China has it.


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