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 –

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).

Beauty & physics, liberal arts & liturgy

Catholic artist and educator David Clayton makes connections between science, aesthetics, classical education, and then, for good measure, liturgy:

In excellent his book, Modern Physics and Ancient Faith, describing the consistency between the Faith and the discoveries of science, Stephen M Barr describes the scientific investigation of a grouping of sub-atomic particles which he refers to as a ‘multiplet’ of ‘hadronic particles’. He describes how when different properties, called ‘flavours’ of ‘SU(3) symmetry’, of nine of these particles were plotted mathematically, then they produced a patterned arrangement that looked like a triangle with the tip missing.

‘Without knowing anything about SU(3) symmetry, one could guess just from the shape of the multiplet diagram that there should be a tenth kind of particle with properties that allow it to be placed down at the bottom to complete the triangle pattern. This is not just a matter of aesthetics, the SU(3) symmetries require it. It can be shown from the SU(3) that the multiplets can only come in certain sizes….On the basis of SU(3) symmetry Murray Gell-Man predicted in 1962 that there must exist a particle with the right properties to fill out this decuplet. Shortly thereafter, the new particle, called the Ωˉ was indeed discovered.’

This result would have been of no surprise to anyone who had undergone an education in beauty based upon the quadrivium, – the ‘four ways’ – the higher part of the education of the seven liberal arts of education in the middle-ages[1]. The shape that Murray Gell-Man’s work completed was the triangular arrangement of 10 points known as the tectractys. As described in my previous articles for the New Liturgical Movement, this is the triangular arrangement of the number 10 in a series of 1:2:3:4. 1, 2, 3 and 4 are the first four numbers that symbolize the creation of the cosmos in three dimensions generated from the unity of God; and notes produced by plucking strings of these relative lengths we can construct the three fundamental harmonies of the musical scale. . . .

‘The traditional quadrivium is essentially the study of pattern, harmony, symmetry and order in nature and mathematics, viewed as a reflection of the Divine Order. When we perceive something that reflects this order, we call it beautiful. For the Christian this is the source, along with Tradition, that provides the model upon which the rhythms and cycles of the liturgy are based. Christian culture, like classical culture before it, was also patterned after this cosmic order; this order which provides the unifying principle that runs through every traditional discipline.  Literature, art, music, architecture, philosophy –all of creation and potentially all human activity- are bound together by this common harmony and receive their fullest meaning in the liturgy…

When we apprehend beauty we do so intuitively. So an education that improves our ability to apprehend beauty develops also our intuition. All creativity is at source an intuitive process. This means that professionals in anyfield including business and science would benefit from an education in beauty because it would develop their creativity. Furthermore, the creativity that an education in beauty stimulates will generate not just more ideas, but better ideas. Better because they are more in harmony with the natural order. The recognition of beauty moves us to love what we see. So such an education would tend to develop also, therefore, our capacity to love and leave us more inclined to the serve God and our fellow man. The end result for the individual who follows this path is joy.’

When the person is habitually ordering his life liturgically, he will tap into this creative force, for he will be inspired by the Creator. Meanwhile all those multiplets of hadronic particles in the cosmos will be giving praise to the Lord.

via The Way of Beauty.

HT: Cathy Sneidman

The mathematical part of classical education

Those who are bringing back classical education as an alternative to the deadends evident in John Dewey’s progressive education are familiar with the trivium: grammar, logic, and rhetoric, the three liberal arts that lead to a mastery of language. The other four liberal arts, the quadrivium, though, gets short shrift: arithmetic, geometry, music, and astronomy.

Many people think that the liberal arts is just another word for the humanities, forgetting the quadrivium completely. Dorothy L. Sayers, whose essay “On the Lost Tools of Learning” was a major catalyst for the revival of classical education, thought that the quadrivium represented “subjects” that would be learned after the trivium provided the tools for doing so. In this she was just wrong. The quadrivium are “arts”; that is, powers of the human mind. They are essentially mathematical, even in the way music was approached. Thus, classical education embraces the two spheres that educators recognize are necessary for education: language and mathematics.

Anyway, my daughter, who has been studying Boethius, the great systematizer of the quadrivium, explained to me the connections between the arts of the quadrivium, in a way that also helped me see the way mathematics really does provide a unifying model for the order and design that underlies all existence.

arithmetic = numbers
geometry = numbers in space
music = numbers in time
astronomy = numbers in space and time

Do you see why music is numbers in time? And why astronomy is numbers in space and time?

Now what we need is to bring mathematical education back from the dead–it’s telling that progressive education, for all its claim of being scientific and all, is failing most dramatically precisely in teaching science and mathematics–by coming up with a classical way of teaching it. Does anyone have any ideas? (And by this I don’t mean just teaching it more effectively or traditionally, such as Saxon Math. That and similar methods still lift numbers out of any context, which is not the classical way.)

HT: Joanna