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Gonville and Caius College, Cambridge, England

52° 12′ 21.4″ N, 0° 7′ 3.34″ E

Stained-Glass Scientists

The first thing you need to know about Gonville and Caius College is how to pronounce it. Gonville is easy, no surprises there. But Caius is another matter. John Keys was the second founder of the college; in 1529 he entered what was then Gonville Hall at just 18 years old. He later became a Fellow of the college, and finally Master (after having spent a chunk of his own fortune restoring it).

Along the way he Latinized his last name, turning Keys into Caius without changing the pronunciation. Thus, Gonville and Caius is pronounced “Gonville and Keys”; within Cambridge the college is usually referred to as simply Caius.

Caius has an illustrious list of alumni. They include William Harvey (who discovered blood circulation), George Green (mathematician), John Venn (who popularized the Venn diagram), Charles Sherrington (Nobel Prize–winning neurophysiologist), R. A. Fisher (probably the greatest statistician ever), Sir James Chadwick (who discovered the neutron), Francis Crick (DNA; see Chapter 71), and Stephen Hawking.

Six of these alumni are honored with stained-glass windows representing their greatest contribution in the college Hall where students and Fellows eat.

John Venn’s famous Venn diagram (Figure 47-1) is used to show intersecting sets of items and is represented by three colored circles. Each circle overlaps the other two, and all three overlap in the middle with appropriate changes in color.

R. A. Fisher was not only a great statistician, but also a celebrated geneticist and evolutionary biologist. His window shows a 7×7 Latin Square that uses colors instead of the typical numbers (Figure 47-2). A Latin Square is a square grid in which numbers are placed so that each number only appears once per row and column. (Many people will recognize this as a sort of Sudoku.)

Figure 47-1. Venn’s window diagram; courtesy of Derek Ingram,
Gonville and Caius College

Figure 47-2. Fisher’s Latin Square;
courtesy of Derek Ingram, Gonville and Caius College

Fisher used Latin Squares in statistical analysis and in particular in agriculture. By dividing a field into a square grid, it’s possible, for example, to try out seven different fertilizers using a 7×7 Latin Square. By associating one fertilizer with each of the seven different numbered squares, it is possible to use the square to eliminate differences in soil, drainage, sunlight, etc. because each fertilizer will be applied once per row and column.

Francis Crick’s window shows the structure of DNA. When the window was installed, Crick insisted that it not be visible from the outside at night so that the DNA would only be seen coiling in the right direction.

George Green’s window shows something that at first glance looks like it might be bacteria, but it’s in fact a representation of Green’s Theorem (which you’ll be familiar with only if you studied calculus to a high level). Green’s Theorem shows how an integration over an enclosed area (the bacteria-like shape in the window) can be turned into an integration over just the line that forms the shape’s boundary. Although this may seem totally abstract, it is used directly in the planimeter (see page 67), which can calculate the area of a shape on a graph or map just by tracing around its edge.

Sir James Chadwick’s window shows an alpha particle (a helium nucleus) hitting a beryllium atom and causing the emission of a neutron (and creation of a carbon atom).

Finally, there’s Charles Sherrington’s window, which shows “two excitatory afferents with their fields of supraliminal effect in a motor neuron pool of a muscle.” It looks like some sort of plant, but it actually shows the motor neurons that control muscle movements. Sherrington won the Nobel Prize in 1932 (with Edgar Douglas Adrian) for discovering the function of neurons.

Gonville and Caius is open to the public (and free of charge), but unfortunately the public does not have access to the Hall where the windows are installed. To see them, you’ll have to make friends with a Cambridge University student who’s willing to accompany you, or make do with a virtual tour.

The college is right next to Trinity College, where Newton lived (see Chapter 69), and a short walk from the Eagle Pub, where Crick and Watson announced the “secret of life” (see Chapter 71).

Practical Information

The website of Gonville and Caius College is at http://www.cai.cam.ac.uk/, and you can take an interactive tour at http://www.cai.cam.ac.uk/map1.php.

Set Theory and Transfinite Numbers

To mathematicians, a set is a collection of objects with no duplication. For example, a mathematician might state that U is the set of colors on the American flag (red, white, and blue), I is the set of colors on the Italian flag (red, white, and green), and J is the set of colors on the Japanese flag (red and white). The objects inside a set (in this case, the colors) are called its elements.

A Venn diagram shows the relationship between two sets. Each set is represented by a circle containing the set’s elements, and where the sets have common elements, the circles overlap (this is called the intersection of the sets). A Venn diagram of the sets U and I shows that the common colors are red and white (Figure 47-3).

Figure 47-3. Venn diagram of U and I

All the colors of the American and Italian flags taken together form another set (with elements red, white, blue, and green). When two sets are merged together, the resulting set is called their union.

It’s also possible for one set to be entirely contained within another. For example, J is entirely contained inside U, so we can say that J is a subset of U. In the Venn diagram, the circle for J is completely inside the circle for U (Figure 47-4).

Figure 47-4. Venn diagram of U and J

The main creator of set theory, which formally introduced the ideas of sets, subsets, intersections, and unions, was the German mathematician Georg Cantor. Cantor used set theory to answer questions about infinity, and came up with the idea that there’s more than one infinity and that some are bigger than others. This was quite a controversial idea at the time.

The sets U, I, and J are all finite; that is, they have a finite number of elements (U and I have three elements, and J has two). But it’s possible to create a set that has an infinite number of elements. For example, the set ℕ is the set of all natural numbers (all whole numbers—0, 1, 2, 3, …). Obviously there is an infinite number of them.

The set ℤ is the set of all integers—it’s the set ℕ plus all the negative whole numbers as well (0, 1, –1, 2, –2, …). One of Cantor’s questions was, “Are there more integers than natural numbers?” or equivalently, “Are there more elements in ℤ than in ℕ?” The answer is no—they both have the same number of elements.

To establish this, mathematicians use a one-to-one correspondence. If you watch a child count a bag of apples, she’ll point to each apple in turn, counting out loud (1, 2, 3, etc.) until she reaches the last apple. The child has set up a one-to-one correspondence between the apples and the numbers: each apple is assigned a single number in the sequence 1, 2, 3, etc.

The same thing can be done for infinite sets. A set is countable if there’s a one-to-one correspondence with the natural numbers. If there’s a way to assign a natural number to each element of a set, then the set has the same size as the natural numbers (the same number of elements).

So, Cantor’s question about ℤ comes down to, “Can a one-to-one correspondence be made between ℤ and ℕ?” Doing so is fairly easy: first connect 0 in ℕ to 0 in ℤ. Then assign each even number to the corresponding number divided by –2: so 2 in ℕ is assigned to –1 in Z, 4 is assigned to –2, 6 to –3, and so on. That gives a one-one correspondence between the even numbers and the negative numbers. Finally, the odd numbers in ℕ are each assigned a positive number in ℤ by adding 1 and halving the result so: 1 in ℕ is assigned to 1 in ℤ, 3 is assigned to 2, 5 is assigned to 3, and so on.

So ℤ and ℕ have the same number of elements (albeit an infinite number). Cantor’s next question was, “Are all sets countable?” The answer to that is no. Cantor’s brilliant argument is called ”diagonalization,” and shows that the set of real numbers (the set containing every number, including all the fractions and decimal numbers) is not countable.

Suppose there was a one-to-one correspondence between ℕ and ℝ (the set of real numbers). It would therefore be possible to assign each natural number to a real number. Cantor’s diagonalization is best understood with a picture of this imaginary one-to-one correspondence. In Figure 47-5, the natural numbers are on the left and each corresponding real number is on the right.

Figure 47-5. Cantor’s diagonalization

There’s an easy way to get a contradiction from this by making a new number different from all the others. In the figure, the first digit of the first number is 1, the second digit of the second number is 9, the third digit of the third number is 2, and so on. In this way, it’s easy to build up a new number (such as the number at the bottom of Figure 47-5) that differs digit by digit from every number in the table, just by picking digits that differ from the digits on the diagonal.

But that means there’s a real number that’s not in the table, and that means that the original assumption (that ℕ and ℝ have a one-to-one correspondence) is false. And that means that ℝ is not countable and has more elements than ℕ. So Cantor had established two different infinite numbers—the number of elements in ℕ and the number of elements in ℝ. He called these new numbers “transfinite,” because the idea of having more than one infinity was, at the time, controversial (especially to theists who believed in an infinite God).