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Case Study 2: Galileo's Integers

Galileo Galileo Galilei (1564 – 1642 CE)

In the 1600s, while a professor of mathematics in Padova, Italy, Galileo Galilei attempted to provide a response to a paradox that had been known since medieval times: the so-called paradox of the two circles.

According to Euclidean geometrical theory, a point is the most basic constituent of any shape, and a line is a series of points. A circle, therefore, is an endless series of points. If we take two circles, viewing both as constructed out of an infinite series of points, then reflection reveals that the two circles must be infinite yet not quantitatively infinite in the same way.

The medieval philosopher and Franciscan Friar Duns Scotus, for example, showed that we can have two sets of infinite points, mathematical points are without dimension and are the building blocks of lines, if we take two circles, one surrounding the other, the lines used to make up this diagram show that the lines composing the circles are the same size (without beginning and without end and so holding infinite points) but, since one circle is larger than the other, also different in size.

Scotus's Cricles

As witnessed in the picture above, the outer circle is clearly larger than the inner. This leads to the result that point A’ on the large circle corresponds exactly to point A on the smaller circle and point B’ to B. At the foundations of Galileo’s solution to the paradox was a structurally similar paradox involving the natural numbers that he later articulated in print in his book Dialogues Concerning Two New Sciences.

What is known as Galileo’s paradox proceeds as follows: If we take the natural numbers as a set {1,2,3…}, we are led to believe they are infinite (Galileo provides no proof, but this was not an uncommon assumption and it was later logically supported through axiomatic investigations of arithmetic). According to Galileo, many of the natural numbers have squares, i.e. products resulting from the even multiplication of numbers into themselves.

Thus, we have another set, the squares {1, 4, 9, 16…} and Galileo writes:

[if] I inquire how many roots there are, it cannot be denied that there are as many as the numbers because every number is the root of some square. This being granted, we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said that there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers, Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of all the numbers; up to 10000, we find only 1/100 part to be squares; and up to a million only 1/1000 part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers taken all together.

In other words, the infinite series of natural numbers is equal in size to the infinite series of squares, even though it seems that there should be more natural integers and less perfect squares. Galileo concludes:

So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities.

We can see that Galileo’s paradox is important since it anticipates later mathematical discoveries by Georg Cantor. Furthermore, in applying the same logic to Duns Scotus’s circles, Galileo concluded that although the number of points used to construct the two circles was the same size (infinite) one of the circles appears larger because of infinitely small gaps. Obviously there would have to be an infinite number of these infinitely small gaps. Galileo accepted this and it showed, to his mind, yet another case of the limits that affect us when we try to think infinity. According to Galileo we can’t help thinking about the infinite, but it seems to transcend our (ultimately finite) powers of understanding.