Dear Jen,
I have almost no formal education in mathematics, so bear with me! I’ve spent a lot of time recently with Plato’s Meno, in which Socrates talks to a slave about how to find half the area of a square. The solution is to use the diagonal. We know the diagonal of a square with the sides of 1 is √2, which is a so-called ‘irrational number’. √2 written in in decimal form, is a never-ending sequence of numbers [1.41421…].
For a long time I wondered how an indefinite sequence of numbers (indefinite in its never-endingness) could equal the length of a diagonal, which is definite. The standard answer to the query is that the never-endingness of [1.41421…] is a matter of its representation, not its essence. If I represent the same number as √2, rather than the decimal form, there is no problem with never-endingness. The problem disappears!
Or does it?
First, in what sense is [1.41421…] never-ending? On what basis do we assert its actual infinitude? Perhaps it ends somewhere down the line that we haven’t discovered yet.
Second, is √2 a number? It seems to me less a number than a process to arrive at a number. Is 42 a number? Wittgenstein says this about Cantor’s infinity stuff:
Suppose someone says, “Show me a number different from all of these,” and as an answer he’s given Cantor’s diagonal rule. Why shouldn’t he say, “But that’s not what I meant. You haven’t given me a number. You have merely given me a rule in words for the step-by-step construction of numbers that are different from each of these successively.”?
Couldn’t I say that √2 is a rule for the construction of a number to define the length of the diagonal of the square? And not the length itself?
Another problem: couldn’t I say that how a number can be represented is an important part of its essence? The diagonal of the square with the side lengths of 1 is represented in decimal form by a sequence whose ends is hitherto undiscovered (called by many ‘irrational’ and never-ending). If representation and essence are not so easily split, changing our representation of the diagonal from [1.41421…] to √2 doesn’t help us much.
These essential differences in how numbers can be represented are another way of talking about incommensurability. (The Pythagoreans threw men off ships for daring to utter aloud the incommensurability of the side and the diagonal of the square.)
Wittgenstein again:
Should we avoid the word “infinite” in mathematics? Yes, where it appears to give a meaning to a mathematical procedure instead of getting a meaning from it. And it would be silly to become disappointed if we find nothing infinite in arithmetic or mathematics. However, it is not silly to ask what gives the word “infinite” its meaning for us. Then we can go on to ask about its connection, if any, with these mathematical calculations in which it appears to play a part.
You know more about all of this than I ever will, so I beg your forbearance in the face of my questions. I know you’ll say that I should learn the basics first. But heaven help me, Wittgenstein made me an ultrafinitist!
vale basilice,
D