Completing the infinite

Anthony writes "An infinite procedure is, by definition, a procedure which can never be completed". Well no. By definition, an infinite procedure is one that can never be completed in a finite amount of time. To say it can never be completed at all begs the question, for it presumes that all processes are finite.

Cantor: If one considers the arguments which Aristotle presented against the real existence of the infinite (vid. his Metaphysics, Book XI, Chap. 10), it will be found that they refer back to an assumption, which involves a petitio principii, the assumption, namely, that there are only finite numbers, from which he concluded that to him only enumerations of finite sets were recognizable. [Punktmannigfaltigkeiten § 4 p. 104-5]

You ask what 'finite' means, which Russell answers as follows.

§ 183 Of all the philosophers who have inveighed against infinite number, I doubt whether there is one who has known the difference between finite and infinite numbers. The difference is simply this. Finite numbers obey the law of mathematical induction; infinite numbers do not… It is in this alone, and in its consequences, that finite and infinite numbers differ.

The principle may otherwise be stated thus: If every proposition which holds concerning 0, and also holds concerning the immediate successor of every number of which it holds, holds concerning the number n, then n is finite; if not, not. This is the precise sense of what may be popularly expressed by saying that every finite number can be reached from 0 by successive steps, or by successive additions of 1. This the principle which the philosopher must be held to lay down as obviously applicable to all numbers, though he will have to admit that the more precisely his principle is stated, the less obvious it becomes. [The Principles of Mathematics]

Mathematical existence

There's a tendency to get into all sorts of philosophical quicksand when talking about existence – does it mean physical existence, or space-time existence, is there a sense in which Harry Potter or Frodo have a fictional existence, etc. But let's keep it simple. When I ask whether the square root of 2 exists, all I am asking is whether there is a positive number such that when you multiply it by itself, the product is the number 2. In that way I haven't used the word 'existence' or 'exist' at all. Of course, I used the expression 'there is', but people don't seem to find the verb 'is' problematic in the same way as the verb 'exists'. So, is there a square root of 2?

This brought us to a discussion of sequences of decimals. We agreed that such sequences exist, i.e. that some things are sequences of decimals, or some sequences are of decimals, and we agreed that these sequences can be multiplied. Then we agreed that these sequences can be infinite, and that infinite sequences can also be multiplied. Finally, we agreed that (if we bought the other stuff), there is at least one infinite sequence such that when multiplied by itself, the product is 2 (or rather, the product is 1.999999… which you can verify on an Excel spreadsheet). So, there is a square root of 2. Or, if you like, the square root of 2 'exists'.

Is that it? Of course, we had to buy a couple of ideas. First, that some things are numbers. Some things are chairs, some things are tables, some things are stars or planets, some things are or may be angels (pace Anthony, who does not believe that any things are angels). And some things are numbers. Note my avoidance of the word 'exists'. Second, some numbers correspond to finite sequences of decimals, others to infinite sequences. Do we buy that? Time to read some more Ockham. More later.

On the existence of root two

Mathematicians are fond of defining things into existence. "If you say it exists, then it exists". This infuriates philosophers, who see existence as something that has to be proved.  You can say that p, and it can thus be true that you say that p. But that does not entail p.

Nonetheless, mathematician (and Fields medal winner) Tim Gower has an entertaining dialogue on whether you can prove the existence of the square root of two.

I like the idea that for any non-rational number you can define an ever-decreasing interval that appears to converge on the number.  For example, take any finite decimal expansion of root 2.  This will be slightly lower than the fully expanded number. Then add 1 to the digit at the very right hand side.  This will give a number slightly higher than the fully expanded number. Thus for any finite expansion whatever, you can define an interval which contains root 2.  For example, the interval

[1.4142135623, 1.4142135624]

Furthermore, for any such interval you can find the next digit in the expansion, and define in interval which lies completely inside the first interval, and which also contains root 2. For example, we know that the next digit in the expansion above is 7.  Thus  the interval

[1.41421356237, 1.41421356238]

contains root 2, and lies inside the first interval.

Thus we can prove the existence of an infinite set of intervals each of which contains root 2, and each of which can be further subdivided into another such subinterval inside that, and so on and so on.  But does that prove the existence of root 2 itself, which is not an interval, and which is 'indivisible'?  That is precisely where I lost Gowers' argument (the bit right at the end, where the disputant appears to accept the idea that you can define something into existence).

[Added as an afterthought.  I have a paper somewhere by Bolzano, written in 1917, where he attempts to prove by logic the intermediate value theorem.  It is blatantly fallacious, although a part of it eventually turned into the Bolzano-Weierstrass theorem.  This must be connected in some way, but it's so long ago I've forgotten most of it.  Could we not say, for example, that there are two sequences implied in the example above, one of increasing numbers that approach root 2 'from below', and the other of decreasing numbers which approach if 'from below'? ]

More Holmes

There's a in-some-way-similar discussion about fiction over at Peter Smith's residence.  In this case, in relation to the truth of mathematical statement and the, er, 'ontological status' of numbers and thingies.  I've only just caught hold of it, and now Easter looms.