12th September 2010

Chovot HaLevavot's Sha’ar HaYichud

I posted the following as a comment on Stephen Hawking’s Rosh Hashanah Gift By Yitzchok Adlerstein. It was originally supposed to be a tangential thought on something he said about Chovot HaLevavot's Sha’ar HaYichud. It seems to actually be more relevant to the issue of science and Torah than I had originally anticipated.

Chovot HaLevavot's Sha’ar HaYichud is incorrect in its mathematical proof of God's unity. In chapter 5, he introduces 3 mathematical propositions on which he rests his proof. One of these propositions is that "That the infinite should have parts is inconceivable. ... Let us assume that a thing is actually infinite, and that we take a [finite] part from it. The remainder will undoubtedly be less than it was before. If this remainder is infinite, one infinite will be greater than another infinite, which is impossible. If this remainder is finite, then when we put the part we took back together with the finite remainder, the result should be finite [thus contradicting the premise]."

There are two problems with this: he assumes that the size of an infinite quantity decreases when you take a subset of the infinite quantity. (In informal terms) when there is a one-to-one mapping between two infinite sets, they are the same size. There is a one-to-one mapping between the even integer and the set of all integers. Each integer can be doubled to give an even integer, even though the even integers are a subset of the set of all integers. Thus demonstrating that the size of an infinite quantity decreases when you take a subset of the infinite quantity.

The other problem is that there are different sizes of infinity. The integers may be listed (in an infinitely long list) without skipping any numbers in the middle, however the real numbers may not. (If they could, the positions in the list would be a mapping to the set of all integers.) No matter how many real numbers you list, you can always find another real number that should be in the list. So the real numbers are a larger infinity than the integers.

This is not to say chas v'shalom that God is a smaller infinity or a larger infinity (there are an infinite number of sizes of inifinity, so if we assign God to one of them, there would have to be a bigger infinity than Him) -- rather this is to demonstrate the futility of proving God through mathematical reasoning in the first place. Which, if you think about it, actually makes sense. Since we have no Human language to describe God as he is (all the anthropomorphisms we do use are borrowed terms and analogies), formal mathematics shouldn't have the ability to describe or define Him either.

I haven't worked out the details of what this says about the validity of science in dealing with issues like creation, but my feeling is that it does imply that science can't foreclose God's existance...

(I hope my informal explanations of the mathematical arguments are understandable. For mathematically rigorous versions of the arguments I have made informally here, see the following Wikipedia entries Hilbert Hotel, Countable set, Aleph number, Cardinality of the continuum.)