*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.)