integers.

Could you diagram that one for me ...

and whole number arithmetic went their separate ways

It's not hard to see that one geometric square can be cut apart into four pieces and reassembled as two identical smaller squares (so it's natural to say the larger square is twice one of the smaller ones): draw the two diagonals of the larger square, then cut along them; you'll get four pieces (identical right isosceles triangles); take two and glue them together along their hypotenuses, then do the same with the other two. Voila! Two identical smaller squares from a larger one from a larger one!

Now, if you are a atomist, it is natural for you to hope that the squares are actually little square arrays of tiny atoms:

oo
oo

ooo
ooo
ooo

oooo
oooo
oooo
oooo

And since one geometric square can be twice another, the number of atoms in the bigger square should be twice the number of atoms in a smaller square. So you start looking for a square number that's twice another square number, and then youhave a lot of "close but no cigar" moments:

Two of these

oo
oo

make up one of these

ooo
ooo
oo

Oops! There's an atom missing!

Two of these

ooooo
ooooo
ooooo
ooooo
ooooo

make up one of these

ooooooo
ooooooo
ooooooo
ooooooo
ooooooo
ooooooo
oooooooo

Oops! There's an atom too many!

There's actually an endless supply of near misses like that, with an atom too few or an atom too many. Here are the first few near misses:

(3 x 3) - 1 = 2 x (2 x 2)
(7 x 7) + 1 = 2 x (5 x 5)
(17 x 17) - 1 = 2 x (12 x 12)
(41 x 41) + 1 = 2 x (29 x 29)
(99 x 99) - 1 = 2 x (70 x 70)
(239 x 239) + 1 = 2 x (169 x 169)
&c &c

But surely we can find a square number that's twice another square number, can't we?

Um ... sorry! Nope!

Here's an easy way to see there's no square number that's twice another square number. Suppose there were whole numbers A > 0 and B > 0 with A x A = 2 x B x B. Then there's a smallest such A > 0: we could find it by trying

A = 1 (nope!)
A = 2 (nope!)
A = 3 (nope!)
...

and so on until we found the very first one that worked: A x A = 2 x B x B for some B > 0. Notice that B < A. Now A is either even or odd, and if A is odd then A x A is odd and equal to the even number 2 x B x B. Since no odd number is even, we see that A must be even: say A = 2 x C. Thus 2 x C x 2 x C = A x A = 2 x B x B. Dividing by 2, we get B x B = 2 x C x C. So B is another a square number that's twice another square number. But A was supposed to be the very smallest example, and yet B < A! That's ridiculous, so there's no square number that's twice another square number

Hippasus was not popular with the atomists

And some think it may be my only hope!

is Principia Mathematica ("Mathematical Principles&quot by Alfred North Whitehead and Bertrand Russell -- two names which philosophy students should recognize. It was an attempt to describe a set of axioms and rules in symbolic logic from which all mathematics could be proven.

There is a famous quote on about page 400 of the first volume, following a considerable amount of symbolic logic, "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." With the additional comment, "The above proposition is occasionally useful."

Unfortunately for Whitehead and Russell, there are a number of mathematical objections which can be raised. By the rules of the Principia Mathematica, the Axiom of Infinity (that there exists at least one infinite set) and the Axiom of Choice (that the product of a collection of non-empty sets is non-empty) are conditionals. However, the Axiom of Reducibility (required by Russell's Paradox -- the question about whether the set of sets which are not a members of themselves is a member of itself) says that they are not conditionals. Kurt Goedel's Incompleteness Theorem showed that it is impossible to derive all mathematics from a finite set of axioms. (When I was taking Introduction to Metaphysics, I had an argument with the professor about Goedel's Incompleteness Theorem and the Heisenberg Uncertainty Principle, which I maintained were philosophically significant, because they demonstrated that certain questions are unanswerable and certain facts are unknowable; he disagreed. I still say he was wrong.)