As mentioned in the previous post, the BBC article contains video of Dr. Anderson explaining how his work allows us to evaluate the expression 00. I’ll save you the trouble of having to watch it. Here it is:
We define the number N=0/0. The number N stands for nullity, and is the new number Dr. Anderson claims to have discovered/invented. He uses a Greek letter phi to represent it, but an N will be simpler for our purposes.
He now argues:
00 = 0(1-1) = 01 x 0 -1 = (0/1)1 x (0/1)-1.
Anything to the first power is equal to the thing right back again, and anything to the minus one power is merely the reciprocal of the thing. Consequently, we get:
(0/1)1 x (0/1)-1 = (0/1) x (1/0) = 0/0 = N.
In standard crank fashion, Dr. Anderson sums up his accomplishment immodestly as follows, “So you’ve just solved a problem that hasn’t been solved for 1200 years. It’s that easy.”
Mathematicians have an expression for this sort of thing: “Symbolic nonsense.” You can invent whatever symbols you like and manipulate them according to whatever rules you like, but don’t mistake that for actually saying something about reality.
Actually, though, this little exercise gives me a chance to present one of my favorite examples of symbolic nonsense. I’ll write sqrt(x) to denote the square root of x.
Ponder the following:
(sqrt(-1))2 = -1.
Seems simple enough. But we also have this:
(sqrt(-1))2 = sqrt(-1) x sqrt(-1) = sqrt(-1 x -1) = sqrt(1) = 1.
So which is it? Looks like the complex analysis textbooks might have to be rewritten.
I’ll leave it to some clever commenter to explain precisely what went wrong.
Incidentally, does anyone out there know how to make HTML produce a proper square root sign? I found one website that said sqrt and /sqrt would do it, but that doesn’t seem to be working.