As you may have noticed, the crank behind the "Inverse 19" rubbish in my Loony Toony Tangents post has shown up in the comments. And of course, he's also peppering me with private mail.
Anyway... I don't want to belabor his lunacy, but there is one thing that I realized that I didn't mention in the original post, and which is a common error among cranks. Let me focus on a particular quote. From his original email (with punctuation and spacing corrected; it's too hard to preserve his idiosyncratic lunacy in HTML), focus on the part that I've highlighted in italics:
I feel that with our -1 tangent mathematics, and the -1 tangent configuration, with proper computer language it will be possible to detect even the tiniest leak of nuclear energy from space because this mathematics has two planes. I can show you the -1 configuration, it is a inverse curve
Or from his latest missive:
thus there are two planes in mathematics , one divergent at value 4 and one convergent at value 3 both at -1 tangent(3:4 equalization). So when you see our prime numbers , they are the first in history to be segregated by divergence in one plane , and convergence in the other plane. A circle is the convergence of an open square at 8 points, 4/3 at 8Pi
One of the things that crackpots commonly believe is that all of mathematics is one thing. That there's one theory of numbers, one geometry, one unified concept of these things that underlies all of mathematics. As he says repeatedly, what makes his math correct where our math is wrong is that there are two planes for his numbers, where there's one for ours.
The fundamental error in there is the assumption that there is just one math. That all of math is euclidian geometry, or that all of math is real number theory, or that real number theory and euclidian geometry are really one and the same thing.
Math isn't one thing. It's a toolkit. It's a way of approaching things, a way of abstracting things in a formal way using logic. There isn't one math: there are many different maths. Number theory. Category theory. Set theory. Calculus. Mathematical logic. Topology. They're all math, and they're all different. You can't say that first order predicate calculus is right and first order Bochvar three-valued calculus is wrong. They're different, and they each work in their own setting.
It can be hard to see that this is the fundamental error when you're looking at the babble of a crank, because they make so many errors, it's easy to lose track of the deep ones.
In the case of our latest loonie-toonie friend, it's easy to skip over the deeper error because aside from everything else, he's making dimensional errors. A number isn't a point in a plane by any possible definition, because taken geometrically, a number is one-dimensional, and a plane is two-dimensional. (Yes, you can create a correspondence, by using a pairing function; but the result will not be meaningful. It still doesn't make senseto think of the two-dimensional cartesian plane as being fundamentally a representation of the real numbers.)
But even if you fix the dimensional error: a point in a plane isn't a pair of two numbers. And a plane isn't a grid divided up into squares. There isn't a single plane of numbers, or of pairs of numbers.
So when our crackpot pal says "thus there are two planes in mathematics" - really, you can stop right there. You don't need to read anything else to know that he's a crank. Because in real math, there aren't two planes. In real math, there isn't one plane. In real math, there don't have to be planes at all. Planes are one abstraction that you can build using math. There isn't one ultimate true mathematical plane, of numbers, or of anything else.
Math isn't one complete unified thing.
Frequently, when you discuss stuff like this, people will say something like "of course not, because of Gödel". But it's not just because of Gödel. Back in the days before Gödel, Whitehead and Russell tried to build the complete unified mathematics. They couldn't, because of Gödel. But I'd argue that even if they'd succeeded that in any useful, meaningful sense, it still wouldn't matter. At the level where we actually do math, category theory wouldn't really be sharing that much of a basis with number theory. Differential calculus still wouldn't be sharing a lot with lambda calculus. They're different fields of math, different ways of studying and understanding basic formal abstractions of things we'd like to understand better.