In Why Non-Euclidean Geometry is a Cheat Miles tries to establish that non-Euclidean geometry is a fraud only used to hide physical inconsistencies behind an opaque mathematical formalism. The paper also includes a critique of the complex numbers. As usual we try not to comment much on his metaphysical ramblings and silly ad hominem attacks but rather concentrate on mathematical errors. Therefore we will ignore for time being the whole first part of the paper which is concerned with non-Euclidean geometry and consists only of his personal opinion. The second part concerned with complex numbers proves a lot more interesting because Miles actually does some mathematical derivations.
First of all Miles gives the definition of a complex number. Then he continues to argue that is not a constant.
Wikipedia, the ultimate and nearly perfect mouthpiece of institutional propaganda, defines the absolute value of the complex number in this way:
Algebraically, if z = x + yi
Then |z| = √x2 + y2
Surely someone besides me has noticed a problem there. If i is a constant, there is no way to make that true. That equality can work if and only if i is a variable. But i is not a variable.
Let x = 1 and y = 2
i = .618
Let x = 2 and y = 3
i = .535
Let x = 3 and y = 4
i = .5
But i is a number. A number cannot vary in a set of equations. Letting i vary like this is like letting 5 vary. If someone told you that in a given problem, the number 5 was sometimes worth 5.618, sometimes 5.535 and sometimes 5.5, you would look at them very strangely. I don’t think you would trust them as a mathematician.
While the definition is in fact correct his calculations are not. He equates and solves for . This is clearly wrong because (see definition of complex numbers). (1)
There is one metaphysical quote we want to comment on because it concerns the whole of mathematics.
The fact that i equals anything is a major axiomatic problem, since it can’t equal anything but √-1, and √-1 is nothing.
and a few lines below
“How can you define something that does not exist?” Defining something that does not exist as “something that is imaginary” and then claiming that is a tight definition is a bit strange, is it not?
Without delving too deep into philosophy of math note that mathematics is not concerned with existence in the strict ontological sense. For an object to exist it suffices to write it down and show that it obeys certain required relationships. Sloppily speaking one could say that the (consistent) definition brings an object into existence. Miles’ statement above is clearly vacuous.
In a paragraph below he even states that it was wrong to equate . He blames this (as far as we understand) on the wrong notion of an algebra.
The real reason you cannot solve for i here is that z = x + yi is not algebraic. It is not analogous in form to |z| = √x2 + y2, so the whole “if/then” claim above is false and misleading. The second equation is algebraic, but the first equation is a vector addition. I will be told that vector addition is part of vector algebra, so it must be “algebraic.” But I don’t like that use of the word algebra. In algebra, the mathematical signs like “+” should be directly applicable, without any expansion. In algebra, you should be able to solve for unknowns. As I have just shown, you can’t do that here.
What Miles’ notion of an algebra is clearly escapes us. Also Miles doesn’t seem to like the construction of via ordered pairs. Finally we want to point out one obivous error in his critique of this construction.
He tends to confuse the notion of identity and equality and seems not very proficient in manipulating equations through exchange of symbols for variables.
And another problem: where did he get (0,y) = (y,0)(0,1)? That is just equation finessing. He claims to have gotten it from here
z1z2 = (x1,y1)(x2,y2) = (x1x2 – y1y2, x1y2 + x2y1)
But according to that equation, y can never be in the first position. Look again at the middle part of that triple equation: (x1,y1)(x2,y2). Do you see a “y” in the first position there? No. We need some explanation of (y,0)(0,1), but historically we don’t get it. Then look at the last part of that triple equation:
(x1x2 – y1y2, x1y2 + x2y1)
We need to ultimately find (0,y) there, but the only way you can get 0 in the first position is if x1x2 = y1y2. And the only way to get “y” in the second position is if x1y2 + x2y1 = y.
If the second point is (0,1), as given here, then x2 is zero, which means that
x1x2 = y1y2 = 0
Since y2 is given as 1, then y1 must be 0.
So the correct equation must be
(0,y) = (x,0)(0,1)
And, since x1y2 + x2y1 = y,
then x = y
While the definition of complex multiplication in the second line is indeed correct the errors occurs in the marked line. Miles fails to recognize that and thereby . Instead for unkown reasons he identifies with , which leads to an equation but not an identity.
From what we have said it should become clear that Miles’ “revolutionary” results stem from simple errors in his calculations and not inconsistencies in the axioms of complex numbers.
(1)Note: Miles goes on by argueing that he should be able to solve for because wikipedia says “the equations are algebraic“. If we were to write down this would indeed be an algebraic equation in similar to the equation or where is a constant. This does not save his argument because he mistook the axioms of complex numbers anyway.