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.

I have two comments here:

1) Mathematicians always assume that somehow they do not have to adhere to the rules of physics, like for instance being able to reduce numbers to tautologies. Nothing is farther from the thruth however. Every mathematical definition rests on a physical tautology. This is also where complex numbers go wrong. For instance, analyze the identity x^2 = 2. There can only be a physical (tautological) definition of it since the mathematical definition has to rely on a neverending series.

2) Now apply that to x^2 = -1. Physically there are only positive numbers (unless you take the wrong perspective). Imagine a circle of radius -1 as an example. Now, that is why mathematicians have to rely on its definition as “imaginary”. Any other representation of the use of i brings us back to 1) since it relies on geometry.

That is the origin of Miles’ statement “√-1 is nothing”.

Dear Steven,

1) Your claim that “every mathematical definiton rests on a physical tautology” is completely unfounded and contrary to the axiomatic method. I don’t even understand what you mean by “physical tautology” and think the concept is ill-defined.

2) It seems to me what you want to say is something in the following lines: If a concept cannot be imagined it does not exist. Please note that the term “imaginary” has nothing to do with imagination. Furthermore I think the ontology of mathemathical objects has sufficently been treated in the post. If you want to learn more about the philosophy of math I recommend Hermann Weyl’s book “Philosophy of Mathematics and Natural Science” which can be accessed partially at http://books.google.com/books?id=565kXGJPkiYC&printsec=frontcover&dq=philosophy+of+mathematics+weyl&hl=de&ei=ApOjTLKgM4j14AaJmc3yAg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCsQ6AEwAA#v=onepage&q&f=false.

So do I.

1) First of all, what are these “rules of physics” that mathematics has to adhere to? Can you name one of them? To me it seems rather that mathematics doesn’t need to rely on physics at all. Physics undeniably uses it as a tool and in turn inspires mathematics with new ideas and with its intuition. But as mathematics is its own subject, it only relies on the rules it imposes on itself.

Second of all you are wrong when you say that the square root of 2 “relies” on a neverending series. It is true that if you were to write down the square root of 2 in the decimal number system you would have to write an infinite number of digits. However, that is just a representation of the mathematical object that satisfies the equation x^2 = 2. You are confusing notation with the object itself.

2) I don’t even know what you mean by “Physically there are only positive numbers”. Would you clarify that because I think it’s an interesting point?

Furthermore I can’t really “imagine” a circle with a negative radius since a radius is a distance and our everyday world physical distances are positive (but I’m guessing that was your point). However, in mathematics it’s easy to make up rules that allow for such things and play with them wether they have any immediate connection to reality or not (even though they often do!).

Steven:

I feel you misunderstand a great deal of simple mathematics. I think the technical details have been well addressed above, however something needs to be said about formalism. The formalist school of thought, primarily founded by David Hilbert, directly addresses your idea that all mathematics somehow requires some underlying physical model. The basic idea here is that these symbols we write down do not have to have any physical interpretation at all. A set of axioms are created along with rules of inference, n-ary function symbols, n-ary relation symbols, a symbol for each element in the universe of discourse, and the universal and existential quantifiers constitute a language, and with this language theorems are proved. 1 + 1 = 2 does not have to be interpreted as counting objects or anything like that, + is a binary function and = is a binary relation symbol, = does not even have to mean ‘equals’, strictly speaking an n-ary binary relation is a set of ordered n-tuples contained in a set. Have you ever seen the definition for a vector field or a group where they typically say something like “a set equipped with two binary functions called ‘+’ and ‘X'”, nowhere does it say these symbols imply addition or multiplication although they are typically interpreted this way. The point of the school of thought is that given our axioms, stated formally, and rules on inference, we just make marks on a piece of paper using these rules and the ONLY requirement is internal consistency.

Here is an example n = sqrt(3), now according to your school of thought sqrt(3) cannot exist because as a non repeating non-terminating decimal it could never be written down and thus cannot be completely determined. However, following the axioms for real numbers (real numbers defined by Dedekind cuts) and a few rules of inference I have n^2 = (3^1/2)^2 = 3^(2/2) = 3^1 = 3, and BOOM internal consistency. I could also prove the existence of sqrt(3) via continuity of the function f(x) = x^2 and the intermediate value theorem but I think that would be beating a dead horse. Even more to the point on formalism sqrt(3) doesn’t have to be a number, it can just be a symbol devoid of meaning such that the rules for manipulating marks on a piece of paper are being used properly. For a deeper explanation I can point you to “A friendly Introduction to Mathematical Logic” by C. Leary. I will also state that if you argue against formal logic you must also abandon all work done by MM as he himself implicitly uses it when making any deduction or making substitutions and using “=” to indicate they follow from the previous line.

One final point to make to all others that might read this. It is my hypothesis that Steven Oostdijk IS Miles Mathis. It seems that anywhere someone attempts to discredit any paper by Miles “crank” Mathis, “Steven Oostdijk” shows up to faithfully defend him like a good sock puppet should. He seems intimately aware of the minutia of MM’s every paper, which I have been told there are over 1200 of, this seems a feat of super human memory. I have also seen on MMs own site him acknowledge the fact he uses various different pen names, it appear Steven Oostdijk is one of them.

I also apologize ahead of type for any typographical mistakes, I type fast and quite honestly won’t take the time to proof read my posts on an internet forum.

”Steven:

I feel you misunderstand a great deal of simple mathematics.”

That TOTALLY doesn’t remind me of anyone. No one at all. Nope. Can’t think of a single person that reminds me of. Strange.

Oh yeah, on the issue of a circle with a negative radius, it cannot exist. If you choose to work with cartesian coordinates, the standard way to define a circle is +sqrt(x^2 + y^2) = r (or an equivalent form). Now by simple properties a square can never be negative let alone the sum of two squares, thus r and r^2 can never be negative.

Now lets look at polar coordinates, the circle with radius r is given by f(theta : theta in [0,2pi)) = (r*cos(theta), r*sin(theta)). Now r can be a negative number in this parametrization, but can the radius be negative in this situation? Lets see, lets pick a point on the circle and see how far it lies from the origin, lets assume t is given as some value in [0,2pi), clearly we are in a metric space equipped with the standard Euclidean metric. So d(0,f(t)) = + sqrt((r*cos(t))^2 + (r*sin(t))^2) = sqrt(r^2(cos(t)^2 + sin(t)^2)) = +sqrt(r^2 * 1) = |r| > 0. Do I have to get into cylindrical or spherical coordinates as well to convince anyone?

I know I’ve just posted two comments, but I require some clarification. What exactly is meant by “being able to reduce numbers to tautologies”. As stated above this is an ill defined idea. By using the word tautology you are automatically placing the discussion in the realm of formal logic, so I will respond within this framework. Let us assume we have a set of axioms ‘strong enough’ to describe the natural numbers, let us for example use the Peano axioms. The Peano axioms are:

1. For every natural number x, x = x. That is, equality is reflexive.

2. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.

3. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.

4. For all a and b, if a is a natural number and a = b, then b is also a natural number. That is, the natural numbers are closed under equality.

5. 0 is a natural number.

6. For every natural number n, S(n) is a natural number.

7. For every natural number n, S(n) = 0 is False. That is, there is no natural number whose successor is 0.

8. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.

9. If K is a set such that:

* 0 is in K, and

* for every natural number n, if n is in K, then S(n) is in K, then K contains every natural number.

NOTE: S(n) is the successor function, as in S(n) = n + 1

Now Mathis himself claims all numbers are based upon the natural numbers, so these axioms cannot be a point of contention as they are what define the natural numbers. Now show me somewhere, anywhere, in these axioms where any sort of physical realization occurs, or is necessary for that matter.

It is true that in ancient Greece the notion of number was directly tied to the notion of length of a line segment ( a ‘physical’ quantity), however the concept has matured and the Peano axioms are now an accepted foundation.

Now again within the framework of formal logic, you want to keep tossing the word tautology around, if a theorem can be proven within a consistent framework, then it IS a tautology WITHIN that framework. Construct a proof, then reduce the statements in it to formal logical statements, then construct a truth table. What you will find is that when you assign the value true to the axioms, the statement of the proved theorem must have truth value true, and is thus a tautology within that framework.

I will add one more point regarding a circle with negative radius. If you took some time to read even very introductory real analysis, you would learn what metric space is. Put simply it is a space where one can assign some notion of distance between two points in the space. Formally a metric space is an ordered pair (X,d) where X is a set of ‘points’ (do not have to be points in the Cartesian sense, the points could be people, functions, matrices, whatever) and a function d;X*X -> R. d must satisfy three requiemwnts to be a valid metric: 1) d(x,y)>=0 and d(x,y) =0 if and only if x=y 2)d(x,y) = d(x,y

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