Dear Michael,

I read your essay; the relation between physics and mathematics, especially at the foundational level, is a very interesting topic. Your essay raises some questions, though.

First a comment: on page 2, you wrote: "In Pure Mathematics real number may be regarded as the measure of a displacement." Actually, no. In pure mathematics, a real number is usually regarded as a...

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Dear Michael,

I read your essay; the relation between physics and mathematics, especially at the foundational level, is a very interesting topic. Your essay raises some questions, though.

First a comment: on page 2, you wrote: "In Pure Mathematics real number may be regarded as the measure of a displacement." Actually, no. In pure mathematics, a real number is usually regarded as a so-called Dedekind cut, which is an intersection of a (possibly infinite) number of intervals. There are other definitions as well, but the point is that in pure mathematics one does't use measures or displacements to define real numbers. It is rather the other way around: one uses real numbers to define a measure, which usually can be seen as a function from some set to the real numbers.

Then some questions.

1) You write that "Einstein's equation s² = x² * y² * z² is able to produce some unsolved number-theoretical problems" (I used the *-symbol for addition). As an example, you state that for odd integers x and y, no integers s and z exist that satisfy the equation. But why is that important? The terms in the equation, to which you refer as Einstein's equation (by the way: why?), are namely real numbers, not integers. That means that for any real numbers x, y, and z there is always real number s that satisfies the equation. That real number s doesn't have to be an integer. So where is the problem?

2) You write that "a displacement [ x', x'', x''', y ] is simply another way to say that there exists a kind of ( x' * x'' * x''' * yi ) complex number, or a new kind of complex number" (again with the *-symbol for addition). The question is: why do you need a new kind of complex number? The point is that a spatiotemporal displacement < x, y, z, t > can perfectly be represented by an element (vector) of the set R⁴ = RxRxRxR (where R is the set of real numbers). Of course one can also represent the same spatiotemporal displacement by a quaternion t * x.i * y.j * z.k, where a quaternion is (loosely speaking) a four-dimensional complex number. But there is no need to do that. All calculations, that can be done with the quaternions, can also be done with vectors. So why is any of this related to a wrong mathematical assumption in physics, the topic of your essay?

Best regards, Marcoen

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