The fact I discussed in my latest post means that the middles of two orthogonal systems (like a Cartesian cube and a polar cylinder) are paradoxically contradictory concerning whether they can be specified with rational numbers or not. The middle of one of these systems ought to be possible to specify with a rational number in the other system, but since it is the middle also of this system, it can’t be, but ought to be possible to specify with a rational number in the other system, but since it is the middle of this system, it can’t be, and so on…
The question is what this orthogonal carousel is: is it a chase for rational or irrational numbers? The answer is that the goal for our search resides in irrational numbers themselves. The reason that irrational numbers can’t be specified with rational numbers is simply that each of them actually is two rational numbers – the middle of a cube and the middle of a cylinder – and that they thus are two rational numbers for one and the same thing. The numbers themselves only appear “irrational” because they denote the same thing.
Irrational numbers are thus the opposite to rational numbers in that whereas rational numbers denote the number of different things, irrational numbers instead denote different numbers of a specific thing. Irrational numbers are thus merely an orthogonal consequence of numbering things. If we number things, then there must not only be different numbers of things, but also different things of numbers.
Irrational numbers are thus not “natural” (in the meaning of realism), but rather the back side of numbering. This side would actually not even had existed if realism would have been correct, but now that it isn’t, it does. And, this side tells us that that mathematics can’t catch the middle, but can only avoid it by using irrational numbers. Irrational numbers are thus the turning point for a consistent system (which thus avoids the middle).
Realists, like cladists and particle physicists, do not acknowledge this relation between rational and irrational numbers, but instead claims that there is a meeting place between them, which they call “the tree of life” and Higgs particle”, respectively, although such a meeting place would eradicate the difference between them. This claim is thus just as stupid as claiming that there is a solution to every problem consisting of denying the problem.
Th question is thus: can we solve every problem by denying it (as realists like cladists and particle physicists claim)?