Great Again wrote:Mathematics is not free of irrationality. But it seems to be the last discipline which is still able to integrate, to include, to control parts of the irrational. Physics has already given up.

Nevertheless, mathematics has problems. And these problems started at the same time as the problems of physics - with the difference I mentioned above.

With mathematics one can do almost everything - thus also nonsense.

And even if only since ca. 1800 the idea of

multidimensional spaces - the word would have been better replaced by a new one - became the extended basis of analytic thinking, the first step to it was done at the moment when the powers, actually the logarithms, were detached from their original relation to sensually realizable surfaces and bodies and -

using irrational and complex exponents - were introduced into the field of the functional as relation values of a quite general kind. Whoever can follow here at all, will also understand that already with the step from the notion of a^3 as a natural maximum to a^n the unconditionality of a space of three dimensions is cancelled.

Once the spatial element of the point had lost the still optical character of a coordinate section in a vividly imaginable system and had been defined as a group of three independent numbers, there was no longer any inner obstacle to replace the number 3 by the general n. The concept of dimension is reversed: no longer dimension numbers designate optical properties of a point with respect to its position in a system, but dimensions of unlimited number represent completely abstract properties of a number group. A reversal of the dimension concept occurs: no longer do dimension numbers denote optical properties of a point with respect to its position in a system, but dimensions of unlimited number represent completely abstract properties of a number group. This number group - of n independent ordered elements - is the

image of the point; it is

called a point. An equation logically developed from it is called a plane, is the image of a plane. The epitome of all points of n dimensions is called a n-dimensional space. (From the point of view of set theory, a well-ordered set of points, without regard to the number of dimensions, is called a body, a set of n-1 dimensions is called a surface in relation to it. The "boundary" (wall, edge) of a point set represents a point set of lesser power). In these transcendental spatial worlds, which are no longer in any relation to any kind of sensuousness, the relations to be found by the analysis dominate, which are

in constant agreement with the results of experimental physics.

Only in this sphere of number thinking, which is still accessible only to a very small circle of people, even formations like the systems of hypercomplex numbers (for example the quaternions of vector calculus) and at first quite incomprehensible signs like infinite^n get the character of something real.

In the sharpest contrast to the older mathematics, set theory no longer understands the singular quantities, but the

epitome of morphologically somehow similar quantities, for example the totality of all square numbers or all differential equations of a certain type, as a new unit, as a new

number of higher order and subjects it to new, formerly completely unknown considerations concerning its power, order, equivalence, countability. The "set" of rational numbers is countable, that of real numbers is not. The set of the complex numbers is two-dimensional; from this follows the notion of the n-dimensional set, which also classifies the geometric domains into the set theory. One characterizes the finite (countable, limited) sets with respect to their power as "cardinal numbers", with respect to their order as "ordinal numbers" and establishes the laws and modes of calculation of them. Thus, a last extension of the function theory, which had gradually incorporated the entire mathematics into its formal language, is in the process of realization, according to which it proceeds with respect to the character of the functions according to principles of the group theory, with respect to the value of the variables according to set-theoretical principles.

The unnoticed goal towards which all this strives and which every genuine natural scientist in particular feels as an urge within himself, is the working out of a pure, numerical transcendence, the perfect and complete overcoming of the sight and its replacement by a pictorial language incomprehensible and inconceivable to the layman.

Having reached the goal, the immense, more and more non-sensual (nonsensical), more and more translucent fabric, which spins around the entire science, finally reveals itself: it is nothing else than the inner structure of the word-bound understanding, which believed to overcome the appearance of the eye, to detach "the truth" from it.