The problem of the parallel axiom is not a single, isolated geometric problem––consciously or unconsciously, researchers attacking it had to confront the Greek tragical worldview it embodies. Indeed, the significance of axioms lies beyond their technical statement––they express the worldview of a research community and the laws of its spiritual, psychical, and physical reality in a self-evident and clear form. The parallel axiom focuses on the problem of an infinite straight line––a formulation of the struggle between the Greek logos and ananke in simple geometric language. The straight line is a symbol for freedom: if something is capable of rectilinear motion in a region, then it is free of the region’s disturbing influence, it can move unimpeded, keeping its direction. The infinite straight line raises this freedom to a higher power, indeed, an infinite power, and thus it represents the power of logos. In contrast, the parallel axiom proclaims the power of the relentless, unapproachable ananke. It declares an eternal gap between the two straight lines representing the pure world of transcendent ideas and the world of finite individuality respectively. In addition, it declares that the direction of the two lines is identical, that is, they should point to a common point, which, in turn, is declared non-existent. Thus, the axiom implies, the meeting point of the two worlds is separated from both, and the whirl of contradictions spanned by them is inaccessible for research. These contradictions can only be resolved by a community of researchers which is unceasingly conscious that it lives on the boundary of the two worlds, and never declares the whirl induced by their conflict nonexistent. The revolution of hyperbolic geometry is the revolution of subjectivity. It is the revolution of the researcher who regards the system of contradictions hidden in the parallel axiom as his own contradictions, as contradictions implied, and, in the best case, resolved by his own existence.