VI. The Present State of Philosophy and Science

Now I do not want to get into a discussion here of certain contributions – or the characteristics of these – that can be applied to the field of knowledge from a mathematical point of view. I will only point out the symptomatic nature of these things, pointing out that it is precisely in the field that has retained its wonderful inner strength, in the mathematical field, that it has become most apparent how the thinking of the 19th century has matured to the point of bursting the shell that seals off the human subject of knowledge from the transcendental world. And even if they are only hypotheses, sometimes boldly asserted, which have been pursued purely mathematically, we must take what has happened in the mathematical field as an expression of the longing of human knowledge to go beyond the sensual world. And there we have seen this longing realized precisely in the mathematical field. After all, mathematics in its forms, where it is called geometry, has held certain things to be unshakable since ancient Euclid! Who could have believed, for example, that there is anything more unshakable than the theorem that the three angles of a triangle equal 180 degrees, or the other theorem that if you have a straight line here and a point next to it, then in the sense of Euclidean geometry you can only draw a single parallel to the straight line through this point. That is, in the sense of Euclidean geometry, the sum of the angles of a triangle is equal to 180 degrees, and in its sense, only one parallel can be drawn to this line. This follows from the assumptions of Euclidean geometry. Who could now believe that this could be different! Nevertheless, and this is the significant thing. As I said, I could say many things for and against the content, but I am only addressing the symptomatic aspect of it, the longing to get out of the sensual realm; I just want to characterize it. The strange thing is that in the 19th century we saw the emergence of geometries other than Euclidean geometry. So it tried to crystallize out of itself a geometry — or geometries — that would be valid apply to something other than our ordinary sensory space, because for this space it is true that the three angles of a triangle together add up to 180 degrees, and that through a point next to a straight line you can only draw a parallel. And in the 19th century, we acquired geometries that do not want to apply only to our sensory space, no. Riemannian and Lobachevskyan geometry are two real geometries, born out of human thought according to strict mathematical laws.