Summary: The Crisis of European Sciences and Transcendental Phenomenology (page 4)

Yet this is not self-evident: bodies are not geometrically pure figures. Moreover, we cannot even imagine geometric space—imagination can only transform sensible forms into other sensible forms ¹. The geometer seeks limit forms around which things in the surrounding world oscillate: they are more or less straight, more or less circular, and so on.

With the emergence of pure geometry, an ideal praxis became possible—a pure thought adhering to the realm of pure limit forms, where exactitude is attained. The origin of pure geometry lies in the art of measurement (or surveying), which, through a reversal from practical interest to purely theoretical interest, was idealised and transformed into geometry ².

Galileo observed that wherever such a method has been developed, it has enabled us to overcome the relativity inherent in subjective apprehensions, which is essential to the world of empirical intuition. Through it, we obtain an identical, non-relative truth. Here, we come to know a true being ³.

However, this mathematics engages with the corporeal world only as a mere abstraction.

Scientific knowledge of the world can only be possible if we succeed in finding a method for systematically constructing the world a priori. But how?

Mathematics has already paved the way, through spatio-temporal figures.

By idealising the world of bodies—focusing on what, within it, falls under the spatio-temporal figure—mathematics has created ideal objectivities and, for the first time, established an objective world.

At the same time, through its connection to the art of measurement, mathematics has demonstrated that one can descend from idealities to empirical intuition.

Galileo then had an insight: could this not be possible for the concrete world as a whole? To achieve this, it would be necessary to extend the method of measurement to all real properties and causalities within the world of intuition.

Reality would thus become objective knowledge when all aspects of reality, including sensible qualities that are not directly mathematizable, nonetheless become so indirectly.

But what does indirect mathematisation mean? And why is the direct mathematisation of sensible qualities impossible? After all, there is a magnitude to cold and hot, rough and smooth. What we experience as colour or sound, for instance, is for us, in physical terms, a system of sound vibrations, heat waves—in short, pure processes within the world of forms. Yet for Galileo, this was not self-evident.

Galileo's idea is that everything that manifests itself as real in sensible qualities must have its mathematical index in the processes of the sphere of form. This is what makes indirect mathematisation possible.

Some experiments were already possible in the pre-scientific era. Certain sensible qualities had been quantified. Thus, the Pythagoreans demonstrated the relationship between the pitch of sounds and the length of strings.

Galileo effectively identified causal connections in experience that could be expressed mathematically in formulas. These formulas articulate general causal relationships—what we now call "laws of nature."

The Galilean idea is a hypothesis, and indeed, a hypothesis of a very surprising nature. Admittedly, this hypothesis is confirmed ad infinitum by the progress of science, yet it remains a hypothesis nonetheless. The entire science of nature is infinitely hypothetical and infinitely confirmed.

The arithmetisation of geometry implies that geometric shapes are conceived as exactly measurable. This transformation virtually erases geometry itself. Temporal figures, which present themselves as pure intuitions, are reduced to mere numerical forms or algebraic structures: In algebraic calculus, one computes first, and only at the end does one recall that numbers must represent magnitudes ⁴. Leibniz was the first to glimpse the universal idea of algebraic thought in the highest sense of the term—the idea of a mathesis universalis, as he called it ⁵.

It is only in our own time that this idea has finally come close to systematic realisation.

¹ ibid., p.29
² ibid., p.33
³ ibid., p.44
⁴ ibid., p.52
⁵ ibid.