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

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

With the emergence of pure geometry, an ideal form of praxis became possible—a pure form of thought operating entirely within 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 shift from practical to purely theoretical interest, was idealised and transformed into geometry ².

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

However, this mathematics engages with the corporeal world only at the level of abstraction.

Scientific knowledge of the world is only possible if we succeed in finding the systematic a priori construction of the world. 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 spatio-temporal form—mathematics has created ideal objects and, for the first time, established an objective world.

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

Galileo then had an insight: could this not also 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 causal relations within the world of intuition.

The world would thus become an object of genuine knowledge when all its aspects, including sensible qualities that are not directly amenable to mathematical formulation, can nonetheless be treated mathematically in an indirect way.

But what does indirect mathematisation actually mean? And why is it impossible to mathematise sensible qualities directly? After all, cold and hot, rough and smooth all seem to involve degrees. What we experience as colour or sound is, in physical terms, a system of vibrations or 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 appears as real in sensible qualities must have a mathematical correlate in formal processes. This is what makes indirect mathematisation possible.

Some forms of experimentation 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 identified causal connections in experience that could be expressed as mathematical formulae. These formulae articulate general causal relationships—what we now call 'laws of nature'.

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

The arithmetisation of geometry implies that geometric shapes are conceived as precisely measurable. This transformation effectively erases geometry itself. Temporal figures, which appear as pure intuitions, are reduced to mere numerical forms or algebraic structures: In algebraic practice, 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.

Only in our own time has this idea come close to systematic realisation.

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