1. XI.20. The Mathematization of Nature Philosophy 157 G. J. Mattey ©2002

2. The Crisis of European Sciences Science does not meet the needs of humanity “Merely fact-minded sciences make merely fact- minded people” (§2) The questions of the meaningfulness of human existence are not relevant These questions concern the human being as a free being, rationally shaping himself and his surrounding world Even “humanistic” sciences exclude all questions of value

3. The Big Question Modern history teaches that the shapes of the spiritual world and the norms by which we live appear and disappear with no rational meaning “Can we live in this world, where historical occurrence is nothing but an unending concatenation of illusory progress and bitter disappointment?”

4. Revisionist History The goal is to uncover the prejudices on which this view of humanity is based These prejudices are characteristic of “modern” philosophy, which overturned the ancient philosophy that gives man a purpose The leaders of this movement, notably Galileo and Descartes, did not understand the significance of their revolution In rejecting their rationalism, we must be careful not to substitute a new irrationalism (§5)

5. The Ancient and the Modern Ancient philosophy was naïve and teleological— interested in the lived human world and in human ends It provided us with logic, mathematics and natural science to serve these interests The ancients could not conceive ideal space and formal mathematics The first step toward modern philosophy is Galilean mathematical natural science

6. Mathematical Natural Science The ancients, following Plato, believed that nature participates in the ideal Galileo held that nature itself is ideally mathematical This solves the subjectivity of “my” world Pure mathematical shapes, which can be constructed ideally, are the intersubjective, real, contents of appearances

7. “Pure Geometry” Ancient mathematics was available for Galileo to apply to pure spatio-temporal shapes in general It was ideal, yet practically applied We ordinarily do not distinguish the ideal from the empirical in mathematical thinking Galileo did not recognize how the two come together

8. Geometry and Bodies We do not intuit pure geometrical shapes, only inexact ones, in ordinary perceptions The relations of “identity” and “likeness” in ordinary experience are rough The pure shapes of geometry are the limit which we approach as we become more exact “Limit-shapes” are the resulting ideal objects of geometry (and similar structures for time)

9. Intersubjectivity The pure objects of geometry are not subject to the relativity of experience They are available for all investigators and objects of investigation They allow new shapes to be constructed They are applied to experienced things through measurement

10. Causality Geometry applies only to forms, not to the specific sense-qualities such as color These qualities are understood through the typical behavior of bodies—their “habits” Things generally continue in the way they have up until now (Hume) The empirical world has an “empirical over-all style” Things are bound together through causal relations

11. Indirect Mathematization How can a science of pure forms apply to the material qualities related by causation? Galileo’s solution: treat sense-qualities as themselves mathematical shapes A clue: the ancient Pythagorean recognition that tone is based on the length of a string The bold hypothesis of the Renaissance was to generalize this kind of observation

12. Mathematizing Causality Galileo found mathematical formulas that express causal relations—laws of nature This allows predictions to be made about the course of our experience The formulas are then taken as the “true being of nature itself” Ultimately, the formal structures as such (as in logic and set theory) are the focus (Leibniz)

13. Empty Formalization At the highest level of generality, the formal structures are empty of meaning The pure technique of science is like the rules of card games The “lived-world” is not touched by the formalism, except insofar as it enables predictions The living world is “clothed” in formalism

14. Objectivism vs. Transcendentalism A false consequence of formalism is that the sense-qualities are purely subjective How can the material element of experience be accommodated? (Leibniz, Kant) Only through phenomenological investigation of the “lived world” The transcendental is placed before the “objective” that is described by the formalism