XI.20. The Mathematization of Nature

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