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

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xi 20 the mathematization of nature

XI.20. The Mathematization of Nature

Philosophy 157

G. J. Mattey

©2002

the crisis of european sciences
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
the big question
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?”
revisionist history
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)
the ancient and the modern
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
mathematical natural science
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
pure geometry
“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
geometry and bodies
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)
intersubjectivity
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
causality
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
indirect mathematization
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
mathematizing causality
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)
empty formalization
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
objectivism vs transcendentalism
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