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• 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

• 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

• 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

• 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

• 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”

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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