The Mathematization of Nature: Galileo, Husserl, Mandelbrot

Afamous passage in Galileo’s 1623 polemic Il Saggiatore [The Assayer] gives the medieval topos of the “book of nature” a decisive new turn:

Philosophy is written in that great book which is continually open before our eyes (I speak of the universe), but no one can understand it who does not first learn the language, and come to know the characters, in which it is written. It is written in mathematical language, and the characters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to understand a word; without these, one is wandering vainly in a dark labyrinth.

(237–8, trans. mod.)

Following immediately on Galileo’s belittlement of a Jesuit opponent as someone who “believe[s] one must support oneself upon the opinion of some celebrated author” rather than rely on one’s own observations and reasoning, and who perhaps thinks philosophy is a “book of fiction” like the Iliad or Orlando Furioso (though the actual “books of fiction” Galileo has in mind but refrains from naming are no doubt the Bible and Aristotle), these celebrated sentences announce what historians of science now speak of as “the mathematization of the world-picture”¹ and what Husserl had in already in 1936 analyzed at length in the unfinished Crisis of European Sciences and Transcendental Phenomenology as Galileo’s “mathematization of nature” (23–59). The transformation of what Galileo and Newton still referred to as “natural philosophy” into what we call “science” depended on the promotion of mathematics, and in the first instance geometry, from an interrelated set of methods for generating and assessing claims about delimited aspects of the natural world, to a unified system of laws generating the phenomena themselves and governing the totality of their interrelationships. In Husserl’s formulation “nature itself is idealized under the guidance of the new mathematics; nature itself becomes—to express it in a modern way—a mathematical manifold” (23).

In Husserl’s account of this world-historical transformation, Galileo’s role is simultaneously that of “a discovering and a concealing genius [entdeckender und verdeckender Genius]” (52). The discovery is of modern physics, of an understanding of physical nature as “mathematical nature… according to which every occurrence in ‘nature’… must come under exact laws,” in a way for which the axioms and internal coherence of geometry are paradigmatic. What is at the same time concealed or forgotten in the way Galileo inherits geometry is that “[e]ven ancient geometry was… removed from the sources of truly immediate intuition and originally intuitive thinking… The geometry of idealities was preceded by the practical art of surveying, which knew nothing of idealities” (49). If for Galileo it is the laws of mathematics which are real but hidden from us by the phenomenal appearances they underlie so long as we are unable to read those appearances, for Husserl it will be the phenomenal “life-world” which is the underlying reality and mathematics the more or less well-fitting “garb of ideas” with which we “dress it up” (51).

Writing in 1982, Benoit Mandelbrot opens The Fractal Geometry of Nature with a pronouncement that might as well be a response to Galileo: “Clouds are not spheres, mountains are not cones, coastlines are not circles, nor does lightening travel in a straight line” (1). There is a connection to be drawn between this statement and Husserl’s diagnosis of ancient geometry as already “removed from the sources of truly immediate intuition.” But the thrust of Mandelbrot’s work is not to contrast the idealizations of mathematics with the actual irregularities of what Husserl calls the “life-world,” but rather to develop a geometry and mathematics that would embrace irregularity and to thereby “renew the Geometry of Nature” (28). A condition of this development, however, is the relinquishment of the assumption that length is in principle determinable. In Mandelbrot’s paradigmatic example, “All measurement methods ultimately lead to the conclusion that the typical coastline’s length is very large and so ill-determined that it is best considered infinite. Hence, if one wishes to compare different coastlines from the viewpoint of...