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Synopsis of Judith Miller, ‘Métaphysique de la physique de Galilée’

[‘The Metaphysics of Galileo’s Physics’]

CpA 9.9:138–146

Judith Miller’s single contribution to the Cahiers appears in volume nine, on the genealogy of the sciences. It considers the metaphysical differences that underlie the distinction between post-Copernican and pre-Copernican (i.e. Aristotelian) approaches to science, as presented in Galileo’s landmark Dialogue Concerning the Two Chief World Systems (1632).1 Whereas the older Aristotelian ontology privileged the distinctive and ‘proper’ qualities of self-sufficient substances, Galileo’s new physics involves a relational (and thus proto-structuralist) metaphysics.

Galileo’s dialogue is organised as a series of discussions between two philosophers, Simplicio and Salviati, mediated by an informed and inquisitive layman, Sagredo. Modelled after one Galileo’s severest critics, the character Simplicio is an Aristotelian defender of the old Ptolemaic cosmology. Salviati defends Copernicus, and serves as a mouthpiece for Galileo’s own views. Sagredo (named after a friend of Galileo’s) engages in the discussion with an open mind, and as you might expect, comes to agree with Salviati-Galileo’s position. Galileo summarises the main aims of the text in its prologue:

First, I shall try to show that all experiments practicable upon the earth are insufficient measures for proving its mobility, since they are indifferently adaptable to an earth in motion or at rest [...]. Secondly, the celestial phenomena will be examined, strengthening the Copernican hypothesis until it might seem that this must triumph absolutely [...]. In the third place, I shall propose an ingenious speculation [whereby...] the unsolved problem of the ocean tides might receive some light from assuming the motion of the earth.2

Judith Miller begins with a characterisation of the division that separates modern (i.e. post-Copernican) from ancient science as a ‘revolution’, an ‘epistemological break’ in Bachelard’s sense of the term. ‘The theory of knowledge, the definition of reality, ontology, are all turned upside down’ (CpA 9.9:138). ‘Dialogue’ between two such deeply divided positions is only possible to the degree that a suitable mediator (Sagredo) is able to accomplish the transition from one side to the other, by directly engaging with the ‘epistemological obstacles’ that block access to the new ‘knowledge of the truth’ (138). Sagredo’s role in the dialogue will be to overcome the resistance of the Aristotelian Simplicio on the latter’s strongest and most familiar terrain: logic and philosophy. It is ‘strategically crucial’, Miller says, to read the Dialogue not only as a discussion about science but as a ‘work of philosophy’, in a sense informed by Alexandre Koyré’s influential reading of the modern scientific revolution as a ‘crisis of the European consciousness’.3

The basic metaphysical disjunction at issue here manifests itself in a number of ways; Miller considers first the question of speed, and then the definition of a straight line.

Galileo’s approach to speed violates Aristotle’s fundamental principle of identity and non-contradiction. Simplicio insists that one body can only be said to move faster than another when it traverses more space than the second body in the same amount of time (or the same space in less time). Salviati-Galileo shows this to be a restricted and inadequate conception. He compares the speed with which a rolling ball falls vertically to the ground (i.e. along a line perpendicular to the ground) to the speed with which it rolls to the ground down an inclined plane, and observes that, if they both begin from a stationary position, a moving body descends more rapidly along the perpendicular line; on the other hand, he also shows that in certain circumstances the speeds of two such descending bodies can the same. Like Simplicio, the layman Sagredo initially refuses to accept this, but his willingness to consider the problem in logical terms allows Salviati-Galileo to persuade him to agree to his new, more ‘universal’ conception of speed: the speeds of two moving bodies are ‘said to be equal when the spaces they traverse stand in the same proportion to each other as the times required to traverse them’ (140).

The Galilean method here, in other words, doesn’t restrict itself to the careful observation of diverse empirical realities: it ‘transgresses the fundamental principle of logic, at the very moment when it seeks to introduce, into a science that seems reluctant to accept it, the rigour of mathematical demonstrations’. Where Aristotle saw in non-contradiction and the necessity of syllogism the basis of all demonstration, Galileo thus ‘proclaims the transcendence of the real [du réel] with respect to the principles of contradiction and identity’ (141). The main epistemological obstacle confronting the new science is an ontology that conceives of beings in terms of substance and identity, such that, in the case of a comparison of speeds, one of the terms at issue (either space or distance) must remain the same. With Galileo, by contrast, a being comes to be defined not in terms of its intrinsic or self-identical properties but ‘as a complex of relations entertained by elements without attributes of their own’ (143).

Miller reiterates the point in her second section, ‘The Line’. The argument here concerns the status of three-dimensional space. Aristotle attributes the perfection of cosmic space to the substantive but contingent ‘fact of its tridimensionality’ (143); Salviati-Galileo demands a demonstration that cosmic space must necessarily have three and only three dimensions.

Given two points in space, the first dimension is constructed on the basis of the straight line that connects them, since of all the possible connecting lines this shortest line alone, as Sagredo accepts, is ‘unique and certain [un et sûr]’ (144). Miller argues that this first demonstration already illustrates the three fundamental principles or decisions of Galileo’s metaphysics. (1) When a problem offers an infinite number of possible solutions, the scientist should select the one which is uniquely necessary, which establishes that ‘the real is, and cannot be otherwise’. (2) This insistence on unique necessity nevertheless avoids reliance on the logical (Aristotelian) principle of identity insofar as the transcendence of reality with respect to the logical demonstration (i.e. its transcendence over the principle of non-contradiction or identity) is conditioned by the ‘relational definition of being, i.e. by the definition of the real by the place that determines its identity.’ For the Aristotelian, it is the being-straight of the line that determines its significance; for Galileo, the latter depends on what it relates. A straight line is the shortest way of measuring the distance between two points, but not necessarily the shortest way of measuring the distance between two other parallel lines (since in that case only a perpendicular rather than oblique line will qualify as the shortest). Depending on what it relates, ‘a straight line both is the shortest and is not the shortest’ (144). (3) What determines the identity of a thing (e.g. the shortest line) thus depends on the set of terms it relates.

Taking these three principles together, Galileo suggests that ‘unicity is necessity, necessity is relation, and relation is identity, in violation of the principle of contradiction. It is according to this sequence that the principle of contradiction is transgressed and the metaphysics of Galileo’s physics is established’ (145).

By contrast, Simplicio betrays the fundamental principles of the Aristotelian ontology when he suggests that construction of the second dimension of space (the plane) will rely on ‘the straight line rather than curved lines, since curved lines are not suitable [impropres] for this purpose’. This ontology relies on the substantial or essential ‘propriety’ of a being’s qualities, independently of any relation with other beings. Such essential properties belong to the given or contingent nature of a thing, and cannot be demonstrated as necessary - and since some of these properties are incompatible with others, logic helps to recognise and avoid contradictory combinations (145).

Against Aristotle, Galileo demands that ‘every proposition in physics be the object of a proof’ rather than a mere assertion, and this integration of physics into the ‘demonstrative sciences’ implies its ‘radical transformation’. Salviati-Galileo can prove that space must have three and only dimensions because (on the then-uncontroversial assumption that Euclidean geometry is a true representation of space) only three distinct right angles can be constructed at any given point (146).

Galileo insists in this way that the book of nature is ‘written in the language of mathematics’, rather than made up of substantial forms. The physics that can read such writing rests, Miller concludes, on a ‘metaphysics of relation’, a ‘metaphysics [that] opts for the “structuralist hypothesis”’ in Louis Hjelmslev’s sense of the term: ‘the structuralist hypothesis demands that we define magnitudes by relations and not the reverse.’4

References to this text in other articles in the Cahiers pour l’Analyse:

  • François Regnault, ‘Dialectique d’épistémologies’, CpA 9.4.

English translation:


Primary bibliography

Selected secondary sources:

  • Gaukroger, Stephen. The Emergence of a Scientific Culture: Science and the Shaping of Modernity 1210-1685. Oxford: Oxford University Press, 2009.
  • Koyré, Alexandre. Etudes Galiléennes. Paris Hermann, 1939. Galileo Studies, trans. John Mepham. Atlantic Highlands, N.J.: Humanities Press, 1978.
  • ---. From the Closed World to the Infinite Universe. Baltimore: Johns Hopkins University Press, 1957. Online at
  • Shapin, Steve. The Scientific Revolution. Chicago: University of Chicago Press, 1996.
  • Wallace, William A. Galileo’s Logic of Discovery and Proof: The Background, Content and Use of His Appropriated Treatises on Aristotle’s Posterior Analytics. Dordrecht: Kluwer Academic, 1992.


1. For more information, see ‘Galileo Galilei’, Stanford Encyclopedia of Philosophy,, and ‘Dialogue Concerning the Two Chief World Systems’, Wikipedia,

2. Galileo, ‘To the Discerning Reader’, Dialogue Concerning the Two Chief World Systems, 6.

3. Koyré, From the Closed World to the Infinite Universe, 1. As Koyré describes it, the scientific revolution served to disrupt and evacuate a formerly centred and oriented world, such that man ‘lost his place in the world, or more correctly perhaps, lost the very world in which he was living and about which he was thinking, and had to transform and replace not only his fundamental concepts and attributes, but even the very framework of his thought’ (4).

4. Louis Hjelmslev, ‘Linguistique structurale’ [1948], in his Essais linguistiques, 24.