Definitions more geometrarum and Newton's scholium on space and time

https://doi.org/10.1016/j.shpsb.2020.05.005Get rights and content

Highlights

  • Newton's views on space evolve between the first and last editions of the Principia (1687–1726).

  • Newton uses "definitions" to distinguish between two methods of inquiry: natural philosophy and the "manner of geometers."

  • Newton's views change in response to critique concerning the empirical warrant of his conception of body and space.

Abstract

Newton's Principia begins with eight formal definitions and a scholium, the so-called scholium on space and time. Despite a history of misinterpretation, scholars now largely agree that the purpose of the scholium is to establish and defend the definitions of key concepts. There is no consensus, however, on how those definitions differ in kind from the Principia's formal definitions and why they are set-off in a scholium. The purpose of the present essay is to shed light on the scholium by focusing on Newton's notion and use of definition. The resulting view is developmental. I argue that when Newton first wrote the Principia, he viewed the scholium's definitions as items of “natural philosophy.” By the time of the third edition, however, he came to view their methodological status differently; he viewed them as belonging to the more qualified “manner of geometers.” I explicate the two methods of natural inquiry and draw out their implications for Newton's account of space.

Introduction

Newton's Principia begins with eight definitions and a scholium, the so-called scholium on space and time. For much of its history, the scholium was thought to argue for the existence of absolute space by providing empirical evidence that each body had a unique, true state of motion, and assuming that this motion must be motion relative to an immovable, insensible, all-encompassing geometrical structure — absolute space (Mach, 1919, Ch. 2.6; Reichenbach, 1957, p. 34). There were notable exceptions, of course (like Kant and Euler), but this reading of the scholium was largely entrenched as orthodoxy. It is no longer. One of the more influential breaks with orthodoxy was made by Stein (1967). Stein argues that Newton did not frame the scholium in order to provide empirical evidence for the existence of absolute space. Rather, Newton's purpose was to examine the dynamical presupposition of the mechanics his time and show that, in order for those presupposition to have any purchase on the real world, we must adopt the notions of absolute space, time, and motion as he defines them.

Stein's reading has been developed by Laymon (1978), DiSalle (2006), Huggett (2012), and others, albeit with important modifications.1 Rynasiewicz (1995a; 1995b; 2019) has also reached some corroborating conclusions, although through an independent line of reasoning.2 There are significant debates here.3 But there are also points of agreement. Details aside, the scholars just cited mostly agree that Newton's purpose in the scholium was to show that motion, as treated by the mechanics of his time, could not be “adequately defined as some distinguished form of motion relative to other bodies, but [had to be] analyzed instead in terms of an absolutely immobile space” (Rynasiewicz, 1995a, p. 135, emphasis added). They also agree that Newton's arguments were grounded in the details of actual mechanical practice, not independently motivated philosophical considerations. According to one version of this view, the mistake of earlier interpretation was to presume that “what [Newton meant] by space, time, and motion, and what [he meant] by claiming that they are ‘absolute,’ [was] already established on purely philosophical grounds, so that [the scholium merely showed] what physics has to say about these philosophical concepts” (DiSalle, 2016, p. 37). What they overlooked was that “Newton was not taking any such meanings for granted, but defining new theoretical concepts within a framework of physical laws” (ibid.). Although the details of this particular analysis are contentious, there is consensus on one claim it captures well: that in the scholium Newton engaged in a highly nuanced conceptual exercise, that of establishing and defending definitions. He did so within a framework of already-accepted mechanical principles, but he was establishing and defending definitions nonetheless.

I'll call this shared position “the current interpretation.” It is surely correct. However, because it highlights the methodological importance of defining, it raises questions about Newton's notion and use of definition that were not raised, perhaps not even noticed, by the older, Machian interpretation. Consequently, these questions have been less attended to in the literature. The purpose of this paper is to stress their importance, attempt to answer them, and by so doing shed more light on the scholium. The resulting view of the scholium is developmental. I argue that when Newton first wrote the Principia, he viewed the scholium's definitions as items of “natural philosophy.” By the time of the third edition, however, he came to view their methodological status differently; he viewed them — or, at least, by his own lights had good reason for viewing them — as belonging to the more qualified “manner of geometers.” My discussion focuses on this change insofar as it pertains to Newton's account of space.4

The paper runs as follows. In §2, I articulate four questions about "definitions" in the scholium on space and time. The first concerns the methodological difference between formal definitions and definitions in scholia, generally speaking. The second and third questions concern the meaning of a small caveat in the first paragraph of the scholium on space and time, and its later deletion. The fourth concerns the significance of a set of formal definitions Newton drafted for Book III of the third edition of the Principia and the notion of definition they embody. In §3, I review some existing answers. In §§4–7, I provide my own, using a developmental narrative. In §4, I distinguish between formal definitions and definitions in scholia by means of Newton's distinction between two methods of physical inquiry: the “manner of geometers” and the natural-philosophical manner. In §5, I argue that when Newton first wrote the Principia, he believed that he was providing an account of space in the natural-philosophical manner, one that described spatial ontology as it truly was; no caveats, additions, or deletions. In §6, I show that his belief in the natural-philosophical status of his account of space depended on implicit assumptions about the nature of body. When those assumptions came under attack in the 1710s, he responded by proposing the new definitions for Book III mentioned above, and articulated their methodological status in terms of “the manner of geometers.” In §7, I conjecture that because the new definitions were directly connected to Newton's account of space, and because they were methodologically parallel to the scholium's definitions, Newton's engagement with them should have prompted him to reevaluate the methodological status of the scholium's definitions. While there is no direct evidence that he did so, the hypothesis allows us to answer the four questions with which we began, and the circumstantial evidence for it is compelling. Consequently, I argue that by the mid-1710s, Newton came to adopt — or, at least, by his own lights had good reason for adopting — a more epistemically guarded position regarding his account of spatial ontology, one characterized by the “manner of geometers.” He no longer believed that his account of space was unqualifiedly true, but that it was only true given assumptions implicit in his own physics.5

Section snippets

Defining in the Principia: four questions

The first and primary question is this: If Newton's strategy in the scholium was to define key concepts, why do so in a scholium? Why not present the definitions formally, alongside other “definitions”? The definitions's locus is not incidental to their purpose.

To see why this is not a red herring, consider this. As is well known, works written in the geometrical style, like the Principia, traditionally consisted of definitions, axioms and postulates, problems, theorems, and lemmas. Each played

Some answers

Let's start with a common answer to B, as it is the quickest to dismiss. It is possible that Newton only demurred from defining the genera time, space, place, and motion in the scholium. He did, however, intend to define their species, absolute and relative. He expected his readers to track the distinction, and so didn't need to clarify further. Rynasiewicz (1995a) entertains this reading, but rejects it. He notes that “in turning to locus and motus [Newton] does provide what, according to any

“Definitions,” scholia, and the mathematical/physical distinction

Instead of analyzing the scholium's arguments to arrive at its sense of “definition,” I'll privilege Newton's explicit statements on definitions and scholia. Newton explicitly reflects on the nature of definitions in mechanics in two notable places. The first is DG (examined in this section), the second is the draft definitions to E3 (examined in §6). As we'll see, although they are separated by at least thirty years, they contain almost identical ideas. In this regard, DG can serve as a guide

Treating space natural-philosophically (pre-1710s)

The distinction between the mos geometrarum and the natural-philosophical manner suggests an answer to A; namely, that by treating space, time, and motion in a scholium Newton meant to treat them natural-philosophically. Let's focus on space. It is clear that at least in DG, E1, and E2 (more on E3 shortly), Newton thought he was treating space as it really is, with all attendant claims to truth.

In DG, the asymmetry between Newton's account of space and his account body bears this out. Although

Interlude: draft definitions of body and void

We have just seen that the natural-philosophical status of Newton's account of space relied on an intricate chain of reasoning that bottomed out in empirical evidence concerning the existence of empty spaces. That empirical evidence, however, was not without objection. Newton reasoned that differences in resistance and specific gravity show that “[different] quantities of matter [are] contained in equal spaces,” and that this can only happen if equal spaces contain different proportions of full

Treating space more geometrarum (post-1710s)

Having answered A, B, & D (§§4, 5, & 6), we can hazard an answer to C; why did Newton remove non definio from the scholium? The answer relies on the tight-knit connection between Newton's conception of space and his experimental evidence concerning its emptiness. As we saw in §5, Newton reasoned from the phenomena of rise and descent and differential resistance to the existence of empty space, to space's unique “manner of existing,” to the certainty and natural-philosophical status of his

Conclusion

To sum, this essay has weaved together a story that touches on several interpretive issues: Newton's use of “definition,” his physical/mathematical distinction, the nature of his methodological suppositions in E3's draft definitions and the scholium on space and time, and their interrelations. It has also provided a developmental account of Newton's thought in order to answer the four main questions that made up this essay's ‘skeleton.’ To reiterate: I've argued that Newton drew a principled

Acknowledgments

Many thanks to Katherine Brading, Mary Domski, Donald Rutherford, and Kathryn Tabb for their comments on an early draft of this paper. Thanks also to the anonymous referees of this journal and to John Martin and Marius Stan for unpaid Latin consults.

References (59)

  • K. Brading

    Newton’s law-constitutive approach to bodies: A response to Descartes

  • K. Brading

    Time for empiricist metaphysics

  • I.B. Cohen

    Introduction to Newton’s Principia

    (1971)
  • I.B. Cohen

    A guide to Newton’s Principia

    The Principia: Mathematical principles of natural philosophy

    (1999)
  • V. De Risi

    The development of euclidean axiomatics

    Archive for History of Exact Sciences

    (2016)
  • R. Descartes

    Principles of philosophy, translation of Principia philosophiae of 1644 with additional material from the French translation of 1647. Translated by V. R. Miller and R. P.Miller

    (1983)
  • R. DiSalle

    Understanding space-time: The philosophical development of physics from Newton to Einstein

    (2006)
  • R. DiSalle

    Newton’s philosophical analysis of space and time

  • M. Domski

    Newton’s mathematics and empiricism

  • S. Ducheyne

    The main business of natural philosophy: Isaac Newton’s natural-philosophical methodology

    (2012)
  • S. Ducheyne

    Newton on action at a distance

    Journal of the History of Philosophy

    (2014)
  • J. Earman

    World enough and space-time: Absolute vs. relational theories of space and time

    (1989)
  • G. Gorham

    Newton on God’s relation to space and time: The Cartesian framework

    Archiv für Geschichte der Philosophie

    (2011)
  • R. Greene

    The principles of the philosophy of the expansive and contractive forces

    (1712)
  • N. Guicciardini

    Isaac Newton on mathematical certainty and method

    (2009)
  • W. Harper

    Isaac Newton’s scientific method: Turning data into evidence about gravity and cosmology

    (2011)
  • J. Harris

    Lexicon technicum, or, an universal English dictionary of arts and sciences

    (1708)
  • N. Huggett

    What did Newton mean by ’absolute motion’?

  • A. Janiak

    Newton as philosopher

    (2008)
  • Cited by (2)

    • Constituting the ‘object’ of science in Newton's Principia: the many faces of Janus

      2022, Studies in History and Philosophy of Science
      Citation Excerpt :

      Apart from what Newton himself thought, I consider the formulations in the Principia to be compatible with the view of Robert DiSalle when he writes that ‘conceptions of space and time are not arbitrary metaphysical hypotheses appended to otherwise empirical physics; they are assumptions implicit in the laws of physics’ (DiSalle, 2004, p. 34).10 Tackling questions raised by the fact that Newton explained the meanings of ‘absolute’ space and time in a Scholium, in contradistinction to his previously stated Definitions, Biener (2020) explains it in terms of Newton's intellectual development and clarifies the sense in which space and time are indeed defined there. From another perspective, specifically in relation to space, DiSalle (2020) connects Newton's introduction of ‘absolute space’ into his theoretical framework with his ‘emerging understanding of the relativity of motion’, towards a dynamical treatment of motions in a system.

    View full text