Abstract
We examine the contrast between the “Newtonian style” and the Cartesian, hypothetico-deductive method in order to expand on George Smith's account of working hypotheses and theory-mediation. We stress the pivotal role of theory-mediation in turning experience into well-defined phenomena and introduce the complementary notions of conditional and independent evidence. Conditional evidence is evidence in favor of an hypothesis that is based on phenomena that would not be constituted as phenomena without the hypothesis in question. Direct evidence is evidence based on phenomena that are independently available. We use the distinction to interrogate the role of mathematical representation as the most basic constitutive posit of the Newtonian style, as the working hypothesis that enables all further theory-mediation. Our primary examples are Newton's struggles with motion in fluids and his defense of the Laws of Motion.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The General Scholium was added to the second edition Principia (1713) and retained in the third (1726). A similar though less forceful dismissal of “hypotheses” can be found in Newton’s correspondence. See Shapiro 2004 for discussion of Newton’s use of the term “experimental philosophy” to separate his program of natural philosophy from the “hypothetical” natural philosophy that he associates with Descartes and Leibniz.
- 2.
The calculations that Newton uses to establish the incompatibility between Descartes’ Vortex Hypothesis and Kepler’s Area Rule rest on incorrect assumptions, as first pointed out by George G. Stokes in a paper of 1845. See Cohen 1999, Chapter 7 and Smith 2005, Notes 4 and 5 for further discussion of Newton’s errors.
- 3.
In the third edition Principia, for instance, Newton holds a three-mechanism view such that the overall resistance acting on a body moving through a fluid is produced by the fluid’s density, internal friction, and tenacity (or absence of lubricity, or slipperiness), and in this case, the tenacity is taken to be independent of the moving body’s velocity, whereas the internal friction is taken to be proportional to the body’s velocity. See the Scholium to Book 2, Section 3 (Newton 1999, 678–678) and Smith’s contribution to Cohen 1999 (pp. 188–194).
- 4.
Although Smith’s account of Newton’s reasoning in Book 2 isn’t cast in exactly the same terms, our rendering of Newton’s ‘fundamental assumption’ of fluid resistance captures the additive view of fluid resistance that Smith attributes to all editions of the Principia (See Smith 2000, 2001a, 2005). Our rendering is also sufficiently general to accommodate the fact that in different editions of Book 2 Newton treated the overall resistance encountered by a body as arising from different physical mechanisms (Smith 2005, 157).
- 5.
In other texts, including the later editions of the Principia, Newton used the term “tenacity” (tenacitas) to refer to the absence of slipperiness, or lubricity, in the fluid. See Smith’s contribution to Cohen 1999 (pp. 188–194).
- 6.
Pendulum experiments are useful for studying projectile motion since their properties are easier to measure than the properties of free fall (although Newton ended up using free-fall measurements in the second edition of the Principia). Moreover, Newton lacked a fully general solution for projectile motion in resisting media, so constrained motion proved important.
- 7.
To calculate a single swing arc loss, Newton actually measured the loss of 1/8 or ¼ of the overall arc, and divided by the number of swings. He worked out how to express arc loss per swing in terms of pendulum length, and total resistive force as a ratio to bob weight. He used a cycloidal pendulum, where maximum velocity is proportional to overall arc length. For additional details see Propositions 30 and 31 of Book 2 of the first edition Principia, and Smith 2001a, 259ff.
- 8.
The v2 term is common because at high velocities the total resistance varied as nearly v2. See Smith (2000, 130, footnote 19).
- 9.
Smith notes that “[r]egardless of why Newton presented the findings in the way he did… [the] experiments do not begin to allow reliable conclusion to be drawn about contributions to resistance involving an exponent of v less than 2.” Moreover, despite Newton’s efforts to focus on this term in subsequent experiments (some in water), here too “his results were disappointing” (2001a, 262).
- 10.
For a critical discussion of the “hypothetical” explanations that Descartes presents in Part III of the Principles of Philosophy, see McMullin 2008.
- 11.
Other examples from Book 2 could also be used to motivate the same question. Perhaps the most famous among them is Newton’s treatment of the speed of sound in the second edition Principia. In Book 2, Section 8, Newton compares his theoretical value for the speed of sound with the experimental data that had been compiled by William Derham, and he reports a perfect fit between the two (Newton 1999, 778). But to establish this fit, Newton relies on the alleged existence of “vapors lying hidden in the air,” which have a “crassitude” that increases the speed of sound. Newton has no direct evidence of these hidden vapors, just as Descartes has no direct evidence that the insensible parts of fire adapt their shapes to fill the spaces between the parts of matter. Both are posited as generally intelligible but empirically untestable ways to make sense of what has been empirically verified. For the debate over whether Newton’s introduction of the “crassitude” of air is best viewed as a “fudge factor” or as a good faith attempt to account for Durham’s results, compare Truesdell 1970 and Westfall 1973 with Cohen 1999, 361–362.
- 12.
See Smith 2001a, 2002b, and 2005, 2016. In Smith 2002a and 2016, Smith specifically contrasts Newton’s methodology with the Cartesian hypothetico-deductive method as it is described by Christiaan Huygens in the Preface to his Treatise on Light (1690) (Smith 2002a, 139–140; 2016, 189). For discussion of how Newton’s method for establishing universal gravitation departs from a more generally construed hypothetico-deductive model of scientific reasoning, see Ducheyne 2012 and Harper 2011.
- 13.
We are not using “prediction” in a technical sense. The point is simply that however Cartesian theories (or models, qualitative descriptions, etc.) make claims about the world, the Cartesian theoretician seeks agreement between those claims and the world.
- 14.
Not only can the world not tell the Cartesian how to revise her theory, arguably, the world cannot even tell her if the theory is generally true. This is because, at least as Descartes presents it in Part III of the Principles, the hypotheses that are posited to explain observed phenomena are not to be accepted as true, no matter how much empirical evidence might be amassed in their favor. Hypotheses are more or less acceptable depending on their consistency with Descartes’ metaphysically derived laws of nature, and depending on their consistency with the phenomena. But in general, every theory that Descartes posits to explain the visible world, including his vortex theory of the heavens, is “to be taken only as an hypothesis {which is perhaps very far from the truth}” (Descartes 1984, 105, III.44). For further discussion, see Domski 2019.
- 15.
- 16.
- 17.
We say “the most reasonable” because theoretical choice is not entirely constrained. Questions regarding the inductive scope of accepted claims—how they are “rendered general by induction”—remain open. A more guarded reading of Newton’s claim that gravity is “deduced from phenomena” is that within the theoretical framework of the laws of motion, given the then-current best measurements of planetary and terrestrial motions, and within the range of then-current measurements, the world constrains further theoretical choice to such an extent that accepting the law of gravitation within that range is the most reasonable choice. As Newton notes in the General Scholium, his evidence shows that gravity extends “as far out as the orbit of Saturn, as is manifest from the fact that the aphelia of the planets are at rest, and even as far as the farthest aphelia of the comets, provided that those aphelia are at rest” (589). See also Harper 2011 and Biener 2016.
- 18.
We say “seem to be” because we believe the traditional reading of Descartes is not entirely correct. As we read Part III of the Principles, and as noted in Note 14 above, Descartes is not offering his hypotheses as true and final pronouncements about the workings of nature. They are explanatory devices that, he says, could be “very far from the truth,” and consequently, they are revisable and essentially different in kind from the three laws of nature of Part II, which are final and firm pronouncements about the true workings of nature. For further discussion of the “truth” that Descartes associates with his laws of nature, see Domski 2018.
- 19.
For more general discussions of the problematic and sometimes mistaken reasoning that can be found in these editions of Book 2, Section 7, see Truesdell (1970), Westfall (1973) and the Smith essays cited herein, especially Smith 2005. Rouse Ball’s An Essay on Newton’s “Principia” (1893) is also a noteworthy commentary on Book 2. Ball makes no attempt to hide the mistakes of Section 7, but also finds there “much that is interesting in studying the way in which Newton attacked questions which seemed to be beyond the analysis at his command” (Ball 1893, 99; cited in Cohen 1999, 181).
- 20.
Newton also briefly discusses motion in an “elastic” fluid, which is a rare fluid in which there are repulsive forces between the particles.
- 21.
Newton discusses other shapes in the scholia to Propositions 37 and 38. In what follows, we limit our discussion to the case of moving spheres since it is most relevant to Newton’s purported refutation of Descartes’ Vortex Hypothesis—a hypothesis that concerns the motion of (roughly) spherical bodies through the heavens.
- 22.
- 23.
In the first edition of the Principia, Newton claimed that a stationary body suspended in the flowing efflux stream of water would support the full weight of the cylinder of water above it. He also claimed in the first edition that the water escaping the vessel in the efflux problem would have a velocity equal to a body falling from the full height of the tank. In the second edition, both of these values are cut in half, and as a result, in the second edition, Newton lowered the value for the inertial resistance on the front face of the body by a factor of four (Smith 2000, 119–121). See Smith 2001a for a detailed account of Newton’s first edition treatment of the weight supported by a resting body in a moving fluid, and also for an English translation of the relevant portions of Book 2, Section 7 from the first edition.
- 24.
Smith explicitly considers this possibility in Smith 2005.
- 25.
In fact, in 1752, Jean d’Alembert showed that the drag force on a body moving with constant velocity in an incompressible and inviscid fluid is zero.
- 26.
The quoted inference may seem unbelievably terse, but Newton’s justification of it and its premises is, in fact, unbelievably terse, no more than a few sentences.
- 27.
Or, as Smith puts the point: “Newton’s vertical-fall data for water and air provide no real evidence for his theory of resistance in inviscid fluids. To whatever extent Newton’s presentation gives an impression that these data do constitute evidence for the theory, that presentation is misleading. The only element of good science here appears to be the vertical-fall data themselves” (Smith 2005, 145).
- 28.
Smith explicitly considers these options in Smith 2000, 122–123.
- 29.
Newton offers a reconstructed history to suit his needs, not one we should take as fact. First, he was in no position to comment on Galileo’s processes of discovery. He knew little of Galileo (Cohen 1992). Second, Newton would have known that the laws of Wren, Wallis, and Huygens were based on a variety of principles. Those principles can be recovered within the framework of Newtonian laws, but they are not identical to them. Why Newton didn’t make this explicit is a complicated question (see Biener and Schliesser 2017). For our purposes, it is only important to notice that he portrayed the evidence generated by Galileo, Wren, Wallis, and Huygens as conditional. For more on the Royal Society competition, see Jalobeanu 2011. For further discussion of mathematical certainty in the reasoning that Newton attributes to Galileo, Wren, Wallis, and Huygens, see Domski 2018.
- 30.
Newton’s thought experiment is actually more complicated, but the gist is the same. See Newton 1999, 427–428.
- 31.
- 32.
The following is indebted to Miller 2017.
- 33.
We will shortly discuss a case where the definition implicit in the choice of mathematical representation does not match a philosopher’s stated definition.
- 34.
The parallelogram rule is in the first corollary to the Laws (Newton 1999, 417).
- 35.
We are simplifying a bit. For Descartes, the direction of motion is called a ‘determination’, and there is considerable debate about how it relates to the force of motion (see McLaughlin 2000). Nevertheless, he writes that “The first part of this law is proved by the fact that there is a difference between motion considered in itself, and its determination in some direction; this difference makes it possible for the determination to be changed while the quantity of motion remains intact” (Principles, Book II, Art. 41; Descartes 1984, p. 62).
- 36.
Principles, Book II, Art. 37, 39 (Descartes 1984, 59–60).
- 37.
See letter to Clerselier, 17 February 1945 (Adam and Tannery 1974–1986, IV, 183–188).
- 38.
To be more precise, Newton’s purpose in his exposition of the parallelogram rule is to show how to compose forces. But the construction does so by treating the effects of forces, i.e., motions (Newton 1999, 417).
- 39.
See Miller 2017.
- 40.
We thank the editors of this volume for this point.
References
Adam, Ch., and P. Tannery. 1974–1986. Oeuvres de Descartes. 2nd ed. Paris: Vrin.
Biener, Z. 2016. Newton and the ideal of exegetical success. Studies in History and Philosophy of Science, Part A 60: 82–87.
Biener, Z., and E. Schliesser. 2017. The certainty, modality, and grounding of Newton’s laws. The Monist 100: 311–325.
Cohen, I.B. 1992. Galileo and Newton. In Atti delle celebrazioni Galileiane (1592–1991), 181–208. Trieste: Lint.
———. 1999. A guide to Newton’s principia. In Newton, the principia, 1–370. Berkeley: University of California Press.
Descartes, R. 1984. Principles of philosophy. Trans., with explanatory notes by V.R. Miller and R.P. Miller. Dordrecht: D. Reidel.
———. 2001. Discourse on method, optics, geometry, and meteorology. Trans. P. Olscamp. Indianapolis: Hackett.
DiSalle, R. 2002. Newton’s philosophical analysis of space and time. In The Cambridge companion to Newton, ed. I.B. Cohen and G.E. Smith, 1st ed., 33–56. Cambridge University Press.
Domski, M. 2018. Laws of nature and the divine order of things: Descartes and Newton on truth in natural philosophy. In Laws of nature, ed. W. Ott and L. Patton, 42–61. Oxford University Press.
———. 2019. Imagination, metaphysics, mathematics: Descartes’s argument for the vortex hypothesis. Synthese 196: 3505–3526.
Ducheyne, S. 2012. “The main business of natural philosophy”: Isaac Newton’s natural-philosophical methodology. Dordrecht: Springer.
Harper, W. 2011. Isaac Newton’s scientific method: Turning data into evidence about gravity and cosmology. Oxford University Press.
Jalobeanu, D. 2011. The Cartesians of the Royal Society: The debate over collisions and the nature of body (1668–1670). In Vanishing matter and the laws of motion: Descartes and beyond, ed. D. Jalobeanu and P. Anstey, 103–129. New York: Routledge.
Maffioli, C. 1994. Out of Galileo: The science of waters, 1628–1718. Rotterdam: Erasmus.
McLaughlin, P. 2000. Force, determination and impact. In Descartes’ natural philosophy, ed. S. Gaukroger, J. Schuster, and J. Sutton, 81–112. New York: Routledge.
McMullin, E. 2008. Explanation as confirmation in Descartes’s natural philosophy. In A companion to Descartes, ed. J. Broughton and J. Carriero, 84–102. Malden: Blackwell.
Miller, D.M. 2017. The parallelogram rule from pseudo-Aristotle to Newton. Archive for History of Exact Sciences 71: 157–191.
Newton, I. 1999. The mathematical principles of natural philosophy. Trans. and ed. I.B. Cohen, and A. Whitman. Berkeley: University of California Press.
Reid, J. 2012. The metaphysics of Henry More. Springer.
Shapiro, A. 2004. Newton’s experimental philosophy. Early Science and Medicine 9: 185–217.
Smeenk, Ch., and G.E. Smith. ms. Newton on constrained motion.
Smith, G.E. 2000. Fluid resistance: Why did Newton change his mind? In The foundations of Newtonian scholarship, ed. R. Dalitz and M. Nauenberg, 105–136. Singapore: World Scientific.
———. 2001a. The Newtonian style in Book II of the Principia. In Isaac Newton’s natural philosophy, ed. J. Buchwald and I.B. Cohen, 249–313. Cambridge: MIT Press.
———. 2001b. Comments on Ernan McMullin’s ‘The impact of Newton’s Principia on the philosophy of science’. Philosophy of Science 68: 327–338.
———. 2002a. The methodology of the Principia. In The Cambridge companion to Newton, ed. I.B. Cohen and G.E. Smith, 1st ed., 138–173. Cambridge University Press.
———. 2002b. From the phenomenon of the ellipse to an inverse-square force: Why not? In Reading natural philosophy: Essays in the history and philosophy of science and mathematics, ed. D. Malament, 31–70. Peru: Open Court.
———. 2005. Was wrong Newton bad Newton? In Wrong for the right reasons, ed. J. Buchwald and A. Franklin, 127–160. Springer.
———. 2014. Closing the loop: Testing Newtonian gravity, then and now. In Newton and empiricism, ed. Z. Biener and E. Schliesser, 262–351. Oxford University Press.
———. 2016. The methodology of the Principia. In The Cambridge companion to Newton, ed. R. Iliffe and G.E. Smith, 2nd ed., 187–228. Cambridge University Press.
Truesdell, C. 1970. Reactions of Late-Baroque mechanics to success, conjecture, error, and failure in Newton’s Principia. In The Annus Mirabilis of Sir Isaac Newton, 1666–1966, ed. R. Palter, 192–232. Cambridge: MIT Press.
Westfall, R.S. 1973. Newton and the fudge factor. Science 179: 751–758.
———. 1980. Never at rest: A biography of Isaac Newton. Cambridge University Press.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Biener, Z., Domski, M. (2023). Working Hypotheses, Mathematical Representation, and the Logic of Theory-Mediation. In: Stan, M., Smeenk, C. (eds) Theory, Evidence, Data: Themes from George E. Smith. Boston Studies in the Philosophy and History of Science, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-031-41041-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-031-41041-3_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-41040-6
Online ISBN: 978-3-031-41041-3
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)