Abstract
Original explorers often see a puzzling conceptual landscape more vividly than jaded later travelers. This essay surveys several ways in which Descartes and Leibniz recognized descriptive problems within applied mathematics more clearly than later commentators have appreciated.
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Notes
Viz., that fluid velocity is inversely proportional to pressure.
Shtern (2012). Such circumstances often qualify as “weak solutions” of the relevant governing equations.
He also called such curves ‘imaginary,” meaning that their shapes could be pictured within the imagination only and not describable by a purely intellectual rule. See Domski (2009). Our “crank/rocker” comment suggests an interesting gloss upon Cartesian “indefiniteness.” In the absence of disturbances supplied by freely willing agents, the Cartesian inorganic world will grind on in the manner of a gigantic piece of connected clockwork. In such a setting, the two controls of our little spiral drawing device will again become connected to one another in “one degree of mobility” fashion, but this time the mechanical entanglement transpires through the entire “system of the world” and can’t be isolated within any palpable crank/rocker mechanism. In such circumstances, our finite minds can’t adequately represent the spiral motions to our intellectual facilities due to the infinite complexity of the transmissive machinery involved. Perhaps the “indefiniteness” in the fluid flow case can be approached similarly: as the fluid passes, the machinery of the “system of the world” causes the walls of the enclosing pipe to flex imperceptibly in a manner that allows the fluid matter to rearrange its shapes without ever reducing to a truly infinitesimal “dust” in the process.
I stress that I am unaware of any textual evidence in favor of this gloss on “indefiniteness” but it strikes me as a suggestion very much in the Cartesian spirit.
See Engelsman (1984). The barriers trace to early unclarities in the understanding of differentials and explain why Leibniz was not able to advance to a full PDE modeling of a dynamically loaded elastic material, of the sort required in billiard ball impact.
These distinctions relate in turn to the differences between “elliptic and hyperbolic modeling circumstances” discussed in Wilson (2017d).
Indeed, one often encounters the phrase “billiard ball physics” employed as a synonym for “Newtonian classical physics.” In Wilson (2006), Chapter 9, I discuss how intimately Hume’s dissatisfactions with causation trace to his viewing impulsive ball impact as embodying the essence of Newtonian science.
Rohault (1723), p. 81, articulates the implausible Cartesian contention that idealized colliding balls never distort. To this, the Newtonian Samuel Clarke objects that some “force of elasticity” must be invoked without explaining how it arises (Newton sometimes wrote of immutable atoms being surrounded by a cushioning atmosphere of “fluid”). For more data on this interesting debate, see Scott (1970)). Leibniz’ approach is far closer to a modern continuum physics conception of strain energy. For a fuller discussion of all issues surveyed in this section of the essay, see my Wilson (2017c).
See also the somewhat different rendering of this text in the Ariew and Garber translation, p. 124. In this particular passage, Leibniz is also characterizing his earlier views on impact, before he realized that coherent underpinnings of a “metaphysical sort” for the underlying elastic responses needed to be provided. On the reading I suggest, Leibniz continues to believe that physics operating in an efficient causation mode will inevitably exploit descriptive cutoffs of this same sort but he now maintains that we won’t be able to understand the elastic response fully unless we switch to a descriptive framework in which our billiard balls are further credited with a teleological capacity that allows them to generate restorative forces in proportion to the degree they have been displaced from their natural rest conditions. In particular, Leibniz’ modeling of a flexible beam is explicitly constructed around the restorative teleologies that we typically eschew when we approach a physical system in efficient causation mode (I append this comment in response to a query from an anonymous referee).
More exactly, the resistive tractions against change of shape induced by the stored strain energy.
More exactly, in the different “constitutive principles” they obey, which frequently depend upon other parameters than simple stress and strain.
To be sure, skeptics like Locke worried that we might permanently lack the concepts required to truly rationalize compressive restoration, in the same manner that the mechanisms of cohesion and fracture might forever elude our understanding. For general background, see Rowlinson (2005).
“Letter to Jacob Bernoulli,” 1702 quoted in Meli (1993), p. 55.
Of course, many considerations lie behind Leibniz’ remarks on the decompositional nature of the continuum. It strikes me that the fact, plainly evident to Leibniz, that the successful employment of differential equations within mathematical physics demands that we model lower scale events in an artificial, smoothed-over manner casts a helpful light upon those broader considerations (again, a response to an anonymous referee).
I find it surprising that few commentators on Leibniz’ theory of matter observe that he and the Bernoullis developed the first viable models of flexible materials such as springy beams. To do so he needed to break his problems into one dimensional subproblems: first locate the beam’s neutral axis and bending moment and then compute its equilibrium configuration when loaded with rocks.
His “complementarity” sometimes seems to refer to physical modelings where different modes of mathematical description condition one another in useful, but not wholly amalgamated, ways.
I am grateful to the Templeton Foundation (Robert Batterman, principal investigator) for support on this project and to Alan Nelson for helpful comments.
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Wilson, M. What I’ve learned from the early moderns. Synthese 196, 3465–3481 (2019). https://doi.org/10.1007/s11229-017-1361-8
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DOI: https://doi.org/10.1007/s11229-017-1361-8