Abstract
To illustrate the view that a speaker can have a partial understanding of a concept, Burge uses the example of Leibniz’s and Newton’s understanding of the concept of derivative. In a recent article, Sheldon Smith criticizes this example and maintains that Newton’s and Leibniz’s use of their derivative symbols does not univocally determine their references. The present article aims at challenging Smith’s analysis. It first shows that Smith misconstrues Burge’s position. It second suggests that the philosophical lessons one should draw from the practice of the historians of philosophy are more ambivalent than what Smith thinks.
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Notes
See footnote 15 below.
In this introduction, I speak indifferently of the determination of the concept of derivative and of its extension. I clarify this point below in Sect. 2.
As he makes clear, Smith does not approve Burge’s use of the intensional terminology; see his (2015, footnote 4, p. 3).
The word “derivative” is not used by Newton. So far as I know, it is Lagrange who systematized its use in his Traité from 1797. One finds different notations in Newton, but the most common one is the notorious dot notation. See Sect. 4.
Smith mentions Berkeley’s criticism, according to which one cannot both at the same time consider that h= 0 in \(\frac{{\left(x+h\right)}^{2}-x^{2}}{h}\), and that h = 0, which is required to set the derivative of x2 to 2x.
When used locally, the reference of “derivativeN” is a little more complicated: it is a pair of couples each consisting of a function and a point on the curve defined by the function: (f’(x), (a, f’(a))), (f(x), (a, f(a)).
One can easily understand the reasons that lead Smith to restrict the description of the Newtonian use of the derivative in this way. Taking into account the “contexts of justification” would have been even more unfavorable to Burge’s thesis according to which the Newtonian and Weiestrassian concept of derivative are identical. It would be because he wants to give initial plausibility to the thesis he is opposing to that Smith makes this restriction. Nevertheless, considering such a restricted use as the canonical one has, as we shall see, consequences that Smith may not have fully appreciated.
(Smith 2015, p. 22): “As a brief sketch: when it comes to processes modelled by hyperbolic partial differential equations, one will typically use the distributional notion of the derivative since “Weierstrass” notion can become inapplicable, for example in the case of shock waves. However, if one knew that shocks would not form, one would probably stick with “Weierstrass” notion because of its comparative simplicity. To use another example, various approaches to including dissipation into a Lagrangian treatment of mechanics involve the ‘fractional calculus’ …. Within that approach to dissipative physics, that notion is, perhaps, optimal, but it has rather limited use elsewhere. When exploring the (local) symmetric properties of functions, one will appeal to the symmetric derivative …. For various reasons, the strict derivative proves more useful within the context of Hensel’s p-adic analysis …. In short, each derivative concept is optimal for some purposes, but none is overall globally optimal.”.
There is a complication here: instead of rejecting suppress (PW) and (PS), Smith seems to admit both. But this cannot be the case. What Smith is claiming is that there is no point of conjecturing how Newton would have evaluated its derivative for the function absolute value at the origin – he cannot mean that Newton would have both accepted and negated that the value of such a derivative is zero. As far as I can tell, the idea is that “derivativeN” has only a partial extension, an extension which is defined for all curves that Newton actually considered, and not for the others. The first road (claiming that Newton’s concept can be defined in different non-coextensive ways) is open to Smith only because he severs the connection between a concept and its extension.
Field’s example is Newton’s mass. His two theses “(HR) Newton's word ‘mass’ denoted relativistic mass” and “(HP) Newton's word 'mass' denoted proper mass” corresponds to our (PW) and (PS). See his (1973, pp. 466 and 473).
To maintain that Weierstrass did prove Newton’s theorems like (TN), one does not have to endorse (PW): “all one needs to do to justify Newton’s belief … is to give a rationale for dividing by i when Newton divided by i and dropping i out of the calculation when Newton dropped it out” (Ibid. p. 33). Weierstrass’ derivative, as the symmetric derivative (along with the other permissible candidates), provide such a rationale. Thus, even if Newton’s belief was indefinite, “Weierstrass justified it in one of the ways that it could be justified”, and “this is so even though Newton and Weierstrass did not share a derivative concept” (Ibid. p. 33). Smith’s proof-centered perspective is different from Field’s semantical one, but the general strategy is the same: to show that indeterminacy does not throw us out of rationality.
For simplicity, I leave aside the fact that Smith also considers that Newton’s canonical use contains the application of the derivative to a curve at a point.
As Guicciardini says (2009, p. 288): “Newton’s fluxional analysis is not to be conflated either with Leibniz’s calculus or with later developments achieved by mathematicians such as Euler or Lagrange.” There are thus at least four notions that should be distinguished: Newton’s, Leibniz’, Euler’s and Weierstrass’ derivative.
In addition to Newton’s, Leibniz’s and Weierstrass’ derivative, Burge refers to Weierstrass’ and Frege’s concepts of natural number.
(Quine 1960, p. 28): “Translation between kindred languages, e.g., Frisian and English, is aided by resemblance of cognate word forms. Translation between unrelated languages, e.g., Hungarian and English, may be aided by traditional equations that have evolved in step with a shared culture. What is relevant rather to our purposes is radical translation, i.e., translation of the language of a hitherto untouched people. The task is one that is not in practice undertaken in its extreme form, since a chain of interpreters of a sort can be recruited of marginal persons across the darkest archipelago.”.
Ibid., p. 27: “Manuals that translate one language into another can be set up in divergent ways, all compatible with the totality of speech dispositions, yet incompatible with one another.”.
Ibid. p. 40: “In practice, of course, the natural expectation that the natives will have a brief expression for ‘Rabbit’ counts overwhelmingly. The linguist hears ‘Gavagai’ once, in a situation where a rabbit seems to be the object of concern. He will then try ‘Gavagai’ for assent or dissent in a couple of situations designed perhaps to eliminate ‘White’ and ‘Animal’ as alternative translations, and will forthwith settle upon ‘Rabbit’ as translation without further experiment— though always in readiness to discover through some unsought experience that a revision is in order.” But (Ibid. p. 51): “Who knows but what the objects to which this term applies are not rabbits after all, but mere stages, or brief temporal segments, of rabbits? In either event the stimulus situations that prompt assent to ‘Gavagai’ would be the same as for ‘Rabbit’. Or perhaps the objects to which ‘Gavagai’ applies are all and sundry undetached parts of rabbits; again the stimulus meaning would register no difference. When from the sameness of stimulus meanings of ‘Gavagai’ and ‘Rabbit’ the linguist leaps to the conclusion that a gavagai is a whole enduring rabbit, he is just taking for granted that the native is enough like us to have a brief general term for rabbits and no brief general term for rabbit stages or parts.”.
(Field 1973, p. 480–1): “By modifying the program of referential semantics in the way I have suggested, we come to rather different conclusions about indeterminacy from those reached by Quine. Quine thinks the existence of indeterminacy shows that scientific terms are “meaningless [and denotationless] except relative to [their] own theory; meaningless [and denotationless] intertheoretically.”… Now, what I contest in this argument is the assumption that the semantic relations of denotation and signification are in any interesting sense “relative to the conceptual scheme”; on my view … the existence of referential indeterminacy shows only that the relations of denotation and signification are not well-defined in certain situations, and that if we want to apply semantics to those situations we have to invoke the more general relations of partial denotation and partial signification. But these more general relations … are perfectly objective relations between words and extralinguistic objects.”.
See (Burge 2007b).
See (Burge 2005, p. 261: “The striking element in Frege’s view is his application of this distinction [incomplete versus complete understanding] to cases where the most competent speakers, and indeed the community taken collectively, could not, even on extended ordinary reflection, articulate the ‘standard senses’ of the terms. The view is that the most competent speakers may be in the same situation as the less competent ones in expressing and thinking definite senses which they cannot correctly explicate or articulate. Definite senses are expressed and ‘grasped’ (with merely the weak implication that they are thought with), even though no one may be capable of articulating or explicating those senses (grasping them clearly and analytically).” Burge shares the view he attributes to Frege in the passage.
See (Putnam 1975a, p. 53): “To speak of Einstein’s contribution as a “redefinition” of ‘kinetic energy’ is to assimilate what actually happened to a wholly false model.”.
It would be difficult to consider the word “derivative” as a natural kind term: what would be the equivalent of the physical environment in the mathematical case? Of course, one might endorse a full-fledged Platonism and set a world of abstract kinds, which would give the word “derivative” a reference. Fregean Platonism can be seen as a way to introduce natural species in mathematics. In such a perspective, integers would be kinds of abstract objects, different from other kinds of abstract objects, whose properties mathematicians would discover step by step. This anchoring into an abstract world would ensure the stability of reference of mathematical terms across time. It is crucial to understand that Burge’s theory of partial understanding is an effort to dispense with such an explanation by natural species.
(Unguru 1979, p. 56): “The history of mathematics is history not mathematics. It is the study of the idiosyncratic aspects of the activity of mathematicians who themselves are engaged in the study of the nomothetic, that is, of what is the case by law. If one is to write the history of mathematics, and not the mathematics of history, the writer must be careful not to substitute the nomothetic for the idiosyncratic, that is, not to deal with past mathematics as if mathematics had no past beyond trivial differences in the outward appearance of what is basically an unchangeable hard-core content.”.
Some historians object to this account, showing that the critique of anachronism plays a crucial role in the old internalist historiography. For instance, see (Blåsjö 2021).
Newton’s mature approach is based on Roberval’s insight that a curve must be seen as the result of the composition of different movements – asymmetry, via the reference to motion and time, is thus deeply embedded in Newton’s approach.
As examples, let me mention the recent emergence of a philology of diagrams (Netz 1999), in which geometrical figures are considered an integral part of mathematical material, and in which the alterations of diagrams over time must be studied with as much precision and rigor as those undergone by the textual part. Let me also mention the re-evaluation that the prefaces of some Greek mathematical works have recently undergone (see Mansfeld 1998, Bernard 2003).
One thinks of course of Descartes: “it was synthesis alone that the ancient geometers usually employed in their writings. But in my view this was not because they were utterly ignorant of analysis, but because they had such a high regard for it that they kept it to themselves like a sacred mystery” (Descartes 1984, p. 111). Guiccardini emphasizes that this gap between analysis and synthesis is pervasive in Newton’s Principia (2007, 259: “[Newton] did employ algebra, which for several purposes proved to be a useful analytical tool. But he did not print the Cartesian (or common) analysis but rather the geometrical synthesis, that is, the compositio but not the resolutio.”).
I have not said anything about unwritten historical sources, such as mathematical instruments, models and other machines, which bring a new layer of complexity and heterogeneity into the definition of use.
Burge considers that this reduction is typical of the logical positivism. See (2007a, p. 268): “[the logicist’s key idea] is the view that what there is to be understood, ‘meaning’, is to be reduced to actual procedures that express or constitute actual understanding. That is, meaning is identified with cognitive or theoretical usage. Roughly speaking, it contains two moves: the reduction of what is understood to actual understanding, and the explication of actual understanding in terms of presently articulateable abilities or experiences.”.
(Engelsman 1984, p. 8): “[early partial differential calculus] was called ‘differentiation from curve to curve’, and it did not deal with three space variables of the same character, but with two space variables x and y and with the parameter a, or ‘modulus’ of a family of curve.”.
(Guicciardini 2021, p. 17) takes Whiteside as a typical representative of the “old” historiography that projects contemporary settings into the mathematics of the past.
See (Engelsman 1984, p. 16): “A Puritan transcription just copying the originals is obviously out of the question. There is no sensible point in forcing the reader to work his way through all sorts of ad hoc notations. Moreover, a clear transcription of a given argument also requires that the implicit assumptions about independent and dependent variables are made explicit; it is essential for a thorough understanding of an argument to be clearly informed about such assumptions.”.
See the passage quoted footnote 17.
Note that this criticism of Smith’s approach is also a criticism of Burge’s view. Externalists do adhere to the idea that the use of a symbol is what should guide the determination of the reference (i.e., to (UEQuine)) – externalists argue only for a broadening of what qualifies as use. On this, see (Ebbs 2009).
I thank Karine Chemla, Gary Ebbs, Henri Galinon, Catherine Goldstein, Ivahn Smadja, and two anonymous reviewers for their helpful comments on drafts of the paper.
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Gandon, S. Sheldon Smith on Newton’s Derivative: Retrospective Assignation, Externalism and the History of Mathematics. Topoi 42, 333–344 (2023). https://doi.org/10.1007/s11245-022-09858-2
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DOI: https://doi.org/10.1007/s11245-022-09858-2