Posthumus Vindication and Newton's Concept of the Derivative

In a recent Mind paper, `Incomplete Understanding of Concepts: the Case of the Derivative', Sheldon Smith vividly sets up some classic questions about Newton's concept of the derivative, and how later mathematical work can be seen as vindicating Newton.

However I'm not entirely convinced by Smith's answers to these questions.

Historical Background:

[Smith tells us how] Newton and Leibnitz had certain limited beliefs about the derivative

• that it was "the local rate of change of a function given by the slope of the tangent" so the derivative of x^2 kinda should be 2x
• that it was the limit as i goes to 0 of (f(x+i)-f(x))/i, hence derivative of x^2 was [(x+i)^2-x^2]/i which they thought was =(2xi+i^2)/i=2x+i=2x

but they did not have a very solid justification for the later reasoning (particularly the presumption that one can divide by i in the claim above).

Since then, mathematicians have defined multiple derivative-like notions which all let one defend reasoning like the above more rigorously, but don't always agree:

• the usual: the derivative of f(x) is the function f'(x) such that for every epsilon there is an i such that |(f(x+i)-f(x))/i - f'(x)| < epsilon
• the symmetric derivative: [like the normal definition but with (f(x+i)-f(x-i))/2i in place of f(x+i)-f(x))/i] (note that when f(x)= |x|, the symmetric derivative is 0 whereas the standard definition is undefined).
• a definition using infinitesimals
• a definition which also can apply to generalized functions like the Dirac delta function

Furthermore there is a common intuition that, in providing some of the definitions above and proving things with them, mathematicians like Weierstrass "justified [Newton's and Leibnitz's] thoughts" and that Newton and Leibnitz would have felt "vindicated" by subsequent developments of the derivative.

The questions:

Now, Smith argues that Newton didn't seem to be using any particular one of these modern concepts of the derivative.

• Newton didn't (somehow) implicitly have any of these precise concepts in mind, and which definition of limit he would have preferred to adopt (if he had been told about all of them) might vary with which one he found out about first.
•  There's no single "best sharpening" of what Newton believed/had in mind which must be accepted in limit of ideal science. We just have separate notions of derivative, each of which is mathematically legitimate. Thus we can't say that Newton meant, say, the standard contemporary notion of the derivative because he was conceptually deferring to the results of ideal science.

So he asks:

1. How `` should [one] think about the derivative concepts with which Newton and Leibniz thought''? 
2.  How ``could [Weierstrass] have managed to justify their thoughts even if their thoughts did not involve the same derivative concept as Weierstrass’s''?

Smith's Answers:
I take Smith's answers to the above questions to be as follows:

Q1: What was Newton's concept of the derivative [specifically, how does it effect the truth conditions for sentences]?

A: Newton's concept of the derivative (call it derivative_N) "only has a definite referent" in cases where all acceptable sharpening definitions of his concept agree.  So, for example, if the symmetric derivative and the standard derivative were both acceptable sharpenings, then expressions like `the derivative_N of f(x)=|x|' would fail to refer [or, perhaps, would refer to function which is undefined at 0 so that 'the derivative_N of f at 0' would fail to refer].

Q2: How was Weierstrass able to vindicate Newton, given that his concept of the derivative was different from Newton's?

A: One can vindicate Newton by justifying particular claims Newton made (e.g., about the derivative of x^2). And one can do this giving a proof of the corresponding claim employing Weierstrass's definition, if it also happens to be the case that all other permissible sharpenings of Newton's notion of the derivative would agree on this claim.

A Small Objection: 

I'm not entirely convinced by Smith's account of Newton's concept (Q1) for various reasons. But even if Smith is right about Q1, I think his answer to the vindication question (Q2) is fairly unsatisfying.

For suppose (as Smith seems to presume) Weierstrass vindicated Newton by showing the truth of particular claims he made about calculous -- that, say, what he expressed by saying ``the derivative of x^2 is 2x'' was true. If (as Smith's account of Newton's concept seems to tell us) the truth of this claim requires that all acceptable precifications agree in making ``the derivative of x^2 is 2x'' come out true, how can one adequately justify Newton's claim merely by discovering *one* such precificiation and showing that *it* makes the above sentence come out true?

A fix?

Maybe Smith could solve this problem (while keeping his account of the concept and the, IMO, good idea that vindicating Newton doesn't require assessing all possible derivative-like notions) as follows.

Say that "vindicating Newton's thought"  in the sense we normally care about (the in the sense that seems to have happened, and that, plausibly, Newton and Weierstrass would have cared about) doesn't require showing that some of Newton's specific mathematical utterances expressed truths. Instead, one can do it just  by showing Newton was right to believe some more holistic meta claim like `There is some mathematical notion which makes [insert big collection of collection of core calculous claims and inference methods] all come out true/reliable'.