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'.

Three Projects Involving Dispensing With Mathematical Objects

One of the many unfortunate things about academic fashions is that when a popular project goes out of fashion, superficially similar-looking projects which don't face the same difficulties can be tarred with the same brush.

Many people (myself included) feel cautious pessimism about formulating a satisfying nominalist paraphrase of contemporary scientific theories [one major issue is how to make formulate something like probability claims without invoking abstract-seeming events or propositions]. But we wouldn't want to over generalize.

 In this post I'm going to suggest three different motives for seeking to systematically paraphrase our best scientific theories (and many other true and false ordinary claims) in a way that dispenses with quantification over mathematical objects, and note that the requirements for success in the first (ex-fashionable) project are notably laxer in some ways than the requirements for the others.

[Note: I don't mean to endorse any of the projects below, but I think that 3 and Burali-Forti based versions of 2 are at least interesting.]

Three Motivations for Paraphrasing Mathematical Objects Out of Physics

1. General rejection of abstracta: You deny the existence of mathematical objects because you think allowing any abstract objects are bad. (this is the classic motive)
2. Explaining special features of mathematical practice by rejection of mathematical objects: You deny the existence of mathematical objects because you think that not taking mathematical existence claims at face value is allows for the best account of certain special features of pure mathematical practice, (e.g., by mathematicians’ apparent freedom to choose what objects to talk in terms of/disinterest in mathematical questions that don’t effect interpretability strength, or by the Burali-Forti paradoxes in higher set theory) not to take apparent quantification over mathematical objects at face value. 
3. Grounding math in logic/bringing out a claimed special relationship between math and logic: You may allow the existence of mathematical objects, but you’re moved by the close relationship between an intuitive modal notion of coherence/semantic consistency/logical possibility and pure mathematics to seek some kind of shared grounding and think that the coherence/logical possibility notion looks to be the more fundamental. As a result, it seems promising to seek a kind of "factoring" story, which systematically grounds all pure mathematics in facts about logical possibility, and all applied mathematics in some combo of logical possibility and intuitively non-mathematical facts.

 Distinguishing these motivations/projects matters, because what you are trying to do influences what ingredients are kosher for use in your paraphrases: 

If you have the first motivation (defending general nominalism), you need to avoid quantifying over any other abstracta, including: platonic objects called masses or lengths which don’t come with any automatic relationship to numbers, corporations, marriages, sentences in natural languages, metaphysically possible worlds, and (perhaps) propensities. 

But if you have the second motivation (defending mathematical-nominalism-adopted-to-explain-special-features-of-mathematical-practice), then quantification over abstracta which don’t have the relevant special feature, (e.g., objects in domains where we don't think appear to have massive freedom to choose what objects to talk in terms of or objects which don't give rise to a version of the burali-forti paradox), is fine. 

And if you have the third motivation, then quantification over any objects which aren’t intuitively mathematical-- or aren't mathematical in whatever way you are claiming requires a special relationship to coherence/semantic consistency/logical possibility -- is OK.

Hello World (again)!

As you can see, I haven't posted to this blog for ages.

I've been busy a) enduring the horrors of the job market b) getting a sweet 5 year postdoc (I still can't express how luck and grateful I feel) c) finishing a stack of old papers and d) writing a zillion page monograph to answer a minor technical question about Potentialism and logical possibility which my advisers asked in grad school (and then rewriting all the proofs 3+ times because a grumpy mathematician friend didn't think the prose was clear or concise enough!).

But now that I have time to focus on new research, I'm thinking it might be fun start blogging again. I'm certainly touched by the number of lurkers who still turn up to check this blog out.

Let's see how things go!

Art and the Examined Life

1.
It's a common thought that works of art can (somehow) imaginatively suggest different ways of approaching and experiencing the objective world around us. For example, a character in Rebecca Goldsmith's curious novel The Mind Body Problem says

``The interesting thing about art is your being presented with another's point of view, looking out at the world from his perspective, seeing the dreaminess of Renoir's world the clarity of Vermeer's, the solemnity of Rembrandt's, the starkness of Wyreth's.''

Some philosophers and literary scholars have suggested that engaging with such works of art is valued/valuable because it helps us understand and sympathize with other possible points of view (whether these really were occupied by specific artists or not) -- and might thereby make us nicer to people. Art experiences might make us more inclined to try to be nice because we are more sympathetic to certain people, and they might make us better at actually doing nice things because we understand these people better.

But  (to my knowledge) there's no clear empirical or folk-psychological case that having deep artistic experiences does make people significantly nicer. And I'm inclined to be skeptical.  There's a funny line in C.S. Lewis somewhere about how any inclination to think that art makes people more virtuous will be dispelled by asking an English professor to think about their colleagues. (Lewis was an English professor at Oxford and Cambridge).

2.
I'm gonna suggest a different idea about why we might highly value art for its power to evoke a different way of looking at things (in addition to valuing it as a source of pleasure, a tool for distraction etc.).

Maybe art experiences are valued for expanding our knowledge of how it would be psychologically possible for us to approach the world (including what adopting such approaches would feel like from the inside). In doing so, they help us a) live an examined life and b) choose how to live by expanding our sense what the space of (psychologically accessible) options is like.

For great works of art seem to reveal the possibility of ways of approaching and experiencing life [noticing things, finding projects appealing, reacting emotionally] which it would have otherwise been very hard for us to first personally imagine (imagine Jane Austen reading Nietzsche, or Nietzsche reading Jane Austen). Like Hume's first taste of pineapple (and unlike his first experience of a missing shade of blue) such art experiences expand our knowledge of what kind of experiences it is possible to have.

This has two benefits.

First, art can help us adopt a life we want to live, in approximately the same way that travel or visiting different social scenes does -- by making us aware of regions within the space of possible approaches to life [i.e. the space of options which are at least sufficiently psychologically possible for us to take up that we can imaginatively simulate them/enter into them for a while] is -- hence potentially aware of attractive options which were hitherto overlooking.

Second, I suspect that merely getting this kind of knowledge of psychological/phenomenological possibility space (whether it ever gives us practical benefits or not) contributes more to `living an examined life' (in whatever rough intuitive sense that seems desirable) than many big pieces of awesome philosophical knowledge would (e.g. knowledge of the right solution to the liar paradox).