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'.

Bookclub: 'Compositionality, Understanding, and Proofs'

In the latest Mind, Peter Pagin argues that Dummett's proof theoretic semantics is incompatible with the compositionality - a popular view in philosophy of language.

Compositionalty is the view that the meaning of a sentence is completely determined by the meaning of its parts i.e. for every connective that might be used to build up a sentence, there's a composition function which takes the meanings of whatever components the connective is being applied to, to the meaning of the overall thing you get after applying the connective.

Proof theoretic semantics is the idea that: a) understanding a sentence consists in an ability to recognize (canonical) proofs of that sentence, and b) the meaning of a sentence is "the property of being a proof of that sentence".

Odd as I feel defending Dummett, on any subject, I think Pagin is wrong to say these two things are incompatible.

What compositionality (as stated in e.g. the stanford encylopedia, and "informally" by Pagin himself) requires is that, for each connective phi, there be a function Cphi which takes the < property of being a proof of p, the property of being a proof of q, the property of being a proof of r > to <the property of being a proof of phi(p, q, r))<. But if you accept compositionality at all, this has to be the case, because the property of being a proof of phi(x) can only be different from that of being a proof of phi(y) if x and y are different, and hence the property of being a proof of x is different from the property of being a proof of y. I don't think Pagin would deny this.

The problem is that Pagin seems to think compositionality + proof theoretic semantics requires something more. He writes:

"The combination of proof-theoretic semantics with the requirement of recognizability of proofs comes into conflict with compositionality. For assume that we have a semantic function phi for a language L. A generalized composition function {rho} for phi must then meet two conditions: (i) it must be possible to know the meaning of any complex expression in L by knowing {rho}, the modes of composition and the meaning of simple expressions; and (ii) the condition of being a canonical proof must, for every provable sentence A, be met by some proof that is recognizable by any speaker who understands A."

Note the switch here from the idea that compositionality says there must BE a function, to the claim that it must be possible to learn the meaning of words by KNOWING this function together with various other facts.

Firstly, the very idea of "knowing rho" (where rho is a function) makes me feel itchy and confused. I understand what it is to know *that something is the case* e.g. that a function f takes a certain value on a certain input. And I (kindof) understand what it is to know a person (e.g. I don't know Bill Gates, but I do know my advisor W.G.). But what's the equivalent of being on a first name basis with an abstract mathematical object? Does knowing a function mean being able to compute it? Being able to give a definite description that refers to it? Being able to give two distinct definitions definitions and knowing that they pick out the same function.

My best guess at what Pagin intends here, is that 'knowing rho' = knowing some proposition of the form:", is the composition function for whatever language L is in question'.

But now, note that Pagin's claim doesn't follow at all from the idea of compositionality - that the meaning of a composite sentence completely supervenes on the meanings of the pieces it is composed out of. The claim that a function with a certain property *exists* does not entail that it is possible to *know* such a function exists, or that this function is computable, or that it is possible to know which program computes it! So, compositionality doesn't imply that its even possible to have such knowledge, much less that it's possible to use this knowledge to learn the meaning of various composite expressions.

This distinction is especially crucial to remember in the context of discussing Godel's Thereom. For, remember from the Putnam-Penrose debate that all our reasoning about mathematics might well *be* recursively axiomatizable, it's just that we couldn't use mathematical reasoning to come to *know* what this recursive axiomatization was.

And, alas, Godel is exactly where Pagin is headed. For, his argument turns out to be that, if you could know some concrete specification of the composition function rho, you could mill out a recursive specification of the class C of acceptable proofs in number theory, then you could use this to construct an acceptable proof of the con sentence for C, which is itself a statement in number theory, but (by Godel I) cannot be proved in C. Contradiction.

Pagin's conclusion is that compositionality and proof-theoretic semantics are incomptatible. But, if this argument works, all it really shows is proof-theoretic semantics requires that one could not come to *know* a recursive specification of the composition function phi.

At this point, Pagin might say that the whole point of compositionality is to explain how we can know the meaning of complex sentences, by knowing their parts, so that accepting this point would be bad news for the proof-theoretic semanticist. But note that, we obviously don't understand composite sentences by explicitly breaking them don into parts. So the fact that we could never realize that something was a concrete specification of the composition function for our language, doesn't prevent compositionality from helping explain our linguistic abilities.

Bookclub: Pedersen on Wright's Entitlement

This latest installment is about Nikloj Pedersen's recent Synthese article on Crispin Wright. Pedersen criticizes (correctly in my opinion) certain possible motivation for Wright's idea that we are entitled to assume certain "cornerstone" propositions (like 'I'm not a brain in a vat') just because assuming these is requisite for getting any substantative theory of a given area off the ground. (You can't just point out that accepting ~BIV leads you to have many and no fewer true beliefs than the skeptic if BIV is true, and no fewer true beliefs if BIV is false. For, avoiding false beliefs is presumably also epistemically important, and assuming ~BIV imposes a risk of having many more false beliefs)

Instead he proposes that such cornerstone assumptions have "teleological value" insofar as they are aimed at something of value (namely, true belief), whether or not they actually succeed in producing such true beliefs. But this seems to immediately generalize to all beliefs - not just cornerstone ones.

For, what beliefs aren't aimed at the truth? It's just as true of the person who assumes the existence of a massive conspiracy as of the person who assumes the existence of the external world that they aim at having many true beliefs. Indeed, many people would say that it's a necessary truth, part of what it means for something to be a belief, that in believing that P one is trying to believe the truth.

With the possible exception of cases like the millionaire who bribes you to believe some proposition, all beliefs would seem to aim at truth. Hence it seems that all beliefs inherit teleological justification in Pedersen's sense.

One might be able to make this into an interesting view - all beliefs (not just cornerstone ones) are warranted until one gets active reason to doubt them. Such a position is remeniscent of conservitivism and coherentism. But, from the article Pedersen shows no sign of intending to say that all beliefs are default justified.

bookclub: Gareth Evans on Semantics + Tacit Knowledge I

I just discovered Gareth Evans has a neat article (probably a classic) about the very issues of what a semantic theory is supposed to do, which I've been worrying about recently. I found it so interesting that I'll probably write a few posts about different issues in this article.

The article starts out by paraphrasing Crispin Wright, to the following effect:

If philosophers are trying to state what it takes for sentences in English to be true, there's a very simple schema '"S" is true iff S' (this is called Tarski's T schema) which immediately gives correct truth conditions for all English sentences.

But obviously, when philosophers try to give semantic theories they aren't satisfied with just doing this. So what is the task of formal semantics about?

I think this is a great question. When I first read it I thought:

Perhaps what we want to do is notice systematic relationships between the truth conditions for different sentences in English e.g. whenever "it is raining" is true "it is not the case that it is raining" is false. If you want to make this sound fancy, you could call it noticing which syntactic patterns (e.g. sentence A being the result of sticking "it is not the case that" on to the front of sentence B) echo interesting semantic properties (e.g. sentence A having the opposite truth value from sentence B).

However, I would call this endeavor the study of logic, rather than semantics. So far we have logical theories that help us spot patterns in how words like "and" and "there is" (and perhaps "necessarily") effect the truth conditions for sentences they figure in. There may be similar patterns to notice for other words as well (e.g. color attributions - something can be both red and scarlet but not both red and green) and one could develop a logic for each of these.

We aren't saying what "and" means (presumably if we are in a position to even try to give a logic for English expressions we already know that "and" means and), rather we are discovering systematic patterns in the truth conditions for different sentences containing "and".

So, rule one other thing off the list.

Instead, Wright suggests (and Evans seems to allow) that semantics to go beyond trivially stating the truth conditions for English sentences by "figuring in an explanation or the speaker's capacity to understand new sentences". (I am quoting from Evans, but both deplore the vaguness of this statement).

This sounds initially plausible to me, but it raises a question:

Once we have noticed that attributions of meaning don't require anything deeper than the kinds of systematic patterns of interactions with the world displayed by wittgenstein's builders (maybe with some requirement that these interactions be produced by something that doesn't look like ned block's giant look-up table), the question of how human beings actually manage to produces such behavior seems to be a purely scientific question.

There are just neuroscientific facts, about a) how the relevant alterations of behavior (corresponding e.g. to learning the word "slab") are produced when a baby's brain is exposed to a suitable combination of sensory inputs and b) what algorithm most elegantly describes/models this process.

So, what's the deal with philosopher trying to do semantics? And what does it take for an algorithm to model a brain process better or worse? I'll try to get more clear on these questions, and what Evans would say about them, in the next post.

bookclub: Wright and Hale's comeback

Last time, I blogged about McFarlane's criticisms of Wright and Hale - here's what I think of their response, published in the same journal.

W+H say that the difference between stipulating the axioms of PA vs. stipulating Hume's principle is that the former stipulation fails their requirements by being "arrogant". So, what exactly is supposed to be arrogant about this stipulation:
A(x) x=x iff there is an object 0, and .... (the axioms of PA)
that doesn't also apply to all the instances of the schema below?
the Fs are equinumerous with the Gs iff the number of Fs = the number of Gs

I confess that I'm not sure exactly what W+H's answer to this, even after multiple readings of the paper. But here are some things I see in the paper that don't seem to work as explanations for why Hume's principle escapes arrogance:

1. Specification of truth conditions for all atomic sentences:
One criterion that comes up is that the latter collection of stipulations gives truth conditions to all atomic sentences in the language of numbers, in terms which someone who doesn't yet understand number talk can understand.

But then I don't see why the advocate of just stipulating the PA axioms couldn't break their stipulation down into a similar infinite series of stipulations, each of which is equated to a logical truth as follows and so on for all the countably many atomic sentences derivable from PA e.g.
A(x) x=x iff 0 is a number
A(x) x=x iff 1 is a sucessor of 0
..
You might worry that we can't intend to make any such stipulation, but note that both series of axioms will be recursively axiomatizable, (we aren't enumerating all the truths of arithmetic, just all the ATOMIC truths). So it's hard to see how we could be capable of intending all the instances of W+H's schema, but not the PA schema.

2. RHS can generate understanding of the new terms
Somehow W+H think that saying 'Let 'the numbers' name to some cannonical collection of objects which relate to each other in such a way that there's a "0 object" it stands in the successor relation to other objects etc' would not suffice to let someone who did not previously have the concept of number understand it, whereas hume's principle does.

But I don't buy the argument for this. All W+H say is that a) something else, namely the ramisfication of the PA stipulation, would have to be couched in second order logic and hence presuppose something like understanding of the numbers and b) the PA stipulation just adds to the ramsification by giving labels to the particular objects that stand in the relations stipulated by the ramsification.

They then seem to conclude that making the stipulation of the axioms of PA cannot suffice for understanding. This would be a plausible argument if they first showed that the ramsification doesn't suffice to express the concept of number, and then argued that the straight version doesn't add anything. But what they actually argued was that the ramsification would be incomprehensible to someone who didn't already have something more powerful than the concept of number. So, the fact that the straight stipulation of the PA axioms doesn't *add anything* to the ramsification, doesn't suffice to show that it couldn't be used to give someone understanding of the concept of number.
(which basically asserts that there are things satisfying the PA axioms using the language of second order logic).

W+H's argument seems to be exactly analogous to saying:
The stipulation 'a bachelor is a unmarried man' doesn't suffice to introduce someone to the concept of bachelorhood, because it doesn't add anything to 'a bachelor is an unmarried man who is either a happy bachelor or a sad bachelor' and that statement cannot be used to introduce someone to the concept of bachelorhood.
And this is surely not a good argument.

p.s. I don't think the notion of 'what it takes to gives someone a concept who doesn't previously have it' is well defined - like psycholgoically people can know related concepts, be susceptable to conditioning in various ways etc. Just giving some people one example of gouchery would suffice to make them understand the word, whereas you could say any number of things to a rock, and this wouldn't teach the rock the concept.

bookclub: McFarlane contra Wright and Hale

I just finished reading a McFarlane's article in the newest Synthese, about neo-Logicism. He argues that the ideas which Wright and Hale put forward in favor of our license to adopt Hume's principle as a stipulation, would seem to equally support just directly laying down the axioms of PA.

This seems 100% right to me, and it occurs to me that my thesis proposal about how a priori knowledge is possible, can be seen as taking this strategy.

However, I wouldn't say that we literally do stipulate the axioms of PA (would anyone say this? I mean it seems like an obvious psychological/historical fact that people generally don't make any such stipulation). Rather, we come to find these statements (and others obvious) which then functions like a stipulation in that it helps determine the meaning of our words in such a way as to make it likely that these statements we find obvious will express truths.

Just saying this doesn't completely dispel worries about the epistemology of math though. For not all stipulations are OK - some stipulations, like those for "tonk" would not lead to a practice that counted as reliable reasoning. So there are two remaining questions.

1. Given that some stipulations are bad, how do we manage to make good stipulations so often?
2. Given that some stipulations are bad, how can we be justified in making any stipultion?

I think the answer to 1 is a combo of quinean/millian theory revision with a nudge from nature, together with facts about the relative profusion of abstract objects as targets for our stipulations.

I think the answer to 2 is that we can't help but start from what we find obvious and reason in ways that seem compelling, so we have prima facie warrant to do this - in the absence of any reason to think to doubt that these feelings of obviousness could be reliable in a given sphere.