Was Berkeley Right about Mathematics?

It is pretty common in discussions of the history of mathematics for authors to mention Berkeley's attack on the calculus, usually with some derision, though in the more sophisticated authors it is a balance of derision and respect (or, at least, empathy).

For example, here is a nice summary of the episode by Penelope Maddy:

In the late 1600s, in response to a number of questions from physical science, Sir Isaac Newton and Gottfried Wilhelm von Leibniz simultaneously and independently invented the calculus. Though the scientist's problems were solved, the new mathematical methods were scandalously error-ridden and confused. Among the most vociferous and perceptive critics was the idealist Berkeley, an Anglican bishop who hoped to silence the atheists by showing their treasured scientific thinking to be even less clear than theology. The central point of contention was the notion of infinitesimals, ridiculed as 'the ghosts of departed quantities'. Two centuries later, Bolzano, Cauchy, and Weierstrass had replaced these ghosts with the modern theory of limits. (Realism in Mathematics, p. 22)

What seems funny about Berkeley's critique is the apparent confidence with which he proffered it, in contrast to the rigorous development the calculus would undergo in the following centuries. What extracts a measure of respect is the fact that, at the time he wrote, Berkeley was not wrong.

But if, on a charitable reading of Berkeley, his point wasn't forever to cast doubt on the calculus, but rather to point out that the metaphysical and logical foundations of mathematics are not more justified than the suppositions of faith, it is interesting (and somewhat amusing) to ask whether Berkeley has actually been disproved in this.

Obviously, mathematics as a science is certain. Berkeley never denied it. But that's different from saying the metaphysical and logical foundations of mathematics are certain. These latter are characterized by disputes without termination. They involve paradoxes and uncertainty. And they require the acceptance of assertions whose justification and warrant are, by the admission of philosophers and mathematicians, puzzling.

For example, the most common picture throughout the 20th century was that set theory serves as a foundation for mathematics. However, touring the landscape of views on this matter will make you dizzy.

Here are a few questions that are recognized as utterly central yet by all appearances unresolved: Is there only one correct set theory, objectively true to the exclusion of all others? What are sets? Are they physical or non-physical? If they are non-physical, how can we even possibly know about them? If they are physical, which physical things are they, and are we sure there are enough of them to serve as a foundation for mathematics? (And if there are, how could we know that?)

Of course, answers to these questions have been proposed – some would claim successfully. Maybe so. But many theologians say the same about their solutions to puzzles about the mysteries of the faith, and a comparison of their theories with those of philosophers of mathematics might well show that the former aren't any worse off than the latter.

Again, Berkeley's point is just that the theological mysteries are not more doubtful than the foundations of our beloved mathematics. He gets teased for not realizing that his objections to the calculus would be answered a hundred years later, but there are questions similar to his that are still unanswered by logicians and philosopher of mathematics. So, joke's on them!

Sober on Essentialist Explanations in Science

"My suggestion is that the gas's kinetic energy explains what it is for the gas to have the temperature it does. It does not explain why the gas has the temperature at the time in question. The relationship here is much like the one between a dispositional property and its physical basis. If zebras differ in fitness because of their leg structure, then there is a sense in which leg architecture explains fitness relationships. The relationship is not causal; it does not explain why those fitness values came into existence. Science not only explains why certain states of affairs and events come into existence; it also seeks to explain the nature of those events."

--Elliott Sober, The Nature of Selection, p. 75.

Note that explanation is hyperintensional, and that not just any necessarily coextensive term would work equally well in place of 'the gas's kinetic energy' here.

Laplace and Mereology

"We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes." -- Laplace

My understanding is that Laplace's demon was primarily intended as a heuristic for understanding microphysical determinism. But there is a serious question (noted by or implicit in the work of a number of philosophers of science, e.g., Sober, Loewer, Albert) as to whether, in fact, "nothing" about the physical universe would escape the demon's knowledge. There is some ambiguity in what is meant by "submit these data to analysis," but intuitively it seems that the demon's abilities would only allow him to know all of the facts describable in microphysical terms. There is a real question then about whether this would amount to knowledge of all of the facts, including about mereologically composite objects, let alone about minds.

Would we need to add to the demon's capacities, then, some sort of additional "translation manual" to tell him which macro-states are being realized by the micro-physical states? And would this amount to a problem for a reductionist picture of the world? Maybe, maybe not.

Thomistic Moderate Realism Reduces to Armstrongianism or Platonism

I've always had difficulty understanding Aquinas's "moderate realist" view on universals, at least as that view is expounded by his interpreters. It seems that they want to have their cake and eat it too: Thomists both believes that universals are "objective" and "extramental" in some sense. But they also are not extreme realists, i.e., (modern) Platonists. They also seem to say things that make it sound as if universals are merely conceptions; if taken at face value, that reduces to conceptualism or nominalism of some sort.

Here's one way of bringing out the problem.

Assuming there are universals, then:

• 1. Either (a) universals exist extramentally or (b) they do not exist extramentally.
• 2. If (b), then nominalism or conceptualism, QED.
• 3. If (a), then either (i) they only exist in the objects that have them, or (ii) they sometimes exist outside of the objects that have them.
• 4. If (i), then that is Armstrong's view.
• 5. If (ii) then that is Platonism.
• 6. So if (a) is true, then either Armstrong's view or Platonism is true, and so moderate realism either reduces to Armstrong's view or Platonism.

6 already seems to disambiguate the Thomistic view in a way that makes it unacceptable, and does not let it have all of the desirable qualities it is supposed to have.

• 7. If Armstrong's view holds, then universals depend on the objects that have them, and therefore cease to exist if the objects do.
• 8. But if the universals cease to exist, then statements about non-existent things, like "dinosaurs are big creatures", do not require universals to be true, and so universals are superfluous.
• 9. So if Armstrong's view holds, then universals are superfluous.

That only leaves Platonism, and there are huge problems with Platonism.

To be fair, I am leaving much unsaid here, and I am not making any distinctions within "Platonism." But ultimately I think this is basically correct; the way Platonism is construed in modern times basically just is the view that there are non-mental, objective, necessarily existent, universals.

Naturalness vs. "Arbitrariness" and "Simplicity" in Mereology

There are three answers one can give to the question: "When does composition occur?"

(1) Always.
(2) Sometimes.
(3) Never.

The first view is something like David Lewis's view: Any plurality of objects composes a third. Hence, for example, there is an object consisting of Barack Obama, my left leg, an orange, and half of the beach in Santa Monica. (We could call it the BLOB.) This is the view encompassed in classical mereology.

The third view is something like Peter Van Inwagen's view in 'Material Beings'. This view holds that (with maybe a few very specific exceptions), there are not, literally, any composite objects. There are just "simples" -- atoms in the void, physically proximate to each other and arranged in various ways.

The second view encompasses all other possibilities. One of these possibilities is "common sense" ontology, or something like it. One such view might hold that things like physical organisms, tables and chairs, rocks, planets, stars, maybe even galaxies, etc. are composite objects. But not just any plurality of things constitutes an object on this view; for instance, there is definitely no object such as the BLOB.

One argument (I think due to Van Inwagen) says that (2) can be ruled out rather easily because it is arbitrary and/or overly complicated. Hence, we must choose between (1) and (3).

However, it seems to me that people who hold to (2) might argue that their ontology only encompasses what is natural; just as there is a distinction between natural properties (like 'having mass') and gerrymandered properties (like 'being Barack Obama-or-my leg-or-an orange-or-half of Santa Monica Beach'), there may well be a distinction between natural composites and gerrymandered composites. And just as one might choose to privilege the natural properties by saying they are the only ones that exist (as D.M. Armstrong does), so one might choose to privilege the natural composites by saying they are the only ones that exist.

Obviously more needs to be said than this and this view would need to be fleshed out. But I'm more interested in the methodological question, and all I need granted is that it is a distinction one could coherently use so as to avoid (1) or (3).

Now, people like Van Inwagen might (probably, would) respond to this view by claiming that it is arbitrary, that it multiplies distinctions, that the notion of "naturalness" is mysterious and vague, and so on.

But I think it is worth noting here an "arbitrariness" in this objection: Claiming that some entity is more natural than another (or, by extension, that one's theory is more natural) is no more mysterious than claims that that (2) is arbitrary and complex, and that (1) and (3) are non-arbitrary and more simple. Defining the sense in which (1) and (3) are "non-arbitrary" and "simpler" is no easier than defining the sense in which (2) is "more natural."

Frankly, simplicity and arbitrariness, as used in this way, seem to me to be just as bad off as the other notions that anti-hyper-intensionalists use as criteria of theory choice; they are themselves hyperintensional notions in fact. That's not to say that they are bad off -- I do think there is an intuitive sense in which theories can be "simpler" and "less arbitrary" than other theories. But it is arbitrary to use "arbitrariness" and "simplicity" as criteria for selecting between metaphysical theories, and then pretend you don't know what it means when one says that his theory is more "natural" than others or, relatedly, posits entities that are "more natural."

Pragmatics at Home

Here's an interesting case of implicatures I noticed the other day when discussing what movie to watch with my wife. (>> means "pragmatically implies" and *>> means "does not pragmatically imply")

Case One:
W: Do you want X?
H: Only if you want it.
>> I don't want it.
*>> I do want it.

Case Two:
W: Do you want X?
H: Not if you don't.
>> I do want it.
*>> I don't want it.

What is strange about this case is that, presumably, the husband H's responses in both cases are logically equivalent to each either; assuming I'm parsing them right, they both say "I want X only if you want X" or, equivalently, "I do not want X if you do not want X." (***See bottom of page for an explanation, if this isn't clear.)

But the response in Case One (at least sometimes) implies something different than the response in Case Two.  (I say 'at least sometimes', because, as with many implicatures, it may depend somewhat on the sonic properties of one's utterance too -- i.e., the way one pronounces the words.) But this seems to imply that the implicature is "detachable" in Grice's sense (see the bottom of p.57 and ff., here).

However, according to classical Gricean pragmatics, conversational implicatures are non-detachable; hence, if these were conversational implicatures, they would both imply the same things (which they don't). So it seems that they must be conventional implicatures. (See here on that distinction.) That's sort of weird though, because conventional implicatures are usually associated with syncategorematic expressions that do not contribute any additional truth-conditional meaning to the sentence (for instance, "however," "but," "even though," "nevertheless," etc.).

Also, these implicatures seem to be more like conversational implicatures than conventional ones, since they do seem to sort of follow from something like Grice's Maxim of Manner (or, better, Levinson's M-Principle); i.e., saying something in an equivalent but roundabout way implies a non-standard meaning. For depending on whether one uses the double negation form or not you get a different implicature. However, it doesn't *quite* fit this rule I think, because it doesn't seem like either of the response from Case One or Case Two is more "roundabout" than the other; in other words, the responses in both cases seem to be symmetric as far as the "oddness" of their phrasing goes.

Anyway, kind of an interesting case. FYI, in the actual situation, I *did* want it, but I didn't want it if she didn't. : - )

***
To see the equivalence, note that all of the following are equivalent:

• I want it only if you want it.
• I only want it if you want it.

(These are clearly equivalent. For consider the following:
x goes to the store only if x is hungry.
x only goes to the store if x is hungry.)

• If you do not want it, I do not want it.
• I do not want it if you don't want it.

Modernizing (and Medievalizing) Analyticity

A project I've been hoping to do eventually is to re-work the concept-containment notion of analyticity. Something has always seemed to me intuitive about it, at least in the "Bachelors are unmarried" sorts of cases. (From the 19th century onward, for various reasons, the analytic truths seem to have become equated with the logical truths; this seems wrong to me.)

I'd also like to see whether the notion of analyticity comes up at all in medieval philosophy, and whether any of the tools of medieval philosophy could be of use for this project (or, what would be just as interesting, whether they would find the concept-containment notion of analyticity hopelessly confused).

There are several problems for the concept-containment approach, but a big one is this:

1. Forms of Analytic Judgments: Kant (sometimes) defined analyticity in terms of conceptual containment, roughly as: A judgment of the form 'A are B' is analytic iff the concept B is contained in the concept A.

The worry is that there are many judgments that do not seem to have this form, but where they seem to also be true in virtue of 'meaning' or 'concepts' (or something close). Some examples, taken from SEP:

(11) If Bob is married to Sue, then Sue is married to Bob.
(12) Anyone who's an ancestor of an ancestor of Bob is an ancestor of Bob.
(13) If x is bigger than y, and y is bigger than z, then x is bigger than z.
(14) If something is red, then it's colored.

Other examples, from Jerrold Katz:

(15) Mary walks with those with whom she herself walks.
(16) Mary walks with those with whom she herself strolls.
(17) Poor people have less money than rich people.
(18) Rich people have more money than poor people.

It's not totally clear how all of these examples can be analytic (assuming they all are) if we take analytic truth to mean that the predicate-concept is contained in the subject-concept.

I don't have a worked-out answer to this yet, but I suspect that some of the work in cognitive linguistics might be helpful. In fact, Katz' own work might be helpful here, though from what I understand Katz is a Platonist (rather than a "conceptualist") about meaning, and so it would have to be properly adapted.

In addition to these strategies, where medieval philosophy might be useful here is in seeing how we might, in fact, be able to reformulate all "basic" sentences and then parse them out so that they technically obey the constraint of "subject-copula-predicate" form.

As Terence Parsons points out, medieval scholastic Latin is unique in that it is a natural language and yet there is no distinction between a sentence's ordinary surface grammar and its logical form.

Now, if one had a close enough association between words and mental concepts, one might then be able to get a nice conceptualist-type semantics going. And it seems certain medieval thinkers did have precisely this sort of close association between mental concepts and meaning, viz., in the theory of "subordination" (I'm thinking of Buridan and Ockham right now).

This suggests that we could translate basic ordinary-language sentences into medieval subject-copula-predicate sentences, and the concept-containment idea could become more useful again.

I'm not sure it will be as simple as that or that this will solve everything, but I suspect it will make things easier.

There are some other issues for a conceptual containment theory of analyticity too:

2. Conceptual Containment: There are really at least two problems here: (a) a theory of concepts, and (b) a notion of containment. Can we give a plausible and clear theory of both? (And, what would be even better: Can we give a theory of both that is mathematizable and subject to rigor and computation once we are given a case?) And how neutral can we be here with respect to different theories of concepts and containment?

How can medieval philosophy help here? Medieval philosophy certainly contains much discussion of concepts, and that should certainly be useful.

As for the notion of containment, I can't help but think of Scotus' theory of "repugnance" and "non-repugnance" by which he assesses the modal status of basic propositions -- the kinds of propositions concept-containment seems to be after (see page 162, here). What's interesting is that Scotus seems to define repugnance as a relation holding between terms. That seems to make it a semantic relation rather than a conceptual relation for Scotus; moreover, the relation's holding is said to be grounded in "notae," which are in some sense objective features of the external world.

So this isn't exactly concept-containment analyticity; but still, in terms of its formal/logical properties, non-repugnance/repugnance behaves similarly to concept-containment/exclusion. (This seems to be an instance of a more "externalist," metaphysical picture of containment and exclusion relations; I get the sense that this sort of externalism is the norm in medieval philosophy, especially pre-Nominalism, though even after that as well.) Moreover, like repugnance-relations, concept-containment relations are supposed to ground the modal status of propositions, and moreover, they both seem to deal with the same sorts of propositions. So I can't help but think Scotus will be helpful here, even if he probably wouldn't have a view of analyticity like Kant's.

3. Rigor: How would a rigorous conceptual semantics go? Can we make it as formal, precise and mathematical as the non-conceptualist semantics have been? Some work has already been done on this sort of thing, but there's more to be said. Again, I wonder if the medieval logic might help us here, since Parsons and others have shown it to be entirely rigorous and fit for mathematical treatment. Medieval logic at its height was generally far more sophisticated than anything after it -- certainly more sophisticated than anything Kant did on logic -- and so I can only imagine it will make this project easier.

4. Externalism: Although the conceptual-containment approach seem plausible in certain cases, so does semantic externalism. Gillian Russell, a professor here at UNC, has done some brilliant work on updating analyticity to take these post-Kripkean insights into account. However, she tends toward the more externalist side of things, and I'd like to see whether we can salvage more of the connections between meaning, analyticity and concepts than she does, while still giving a reasonable account of the externalist insights (something I worry cognitive semantics a la Peter Gardenfors hasn't quite done -- see 4.1 here, for instance).

One thing I worry about in trying to find analyticity in medieval philosophy is that medieval philosophy seems to tend much more toward externalism about semantic content (though this is only a hunch I get -- I can't point to anything specific). But maybe I'm wrong and there is room for analyticity in medieval philosophy; and even if not, the project may still be worthwhile, since maybe we will find new reasons to either abandon or revise our conception of analyticity in interesting ways we've never thought of.

So I plan to do some research on medieval logic and semantics in the next few weeks, and maybe this will help my concept-containment project. Moreover, I think it is an interesting historical project in its own right to see whether something like analyticity can be found (or reformulated) in terms of medieval semantic categories.