*Mind*, Mark Colyvan launches a new offensive against fictionalists like Azzouni, Melia and Yablo. They present a non-platonist interpetation of the language of mathematics and science that, they argue, does not require the "hard road" that Field took. Recall that Field tried to present non-mathematical versions of our best scientific theories. As Colyvan describes the current situation, though, "There are substantial technical obstacles facing Field's project and thse obstacles have prompted some to explore other, easier options" (p. 2). Colyvan goes on to argue that, in fact, these fictionalists do require the success of Field's project if their interpretations are to be successful.

I like this conclusion a lot, and it is actually superficially similar to what I argued for in my 2007 paper "A Role for Mathematics in the Physical Sciences". But what I argued is that Field's project is needed to specify a determinate content to mixed mathematical statements (p. 269). Colyvan takes a different and perhaps more promising route. He argues that without Field's project in hand, the fictionalist is unable to convincingly argue that apparent reference to mathematical entities is ontologically innocent. This is especially difficult given the prima facie role of mathematics in scientific explanation:

The response [by Melia] under consideration depends on mathematics playing no explanatory role in science, for it is hard to see how non-existent entities can legitimately enter into explanations (p. 11, see also p. 14 for Yablo).I have noted this explanatory turn in debates about indispensability before, but here we see Colyvan moving things forward in a new and interesting direction. Still, I continue to worry that we need a better positive proposal for the source of the explanatory contributions from mathematics, especially if it is to bear the weight of defending platonism.

## 18 comments:

Hi Chris,

I have always taken indispensability arguments to be extremely weak and what you call the explanatory turn seems to expose their flaws. In particular, I don't see which mathematical entities play any explanatory role in science. Could you give some examples in which a mathematical entity (the number seven, a triangle) is used to explain some real-world phenomenon? As far as I can see, in science, mathematical entities are used to describe or represent how things in the world are or behave not to explain anything that happens in the real world. Phenomena are usually explained (causally) by the causal powers of ordinary concrete entities, which mathematical entities are sometimes used to describe.

Gabriele -- like I said at the end of my post I am not sure how to present an account of mathematical explanation that supports platonism. But I think it is fairly uncontroversial that most scientific explanations are highly mathematical, in the sense that they employ mathematical language. Also, we do not have adequate replacements which do not employ non-mathematical language.

As I argued in print, this is not sufficient support for platonism if there are nominalist interpretations of pure mathematics. Still, I agree with Colyvan (and Baker) that explanations are a special issue that might be more promising for the platonist.

You can arrange 16 chairs into a square but you can't neatly arrange 17 chairs that way, because 16 is a square number and 17 is prime.

Four-legged stools sometimes wobble but three-legged stools don't, because 3 points determine a plane.

I say those count as true explanations that appeal to mathematical entities.

Mike -- these are great examples, but the issue is how their explanatory power can be leveraged into an argument for platonism. A determined nominalist might insist, like Gabriele, that all explanations are causal, and so these mathematical entities are not involved in the explanation. Of course, others, like Cheyne I believe, wish to expand the notion of "cause" to include causation from abstract objects.

Personally, I favor an extended notion of explanation that goes beyond efficient causes. But it is hard to see how to argue that these are explanations unless one already accepts abstract objects.

Still, I agree with Colyvan (and Baker) that explanations are a special issue that might be more promising for the platonist.Sorry, Chris, but I still don't see why.

Let me try to make my point more clearly. As you says, many scientific explanations employ mathematical language (and may not be expressible in any other terms but mathematical terms) but this does not mean that mathematical entities are doing any genuine explanatory work so that we have postulate their existence in order to explain the occurrence of the phenomenon.

I suspect all this is based on conflating different senses of explanation. In one sense, an explanation is a linguistic act. In another sense, a set of facts explains some other fact if the latter depends on the former (or something along these lines). Now, mathematics seems to play a role in explanations in the first sense but not of explanations in the second.

So, to use Mike's example, in explaining why this three-legged stool doesn't wobble while that four-legged stool does, I might mention points and planes. But points and planes are not playing any role in making the stool not wobble. The only things that is playing a role is the position of the feet of the stool on the floor. The fact that we can use points to represent the feet and a plane to represent the floor does not make planes and points does not help the stool not to wobble any more than my sliding my hand on the table in explaining how someone slipped and fell helps them falling.

Chris: if someone is committed to rejecting the conclusion, they can always reject a premise. On the other hand, if someone is open to the existence of abstract objects, if they accept that whatever explains causes and they accept that whatever causes exists, it seems to me that I've got a prima facie good example that might convince that person of the existence of the number 16.

Gabriele: of course Newton explained the orbits of the planets in one sense of 'explains' and the gravitational pull of the sun explains their orbits in another sense. The relation between the two kinds of explanation is disquotational. Given a true verbal explanation, then the corresponding worldly explanation obtains. If my verbal explanation of why three-legged stools don't wobble is right, then it corresponds to an explanation in the world.

It's true, I concede, that the stool explanation isn't a great argument for abstract entities: the relevant plane and points might be the floor and the tips of the legs. But it is a pretty good example to suggest that there's such a thing as structural (formal) explanation in addition to efficient causation.

It's true, I concede, that the stool explanation isn't a great argument for abstract entities: the relevant plane and points might be the floor and the tips of the legs. But it is a pretty good example to suggest that there's such a thing as structural (formal) explanation in addition to efficient causation.Mike: I don't think those are examples of two different kinds of explantions at work here (a structral one and a causal one) Your "formal" explanation is just a very abstract (and elliptical) causal explanation. If the stool does not wobble, it is still because its three feet all touch the floor. Now, in this case we are lucky because we can pretty much ignore all the features of the stool except that it has three legs and all features of the floor except that it is largely flat, but if the stool doesn't wobble it is still because it has all three feet on the floor. The fact that this explanation abstracts from so many features of the stool and the floor is what make it general enough to apply to all stools whose feet can represented as three points and all floors that can be somewhat accurately represented by a plane. But what is doing the real work in all those cases--i.e. what makes each of those stools not wobble--it is that all of their feet are touching the floor. In other words, by abstracting most of the details of the particular situation we get to a general explanation that applies to many cases but this is not an explanation that is any different from the causal one--it's just the skeleton of a general causal explantion.

Good, Gabriele. I take it that we agree on a central point, that the explanations I gave are causal explanations. Do we agree that the explanations aren't

efficientcausal explanations? That is, they don't explain by appealing to an object or event prior in time to another object or event, where the first brings about the second?Hi Mike,

I'm afraid we don't. I'm not sure which "non-efficient" causal explanations are left out by you definition of efficient causal explanation. Is the stool explanation non-efficient because the cause is contemporary to the effect? If so, I'm not sure why this distinction is relevant to our discussion. If (as I suspect) something else makes that causal explanation non-efficient according to you, I can't see what that something else is, as I can't see how your explanation of the stability of the stool is not an efficient causal explanation albeit an elliptical one. (This reminds me of the ink well "objection" to the DN model of explanation. Of course, we don't mention the laws of mechanics when we explain why the ink well fell but this is just for pragmatic reasons--we presuppose that our interlocutor knows that if the ink well is hit in a suitable manner it will fall on the floor so the only piece of information we need to provide them with is that it was in fact hit that way.)

Is the stool explanation non-efficient because the cause is contemporary to the effect?Yes, that's sufficient

If so, I'm not sure why this distinction is relevant to our discussion.1. It's useful in discussions like this to find common ground

2.If the non-wobbling of three-legged stools has non-efficient causes, that's interesting, isn't it?

3. It pushes us to the question of what these explanatory, non-efficient causes might be.

I say that they are the shape of the stool and its number of legs. If geometrical and numerical features are causes, and if what is geometrical and numerical is mathematical, and if what is a feature is an object, then it follows that mathematical objects are causes, which is the issue under dispute.

Hi Mike,

1. It's useful in discussions like this to find common groundI agree--I didn't mean to sound confrontational! Sorry if i did. ;-)

2.If the non-wobbling of three-legged stools has non-efficient causes, that's interesting, isn't it?It may be interesting but I still don't think it's relevant to the debate. The reason why the cause of the stability of the stool is not efficient is that the normal force on the feet of the stool is what causes the stool to be stable but does not "precede" the stability. The question, however, is whether any mathematical entities is doing any explaining and I thought we had agreed it doesn't.

3. It pushes us to the question of what these explanatory, non-efficient causes might be.As far as I can see, they are concrete causes just like efficient causes not "abstract" causes. The fact that some causes are contemporary to their effects does not show that they are any less concrete.

I say that they are the shape of the stool and its number of legs.I say it is the fact that all the feet of the stool touch the floor (if it was the number of legs, would the explanation of why four-legged stools not wobble be different from the one of why three-legged stools don't?).

(Btw, I saw you have some papers on on Locke on powers and qualities that sound very interesting. I look forward to reading them).

Gabriele,

I agree--I didn't mean to sound confrontational! Sorry if i did. ;-)No, you've been unfailingly polite. I just meant to explain why I though it matters and to establish common ground.

As far as I can see, they are concrete causes just like efficient causes not "abstract" causes.The mathematical/non-mathematical distinction is somewhat independent of the abstract/concrete distinction. The roundness of a peg might be thought of as particular concrete shape that exists in space and time. Someone who thought that that particular shape keeps it from fitting in a square hole may be said to believe in concrete mathematical causes.

I say it is the fact that all the feet of the stool touch the floor (if it was the number of legs, would the explanation of why four-legged stools not wobble be different from the one of why three-legged stools don't?).That's fine as far as it goes, but if someone were to ask you why some four-legged stools wobble but no three-legged stool do, would you be able to explain that fact? And would your explanation appeal to geometrical facts? And would your verbal explanation correspond to a worldly explanation?

(Btw, I saw you have some papers on on Locke on powers and qualities that sound very interesting. I look forward to reading them).Thanks, that's nice of you to say.

Hi Everyone,

A friend of mine directed me to this blog entry and I've found the discussion very interesting.

I just wanted to jump in quickly and give 2 examples that come up in the literature for explanatory mathematics. Sorry if this is already old news!

1) Alan Baker [2005] puts forward an explanation regarding the periodic life cycles of the cicada beetle. It turns out that the life cycle length is always a prime number (I think they are 13 and 17 years) and also different subspecies of the same genus have different prime life cycles. Biologists were able to explain many features of the life cycle of the cicada, but not the prime length of each species.

Two explanations were put forward for this. The first is the fact that the durations are prime mean that their life cycles would intersect as rarely as possible to predators who also exhibit a periodic life cycle. The second is that the subspecies being relatively prime would most greatly avoid possible hybridization.

Baker claims that both these explanations are mathematical in nature, and moreover that the mathematics is indispensable to the actual explanation.

2) Colyvan has put forward several examples in the past, but one recent one is the honeycomb explanation. I don't remember the details, but the question is why bees use the honeycomb shape to tile an area. It turns out that a proof was done recently which shows that the honeycomb shape is the most efficient way to tile an area with the least amount of perimeter. Like Baker, the claim is that the mathematics is explanatorily indispensable.

Of course there are many objections to these examples. Baker has a forthcoming paper which addresses some objections.

The mathematical/non-mathematical distinction is somewhat independent of the abstract/concrete distinction. The roundness of a peg might be thought of as particular concrete shape that exists in space and time. Someone who thought that that particular shape keeps it from fitting in a square hole may be said to believe in concrete mathematical causes.Ok, let me put aside the very important question of what is for the peg to "be round" and grant that the peg instantiates the property of

being round. What I would say is that this property is not a mathematical entity more than any other property (if any) the peg may instantiate (such as, say,being made of wood). Circles might be mathematical entities and so numbers and vectors, but the property ofbeing round(if there is one such thing) is not a mathematical entity more than any other genuine property the peg has. (I guess you'd want to deny this)Those that deny the existence of mathematical entities usually don't mean to deny that there are concrete objects that are spherical in shape (such as this glass marble). They only want to deny that beside such concrete spherical objects there are also abstract spheres.

Now, what I want to deny here is not so much that there are abstract spheres (as opposed to spherical objects) but that if there are any spheres they enter in any genuine explanation of physical phenomena.

Fair enough. It’s easy (and fun!) to come up with examples where mathematical truths explain earthly facts. Usually, people assume that mathematical truths are about abstract objects, but applications of mathematics might give us reason to hesitate about the usual belief. Maybe the proposition that three points determine a plane is really about the bottom of bar stool legs. Maybe the proposition that spheres are convex everywhere is about ball bearings and not about ghostly Euclidean objects. Maybe arithmetic really is about pieces of gingerbread. There are obviously problems with such a view, but I’m happy to leave it at that.

At the risk of sounding pedantic, let me clarify, though, that nothing of what I said implies that there are no mathematical entities or that mathematical claims are not about them.

I was only maintining that the "mathematical" explanations of physical phenomena do not require the exitence of mathmatical entites (just like say Newtonian explanations require the existence of simple pendula, firctionless plaines, and point masses). The mathematical entities are just used to represent physical objects and properties.

I am glad to have found this blog. I have been reading about Indispensability for about 5 years now and it is is good to see I am not the only one amazed at the depths of Indispensability and also frustrated by the inability to prove any a priori existence of abstract mathematical entities behind this indispensability.

I take some comfort in knowing that Godel, the man who dealt the death blow to self-referential systems of mathematical logic and formalism, was also considered "the leading Platonist of modern times" [James Robert Brown 2008]. So there is not only a breed but also a pedigree for those committed to the concepts of unprovability and belief.

-Another Mike

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