Since August, I’ve written two and one half versions of chapter 1 of my thesis. I loved version one, and it came straight from the heart. But as it turns out, writing that way will get you an unpublishable screed rather than some reasonable contribution to the literature.

Version two was significantly toned down, and had a more definite target. I was critiquing the account of ‘generality’ given by Michael Strevens in his 2004 paper (pdf) and 2008 book*. I thought he simply said that the generality of a statement is the number of actual or possible states of a given system that it applies to. I prepared a cutting argument about how ‘number’ of states is a wrong-headed way of thinking about things, because in real-valued systems there are an uncountably infinite number of possible states! And so I hoped to motivate my own view of generality. But as it turned out, I hadn’t read the paper hard enough, and had missed the following:

On the new definition, the degree of abstractness of a model [its degree of generality] is proportional to the number of possible physical systems satisfying the model. More exactly, since a typical model satisfies ranges for one or more real-valued systems, abstractness is proportional to the standard measure (in the mathematical sense) of the set of possible physical systems satisfying the model. (2004, p.170)

Curses! Defining it in terms of measure solved the problem I wanted to raise entirely. It did so in a simple and sexy way. My own solution to this non-existent problem looked clunky and awkward in comparison. It was couched in an ugly and ad-hoc formalism that would probably only ever really make sense to me.

And that’s when it happened. I had a moment of insight into the very heart of academic mediocrity. Having already set aside my work of love, and facing the collapse of my compromise version, I thought to myself:

…maybe if I make the formalism dense enough, no one will notice it’s wrong.

When it sunk in that I had actually thought that, that I had actually felt it, I was struck with equal parts horror and a feeling of liberation. I had, for at least a moment, become the thing I hate most about the academy. But I also knew for a certainty that I would rather walk away from the whole project than accept that state. I quit the academic life for ever that day, for about 20 minutes.

After gathering my wits for a few weeks, and surviving the holiday season, I’m now in the process of writing v.3 of chapter 1. It’s somewhere between v.1 and v.2 in terms of ambition, and it’s going fairly well.

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* *Strevens, M. (2008), Depth: An Account of Scientific Explanation, Harvard University Press, Cambridge, MA.*

Strevens, M. (2004), ‘The causal and Unification Approaches to Explanation Unified – Causally’, *Noûs* **38**(1), 154-176.

One of my chief complaints about nonmathematicians reasoning about spaces of possibilities is this assumption that there is some sort of standard measure on these spaces. There is not, and if Strevens is resorting to it in this way, then I think it’s pretty fatal to his argument. I will have to read his paper to be satisfied of my opinion, but in the meantime I urge you to reconsider your rosy appraisal of his work.

Hey Mr. Bailey! I don’t think he needs to assume that there is a common measure across dimensions of possibility. It’s probably enough, for his purposes, to be able to define measures on single dimensions. You can then say, on a given dimension, that one theory is more general than another (e.g. relativity holds at a greater proportion of velocities than newtonian mechanics).

That leaves open the horribly messy question of comparing incommensurable quantities, for which I’m sure there is no standard measure. But I think the reply is that we make some kind of intuitive or informal judgement about how the tradeoffs between generality in different dimensions of possibility play out, and that’s the best we can do. If there really is no definitive answer about how that should go, then Strevens can’t be responsible for providing one.

But even within a single dimension of probability, there isn’t a canonical measure. How do you say that one subset of the velocities is larger than another subset of the velocities? Well, you could go with the obvious answer, but then suppose we transform to inverse velocities (i.e., travel times); we have the same space of possibilities, parametrized in a different way, such that this space of possibilities now has a different “obvious” measure.

By the way, I would agree that relativity holds on *more* velocities than newtonian mechanics, but this is just to say that the domain of applicability of the former includes that of the latter. It has nothing to do with measures.

But if a single dimension is all he really needs to be able to compare on, isn’t constructing some kind of measure going to be generally sufficient to say whether one theory ranges over a subset of another theory? And if you’ve got that, you can compare them in terms of generality.

Unless you can think of instances in which different measures would actually invert that relation – such that on one measure, theory x ranges over a subset of the values that theory y rangers over, but on another way of constructing the same dimension, y ranges over a subset of x’s range…

I’m not considering that case. I’m considering cases where neither range is a subset of the other, but one range is taken to have greater measure than the other, so one says that that theory is more general. That’s a relation that *can* be inverted by changing the measure, very easily.

I think he’d have to admit that he has not much to say about such cases. Unfortunately for me, my proposed alternative didn’t have a hell of a lot more to say about such matters. I think the problem you’re pointing to is a real one, but one that is pretty well understood by those involved. Consider for instance what Matthewson and Weisberg* have to say on the subject: “As we sometimes measure generality in terms of logical possibility, we will be dealing with infinite sets of possible targets. This means that we cannot always order the generality of models or sets of models according to cardinality, but will have to consider whether they apply to some set of target systems that is a proper subset of another, thus being of lesser generality. This is a less universal measure than we might like, since it restricts us to cases where we are comparing sets that stand in set/subset relations to each other, but to date this is the most comprehensive way we know of to analyze p-generality.”

It’s a kind of half-concession that this is a problem. We’re stuck comparing incomparable things, but when things are comparable then we have clear ways of comparing them.

*Matthewson and Weisberg (2009) “The Structure of Tradeoffs in Model Building”,

Synthese170:169–190“…maybe if I make the formalism dense enough, no one will notice it’s wrong.”

You mean like this?:

http://www.gocomics.com/calvinandhobbes/2013/02/14