Monday, April 5, 2010

Inference to the Best Explanation for Mathematical Claims

Following up my last post, I want to outline an argument for why a reasonable restriction on IBE will block the use of IBE to provide much justification for mathematical claims. The main "mathematical explanations" which are discussed by advocates of this sort of justification, like Baker and Colyvan, involve explanations of patterns observed in the physical world. These examples include the life-cycle of cicada and the shape of the cells of a honeycomb. One of the problems with these explanations is that they bring in complications associated with explanations via natural selection. Another problem is that they may involve mathematical terms in the description of what is to be explained.

To avoid these problems, I will focus on the bridges of Konigsberg case (see here for some background). The explanation could be reconstructed as
(1) The bridges of Konigsberg form a graph of type O.
(2) There is no Euler path through a graph of type O.
(3) Therefore, there is no Euler path through the bridges of Konigsberg.
An Euler path is a circuit through the graph that crosses each edge exactly once. For someone who worries that even this begs the question by using a mathematical term we can offer to extend the explanation to include "(4) Therefore, it is impossible to cross each of the bridges exactly once."

I claim that Sensitivity blocks the use of IBE to support (2). This is because an agent who was genuinely in doubt about the truth of (2) would also have as a relevant epistemic possibility that (2') There is no Euler path through a graph of type O with fewer than 100 vertices. This means that there is an alternative explanation of (3) which employs weaker mathematical assumptions:
(1’) The bridges of Konigsberg form a graph of type O with fewer than 100 vertices.
(2’) There is no Euler path through a graph of type O with fewer than 100 vertices.
(3) Therefore, there is no Euler path through the bridges of Konigsberg.
My conclusion, then, is that this puts the burden on the advocates of using IBE to justify mathematical claims to argue that Sensitivity is incorrect or that some other features of these cases have been overlooked.

9 comments:

Jonathan Livengood said...

Given your use of Sensitivity, I don't think that Sensitivity is all that plausible a restriction on IBE. (I'm also not sure that it is satisfied in the cases of atoms.)

Let me illustrate. Suppose I am pulling balls from an urn. I pull 50 balls from the urn and observe that all of them are white. Here are two explanations.

(1) All balls in the urn being drawn from are white.

(1*) Some of the balls in the urn being drawn from are white, and the first 50 draws were all white (maybe by dumb luck).

I submit (without argument) that (1) is a better explanation than (1*). But (1) stands to (1*) just as (2) to (2') in your Konigsberg example. It could be that all the o-graphs with fewer than 100 vertices have no Euler paths by dumb luck. So, while appealing to that fact in an explanation makes a weaker commitment, it is not to offer a better explanation.

Does that sound right or am I missing something?

Greg Frost-Arnold said...

My reaction was similar to Jonathan's. (Is it the Pitt HPS hive-mind?)

The existence of a universal force obeying Newton's gravitational law is the best explanation (in 1700) of a variety of phenomena: terrestrial free fall, planetary paths, the tides, comets' paths, the moons of Jupiter, etc.

But what if Newton had said instead that the force was NOT universal, but instead its operation is restricted to our solar system? That seems analogous to your 2'. But we(?)* don't usually think of that as undermining Newton's IBE argument for a universal gravitational force (of course, Newton himself makes this follow deductively in later editions by including the third Rule of Philosophizing).

What's going on here? For an explanation to be BEST, it ceteris paribus needs to be simple. Adding these qualifications (your 2', Jonathan's 1*, my restricted gravitational force) de-simplify the explanation. Thus the competing explanation ('epistemic possibility') is worse along one dimension of explanatory goodness -- the simplicity dimension.

----
* Maybe an anti-realist who pushes the 'transient recurrent underdetermination' line would say exactly this: in 1700, we had no evidence to decide between these two competing theories, so we should remain agnostic between them.

Chris Pincock said...

Jonathan and Greg, thanks for the comments.

On the urn case, I wanted this to be a case where the way in which (1) contributes to the explanation does undermine the relevant epistemic alternatives like (1*). Here it is because (1) shows that the pattern in the draws is not accidental. So, I agree that (1) is a better explanation.

Why do you think (1*) is analogous to (2')? Either the o-graphs with fewer than 100 vertices all lack Euler paths or at least one of them has an Euler path. In the former case, given that these bridges fit into this category, we have a good explanation. In the latter case, it looks like we have an application of IBE that would lead to a false mathematical conclusion. But if this happened, then the inference to (2) would also lead to a false conclusion as well.

Perhaps the issue is the goodness of the explanation when both (2) and (2') are true. My goal is to motivate the point that the difference between (2) and (2') is idle to the way in which the mathematics contributes to this explanation.

On Greg's point about simplicity, I am not convinced that we should label the wider claim (or explanation) as simpler. It projects a regularity into the mathematical domain in an irrational way. In math we can't just check a 100 cases and infer that the universal claim is true because this is simpler. Again, I am trying to take this point and apply it to the mathematical explanation cases.

Hope this helps explain where I am coming from at least. Additional counterexamples are welcome!

Jonathan Livengood said...

I think (1) is to (2) as (1*) is to (2') more or less for the reason you give. For the urns, (1) shows that the pattern in the draws is not accidental; whereas, (1*) does not. Similarly, for the graphs, (2) shows that the claim about the bridges is not accidental; whereas, (2') does not. (Actually, I suppose that some more basic facts about topology show that (2) is true and hence that the Euler's claim about the Konigsberg bridges was not accidental.)

My point is that it isn't in virtue of the fact that all o-graphs with fewer than 100 vertices lack Euler paths that the [Eulerian] Konigsberg bridge system lacked an Euler path. Both of those facts obtain in virtue of more general topological facts, and it is the more general facts that do the explaining. I think this goes all the way back to Aristotle's Posterior Analytics, where he argues that it is not in virtue of being scalene that a scalene triangle has internal angles that sum to 180 degrees, it is in virtue of its being a [Euclidean] triangle. (Of course, Aristotle wouldn't have added the "Euclidean" qualifier!)

Jonathan Livengood said...

Oh, also, one thing that I'm curious about given your earlier comment ... do you take Sensitivity to apply to IBE inferences only in the mathematical domain? I ask because it seems that the force of Greg and my objections depends at least in part on your answer. If you mean for Sensitivity to apply to IBE generally, then it will surely be stripped of most if not all of its power and will be totally unsuited for work in the realism debate.

Chris Pincock said...

Jonathan, thanks for the additional comments.

It is an interesting question about the metaphysical basis of mathematical truths and whether we can articulate a sense in which the instances or restricted versions of a general mathematical truth can be said to hold in virtue of the unrestricted general truth. Ordinary modal tests do not work that well for mathematics. But my view is that we cannot deploy these sorts of metaphysical tests when we consider a candidate IBE. This is because we lack knowledge of the claims which we are doing the explaining with, so we also lack knowledge of the ultimate metaphysical basis of the truth of these claims, if they are true. Analogously, a scientific law A may obtain in virtue of another scientific law B. But for some explanations, it is only the IBE to A that is justified. Or at least that is the sort of restriction I am trying to articulate.

The triangle vs. scalene triangle is a good case for me to think about. My sense of what is going on there is that we have a satisfactory proof of the triangle claim which shows that being scalene is not relevant. In the indispensability argument we cannot take such proofs for granted as they ultimately depend on axioms which someone like Quine only accepts on scientific considerations.

I don't think I understand your last conditional claim. I do intend Sensitivity to apply outside mathematical cases. In particular I hope to argue that we can use it to make sense of scientist's reluctance to believe in atoms prior to Perrin's work. For me cases like caloric and the ether show that we need to place some substantial restrictions on IBE if we are to be realists. Psillos has tried to do this one way, but I hope something like Sensitivity offers another way to defend a limited kind of scientific realism.

Note: Unfortunately the blog was spammed last night with 15 spurious comments from someone writing in Chinese, so I had to turn on comment moderation. But I will do my best to approve genuine comments quickly.

Firionel said...

I am only a humble mathematician by training, and not a philosopher, so I need to ask a couple of questions for clarification (hoping against hope that they will not betray my outright grotesque ignorance on the subject in question):

1 Do I take your argumentation to tacitly assume that there is such a thing as mathematical reality?

2 Doesn't you argument in some way assume that some classifications of objects (mathematical or otherwise) are more natural than others? Or have I got it the wrong way round, an the non-existence of such 'natural' classifications is precisely your argument. (My probably rather confused impression after reading the post several times is something like "'natural' classifications exist for physical objects but not for mathematical ones"; that would seem to be wildly inconsistent.)

3 Isn't this discussion taking a shortcut around the problem of whether there is a valid path of inference from physical phenomena (properties exhibited by physical objects) to mathematical phenomena, much more so since the consensus (at least among mathematicians) would be very much to deny that possibility. (cf. the whole discussion surrounding 'theoretical mathematics'; see Jaffe, Quinn 1993 )

Chris Pincock said...

Firionel,
Thanks for your comment, and sorry for the delay in responding.

On 1, the conclusion of these arguments is supposed to be either that mathematical entities like the natural numbers exist or that there are mathematical facts which make claims like "2+2=4" true. I think that most people engaged with this argument assume that if mathematical claims are true, then they are true in virtue of some objective mathematical reality.

On 2, this is a good question. The general structure of the explanatory arguments is that we can see that some claim is part of the best explanation of some physical phenomena. We use this feature of the claim as evidence that the claim is true. For this sort of "inference to the best explanation" to be plausible, I think that the way things are described must involve "natural" classifications. But I do not intend to make these natural classifications any harder to find for mathematics than in physics.

On 3, I am trying to argue that it is very difficult to use the way in which mathematics helps in science as evidence for the truth of mathematical claims. So, I am on your side here. Thinking back to the reaction to Jaffe/Quinn, I believe that this also fits with the reactions of many mathematicians to their "theoretical mathematics". Most mathematics is just not done this way because purely mathematical reasoning is valued over these sorts of considerations. It would be good for my purposes to link up my concerns with the concerns expressed in these debates.

Firionel said...

Hi,

just a short "Thank you!" note for that vastly enlightening answer.

And don't worry, if I'm posting comments with convoluted questions on a months old blog post, how self-centered would I have to be to expect a prompt reply?

Keep up the good work.