Part of Pincock’s article struck me as quite similar to Weisberg’s robustness analysis and I wanted to see if people thought these similarities are actual, relevant, or important.

Pincock argues that even his revised indispensability argument falls short of allowing us to accept the conclusion that scientific practice gives us good reason to believe the truth of some mathematical claims. This is because the argument is question begging. The only way to avoid this question begging is to justify the mathematical claims independently from their indispensable explanatory role in science. “I want to argue that our justification for a given mathematical claim cannot depend entirely on its contribution to a scientific explanation. This is because the claim must be independently justified for it to make this contribution to the explanation… explanatory contributions can only provide additional boosts in justification for a belief that was already substantially justified.” (9)

This sounds a bit like Weisberg in “Robustness Analysis” when he defends RA against O&S’s claim that RA is a form of non-empirical confirmation. He argues that RA is not non-empirical confirmation because the models which are used to identify robust theorems are themselves already low-level confirmed. That is, the models used in RA already have low-level confirmation and as such the resulting robust theorem is also confirmed. By low-level confirmation, Weisberg means that theorists’ confidence in models was “minimally established by demonstrating that the relevant mathematics could be deployed to make correct predictions. It may have been investigated explicitly by mathematicians.” (740)

It seems that in both cases the truth of the mathematical claim or the confirmation of the model is first necessary to support a scientific explanation or confirmed robust theorem. Pointing to a scientific explanation or robust theorem to prove the truth the mathematical claim or low-level confirmation of a model is to put the cart before the horse. Pincock and Weisberg then seem to be working with a similar structure here but at maybe opposite ends. Does this sound right, am I confounding these two arguments, or is this just trivial?

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Hi Leo,

I really like your post. Great question. I think there is surely a connection between the two in the sense of needing some independent justification for using a particular confirmational strategy. Weisberg thinks robustness does provide confirmation so long as we first accept low-level confirmation concerning mathematics ability to accurately represent the world. Unfortunately, I’m not really sue what Weisberg means when he claims that the ability to make predictions or the ability of previous models to accurately represent the world is somehow supportive of robustness as a confirmational strategy? Does he mean that we have some reason to think one of them is true just because mathematics is sometimes adequate at representing concrete phenomenon? I’m afraid I don’t understand.

In the literature for this week, things seem a bit different. Rather than simply requiring support for a confirmation strategy, here I think there are two wrorries, one about a claims ability to contribute to an explanation, and one about using IBE to confirm mathematical claims. First, a scientific explanation is supposed to include claims that have some justification behind them. This isn’t so much an issue about what is needed to confirm a claim as whether or not a given claim is able to be part of an explanation. Pincock points out that claims must have independent warrant in order to contribute to scientific explanations. Often this justification is provided by IBE. The second question is therefore whether merely being part of our best explanations is sufficient to provide the needed independent warrant for mathematical claims as it is often claimed to do for unobservables. Here Pincock argues that IBE is insufficient for providing the idependent justification needed for mathematical claims to contribute to scientific explanations.

Thus, I think the independent warrant is playing two slightly different roles here. In Weisberg’s case it is being used to justify an entire confirmational strategy. In Pincock’s case, the inpendent warrant isn’t postulated to justify the strategy but it is required to play an explanatory role. Usually, IBE works perfectly well for unobservable entities (unless you are van Fraassen) because they receive justification from the role in our best explanations. Still, there is a question about whether or not this confirmational strategy is available for mathematical claims without independent justification. Pincock says no because mathematical claims have unique features that make it impossible to strightforwardly apply IBE to them in the traditional way.

One last thought. I’m worried that about the following. IBE presumably works only when a claim makes a contribution to our best explanation. Contributing to that explanation, however, requires some independent justification. Yet if IBE is responsible for providing the confirmation required for a claim to make that contribution then things seem a bit circular. That is, IBE provides the justification reuired to make the contribution that is required to be justified by IBE. Something must be incorrect here.

I think what migh be going on is that contribution to a (the) scientific explanation requires independent justification (the way Hempel thought explanations had to be true), but being justified by IBE does not require independent justification because the explanations on offer are only potential explanations. Therefore, IBE does not require prior independent justification of the claims in potential explanations, but instead provides (some) claims within those potential explanations with the justification required to contribute to a scientific explanation. I’m afriad I’ve confused something here so I’ll stop typing, help?

Just to add a clarification note. I’m not sure it is relevant, but it might be: there are ways other than robustness to independently justify a proposition. Point is, Chris might be thinking something other than robustness when he calls for independent justification.

I am beginning to understand that I am the madman in the room. I do not subscribe to scientific realism and I cannot confer belief in objects (directly or indirectly experienced) based on IBE. But I believe that mathematics has a higher confirmed truth value than scientific theories, and this is why it has explanatory value, and I do not believe in mathematical objects.

The difference between the mathematical parts and other theoretical parts of a scientific theory is that the mathematics has already been accepted and proven through the mathematical method of proof and verification. Before any experience of cicadas I know the truths (or can easily derive them) of prime numbers based upon prior assumptions and my assumption in the validity of logic.

Evidence of phenomena cannot shake mathematical truth. No matter what the universe shows me, I will never stop believing in the truth of the Pythagorean theorem in Euclidean space. No experience can shake my belief in the properties of prime numbers. It is because of this faith in mathematics that is has such helpful power in scientific explanation.

Now, the realists might argue that demonstration of the conformity of phenomena to mathematics might make their case. Perhaps, but mathematical truths are independent of this evidence, and the questions then are which potential set of fundamental assumptions that different maths are based upon are you actually confirming? Is number theory the same in Non-Euclidean space? My intuition says yes, but to know I would have to go back and verify that no number theory poofs are dependent on the parallel postulate.

Which is again verification independent of experience for the mathematical proposition in question. And again IBE has no bearing in my belief in mathematical truths.

One way to link the topics of robustness analysis and mathematical explanation is to consider a case where what are explaining is the very same claim which Michael says can be justified by robustness analysis. So, for example, what is our best explanation of Volterra’s Principle? I don’t think Michael puts things this way, but surely the best explanation involves showing how the principle follows from a wide class of models as we vary the parameters. On the assumption that this wide class of models includes a model which accurately represents this aspect of our real target systems, we have an explanation for why the principle holds so widely.

Now I guess Sober is worried that this explanation might be too mathematical because it involves a priori reasoning. I agree with Michael that it is not entirely a priori, but I am happy to have a priori reasoning playing a major role in this sort of explanation and also in other mathematical explanations.