Pincock, “Mathematical Contributions to Scientific Explanation”

1. Pincock prefers his revised explanatory indispensability argument (EIA) to Baker’s. He acknowledges that his version begs the question. I can see that Baker’s does as well (talking about “mathematical objects” seems to assume that mathematical objects exist). If they equally beg the question, why should we prefer Pincock’s version?

-Pincock thinks that the real question is about the truth value of mathematical claims, not about whether there exist any mathematical objects. So he thinks that his version of EIA better suits the debate. Furthermore, he thinks that (2′) but not (2) is accepted by the indexing people, which seems to be a merit of his version of EIA. (I don’t think this is an important progress, since the indexing people will still obejct to (1′).)

2. Why does Pincock think that pointing out the ways mathematics contributes to scientific explanations proves P1 (i.e., we ought rationally to believe in the *truth** *of any *claim** *that plays an indispensable explanatory role in our best scientific theories) true?

Since P1 is a very general claim, it doesn’t seem sufficient to be proved by some putative contributions of mathematics. Furthermore, I doubt whether talking about the truth of a claim is intelligible. Do we really think that a weird claim that can explain something, say ET robbed my apartment is an explanation for my losing all the valuables in my apartment, proves the claim to be true?

– By ‘truth’, Pincock means truth values. P1, or (1′) is actually quite weak. It just means that we should (rationally) believe that any claim that plays an indispensable explanatory role in our best scientific theories is true. (1′) is supported by inference to the best explanation. In short, (1′) means that we should believe that the best explanation is true. (Actually pointing out the ways matheatics contributes to scientific explanations does not support (1′), I would say. It is because P1 says nothing about mathematical claims. I would say that those contributions are relevant to (2′) only.)

3. What’s the role of inference to the best explanation (IBE)? What kind of claims can we infer from it?

– The role of IBE for Pincock is to prove the truth values of the best explanation. (I guess he’s talking about a binary value theory, that a claim can only be true or false.)

4. Is the replacement test a good one? Is it biased towards mathematical explanations?

– Pincock advocates this test. He admits that there may be some limitations because we need to keep the explanadum and the crucial premise fixed. He regards an explanation as an argument, where the explanadum is the conclusion and the explanans (in his words, “the things that do the heavy lifting”) as the premises.

As for whether the test is is biased, Pincock won’t say so. But he also seems to admit that there is a question of why the explanadum involves mathematical concept or theorem (as in the cicada case). (But in the case, the replacement test needs justification.)

Saatsi , “Discussion: The Enhanced Indispensability Argument: Representational vs. Explanatory Role of Mathematics in Science”

1. What kind of things are we looking for in an explanation? How deep should the explanation go? (Are the answers objective or subjective?) When can we say that a factor is non-explanatory?

– To Pincock, what we look for in an explanation is everything that helps us explain the explanadum. The explanation cannot stop at a level where we can still keep asking what explains the explanan. It seems that he thinks that how deep an explanation go is objective because there is a fact about it, but it is also subjective in the sense that it depends on the explanadum (e.g., if the explanadum involves mathematical concepts or theorems, then probably the explanation needs to be mathematical).

2. Saatsi says that a map is not explanatory but representational. However, it makes sense to say that a map, which he thinks is merely representational, explains why we can go from building A to building B in a certain way. Pincock talks about three ways for mathematics to be explanatory. Can we say that a representational role and an explanatory role are not exclusive? Why does Saatsi think that they need to be exclusive?

– To Pincock, a representational role and an explanatory role are not exclusive. He thinks that a mathematical claim can be explanatory by virtue of being representational of some recurring phenomenon, like the cicada case. Saatsi thinks that the two roles are exclusive because he thinks that explanations can stop at the level where we can explain the explanadum and that epistemolgoical claims for the explanan is not an explanation. (If Pincock is right, then Putnam’s peg/board case cannot stop at macro-level theories and must appeal to micro-level theories. I would say that where we can stop depends on the explanandum. We don’t always need to go to the deepest level to explain anything.)

3. Does it make sense to say that an explanatory physical fact can be reduced to a mathematical claim so that the mathematical claim is explanatory?

– Pincock accepts mathematical explanations but whether this is reduction is a question. He doesn’t need to accept reduction in order to preserve mathematical explanations. It seems that not endorsing a reduction to mathematical explanations is easier to defend than endorsing it. But maybe in some cases it’s all right to accept reduction. So, I guess he will leave it open.