Each of the following two quotes comes from a naturalist, but the first appeals to naturalism to justify realism (about mathematics), while the second appeals to naturalism in support of anti-realism (about science).

In his review of Charles Chihara's

*A Structuralist Account of Mathematics*in

*Philosophia Mathematica*

**13**(2005), John Burgess writes:

"If you can't think how we could come justifiably to believe anything implyingCompare van Fraassen, in

(1) There are numbers.

then 'Don't think, look!' Look at how mathematicians come to accept

(2) There are numbers greater than 10^10 that are prime.That'show one can come justifiably to believe something implying (1)." (p.87)

*The Empirical Stance*(2002):

But [empiricism's] admiring attitude [towards science] is not directed so much to the content of the sciences as to their forms and practices of inquiry. Science is a paradigm of rational inquiry. ... But one may take it so while showing little deference to the content of any science per se.(p.63)Both Burgess and van Fraassen are naturalists about their respective disciplines (mathematics and empirical sciences) -- but they disagree on what the properly scientific reaction to questions like "Are there numbers?" and "Does science aim at truth or merely empirical adequacy?" is.

The mathematician deals in proof. And proof is (at least a large part of) the source of mathematics' epistemic force. The number theorist (e.g.) assumes the existence of the integers and proves things about them; that's what she does qua mathematician. People with the proclivities of Burgess and van Fraassen would agree thus far, I think. But they part ways when we reach the question "Are there integers?" A Burgessite (if not John B. himslef) could say "If you're really going to defer to number theorists and their practice, they clearly take for granted the existence of the integers." A van-Fraassen-ite could instead say: "What gives mathematics its epistemic force and evidential weight is

*proof*, and the number theorist has no proof of the existence of integers (or the set theorist of sets, etc.). Since there is no proof of the integers' existence forthcoming, asserting the existence of the integers (in some sense) goes beyond the evidential force of mathematics. Thus, a naturalist about mathematics should remain agnostic about the existence of numbers (unless there are other arguments forthcoming, not directly based on naturalism)."

Is there any way to decide between these forms of naturalism -- one which defers (for the most part) to the form

**and**content of the sciences, and the other which defers only to the form? (Note: van Fraassen's

*Empirical Stance*takes up this question, but this post is too long already to dig into his suggestions.)

## 5 comments:

Kenny, I can see the nod back to Carnap in the position you attribe to Maddy, but I can't see anything Wittgensteinian about it; as far as I can tell, the later Wittgenstein was pretty overtly anti-realist about mathematical ontology. Well, perhaps not overtly in the Investigations, but certainly in RFM and LFM.

I remember Maddy having a section on Wittgenstein in the '97 book, but I actually can't remember what she says there. Does she give an interpretation that conflicts with what I've said above?

^Attribute

When I started this blog, there were three topics I decided I did not want to cover, no matter how tempting:

(1) Boring details of my personal life (which, upon a moment's thought, is equivalent to "my personal life");

(2) Politics;

(3) Interpretation of middle and later Wittgenstein.

So I'll leave it to you two to figure out whether Maddy's claims mesh with Herr Ludwig's.

I did have a reaction to the end of Kenny's remark, though, that gets to the heart of the original post. Maddy " suggest[s] that we accept the statement "there are numbers", because it is mathematically justified." But (and here I'll speak with my van-Fraassen-ite hat on) when a mathematician actually seeks to

justifya claim she makes, she (usually) either presents a proof or a counterexample to a conjecture. And that sort of justifying activity is not available for the claims "The integers exist" or "There are sets" -- except in the very trivial sense of 'well, that's just one of the axioms' or something similar. And if that counts as 'justification' at all, it's at least very different from the kind of justifications that mathematicians engage in on a day-to-day basis, and which we philosophers find epistemically praiseworthy (or something similar).To put the point in Carnapese (one of my favorite dialects of English): there is an important sense in which the answers to the most basic internal questions such as 'Are there numbers?' are

not justified. They determine the structure of justification for the language we are speaking, but they are not justified in the way theorems or empirical generalizations are justified (viz., via calculation or observation, respectively). We are 'entitled' (or something similar) to claims like 'Integers exist', but not on the basis of calculation or observation; our entitlement (or whatever) comes merely from using that language, with its particular justificatory structure.OK, that last paragraph became ridiculously murky at the end. Perhaps a future post will turn it into something that expresses a moderately clear sense.

Kenny: fair enough, I'll need to look at the Maddy again.

Greg, here's one set of considerations that may favour the Burgessite. There are sentences about, say, numbers which, unlike the controversial 'do numbers exist?'-style questions, we have no trouble recognising as true, but which like those contested sentences aren't amenable to proof by the number theorist. As Fraser MacBride points out (2005. ‘The Julio César Problem’, dialectica 59: 224-5), Caesar sentences like '2 = Julius Caesar' seem to be of just this sort:

'For whilst number theory abstractly describes number in terms of their relationships to other numbers (being the successor of one number, the predecessor of another and so on) number theory does not describe the number 17 as abstract or necessary or any such other feature by which we might hope to directly distinguish concrete, contingent Caesar from a number. So it does not follow straightaway from our grasp of number theory that numbers are a different kind of things to persons.'

One lesson to draw from consideration of such sentences might be that an extreme version of form naturalism (as opposed to form and content naturalism) is in trouble; it sanctions agnosticism about Caesar sentences and the like. The Burgessite looks to me to be better placed here.

What I can't tell is whether the extreme version of form naturalism is your van-Frassenite's. The extreme view sanctions agnosticism about statements about numbers in the absence of practice-internal-proof (so to speak), but you give the van-Frassenite a get out clause: "a naturalist about mathematics should remain agnostic about the existence of numbers (unless there are other arguments forthcoming, not directly based on naturalism)." I haven't a clear grip on what this allows and disallows (vis-a-vis acquiring appropriate evidence for claims about numbers), so I can't tell if the Caesar sentences might create a problem.

Dear Friends of the Fine Arts,

excuse me for addressing you in my clumsy school-English. (I suppose: Everyone pondering about the Caesar-problem has read Frege; therefore he understands German.) In shorthand: Frege puts the question: what about 2=caesar?

If we extend Frege's Begriffsschrift into a store-logic (Speicherplatzlogik), the solution (without giving any lengthy reason) could be: "caesar" is a proper name in level 0 (zero). Another sentence in level 0 could be: "der Wagen des Kaisers wird von 4 Pferden gezogen". A clumsy notation (again without lengthy reasoning) of this sentence would be: (w)-4-Z-()-P. In this case the "4" is an abbreviation for equal structures. The second possibility looks like that: "(4)" = the number of hosses pulling w".

This sort of "4" is stored in level 1, as a proper name for equal structures in level 0. What then are numbers? Abbreviations or proper names in our heads, and why invented? Idleness or laziness or practicability. In the above hinted complex system it is not possible to formulate the identity-sentence "2=caesar", because "2" has its home in level 1, whereas "caesar" ist located in level 0. Identity of proper names always needs the same level (Sprachstufe, step in the hierarchy of object-meta-languages). To cite a German proverb: Eat it, bird, or die.

Greetings from Munich/Bavaria.

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