Access to reference magnets: a bitter pill you’ve already swallowed?

[This post proposes a defense of a famous defense of naive scientific realism against a famous antirealist challenge.  So, sorry that I'll have to speed through certain classics to get to the action in a timely fashion (and I'll try to add explanatory links later). 

Also, Ted Sider may have scooped me re: this proposal in Writing the Book of the World section 3.2 (I go back and forth about how to interpret him). But, regardless of priority, I'm shouting about it on the internet now because I'd like to see more uptake.]

Hillary Putnam raised a model theoretic challenge to the commonplace realist idea that `even in the ideal limit of scientific investigation' certain aspects of our best theory of the world could be wrong. David Lewis responded by invoking "reference magnets" (i.e., intrinsically eligible concepts/joints in nature) as a response to this challenge. The idea is that some concepts are more intrinsically eligible candidates for the meaning of words than others. So, it can be correct to interpret someone as meaining plus vs Kripke's quus, or electron vs. electron-that-an-ideal-observer-starting-on-earth-could-discover, even if this involves attributing them slightly more false beliefs.

Swinging back on behalf of Putnam, Tim Button and Jared Warren press a kind of access problem for fans of reference magnets. Suppose that there really are intrinsically eligible joints in nature as Lewis argues. How could creatures like us have come to recognize where these joints are, well enough to know when there is a single reference magnetic joint for our use of some word to defer to?

I think this challenge is worth taking seriously and may point out a bitter pill* which the realist/friend of reference magnetism must swallow. But I want to suggest that this bitter pill may already be part of the larger bitter pill nearly everyone has already swallowed in taking our intuitions about how to do scientific induction at face value. That is, accepting reference magnets doesn't land us with any more of an access problem than we already face in rejecting Humean skepticism about induction.

To see what I mean, consider Nelson Goodman's picture of scientific induction. When we do scientific induction, we don't treat all concepts equally. We currently take some predicates (and relations and functions etc) to be more projectable than others, e.g., green v.s grue. And (Goodman notes) we do a kind of induction about how to do induction, letting experience and reflection change our beliefs about which predicates are projectable. So (in effect) we dogmatically presume both that certain predicates are more projectable than others, and that that certain ways of letting experience change our beliefs about which predicates are projectable are reliable. And plausibly, taking scientific induction at face value requires doing something like this.

But maybe the friend of reference magnets can say that access to these facts about what's joint carving (in the sense of being specially friendly to induction) is all they need for access to reference magnets. If the reference-magnet-fan's doctrine identifies being an intrinsically eligibile concept in the sense of reference magnetism with being an intrinsically eligible concept for the purposes of scientific induction (as, e.g. Sider does and I essentially want to**), then it seems that accepting this doctrine create any extra intuitive access worries.



*[i.e., Perhaps they must embrace a rather depressing picture of the human condition, on which `justified’ reasoning (to the extent that we have any such thing) involves going along dogmatically assuming that certain methods for detecting intrinsically eligible concepts/reference magnets are tolerably accurate (and then being lucky enough to be right about this)]

**[I think one tiny refinement answering Hawthorne's problem about Europe and the Ural Mountains discussed on pg 39 of WTBOTW is needed, but that this makes no difference to the access problem stuff above. More on this in a later post]

Three Projects Involving Dispensing With Mathematical Objects

One of the many unfortunate things about academic fashions is that when a popular project goes out of fashion, superficially similar-looking projects which don't face the same difficulties can be tarred with the same brush.

Many people (myself included) feel cautious pessimism about formulating a satisfying nominalist paraphrase of contemporary scientific theories [one major issue is how to make formulate something like probability claims without invoking abstract-seeming events or propositions]. But we wouldn't want to over generalize.

 In this post I'm going to suggest three different motives for seeking to systematically paraphrase our best scientific theories (and many other true and false ordinary claims) in a way that dispenses with quantification over mathematical objects, and note that the requirements for success in the first (ex-fashionable) project are notably laxer in some ways than the requirements for the others.

[Note: I don't mean to endorse any of the projects below, but I think that 3 and Burali-Forti based versions of 2 are at least interesting.]

Three Motivations for Paraphrasing Mathematical Objects Out of Physics

1. General rejection of abstracta: You deny the existence of mathematical objects because you think allowing any abstract objects are bad. (this is the classic motive)
2. Explaining special features of mathematical practice by rejection of mathematical objects: You deny the existence of mathematical objects because you think that not taking mathematical existence claims at face value is allows for the best account of certain special features of pure mathematical practice, (e.g., by mathematicians’ apparent freedom to choose what objects to talk in terms of/disinterest in mathematical questions that don’t effect interpretability strength, or by the Burali-Forti paradoxes in higher set theory) not to take apparent quantification over mathematical objects at face value. 
3. Grounding math in logic/bringing out a claimed special relationship between math and logic: You may allow the existence of mathematical objects, but you’re moved by the close relationship between an intuitive modal notion of coherence/semantic consistency/logical possibility and pure mathematics to seek some kind of shared grounding and think that the coherence/logical possibility notion looks to be the more fundamental. As a result, it seems promising to seek a kind of "factoring" story, which systematically grounds all pure mathematics in facts about logical possibility, and all applied mathematics in some combo of logical possibility and intuitively non-mathematical facts.

 Distinguishing these motivations/projects matters, because what you are trying to do influences what ingredients are kosher for use in your paraphrases: 

If you have the first motivation (defending general nominalism), you need to avoid quantifying over any other abstracta, including: platonic objects called masses or lengths which don’t come with any automatic relationship to numbers, corporations, marriages, sentences in natural languages, metaphysically possible worlds, and (perhaps) propensities. 

But if you have the second motivation (defending mathematical-nominalism-adopted-to-explain-special-features-of-mathematical-practice), then quantification over abstracta which don’t have the relevant special feature, (e.g., objects in domains where we don't think appear to have massive freedom to choose what objects to talk in terms of or objects which don't give rise to a version of the burali-forti paradox), is fine. 

And if you have the third motivation, then quantification over any objects which aren’t intuitively mathematical-- or aren't mathematical in whatever way you are claiming requires a special relationship to coherence/semantic consistency/logical possibility -- is OK.

The Sheffer stroke, and the blandest antirealism ever

In college my metaphysics prof said realism was the doctrine that there is a complete true description of the world, though it might take infinitely many sentences to express this description. Do the following incredibly bland considerations about different truthfunctional connectives commit me to "antirealism"?

(1) If you want thinking about propositions to play a role in psychology, then you need to individuate propositions narrowly enough that truthfunctionally equivalent sentences express different propositions.

(2) If you individuate propositions as narrowly as required by (1) then tautologies involving the sheffer stroke will be different propositions than any corresponding sentences which use more standard truthfunctional connectives.

(3) The same argument goes for all the other infinitely many n place truthfunctional connectives.

Conclusion: no language with finitely many basic connectives can express all true propositions. No finite language can be used to give a complete true description of the world. In particular if the a language L has only n place truthfunctional connectives then there will be tautologies using n+1 place truthfunctional connectives that cannot be expressed in L.