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.

New Uses for Conceptual Analysis

Coming up with a systematic way to paraphrase sentences involving some wacky new term W in biology, sociology, psychology, or art criticism, into sentences that are just a logical product of claims about sets, mereological sums of other more commonplace objects, and preserves all our intuitive reasoning about W, is useful in three ways.

a) New Applications for Old Knowledge: Getting a method of paraphrase lets us bring our logical/set theoretic/substantive knowledge about the terms used in the analysis to bear on the new term in question. If the facts about the Ws parallel the facts about sets of such and such kind, set theory may have interesting implications for facts about the Ws.

b) Avoiding Adding Terms Which are "Incoherent" or "Have False Presuppositions": Getting a method of paraphrase may let us prove the consistency and conservativity of reasoning about the wacky new entities. For example if you analyze 'x is bachelor' as 'x is unmarried & x is a man', and then only accept informal reasoning about bachelors that can be reconstructed using this analysis, then it is clear that adding the term 'bachelor' and doing this informal reasoning will not allow you to derive contradiction, or any other new consequences. So, adding informal reasoning about bachelors will do no harm. Here the proof theory (any proof of P which uses the term "bachelor" could be turned into one that doesn't) is so obvious that it's easy not to notice. But the mathematical issues involved in showing consistency and/or conservativity of adopting some term (together with analytic feeling reasoning that goes with that term) can become more interesting when conceptual analysis only provides an *implicit* or recursive definition of the term.

This is valuable, to the extent that you are worried a purported new concept may be 'incoherent' (in the sense that intuitive, analytic feeling reasoning about it literally lets you prove contradiction) or may have bad 'presuppositions' (in the sense that that intuitive, analytic feeling, reasoning using the term allows one to derive new propositions not using that term, which are false)

c) Teaching: Obviously getting a method for paraphrase sentences involving new terminology in terms of old terminology provides a way of teaching the new terminology to people who already understand the old terminology.

Note that none of these purposes require that conceptual analyses be unique. Different analyses of claims about, say, the imaginary numbers, in terms of set theory can each serve this purpose equally well. Nor do these uses for conceptual analysis require that one make any claim about the metaphysical status of the objects in question. It's useful to know you can reconstruct all intuitively acceptable reasoning about the imaginary numbers in terms of intuitively acceptable reasoning about sets, even if you don't want to claim that the imaginary numbers ARE sets or anything like that. Nor, lastly, do they require that the analyses have some kind of psychological reality - that when you are thinking about imaginary numbers you are really somehow implicitly (subconsciously?) considering one or the other paraphrase in terms of sets.

[Hidden agenda: Even if it turns out that Occam's razor doesn't apply to positing special sciences objects like livers, species, trade deficits and languages, so there's no need to look for paraphrases which would allow us to *deny* that such "extra" objects exist, finding Quinean-style parapharases will still be illuminating and useful for other reasons. So we philosophers won't be talking ourselves out of a job :). Also, to the extent that you feel like something substantial is going on when one looks for Quinean methods of paraphrase, this may be because these paraphrases illuminate the structure of our intuitive reasoning about Ws, and let us relate the W facts to facts about objects we understand better - not because there is a serious question about whether the Ws really exist.]

Three jobs for logical structure:

You might think the "logical structure" of a sentence is a way of cutting it up into parts [eg. "John is happy" becomes "is happy(john)"] that does three things:

1. gets used by the logical theory that best captures all the valid inferences.
2. matches the metaphysical structure of the world.
3. explains how we are able to understand that sentence, by breaking it down into these parts, and understanding them.

However, it's not obvious that the method of segmentation which does any one of these things best should also do the others. I don't mean that this idea is crazy, just that it is a bold and substantive claim that logic unites cognitive science with metaphysics in this way.

It's also not obvious that *any* method of segmentation can do 1 or 2.

Task 1 might be impossible to perform because there might not be a unique best logical theory. If we think that the job of logic is to capture necessarily truth-preserving inferences, then second-order logic is logic. Any recursive axiomatization of second order logic will be supplementable to produce a stronger one - since the truths of second order logic aren't recursively axiomatizable. (One might hope though, that all sufficiently strong logics that don't say anything wrong will segment sentences the same way)

Task 2 might be impossible because the world might not have a logical structure to reflect. What do I mean by the world "having a logical structure"? I think there are two versions of the claim:
a. The basic constituents of the world are divided between the various categories produced by the correct segmentation e.g. concepts and objects in Frege's case.
This is weird because "constituents of the world" sound like they should be all be objects. But presumably objects don't join together to produce a sentence, so the kind of expressions used in your chunking up can't all be objects.
Its also weird because it just seems immediately weird to think of the world as having this kind of propositional structure, rather than our just using different propositions with structure to describe the world.
b. The objects that really exist (as opposed to those that are merely a facon de parler), are exactly those which are quantified over by true statements when these are formalized in accordance with the best method of segmentation. To misquote Quine: "the correct logical theory is the one such that, to be, is to be the value of a bound variable in the formalization of some true sentence in accordance with that theory."
So, for example, if mathematical objects can't be paraphrased away in first order logic, but they can using modal logic, the question of whether mathematical objects exist will come down to which (if either) of these of logics has the correct segmentation.

Finally, Task 3 is ambiguous between something (imo) silly and something about neuroscience.
The silly thing is that a correct segmentation should reflect what components *you* break up the sentence "John is happy" into, when you hear and understand it (presumably none).
The neuroscience is `what components does *your brain* break up this sentence, when processing it to produce correct future behavior, give rise to suitable patterns of qualititative experience for you etc.?' This is obviously metaphorical, but I think it makes sense. It seems very likely that there will be some informative algorithm which we can use to describe what your brain does when processing sentences (it might or might not be the same algorithm for different people's brains). And, if so, it's likely that there will be some natural units which this algorithm uses.