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.]

A research blog containing 0th drafts and "open questions" - with a focus on philosophy of math.

## Wednesday, May 19, 2010

## Tuesday, May 18, 2010

### A Depressing Theory of Ceterus Paribus Clauses

We want to say "sugarcubes dissolve in water, ceterus paribus", but what does that mean? Philosophical analysis of the phrase ceterus paribus has proved surprisingly difficult. For example, the quoted sentence doesn't mean that all or most pieces of sugar that actually will be dropped into water will dissolve.

Here's a depressing proposal for how ceterus paribus clauses work. We have a substantive (implicit) theory of what "the normal cases" are like, which is based on human daily life and maybe some random traditions too. We use this when evaluating ceterus paribus sentences to choose which way of making the target sentence true to consider. So, for example, 'ceterus paribus' clauses get filled in so that "dropped eggs break, ceterus parbus" is true, because people tend to hang out in places near the surface of the earth, which don't have thick rugs, so it's part of our substantive theory of what's "normal" that when something is dropped there's a hard surface below it (as opposed to a thick rug, or the empty expanse of space).

Here's a depressing proposal for how ceterus paribus clauses work. We have a substantive (implicit) theory of what "the normal cases" are like, which is based on human daily life and maybe some random traditions too. We use this when evaluating ceterus paribus sentences to choose which way of making the target sentence true to consider. So, for example, 'ceterus paribus' clauses get filled in so that "dropped eggs break, ceterus parbus" is true, because people tend to hang out in places near the surface of the earth, which don't have thick rugs, so it's part of our substantive theory of what's "normal" that when something is dropped there's a hard surface below it (as opposed to a thick rug, or the empty expanse of space).

## Sunday, May 9, 2010

### Miniature Phil Math

Almost everyone agrees that our mathematical talk is practically helpful. Unlike astrology, doing math helps us build bridges. But how is math practically helpful? And does the way in which talking about numbers is practically helpful give us any reason to think numbers actually exist?

In this tiny essay I will propose a theory of how the practice of talking as if there were numbers is helpful. Then, I will say that we can appeal to numbers to explain how this practice is helpful, though there are also other correct explanations for this phenomenon which do not commit themselves to numbers. I will conclude by turning to the question of whether there are numbers. On the basis of the previous section I will propose that we do not *need* to posit the existence of numbers to explain the practical usefulness of our mathematical talk. However, we have another reason to believe in numbers which is the following: We want to make statements like "the number of cupcakes doubles every day" true (under certain circumstances), and the pattern of inferences we make with this sentences is quantificational. But this (being describable by some true sentences associated with a existential pattern of inferences) is the only thing that the many different kinds of non-mathematical objects which intuitively exist have in common.

1. How talking about abstracta like numbers is helpful

Talking about abstract objects, like numbers, is helpful because it lets us economically hypothesize patterns 'in the world around us' as well as patterns that might be described as artifacts of language (patterns in which distinct descriptions are logically or otherwise necessarily equivalent). We can say one sentence (about numbers) that will cause people to be willing to infer infinitely many different sentences that aren't about numbers.

For example, suppose I say: "The number of cupcakes doubles every day" This is a claim that quantifies over numbers and days, in the sense that we might represent it as "Ad An if d is a day, and n is a number, then there are n cupcakes on d there are 2n cupcakes on the day after d. "

Hearing this single sentence will lead my listeners to accept many different statements that do not quantify over cupcakes:

"if Ex7 cupcakes today Ex14 cupcakes tomorrow."

"if Ex8 cupcakes today Ex16 cupcakes tomorrow."

"if Ex7 cupcakes tomorrow Ex14 cupcakes the day after tomorrow."

2. What role do abstract objects play in explaining why talk of abstract objects is helpful?

Now we can ask: what role do various objects play in explaining the success of this talk? We might explain the helpfulness of my statement by saying that it is helpful because it...

- lets us track and predict what cupcakes there are and will be

- lets us track *the pattern in* what cupcakes there are and will be

- lets us track and predict how *the doubling function* relates *numbers*, and then predict what cupcakes there will be when, by relating this to facts about the behavior of the doubling function.

It seems to me that all of these are intuitively decent explanations. I take it that what we have here is a typical phenomenon where the same phenomenon (a war) can be explained by accounts that quantify over various different objects (countries vs. people vs. atoms). However, not much would be lost if we just stuck to giving the first explanation, which does not involve any mention of abstract objects.

3. Are there numbers? A good and bad reason for believing in numbers.

If this story about how math is practically helpful is right, should we believe that there really are objects of the kind talked about in these explanations e.g. patterns in the provenance in cupcakes, or numbers and a doubling function?

I don't think there is an *inference to the best explanation* for the existence of patterns in the provenance of cupcakes, or numbers from the helpfulness of this talk. It's not the case that we *need* to posit abstract objects called "patterns in the provenance of cupcakes" or "numbers" to explain how saying the thing described above could help people cope with the cupcakes around them.

Instead, I think it's reasonable to believe in numbers because we have an intuitively true sentence ("the number of cupcakes doubles every day") which allows a existential pattern of inferences - and playing this logical role is all there is to being an object.

The idea here is that when we look at the variety of different "objects" in the world e.g. electrons, magnetic fields goats, holes, waves, contracts, countries, these different kinds of talk don't seem to have much in common with regard to their relation to the physical world. What they do have in common is the pattern of inferences we make between sentences between them. In each case we accept sentences, such that the inferences with these sentences in are elegantly captured (in first order logic) by something of the form "Ex Fx". Now it turns out that talking about numbers and the doubling function shares this same feature.

In this tiny essay I will propose a theory of how the practice of talking as if there were numbers is helpful. Then, I will say that we can appeal to numbers to explain how this practice is helpful, though there are also other correct explanations for this phenomenon which do not commit themselves to numbers. I will conclude by turning to the question of whether there are numbers. On the basis of the previous section I will propose that we do not *need* to posit the existence of numbers to explain the practical usefulness of our mathematical talk. However, we have another reason to believe in numbers which is the following: We want to make statements like "the number of cupcakes doubles every day" true (under certain circumstances), and the pattern of inferences we make with this sentences is quantificational. But this (being describable by some true sentences associated with a existential pattern of inferences) is the only thing that the many different kinds of non-mathematical objects which intuitively exist have in common.

1. How talking about abstracta like numbers is helpful

Talking about abstract objects, like numbers, is helpful because it lets us economically hypothesize patterns 'in the world around us' as well as patterns that might be described as artifacts of language (patterns in which distinct descriptions are logically or otherwise necessarily equivalent). We can say one sentence (about numbers) that will cause people to be willing to infer infinitely many different sentences that aren't about numbers.

For example, suppose I say: "The number of cupcakes doubles every day" This is a claim that quantifies over numbers and days, in the sense that we might represent it as "Ad An if d is a day, and n is a number, then there are n cupcakes on d there are 2n cupcakes on the day after d. "

Hearing this single sentence will lead my listeners to accept many different statements that do not quantify over cupcakes:

"if Ex7 cupcakes today Ex14 cupcakes tomorrow."

"if Ex8 cupcakes today Ex16 cupcakes tomorrow."

"if Ex7 cupcakes tomorrow Ex14 cupcakes the day after tomorrow."

2. What role do abstract objects play in explaining why talk of abstract objects is helpful?

Now we can ask: what role do various objects play in explaining the success of this talk? We might explain the helpfulness of my statement by saying that it is helpful because it...

- lets us track and predict what cupcakes there are and will be

- lets us track *the pattern in* what cupcakes there are and will be

- lets us track and predict how *the doubling function* relates *numbers*, and then predict what cupcakes there will be when, by relating this to facts about the behavior of the doubling function.

It seems to me that all of these are intuitively decent explanations. I take it that what we have here is a typical phenomenon where the same phenomenon (a war) can be explained by accounts that quantify over various different objects (countries vs. people vs. atoms). However, not much would be lost if we just stuck to giving the first explanation, which does not involve any mention of abstract objects.

3. Are there numbers? A good and bad reason for believing in numbers.

If this story about how math is practically helpful is right, should we believe that there really are objects of the kind talked about in these explanations e.g. patterns in the provenance in cupcakes, or numbers and a doubling function?

I don't think there is an *inference to the best explanation* for the existence of patterns in the provenance of cupcakes, or numbers from the helpfulness of this talk. It's not the case that we *need* to posit abstract objects called "patterns in the provenance of cupcakes" or "numbers" to explain how saying the thing described above could help people cope with the cupcakes around them.

Instead, I think it's reasonable to believe in numbers because we have an intuitively true sentence ("the number of cupcakes doubles every day") which allows a existential pattern of inferences - and playing this logical role is all there is to being an object.

The idea here is that when we look at the variety of different "objects" in the world e.g. electrons, magnetic fields goats, holes, waves, contracts, countries, these different kinds of talk don't seem to have much in common with regard to their relation to the physical world. What they do have in common is the pattern of inferences we make between sentences between them. In each case we accept sentences, such that the inferences with these sentences in are elegantly captured (in first order logic) by something of the form "Ex Fx". Now it turns out that talking about numbers and the doubling function shares this same feature.

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