"The" natural numbers

Benacerraf famously pointed out that there are infinitely many isomorphic mathematical structures that could equally well be the referrent of “the natural numbers”. Mathematicians are generally not bothered by this underdetermination of the concept of “the natural numbers”, precisely because the different structures are isomorphic.

What is more worrying are the infinitely many elementarily inequivalent mathematical structures that, it seems, could count as “the natural numbers”. (This becomes particularly worrying given that we’ve learned from Goedel that these structures give rise to different notions of provability.)

(I suppose this is a kind of instance of the Kripke-Wittgenstein puzzles.)

In response, here is a start of a story. Those claims about the natural numbers that differ in truth value between models are vague. We can then understand this vagueness epistemically or in some more beefy way.

An attractive second step is to understand it epistemically, and then say that God connects us up with his preferred class of equivalent models of the naturals.

Whatever content can be conveyed metaphorically can be said literally

Suppose you say something metaphorical, and by means of that you convey to me a content p. I now stipulate that "It's zinging" expresses precisely the content you conveyed. Technically, "It's zinging" is a zero-place predicate, like "It's raining." And now I say: "It's zinging." The literal content expressed by "It's zinging" is now equal to the metaphorical content conveyed by what you said. A third party can then pick up the phrase "It's zinging" from me without having heard the original metaphor, get a vague idea of its literal content from observing my use of it, and now a literal statement which has the same content as was conveyed by the metaphorical statement can start roaming the linguistic community.

Thus: If you cannot say something literally, you cannot whistle it either. For if you could convey it by whistling, you could stipulate a zero-place predicate to mean that which the whistling conveys.

Objection 1: My grasp of "It's zinging" is parasitic on your metaphor, while the third-party doesn't have any understanding.

Response: Yes, and so what what? I wasn't arguing that you can usefully get rid of metaphor. It may well be essential to understanding the content in question. My point was simply that there can be a statement whose literal semantic content is the same as the content conveyed by your metaphor. Understanding is something further. This is very familiar in cases of semantic deference. (I hear physicists talking about a new property of particles. I don't really understand what they're saying, but I make the suggestion that they call that property: "Zinginess." My suggestion catches on. I can say: "There are zingy particles", and what I say has the same content as the scientists' attribution of that property. But while the scientists understand what they're saying, I have very little understanding.) The third party who hasn't heard the original metaphor may not understand much of what he's saying with "It's zinging." But what he's saying nonetheless has the literal semantic content it does by deference to my use of the sentence, and my use of the sentence has the literal semantic content it does by stipulation. All this is quite compatible with the claim that any decent understanding of "It's zinging" will require getting back to the metaphor. But, nonetheless, "It's zinging" literally means what the metaphor metaphorically conveyed.

Objection 2: The stipulation does not succeed. (This is due to Mike Rea.)

Response: Why not? If I can refer to an entity, I can stipulate a name for it, no matter how little I know about it. I may have no idea who killed certain people, but I can stipulate "Jack the Ripper" names that individual. My stipulation will succeed if and only if exactly one individual killed those people. Similarly, if I can refer to a property, I can stipulate a one-place predicate that expresses that property. (If a certain kind of Platonism is true, this just follows from the name case: I name the property "Bob", and then I have the predicate "instantiates Bob".) In cases without vagueness, contents seem to be propositions, and zero-place predicates express propositions, so just as I can stipulate a one-place predicate to express a property, I should be able to stipulate a zero-place predicate to express a propositions. And in cases of vagueness, where maybe a set of propositions (or, better, a weighted set of propositions) is a content, I should be able to stipulate a similarly vague literal zero-place predicate as having as its content the same set of propositions.

There are many ways of introducing a new term into our language. One way is by stipulating it in terms of literal language. That's common in mathematics and the sciences, but rare in other cases. Another way is by ostension. Another is just by talking-around, hoping you'll get it. One way of doing this talking-around is to engage in metaphor: "I think we need a new word in English, 'shmet'. You know that butterflies in the stomach feeling? That's what I mean." We all understand what's going on when people do this kind of stipulation. For all we know, significant parts of our language came about this way.

Deep Thoughts XLII

No one speaks about what cannot be spoken about.

Expressing and asserting

Consider the following broadly Wittgensteinian line of thought:

1. Sentences like "I love you", "This is scary" and "God is all powerful" express love, fear and awe respectively.
2. Therefore, sentences like "I love you", "This is scary" and "God is all powerful" are not assertions of propositions.

I am happy to grant (1), at least if we qualify the "express" with "typically express". But I think the inference of (2) from (1) is simply a non sequitur, though a tempting one.

Consider this somewhat parallel argument:

3. A birthday cake expresses one's honoring of the years someone has lived.
4. Therefore, a birthday cake is not a piece of food.

It is clear that (4) is a non sequitur. Obviously the right thing to say is not that a birthday cake is not a piece of food, but that a birthday cake is a piece of food that expresses one's honoring of the years someone has lived. By analogy, why shouldn't we say that "I love you", "This is scary" and "God is all powerful" are assertions of propositions which assertions express love, fear and awe respectively? I can express, for instance, love by holding hands, baking a cake or sharing a joke. So why can't I also express love by asserting a relevant proposition, viz., that I have a love for the person? And the same point goes through for the other examples.

In other words, the line of thought (1)-(2) sets up a false dilemma: either these sentences are assertions of a proposition or they are expressions of an attitude. But the natural thing to say is that they are both. It is if anything less surprising that assertions of these very relevant contents should express the attitudes they do than that, say, being on one's knees should express awe or that holding hands should express love.

Taking the sentences in question to be assertions and expressions of the indicated attitudes better fits the data than just taking them to be expressions of the indicated attitudes. For the sentences can be embedded in ways that give purely expressivist accounts great trouble. "If God cannot prevent earthquakes, then God is not all powerful"; "If I love you, then I pursue what I take to be your good"; "Either this is scary or my judgments are completely off."

Wittgensteinian anti-literalism

Consider this Wittgensteinian line of thought about the doctrine of the resurrection of Jesus (similar things can be said about many other doctrines).[note 1] The way to understand what believers say when they confess "Jesus is risen" is to see what role that utterance plays in their lives, both there and then (say, in the liturgy) and more broadly in their lives. Further, the word "risen" in this sense has little currency outside religious contexts, so the notion of a "literal" meaning of the word outside of a religious context is problematic, and hence the role that the utterance plays in lives is determinative for meaning. It is difficult, therefore, to say what the disagreement between literalists and non-literalists about the resurrection is really about.

It seems to me that there is a fairly simple response to this line of thought. For the sake of argument, let's take on board the Wittgensteinian assumption that language gets its meaning from use, and hence to examine the meaning of an utterance one must see what role it plays in a life. But we must not forget that a significant part of the role that an utterance plays is found in its inferential connections with other utterances, both deductive and non-deductive.

For instance, the utterance "Jesus is risen" provides inferential support (one needs to judge on a case-by-case basis whether it's deductive or not) for such claims as "Jesus is not dead", "Jesus's skeleton isn't presently lying in the earth", "Jesus is alive", "People saw Jesus after his resurrection by means of photons reflected from his body", etc. This inferential support is at least as constitutive of the meaning of the utterance as is, say, the utterance's liturgical role. Moreover, the it is also a part of the role of the utterance that it is (non-deductively) supported by such claims as "Jesus's tomb is empty and his body is nowhere to be found", "Jesus was seen eating fish after he died", "Some of those who report Jesus as being risen knew what Jesus looked like", etc.

Perhaps unlike the confession "Jesus is risen", the claims I gave as inferentially connected with this confession make use of ordinary vocabulary and in ordinary ways. These claims are a part of our ordinary non-religious language games. We ask whether someone is alive or dead, whether someone's skeleton is buried in the earth, whether someone's tomb is empty, whether anyone saw someone eating fish, and so on. Now the Wittgensteinian anti-literalist will insist that the inferential connections between "Jesus is risen" and such ordinary claims do not do justice to the religious significance of "Jesus is risen", that for that significance one must pay attention to the liturgical and motivational role of the utterance in the lives of believers, and so on. But the literalist can take all that on board. There is more to the resurrection than is captured by the inferential connections with various ordinary non-religious claims. But there is no less: these inferential connections are an essential part of the constitution of the phrase's meaning, in a way that the non-literalist has a harder time accounting for. What connection, for the anti-literalist, is there between "Jesus is risen" and the fact that some of those who reported seeing Jesus knew what he looked like? But on a literalist reading, the inferential connection is clear.

This gives us both a way to characterize the disagreement—the literalist takes such inferential connections with ordinary claims to play a central part in the constitution of the meaning of "Jesus is risen" while the non-literalist does not—and a reason to favor the literalist reading, since the literalist is operating with a fuller collection of data.

At the same time, the literalist reading of "Jesus is risen" has significant flexibility, because the inferential connections are often probabilistic. The claim that x is risen makes probable that x was seen by means of photons reflected from x's body. But it does not entail it. It may be that x's body no longer reflects photons or that nobody has seen x. Thus, the literalist need not be committed to the claim that Jesus was seen by means of reflected photons, and might even deny it (there is a strong tradition that Jesus's body had some very special properties) but the literalist does need to see the confession "Jesus is risen" as of such a sort as to make the photonic claim a plausible inference. On an anti-literalist reading, however, the photonic claim does not seem to be a plausible inference. And that favors the literalist.

This gives us a literalism without a commitment to any such dubious thing as a "literal meaning".

But there is a lesson for the literalist, too. The literalist should not take the meaning of "Jesus is risen" to be exhausted by the inferential connections to ordinary claims. That, too, would be basing the meaning on merely partial data.

Wittgensteinian views of religious language

Wittgensteinians lay stress on the idea that

1. One cannot understand central worldview concepts without living as part of a community that operates with these concepts.

The non-Christian cannot understand the Christian concept of the Trinity; the Christian and the atheist cannot understand the Jewish concept of God's absolute unity as understood by Maimonedes; the theist cannot understand the concept of a completely natural world; and the non-Fascist cannot understand the concept of the Volk. It is only by being a part of a community in which these concepts are alive that one gains an understanding of them.

Often, a corollary is drawn from this, that while internal critique or justification of a worldview tradition such as Christianity, naturalism or Nazism is possible, no external critique or justification is possible. In fact, there is an argument for this corollary.

2. (Premise) One's evidence set cannot involve any propositions that involve concepts one does not understand.
3. (Premise) Necessarily, if a proposition p uses a concept C, and a body of propositions P is evidence for or against p for an agent x, then some member of P involves C.
4. If x is not a member of the community operating with a central worldview concept C, then x does not have any evidence for or against any proposition involving C. (1-3)
5. (Premise) External critique or justification of a worldview of a community is possible only if someone who is not a member of the community can have evidence for or against a proposition involving a central worldview concept of that community.
6. Therefore, external critique or justification of a worldview of a community is not possible. (4 and 5)

This is a particularly unfortunate result in the case of something like Nazism, and may suggest an unacceptable relativism.

The argument is valid but unsound, and I think unsalvageable. I think that (5) is false, and on some plausible interpretations of (1), (2) and (3) are false as well.

First of all, people successfully reason with scientific concepts that they do not understand, like the concept of a virus or of gravity. They inherit the concept from a scientic community that they are not a member of, and while they do not understands the concept, they get enough about the inferential connections involving the concept that the concept should become useful. Thus, even if I do not really understand the concept of a virus, my evidence set can include facts about viruses that I know by virtue of testimony[note 1], and inferential connections with other facts, such as that if x has the common cold, then many viruses are present in x's body. Thus, (2) is false.

As for (3), I don't know for sure if it's false, but seems quite possible that while C does not occur in one's evidence set, it might occur in one's rules of inference. And there does not seem to be anything wrong with having a concept in one's evidence set that one does not understand.

But perhaps you are not convinced by the critique of (2) and (3). I suspect this is because you take (1) to be more radical than I do. The "cannot understand" in (1) is understood as entailing "cannot operate with"—even the weak sort of grasp that the layperson has of scientific concepts is denied to non-members of a community in the case of central worldview concepts. On this interpretation of (1), (2) and (3) are false. I am inclined to think that this interpretation of (1) is the incorrect one because it renders (1) false. The central worldview concepts of a community do not seem to be significantly different from the central concepts of a scientific community. Still, I see the force of such a beefed-up (1), at least in the case of the concepts of the Christian faith (not so much because of the need for community membership as such, but because of the need for grace to enlighten one's understanding).

In any case, (5) is false on either understanding of (1). The reason is simple. To support or criticize a position, one does not need evidence for or against a position. One only needs evidence for or against the second order claim that the position is true. Often, this a distinction without much of a difference. I have evidence that

7. there is life on Mars

if and only I have evidence that

8. the proposition that there is life on Mars is true.

However, this is so only because in (8) I refer to the proposition under the description "that there is life on Mars." But take a different case. I go to a mathematics lecture. Unfortunately, as I shortly discover, it's in German. I sit through it uncomprehendingly. At the end of it, I turn to a friend who knows German and ask her what she thought. She is an expert in the field and says: "It was brilliant, and I checked that his central lemma is right." I still don't know what the speaker's central lemma is, but I know that it is true. I do not have evidence for the lemma, and it could even be (say, if the talk is in a field of mathematics I don't know anything about) that I don't have the requisite concepts for grasping the lemma, but I have evidence that the lemma is true.

Likewise, it is possible to have evidence for and against the claim that the community's central worldview propositions are true, without grasping these propositions and having evidence for or against them. For instance, while I may not be able to understand what the members of the community are saying in their internal critiques, but I may understand enough of the logical form of these critiques and of the responses to them to be able to make a judgment that the critiques are probably successful. Moreover, even if I do not understand some concept, I may grasp some metalinguistic facts such as that if x is a Gypsy, then x is not a part of anything in the extension of the term "the Volk", or that if the only things that exists at w are the particles of current physics, and at w their only properties and relations are those of current physics, then w is in the extension of the term "completely natural". Given such facts, I can gain arguments for or against the thesis that the central worldview claims of the community are true. Thus, (5) is false.

There is a hitch in my argument against (5). External evaluation of the community seems to require that while I have no grasp of particular central terms, I have some grasp of the larger grammar of sentences used by members of the community and I understand some of the non-central terms in their language. But what if I don't? This could, of course, happen. The community could speak an entirely foreign language that I am incapable of parsing.

I can make two responses. The first is that (5) is a general claim about communities whose central worldview concepts I do not understand, and that general claim has been shown to be false. There could be some radical cases where the outsider's lack of understanding is so complete that external critique or justification is impossible. But such cases do not in fact occur for us. Humans share basic structures of generative grammar and a large number of basic concepts due to a common environment.

The second response is that the behavior of members of the community can provide evidence for and against the correctness of their central ideas. If their airplanes keep on crashing, there is good reason to think their scientific concepts are bad. If they lead a form of life that does not promote the central human goods, there is good reason to think that their ethics is mistaken, while if they lead a form of life that does promote the central human goods, there is good reason to their ethics is sound. Now, of course, I could be wrong. Maybe for religious reasons they want their airplanes to crash and design them for that. Maybe they abstain from some central human goods for the sake of some God-revealed higher good. Maybe they are a bunch of hypocrites, and they aren't really achieving the central human goods. However, such possibilities only show that I cannot be certain in my external evaluation. But the claim that external justification and critique is possible is not the claim that one can achieve certainty in external justification and critique. What I've said shows we can achieve high probability even in cases where the community's language is radically not understandable.

Things might be different if we're dealing with an alien species of intelligent beings. But I suspect we could still come to probabilistic judgments, just somewhat less confident ones.

I think the above considerations not only show that the argument (1)-(6) fails, but that we're unlikely to get any successful argument along those lines.

A counterexample to the Private Language Argument

Wittgenstein's Private Language Argument contends that it is impossible for one to form a language by oneself. Here's a counterexample. You live on an island that has no people other than yourself. You live for forty years there. Then you step in a time machine that was left there by someone else. You go back forty years. You do this 999 times. (Let's assume that the time machine also fixes up your body so you can live for a subjective length of 40000 years.) To an outsider, the island looks populated by a thousand people of remarkably similar appearance. There is a community there. But that community in fact includes only one person, you. So, it's possible to have a community with only one person. But there is no reason why such a community couldn't develop a language, since it functions just like all other communities do.

A fun question: Suppose Marcy and George join you on the island, but they don't do any time travel, and so there you are located in 1000 places on the island, and then there is Marcy and there is George. In elections, should you get 1000 votes, with each of them getting only one—there does, after all, seem to be a sense in which you have a lot more interests—or should each of the three people on the island get exactly one vote?

By the way, the clip below illustrates the wrong way of imagining the scenario. In my scenario, it is a mistake to think of first the island having you in one place, then of it having you in two places, and so on. Over the 40 year period in my scenario, you always are in 1000 places.