What numbers could be

Benacerraf famously argued that no set theoretic reduction can capture the natural numbers. While one might conclude from this that the natural numbers are some kind of sui generis entities, Benacerraf instead opts for a structuralist view on which different things can play the role of different numbers.

The argument that no set theoretic reduction captures the negative numbers is based on thinking about two common reductions. On both, 0 is the empty set ⌀. But then the two accounts differ in how the successor sn of a number n is formed:

1. sn = n ∪ {n}

2. sn = {n}.

On the first account, the number 5 is equal to the set {0, 1, 2, 3, 4}. On the second account, the number 5 is equal to the singleton {{{{{⌀}}}}}. Benacerraf thinks that we couldn’t imagine a good argument for preferring one account over another, and hence (I don’t know how this is supposed to follow) there can’t be a fact of the matter about why one account—or any other set-theoretic reductive account—is correct.

But I think there is a way to adjudicate different set-theoretic reductions of numbers. Plausibly, there is reference magnetism to simpler referrents of our terminology. Consider an as consisting of a set of natural numbers, a relation <, and two operations + and ⋅, satisfying some axioms. We might then say that our ordinary language arithmetic is attracted to the abstract entities that are most simply defined in terms of the fundamental relations. If the only relevant fundamental relation is set membership ∈, then we can ask which of the two accounts (a) and (b) more simply defines <, + and ⋅.

If simplicity is brevity of expression in first order logic, then this can be made a well-defined mathematical question. For instance, on (a), we can define a < b as a ∈ b. One provably cannot get briefer than that. (Any definition of a < b will need to contain a, b and ∈.) On the other hand, on (b), there is no way to define a < b as simply. Now it could turn out that + or ⋅ can be defined more simply on (b), in a way that offsets (a)’s victory with <, but it seems unlikely to me. So I conjecture that on the above account, (a) beats (b), and so there is a way to decide between the two reductions of numbers—(b) is the wrong one, while (a) at least has a chance of being right, unless there is a third that gives a simpler reduction.

In any case, on this picture, there is a way forward in the debate, which undercuts Benacerraf’s claim that there is no way forward.

I am not endorsing this. I worry about the multiplicity of first-order languages (e.g., infix-notation FOL vs. Polish-notation FOL).

Physicalism and "pain"

Assuming physicalism, plausibly there are a number of fairly natural physical properties that occur when and only when I am having a phenomenal experience of pain, all of which stand in the same causal relations to other relevant properties of me. For instance:

1. having a brain in neural state N

2. having a human brain in neural state N

3. having a primate brain in neural state N

4. having a mammalian brain in neural state N

5. having a brain in functional state F

6. having a human brain in functional state F

7. having a primate brain in functional state F

8. having a mammalian brain in functional state F

9. having a central control system in functional state F.

Suppose that one of these is in fact identical with the phenomenal experience of pain. But which one? The question is substantive and ethically important. If, for instance, the answer is (c), then cats and computers in principle couldn’t feel pain but chimpanzees could. If the answer is (i), then cats and computers and chimpanzees could all feel pain.

It is plausible on physicalism (e.g., Loar’s version) that my concept of pain refers to a physical property by ostension—I am ostending to the state that occurs in me in all and only the cases where I am in pain, and which has the right kind of causal connection to my pain behaviors. But there are many such states, as we saw above.

We might try to break the tie by saying that by reference magnetism I am ostending to the simplest physical state that has the above role, and the simplest one is probably (i). I don’t think this is plausible. Assuming naturalism, when multiple properties of a comparable degree of naturalness play a given role, ostension via the role is likely to be ambiguous, with ambiguity needing to be broken by a speaker or community decision. At some point in the history of biology, we had to decide whether to use “fish” at a coarse-grained functional level and include dolphins and whales as fish, or at a finer-grained level and get the current biological concept. One option might be a little more natural than the other, but neither is decisively more natural (any fish concept that has a close connection to ordinary language is going to have to be paraphyletic), and so a decision was needed. And even if (i) is somewhat simpler than (a)–(h), it is not decisively more natural.

This yields an interesting variant of the knowledge argument against physicalism.

1. If “pain” refers to a physical property, it is a “merely semantic” question, one settled by linguistic decision, whether “pain” could apply to an appropriately programmed computer.

2. It is not a “merely semantic” question, one settled by languistic decision, whether “pain” could apply to an appropriately programmed computer.

3. Thus, “pain” does not refer to a physical property.

From tropes to the Trinity

Suppose that each cat has its own catness trope, each dog has its own dogness trope, and so on: each thing has its own essence trope. Thus, whenever we have two cats, we have two individuals—understood as bare particulars, bundles, substances, composite wholes, or in some other way—and two catnesses. Now, the question comes: When we say "There are two cats", is the content that

1. There are two individuals each of which is a cat, or that
2. There are two catnesses?

Whenever one is true, so is the other. Moreover, both statements are pretty natural. On a reference magnetism semantics on which the contents of our assertions are determined by charity plus naturalness, the choice between (1) and (2) will be a close one. In fact, it may be so close that it may be indeterminate whether "There are two cats" means (1) or (2). If naturalness favors either reading, it seems to favor (2) slightly, I think. So the choice between (1) and (2) will be a close one, and the answer will be either indeterminate or in favor of (2).

Likewise, does "Felix" refer to the individual or the catness? If the former, then "Felix sits" has the content that this individual has a sitting trope, and if the latter, then "Felix sits" has the content that this catness is coinstantiated with a sitting trope. Again, both candidates are pretty close to as natural. So again, the question whether names refer to individuals or their essence tropes will be a close one on reference magnetism, and maybe the answer will be indeterminate.

Suppose now that by a miracle there came to be two cat individuals that had the very same catness trope. Why not, after all? There doesn't appear to be anything logically contradictory about the supposition. After all, Siamese twins may share a heart. Why can't they share a trope as well?

So now you assign names, Felix and Tiger, and you say: "Felix is not Tiger." While it may have been indeterminate, or at least was close, whether ordinary cat names referred to catnesses or individuals, in the case at hand there is no indeterminacy or closeness. Charity now requires that at least these two names refer to the individuals, not the shared catness, since there is but one shared catness.

But while it is clear that we should say that Felix is not Tiger, it is less clear whether we should say that there are two cats. It is tempting to argue that Felix is a cat, Tiger is a cat, Felix and Tiger are two, so there are two cats. Certainly, there are two each of which is a cat. But are there two cats? Remember that it was indeterminate or close whether in general "There are two cats" referred to a duality of individuals or a duality of catnesses. If it should turn out to have in general referred to a duality of catnesses, then we should say "There is one cat". But even if it was indeterminate whether "There are two cats" refers to a duality of individuals or catnesses, we might reasonably in this very exceptional case settle on duality of catnesses. There is also a significant consideration in favor of describing the case by saying "There is one cat": it lets one say that there is exactly one catness trope per cat.

But now suppose we have three individuals, the Father, the Son and the Holy Spirit, who have one divinity trope. Then charity will make us say that the Father, the Son and the Holy Spirit are three individuals. Should we say that they are three Gods or that they are one God? If in ordinary cases we take counting to go with essence tropes, then we unambiguously have to go for the "one God" reading. If in ordinary cases, counting is indeterminate between individuals and their essence tropes, then we might in the case at hand simply reasonably resolve the ambiguity by speaking in the "one God" way, which then lets us say, as appears right, that there is exactly one divinity trope per God.

I am not defending this exact theory of the Trinity. One needs to be very cautious talking of individuals that have divinity due to divine simplicity. But I think something like the above is Aquinas' story.

Reference magnetism and anti-reductionism

According to reference magnetism, the meanings of our terms are constituted by requiring the optimization of desiderata that include the naturalness of referents (or, more generally, by making the joints in language correspond to joints in the world, as much as possible) and something like charity (making as many real-world uses as possible be correct).

Suppose we measure naturalness by the complexity of expression in fundamental terms—terms that correspond to perfectly natural things. (In particular, we can't talk of what cannot be expressed in fundamental terms, since reference magnetism would presumably not permit reference to what is infinitely unnatural.) Consider the reductionist thesis that the vocabulary of microphysics is the only fundamental vocabulary about the natural world. If this thesis is true, then our ordinary terms like "conscious" or "intention" or "wrong" are going to be cashed out in terms of extremely complex sentences, often of a functional sort. But I suspect that once these expressions are sufficiently complex, then there will be many non-equivalent variants of them that will fit our actual uses about as well and are about as complex. Consequently, we should expect that the meaning of terms terms like "conscious", "intention" and "wrong" to be highly underdetermined.

If we have reason to resist this underdetermination, we need to embrace an anti-reductionism on which the terms of microphysics are not the only fundamental ones, or else have another measure of naturalness.