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).

A new argument for presentism

Here’s an interesting argument favoring presentism that I’ve never seen before:

1. Obviously, a being that fails to exist at some time t is not a necessary being.

2. If presentism is true, we have an elegant explanation of (1): If x fails to exist at t₁, then at t₁ it is true that x does not exist simpliciter, and whatever is true at any time is possibly true, so it is possible that x does not exist simpliciter, and hence x is not a necessary being.

3. If presentism is false, we have no equally good explanation of (1).

4. So, (1) is evidence for presentism.

I don’t know how strong this argument is, but it does present an interesting explanatory puzzle for the eternalist:

5. Why is it that non-existence at a time entails not being necessary?

Here’s my best response to the argument. Consider the spatial parallel to (1):

6. Obviously, a being that fails to exist at some location z is not a necessary being.

It may be true that a being that fails to exist at some location is not a necessary being, since in fact the necessary being is God and God is omnipresent. But even if it’s true, it’s not obvious. If Platonism were true, then numbers would be counterexamples to (6), in that they would be necessary beings that aren’t omnipresent.

But numbers seem to be not only aspatial but also atemporal. And if that’s right, then (1) isn’t obvious either. (In fact, if numbers are atemporal, then they are a counterexample to presentism, since they don’t exist presently but still exist simpliciter.)

What if the presentist insists that numbers would exist at every time but would not be spatial? Well, that may be: but it’s far from obvious.

What if we drop the “Obviously” in (1)? Then I think the eternalist theist can give an explanation of (1): The only necessary being is God, and by omnipresence there is no time at which God isn’t present.

Maybe one can use the above considerations to offer some sort of an argument for presentism-or-theism.

Some paradoxes of reference

Liar-like:

• one plus the biggest integer that can be expressed in English in fewer than fifty words
• one; two; three; one plus the biggest integer mentioned in this list
• one; two; three; one plus the last integer mentioned in this list
• one plus the last integer mentioned in this list; two; three; one plus the first integer mentioned in this list
• one plus this integer

Truthteller-like:

• one; two; three; the biggest integer mentioned in this list
• one; two; three; the last integer mentioned in this list
• the last integer mentioned in this list; two; three; the first integer mentioned in this list
• this integer
• the square of this integer

More on comparing infinities

Some people don't want to say that there are just as many even natural numbers as natural numbers. But suppose that you and I will spend eternity singing numbers in harmony. You will sing every natural number in sequence: 1, 2, 3, ..., with a long pause for applause in between. And while you sing n, I will sing 2n. We will vary the speed of our singing to ensure that we take equal amounts of time. Clearly:

1. The number of natural numbers = the number of your performances.
2. The number of your performances = the number of my performances.
3. The number of my performances = the number of even natural numbers.
4. So, the number of natural numbers = the number of even natural numbers.

Premises (1)-(3) are obviously true, and I don't understand what "the same number" relation could mean if it's not transitive.

Determinates vs. values

Spot has a mass of 10kg, while Felix has a mass of 8kg. The standard Platonic way to model the facts expressed by this is to say that Spot and Felix both have the determinable property of mass and they also have the determinate properties mass-of-10kg and mass-of-8kg, respectively. But there is another Platonic way to model these facts. Rephrase the beginning statement as: "Spot masses 10kg while Felix masses 8kg." The natural First Order Logic rendering of the English is now: Masses(spot, 10kg) and Masses(felix, 8kg). In other words, there is a relation between Spot and Felix, on the one hand, and the two respective values of 10kg and 8kg, on the other.

The determinate property approach multiplies properties: for each possible mass value, it requires a property of having mass of that value. The value approach, on the other hand, introduces a new class of entities, mass-values. So far, it looks like Ockham's razor favors the standard determinate property approach, since we don't want to multiply classes of entities.

However, the determinate property approach has some further ideology. It requires a determinable-determinate relation, which holds between having mass and having mass m. The mass-value approach doesn't require that. We can define having mass in terms of quantification: to have mass is to mass something (∃x Masses(spot, x)). Moreover, the value approach might be able to greatly reduce the number of values it posits. For instance, mass, length and charge values could all simply be real numbers in a natural unit system like Planck units. If one thinks that the Platonist needs mathematical objects like numbers anyway, the additional commitment to values comes for free. Further, the determinate property approach requires positing either a privileged bijection relation (or set of bijection relations) between mass values and non-negative real numbers or enough mathematical-type relations between mass determinates (e.g., a relation of one mass determinate being the sum of two or more mass determinates) to make sense of the mathematics in laws of physics like F=Gmm'/r².

There is also a potential major epistemological bonus for the value approach if the values are real numbers. Standing in a mass relation to a particular real number will be causally relevant. Thus, real numbers lose the inertness, the lack of connection to concrete beings like us, that is at the heart of the epistemological problems for mathematical Platonism.

All that said, I'm not enough of a Platonist to like the story. Is there a non-Platonic version of the story? Maybe. Here's one wacky possibility after all: Values are non-spatiotemporal contingent and concrete beings. They may even be numbers, contingent and concrete nonetheless.

A problem for easy ontology arguments

Consider this "easy ontology" argument:

1. There are no unicorns.
2. So, there are zero unicorns.
3. So, there is a zero.

This seems fine. Now consider the parallel:

4. Every leprechaun is a fairy.
5. So, the set of leprechauns is a subset of the set of fairies.
6. So, there is a set of leprechauns.
7. If there is a set of leprechauns, it's empty. (There aren't any leprechauns!)
8. So, there is an empty set.

That seems fine as well. So far so good. But now:

9. Every non-self-membered set (set a that isn't one of its own members) is a set.
10. So, the set of non-self-membered sets is a subset of the set of all sets.
11. So, there is a set of non-self-membered sets (the Russell set).

But of course (11) yields a contradiction (just ask if the Russell set is a member of itself).

What to do? One move is to make the easy ontology arguments defeasible. This isn't in the spirit of the game. The other is to add to the premises of the easy ontology argument a coherence premise: that there is a coherent theory of zero, of the empty set and of the Russell and universal sets. The coherence premise will be false in the Russell case but will be true in the other cases. But the point is one that should make us take easy ontology less easily. (I wouldn't be surprised if this was in the easy ontology literature, with which I have little familiarity.)

Nonexplanatory Platonic entities

Benacerraf-style arguments that numbers couldn't be any particular collection of abstract entities (say, some particular set-theoretic construction) because there is a multitude of other constructions that could play the same role will fail if numbers play an explanatory role in the world. And one can imagine metaphysical views on which they play even a physical explanatory role. For instance, charge and mass play an explanatory role, indeed perhaps a causal one, in the world. But a Platonist could think that to have a charge or mass of x units (in the natural respective unit system) is to be charge- or mass-related to the number x. In other words, such determinables are relations, whose second relatum must be a number, and their determinates are cases of that relation for a fixed second relatum.

Now, one can still construct a relation to some set-theoretic isomorph of the numbers that structurally functions just like charge. For instance, if f is an isomorphism from the abstracta relata of charge to some abstract Ss, then we could say that a is related by charge* to y, where y is one of the Ss, precisely when a is related by charge to an x such that y=f(x). But there will be a matter of fact as to whether it is charge or charge* that explains the motion of particles. Surely they both don't—that would be a bogus case of overdetermination.

The point generalizes to other cases of Platonic entities that play an explanatory role—not necessarily a physical one—in the world. For instance, propositions might explain the co-contentfulness of sentences. An isomorph of the system of propositions could play some of the same roles for us, but it would not in fact explain the co-contentfulness of sentences. Compare this case. There is an isomorphism between legal US voters and some set of social security numbers. We can then construct a relation voting* between numbers and candidates such that n votes* for c if and only if the voter with social security number n votes for c. But while one could use facts about voting* to organize our information about elections, it is facts about voting—an action performed by persons, not social security numbers—that in fact explain election outomes.

That said, I think this approach will still tell against the standard set-theoretic constructions of numbers in two ways. First, it will tell against any particular construction. For how likely is it that this construction is the right one? Second, it will tell against anything like the set-theoretic constructions being the numbers. For it seems really unlikely that having a charge of three units is anything like a matter of being related in some way to the set {∅, {∅}, {∅, {∅}}}. So this approach is most plausible if numbers are some kind of sui generis entities.

But, on the other hand, the Benacerraf argument could apply against Platonic entities that play no explanatory role but are merely introduced for our convenience of expression. On some views, possible worlds are like that.

Vagueness and the foundations of mathematics

There are many set-theoretic constructions of the natural numbers. For instance, one might let 0 be the empty set ∅, 1 be {0}, 2 be {1,2}, and so on. Or one might let 0 be ∅, 1 be {∅}, 2 be {{∅}}, and so on. (The same point goes for the rationals, the reals, the complex numbers, and so on.) Famously, Benacerraf used this to argue that none of these constructions could be the natural numbers, since there is no reason to prefer one over another.

My graduate student John Giannini suggested to me that one might make a move of insisting that there really is a correct set of numbers, but we don't know what it is, a move analogous to epistemicism about vagueness. (Epistemicists say that there is a fact of the matter about exactly how much hair I need to lose to count as being bald, but we aren't in a position to know that fact.)

It then occurred to me that one might more strongly take the Benacerraf problem literally to be a case of vagueness. The suggestion is this. Provable intra-arithmetical claims like that 2+2=4 or that there are infinitely many primes are definitely true. Claims dependent on one particular construction of the naturals, however, are only vaguely true. Thus, it is vaguely true that 1={0}. Depending, though, on what sorts of naturalness constraints our usage might put on constructions, it could be that some conditional claims are definitely true, such as that if 3={0,1,2}, then 4={0,1,2,3}.

There are some choices about how to develop this further on the side of foundations of mathematics. For instance, one might wonder if some (all?) unprovable arithmetical claims might be vague. (If all, one might recover the Hilbert program, as regards the definite truths.) Likewise, extending this to set theory, one might wonder whether "set" and "member of" might not be vague in such a way that the Axiom of Choice, the Continuum Hypothesis and the like are all vague.

Vagueness, I think, comes from to our linguistic practices undeterdetermine the meanings of terms. Likewise, our arithmetical practices arguably undetermine the foundations.

The above account neatly fits with our intuition that intra-mathematical claims are much more "solid" than meta-mathematical claims. For the meta-mathematical claims are all vague.

The next step would be to consider what happens when plug the above into various accounts of vagueness. Epistemicism is one option: our arithmetical terminology does have reference to one particular choice of foundation, but we aren't in a position to see what it is. I find promising a theistic variant on epistemicism. Supervaluationism seems particularly neat here. There will be one precification which precisifies things consistently with one foundational story, and another with another. can also consider other options.

There might even be some elements of epistemicism and some of supervaluationism. For there might be facts beyond our ken that say that some foundational stories are false—the epistemicism part of the story—but these facts may be insufficient to determine one foundational story to be right.

That said, I think I still prefer a more ordinary structuralism, though this story has the advantage that it takes the logical form of mathematical claims at face value rather than as disguised conditionals.

Real numbers

For a long time I've been puzzled—and I still am—by this. Our physics is based on the real numbers (complex numbers, vectors, Banach spaces—all that is built out of real numbers). After all, there are non-standard numbers that can do everything real numbers can. So what reason do we have to think that "the" real numbers are what the world's physics is in fact based on?

I think one can use this to make a nice little argument against the possibility of us coming up with a complete physics—we have no way of telling which of the number fields is the one our world is based on.

Can timeless things change?

It seems that the answer has to be negative—isn't the idea utterly absurd? But suppose that an A-theory is true and Fred is a timeless being. Let W be the property of being (timelessly) in a world where a war is (presently) occurring. It seems that on A-theories this is a genuine property, and it was true in 1944 that Fred then had W and it is no longer true in 2008 that Fred has W. So it seems that Fred has changed in respect of W. The B-theorist is apt to deny the existence of such a property as W, and instead talk of the family of properties W_t of being (timelessly) in a world where a war is occurring at t. It was true Fred in 1944 had W₁₉₄₄ and it is true in 2008 that he does not have W₂₀₀₈, but that is not a change, since likewise it was true in 1944 that Fred had not-W₂₀₀₈ and it is true in 2008 that Fred has W₁₉₄₄.

So, if the A-theory is true (or at least if one of those A-theories is true that allow tensed properties like W), it follows that timeless beings change. Of course, the change is extrinsic. But even extrinsic change is puzzling in the case of a timeless being. Look at it from Fred's point of view. Does he or does not have W? It seems both, but that is absurd. In the case of a being in time, we would say that the question is ambiguous—does he have W at what time? But we cannot disambiguate this from Fred's point of view.

Here is something an A-theorist might say. She might say—in fact, I think that on independent grounds she should say it—that at every time, a different world is actual. (Right now, a world without a present world war is actual. In 1944, a world with a present world war was actual.) Then there is no contradiction in Fred's both having and not having W, since since in one world (the 1944 one) he has W and in the other (the 2008 one) he does not.

If we take this route, then the "objective change" that A-theorists are enamored of will be a movement (an orderly one) from one world to another. But Fred undergoes that movement just as much as you and I—in 2005 he was in the 2005 world, and in 2008 he is in the 2008 world—though there is a difference whose significance I am unable to evaluate at present (Fred exists timelessly in both worlds, while you and I exist presently in both worlds). It seems, then, that Fred undergoes objective change, while being outside of time. That seems absurd. Moreover, if we take this route then the following conceptual truth becomes really hard to account for: Nothing outside of time can undergo intrinsic change. But why can't Fred have one set of intrinsic properties in the 1944 world and another in the 2008 world? And if he did, then he would be changing in respect of intrinsic properties.

If the above is right, then it seems that what the A-theorist needs to do is to deny the possibility of timeless beings. This has some interesting consequences. If time began with the big bang, and if we are realists about mathematical entities, then the number 7 is about fifteen billion years old, give or take a couple of billion, and if time were to come to an end, then the number 7 would cease to exist. And once we've allowed abstracta to be in time, why should it be any more absurd to allow them in space? I do not know if these kinds of considerations form knock-down arguments against the view (and hence against the A-theory, if the A-theorist needs to go there), but they are worth thinking about.