A dialectically failing argument for truth-value realism about arithmetic

Truth-value realism about (first-order) arithmetic is the thesis that for any first-order logic sentence in the language of arithmetic (i.e., using the successor, addition and multiplication functions along with the name “0”), there is a definite truth value, either true or false.

Now, consider the following argument for truth-value realism about arithmetic.

Assume eternalism.

Imagine a world with an infinite space and infinite future that contains an ever-growing list of mathematical equations.

At the beginning the equation “S0 = 1” is written down.

Then a machine begins an endless cycle of alternation between three operations:

1. Scan the equations already written down, and find the smallest numeral n that occurs in the list but does not occur in an equation that starts with “Sn=”. Then add to the bottom of the list the equation “Sn = m” where m is the numeral coming after n.

2. Scan the equations already written down, and find the smallest pair of numerals n, m (ordered lexicographically) such that n + m= does not occur in the list of equations, and write at the bottom of the list n + m = r where r is the numeral representing the sum of the numbers represented by n and m.

3. Scan the equations already written down, and find the smallest pair of numerals n, m (ordered lexicographically) such that n ⋅ m= does not occur in the list of equations, and write at the bottom of the list n ⋅ m = r where r is the numeral representing the product of the numbers represented by n and m.

No other numerals are ever written down in that world, and no equations disappear from the list. We assume that all tokens of a given numeral count as “alike” and no tokens of different numerals count as “alike”. The procedure of producing numerals representing sums and products of numbers represented by numerals can be given entirely mechanically.

Now, if ϕ is an arithmetical sentence, then we say that ϕ is true provided that ϕ would be true in a world such as above under the following interpretation of its basic terms:

4. The domain consists of the first occuring token numerals in the giant list of equations (i.e., a token numeral in the list of equations is in the domain if and only if no token alike to it occurs earlier in the list).

5. 0 refers to the zero token in the first equation.

6. The value of Sn for a token numeral n is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a capital S token followed by a token alike to n.

7. The value of n + m for token numerals n and m is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a token alike to n follow by a plus sign followed by a token alike to m.

8. The value of n ⋅ m for token numerals n and m is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a token alike to n follow by a multiplication sign followed by a token alike to m.

It seems we now have well-defined truth-value assignments to all arithmetical sentences. Moreover, it is plausible that these assignments would be correct and hence truth-value realism about arithmetic is correct.

But there is one serious hole in this argument. What if there are two worlds w₁ and w₂ with lists of equations both of which satisfy my description above, but ϕ gets different truth values in them? This is difficult to wrap one’s mind around initially, but we can make the worry concrete as follows: What if the two worlds have different lengths of “infinite future”, so that if we were to line up the lists of equations of the two worlds, with equal heights of lines, one of the two lists would have an equation that comes after all of the equations of the other list?

This may seem an absurd worry. But it’s not. What I’ve just said in the worry can be coherently mathematically described (just take a non-standard model of arithmetic and imagine the equations in one of the lists to have the order-type of that model).

We need a way to rule out such a hypothesis. To do that, what we need is a privileged notion of the finite, so that we can specify that for each equation in the list there is only a finite number of equations before it, or (equivalently) that for each operation of the list-making machine, there are only finitely many operations.

I think there are two options here: a notion of the finite based on the arrangement of stuff in our universe and a metaphysically privileged notion of the finite.

There are multiple ways to try to realize the first option. For instance, we might say that a finite sequence is one that would fit in the future of our universe with each item in the sequence being realized on a different day and there being a day that comes after the whole sequence. (Or, less attractively, we can try to use space.) One may worry about having to make an empirical presupposition that the universe’s future is infinite, but perhaps this isn’t so bad (and we have some scientific reason for it). Or, more directly in the context of the above argument, we can suppose that the list-making machine functions in a universe whose future is like our world’s future.

But I think this option only yields what one might call “realism lite”. For all we’ve said, there is a possible world whose future days have the order structure of a non-standard model of arithmetic, and the analogue to the mathematicians of our world who employed the same approach as we just did to fix the notion of the finite end up with a different, “more expansive”, notion of the finite, and a different arithmetic. Thus while we can rigidify our universe’s “finite” and or the length of our universe’s future and use that to fix arithmetic, there is nothing privileged about this, except in relation to the actual world. We have simply rigidified the contingent, and the necessity of arithmetical truths is just like the necessity of “Water is H₂O”—the denial is metaphysically impossible but conceivable in the two-dimensionalist sematics sense. And I feel that better than this is needed for arithmetic.

So, I think we need a metaphysically privileged notion of the finite to make the above argument go. Various finitism provide such a notion. For instance, finitism simpliciter (necessarily, there are only finitely many things), finitism about the past (necessarily, there are always only finitely many past items), causal finitism (necessarily, each item has only finitely many causal antecedents), and compositional finitism (necessarily, each item has at most finitely many parts). Finitism simpliciter, while giving a notion of the finite, doesn’t work with my argument, since my argument requires eternalism, an infinite future and an ever-growing list. Finitism about the past is an option, though it has the disadvantage that it requires time to be discrete.

I think causal finitism is the best option for what to plug into the argument, but even if it’s the best option, it’s not a dialectically good option, because it’s more controversial than the truth-value realism about arithmetic that is the conclusion of the argument.

Alas.

Some finitisms

I’m thinking about the kinds of finitisms there are. Here are some:

1. Ontic finitism: There can only be finitely many entities.

2. Concrete finitism: There can only be finitely many concrete entities.

3. Generic finitism: There are only finitely many possible kinds of substances.

4. Weak species finitism: No world contains infinitely many substances of a single species.

5. Strong species finitism: No species contains infinitely many possible individuals.

6. Strong human finitism: There are only finitely many possible human individuals.

7. Causal finitism: Nothing can have infinitely many items in its causal history.

8. Explanatory finitism: Nothing can have infinitely many items in its explanatory history.

I think (1) and (2) are false, because eternalism is true and it is possible to have an infinite future with a new chicken coming into existence every day.

I’ve defended (7) at length. I would love to be able to defend (8), but for reasons discussed in that book, I fear it can’t

I don’t know any reason to believe (3) other than as an implication of (1) together with realism about species. I don’t know any reason to believe (4) other than as an implication of (2) or (5).

I can imagine a combination of metaphysical views on which (6) is defensible. For instance, it might turn out that humans are made out of stuff all of whose qualities are discribable with discrete mathematics, and that there are limits on the discrete quantities (e.g., a minimum and a maximum mass of a human being) in such a way that for any finite segment of human life, there are only finitely many possibilities. If one adds to that the Principle of the Identity of Indiscernibles, in a transworld form, one will have an argument that there can only be finitely many humans. And I suppose some version of this view that applies to species more generally would give (5). That said, I doubt (6) is true.

Consciousness finitism

My 11-year-old has an interesting intuition, that it is impossible to have an infinite number of conscious beings. She is untroubled by Hilbert’s Hotel, and insists the intuition is specific to conscious beigs, but is unable to put her finger on what exactly bothers her about an infinity of conscious beings. It’s not considerations like “If there are infinitely many people, you probably have a near-duplicate.” Near-duplicates don’t bother her. It’s consciousness specifically. She is surprised that a consciousness-specific finitist intuition isn’t more common.

My best attempt at a defense of consciousness-finitism was that it seems reasonable to think of yourself as a uniformly randomly chosen member of the set of all conscious beings. But thinking of yourself as a uniformly randomly chosen member of a countably infinite set leads to the well-known paradoxes of countably infinite fair lotteries. So that may provide some sort of argument for consciousness-finitism. But my daughter insists that’s not where her intuition comes from.

Another argument for consciousness-finitism would be the challenges of aggregating utilities across an infinite number of people: If all the people are positioned at locations numbered 1,2,3,…, and you benefit the people at even-numbered locations, you benefit the same quantity of people as when you benefit the people whose locations are divisible by four, but clearly benefiting the people at the even-numbered locations is a lot better. I haven’t tried this family of arguments on my daughter, but I don’t think her intuitions come from thinking about well-being.

Still, I have a hard time believing in the impossibility of an infinite number of consciousnesses on the strength of such arguments. The main reason I have such a hard time is that it seems obvious that you could have a forward infinite regress of conscious beings, each giving birth to the next.

Divine temporalism once again

I’m thinking about my recent argument against divine temporalism, the idea that God has no timeless existence but is instead in time, and time extends infinitely pastwards.

Here’s perhaps a simple way to make my argument go (I am grateful to Dean Zimmerman for suggestions that helped in this reformulation). If infinite time is a central feature of reality, as the temporalist says, then one of the most fundamental things for God to decide about the structure of creation is which of these three is to be true:

1. Nothing gets created.

2. There is creation going infinitely far back in time.

3. There is creation but it doesn’t go infinitely far back in time.

But without backwards causation, a temporal God cannot decide between (2) and (3). For at any given time, it’s already settled whether (2) or (3) is the case.

Now, it seems that the temporalist’s best answer is to deny the possibility of (2). We don’t expect God to choose whether to create square circles, and so if we deny the possibility of (2), God only needs to choose between (1) and (3).

But there are two issues with that. First, creation going infinitely far back in time is the temporalist’s best answer to the Augustinian question of why God waited as long as he did before creating—on this answer (admittedly contrary to Christian doctrine), God didn’t wait.

Second, and perhaps more seriously, there is the question of justifying the claim that (2) is impossible. There are four reasons in the literature for thinking that in fact creation has a finite past:

1. Big Bang cosmology

2. Arguments against actual infinity

3. Arguments against traversing an actually infinite time

4. Causal finitism.

None of these allow the temporalist to justify the impossibility of creation going infinitely far back in time. Big Bang cosmology is contingent, and does not establish impossibility. And if the arguments (ii) and (iii)
are good reasons for rejecting an infinite past of creation, they are also good reasons for rejecting divine temporalism, since divine temporalism would require God to have lived through an actually infinite time. And (iv) also seems to rule out divine temporalism. For suppose that in fact creation follows an infinite number of days without creation. During that infinite number of days without creation, on any day we could ask why nothing exists. And the answer is that God didn’t decide to create anything. So the emptiness of the empty day causally depends on God’s infinitely many decisions in days past not to start creating yet, contrary to causal finitism.

Infinite Dedekind finite sets

Most paradoxes of actual infinities, such as Hilbert’s Hotel, depend on the intuition that:

1. A collection is bigger than any proper subcollection.

A Dedekind infinite set is one that has the property that it is the same cardinality as some proper subset. In other words, a Dedekind infinite set is precisely one that violates (1).

In Zermelo-Fraenkel (ZF) set theory, it is easy to prove that any Dedekind infinite set is infinite. More interestingly, assuming the consistency of ZF, there are models of ZF with infinite sets that are Dedekind finite.

It is easy to check that if A is a Dedekind finite set, then A and every subset of A satisfies (1). Thus an infinite but Dedekind finite set escapes most if not all the standard paradoxes of infinity. Perhaps enemies of actual infinity, should thus only object to Dedekind infinities, not all infinities?

However, infinite Dedekind finite sets are paradoxical in their own special way: they have no countably infinite subsets—no subsets that can be put into one-to-one correspondence with the natural numbers. You might think this is absurd: shouldn’t you be able to take one element of an infinite Dedekind finite set, then another, then another, and since you’ll never run out of elements (if you did, the set wouldn’t be finite), you’d form a countably infinite sequence of elements? But, no: the problem is that repeating the “taking” requires the Axiom of Choice, and infinite Dedekind finite sets only live in set-theoretic universes without the Axiom of Choice.

In fact, I think infinite Dedekind finite sets are much more paradoxical than a run-of-the-mill Dedekind infinite sets.

Do we learn anything philosophical here? I am not sure, but perhaps. If infinite Dedekind finite sets are extremely paradoxical, then by the same token (1) seems an unreasonable condition in the infinite case. For Dedekind finitude is precisely defined by (1).

From causal finitism to divine simplicity

If God is not simple, he has infinitely many really distinct features. Moreover infinitely many of these features will be involved in creation, e.g., because there are infinitely many reasons that favor the creation of this world, and for each reason God will plausibly have a distinct feature of being impressed by that reason. But causal finitism (the doctrine that infinitely many things can't come together causally) rules this out. So divine simplicity is true.

Assuming causal finitism, the thing that one might challenge is the claim that infinitely many of God's features are causally efficacious.

There is an even easier argument for divine simplicity based if actual infinites are impossible. For, surely, either (a) God is simple or (b) God has infinitely many really distinct features. If actual infinites are impossible, that rules out (b).