Truth-value realisms about arithmetic

Arithmetical truth-value realists hold that any proposition in the language of arithmetic has a fully determined truth value. Arithmetical truth-value necessists add that this truth value is necessary rather than merely contingent. Although we know from the incompleteness theorems that there are alternate non-standard natural number structures, with different truth values (e.g., there is a non-standard natural number structure according to which the Peano Axioms are inconsistent), the realist and necessist hold that when we engage in arithmetical language, we aren’t talking about these structures. (I am assuming either first-order arithmetic or second-order with Henkin semantics.)

Start by assuming arithmetical truth-value necessitism.

There is an interesting decision point for truth-value necessitism about arithmetic: Are these necessary truths twin-earthable? I.e., could there be a world whose denizens who talk arithmetically like we do, and function physically like we do, but whose arithmetical sentences express different propositions, with different and necessary truth values? This would be akin to a world where instead of water there is XYZ, a world whose denizens would be saying something false if they said “Water has hydrogen in it”.

Here is a theory on which we have twin-earthability. Suppose that the correct semantics of natural number talk works as follows. Our universe has an infinite future sequence of days, and the truth-values of arithmetical language are fixed by requiring the Peano Axioms (or just the Robinson Axioms) together with the thesis that the natural number ordering is order-isomorphic to our universe’s infinite future sequence of days, and then are rigidified by rigid reference to the actual world’s sequence of future days. But in another world—and perhaps even in another universe in our multiverse if we live in a multiverse—the infinite future sequence of days is different (presumably longer!), and hence the denizens of that world end up rigidifying a different future sequence of days to define the truth values of their arithmetical language. Their propositions expressed by arithmetical sentences sometimes have different truth values from ours, but that’s because they are different propositions—and they’re still as necessary as ours. (This kind of a theory will violate causal finitism.)

One may think of a twin-earthable necessitism about arithmetic as a kind of cheaper version of necessitism.

Should a necessitist go cheap and allow for such twin-earthing?

Here is a reason not to. On such a twin-earthable necessitism, there are possible universes for whose denizens the sentence “The Peano Axioms are consistent” expresses a necessary falsehood and there are possible universes for whose denizens the sentence expresses a necessary truth. Now, in fact, pretty much everybody with great confidence thinks that the sentence “The Peano Axioms are consistent” expresses a truth. But it is difficult to hold on to this confidence on twin-earthable necessitism. Why should we think that the universes the non-standard future sequences of days are less likely?

Here is the only way I can think of answering this question. The standard naturals embed into the non-standard naturals. There is a sense in which they are the simplest possible natural number structure. Simplicity is a guide to truth, and so the universes with simpler future sequences of days are more likely.

But this answer does not lead to a stable view. For if we grant that what I just said makes sense—that the simplest future sequences of days are the ones that correspond to the standard naturals—then we have a non-twin-earthable way of fixing the meaning of arithmetical language: assuming S5, we fix it by the shortest possible future sequence of days that can be made to satisfy the requisite axioms by adding appropriate addition and multiplication operations. And this seems a superior way to fix the meaning of arithmetical language, because it better fits with common intuitions about the “absoluteness” of arithmetical language. Thus it it provides a better theory than twin-earthable necessitism did.

I think the skepticism-based argument against twin-earthable necessitism about arithmetic also applies to non-necessitist truth-value realism about arithmetic. On non-necessitist truth-value realism, why should we think we are so lucky as to live in a world where the Peano Axioms are consistent?

Putting the above together, I think we get an argument like this:

1. Twin-earthable truth-value necessitism about arithmetic leads to skepticism about the consistency of arithmetic or is unstable.

2. Non-necessitist truth-value realism about arithmetic leads to skepticism about the consistency of arithmetic.

3. Thus, probably, if truth-value realism about arithmetic is true, non-twin-earthable truth-value necessitism about arithmetic is true.

The resulting realist view holds arithmetical truth to be fixed along both dimensions of Chalmers’ two-dimensional semantics.

(In the argument I assumed that there is no tenable way to be a truth-value realist only about Σ₁⁰ claims like “Peano Arithmetic is consistent” while resisting realism about higher levels of the hierarchy. If I am wrong about that, then in the above argument and conclusions “truth-value” should be replaced by “Σ₁⁰-truth-value”.)

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 issues concerning eliminative structuralism for second-order arithmetic

Eliminative structuralist philosophers of mathematics insist that what mathematicians study is structures rather than specific realizations of these structures, like a privileged natural number system would be. One example of such an approach would be to take the axioms of second-order Peano Arithmetic PA2, and say an arithmetical sentence ϕ is true if and only if it is true in every standard model of PA2. Since all such models are well-known to be isomorphic, it follows that for every arithmetical sentence ϕ, either ϕ or  ∼ ϕ is true, which is delightful.

The hitch here is the insistence on standard (rather than Henkin) models, since the concept of a standard depends on something very much like a background set theory—a standard model is a second-order model where every subset of Dⁿ is available as a possible value for the second-order n-ary variables, where D is the first-order domain. Thus, such an eliminative structuralism in order to guarantee that every arithmetical sentence has a truth value seems to have to suppose a privileged selection of subsets, and that’s just not structural.

One way out of this hitch is to make use of a lovely internal categoricity result which implies that if we have any second-order model, standard or not, that contains two structures satisfying PA2, then we can prove that any arithmetical sentence true in one of the two structures is true in the other.

But that still doesn’t get us entirely off the hook. One issue is modal. The point of eliminative structuralism is to escape from dependence on “mathematical objects”. The systems realizing the mathematical structures on eliminative structuralism don’t need to be systems of abstract objects: they can just as well be systems of concrete things like pebbles or points in space or times. But then what systems there are is a contingent matter, while arithmetic is (very plausibly) necessary. If we knew that all possible systems satisfying PA2 would yield the same truth values for arithmetical sentences, life would be great for the PA2-based eliminative structuralist. But the internal categoricity results don’t establish that, unless we have some way of uniting PA2-satisfying systems in different possible worlds in a single model. But such uniting would require there to be relations between objects in different worlds, and that seems quite problematic.

Another issue is the well-known issue that assuming full second-order logic is “too close” to just assuming a background set-theory (and one that spans worlds, if we are to take into account the modal issue). If we could make-do with just monadic second-order logic (i.e., the second-order quantifiers range only over unary entities) in our theory, things would be more satisfying, because monadic second-order logic has the same expressiveness as plural quantification, and we might even be able to make-do with just first-order quantification over fusions of simples. But then we don’t get the internal categoricity result (I am pretty sure it is provable that we don’t get it), and we are stuck with assuming a privileged selection of subsets.

Causal Robinson Arithmetic

Say that a structure N that has a distinguished element 0, a unary function S, and binary operations + and ⋅ is a causal Robinson Arithmetic (CRA) structure iff:

1. The structure N satisfies the axioms of Robinson Arithmetic, and

2. For any x in N, x is a partial cause of the object Sx.

The Fundamental Metaphysical Axiom of CRA is:

• For every sentence ϕ in the language of arithmetic, ϕ is either true in every metaphysically possible CRA structure or false in every metaphysically possible CRA structure.

Causal Finitism—the doctrine that nothing can have infinitely many things causally prior to it—implies that any CRA is order isomorphic to the standard natural numbers (for any element in the CRA structure other than zero, the sequence of predecessors will be causally prior to it, and so by Causal Finitism must be finite, and hence the number can be mapped to a standard natural number), and hence implies the Fundamental Metaphysical Axiom of CRA.

Given the Fundamental Metaphysical Axiom of CRA, we have a causal-structuralist foundation for arithmetic, and hence for meta-mathematics: We say that a sentence ϕ of arithmetic is true if and only if it is true in all metaphysically possible CRA structures.

Some stuff about models of PA+~Con(PA)

Assume Peano Arithmetic (PA) is consistent. Then it can’t prove its own consistency. Thus, there is a model M of PA according to which PA is inconsistent, and hence, according M, there is a proof of a contradiction from a finite set of axioms of PA. This sounds very weird.

But it becomes less weird when we realize what these claims do and do not mean in M.

The model M will indeed contain an M-natural number a that according to M encodes a finite sequence of axioms of PA, and it will also contain an M-natural number p that according to M encodes a proof of a contradiction using the axioms encoded in A.

However, here are some crucial qualifications. Distinguish between the M-natural numbers that are standard, i.e., correspond to an actually natural number, one that from the point of view of the “actual” natural numbers is finite, and those that are not. The latter are infinite from the point of view of the actual natural numbers.

First, the M-natural number a is non-standard. For a standard natural number will only encode a finite number of axioms, and for any finite subtheory of PA, PA can prove its consistency (this is the “reflexivity of PA”, proved by Mostowski in the middle of the last century). Thus, if a were a standard natural number, according to M there would be no contradiction from the axioms in a.

Second, while every item encoded in a is according to M an axiom of PA, this is not actually true. This is because any M-finite sequence of M-natural numbers will either be a standardly finite length sequence of standard natural numbers, or will contain a non-standard number. For let n be the largest element in the sequence. If this is standard, then we have a standardly finite length sequence of standard natural numbers. If not, then the sequence contains a non-standard number. Thus, a contains something that is not axiom of PA.

In other words, according to our model M, there is a contradictory collection of axioms of PA, but when we query M as to what that collection is, we find out that some of the things that M included in the collection are not actually axioms of PA. (In fact, they won’t even be well-formed formulas, since they will be infinitely long.) So a crucial part of the reason why M disagrees with the “true” model of the naturals about the consistency of PA is because M disagrees with it about what PA actually says!

A new argument for causal finitism

I will give an argument for causal finitism from a premise I don’t accept:

1. Necessary Arithmetical Alethic Incompleteness (NAAI): Necessarily, there is an arithmetical sentence that is neither true nor false.

While I don’t accept NAAI, some thinkers (e.g., likely all intuitionists) accept it.

Here’s the argument:

2. If infinite causal histories are possible, supertasks are possible.

3. If supertasks are possible, for every arithmetical sentence, there is a possible world where someone knows whether the sentence is true or false by means of a supertask.

4. If for every arithmetical sentence there is a possible world where someone knows whether the sentence is true or false by means of a supertask, there is a possible world where for every arithmetical sentence someone knows whether it is true or false.

5. Necessarily, if someone knows whether p is true or false, then p is true or false.

6. So, if infinite causal histories are possibly, possibly all arithmetical sentences are true or false. (2-5)

7. So, infinite causal histories are impossible. (1, 6)

The thought behind (3) is that if for every n it is possible to check the truth value of ϕ(n) by a finite task or supertask, then by an iterated supertask it is possible to check the truth values of ∀xϕ(x) (and equivalently ∃xϕ(x)). Since every arithmetical sequence can be written in the form Q₁x₁...Q_kx_kϕ(x₁,...,x_k), where the truth value of ϕ(n₁,...,n_k) is finitely checkable, it follows that every arithmetical sequence can have its truth value checked by a supertask.

The thought behind (4) is that one can imagine an infinite world (say, a multiverse) where for every arithmetical sentence ϕ the relevant supertask is run and hence the truth value of the sentence is known.

Realism about arithmetical truth

It seems very plausible that for any specific Turing machine M there is a fact of the matter about whether M would halt. We can just imagine running the experiment in an idealized world with an infinite future, and surely either it will halt or it won’t halt. No supertasks are needed.

This commits one to realism about Σ₁ arithmetical propositions: for every proposition expressible in the form ∃nϕ(n) where ϕ(n) has only bounded quantifiers, there is a fact of the matter whether the proposition is true. For there is a Turing machine that halts if and only if ∃nϕ(n).

But now consider a Π₂ proposition, one expressible in the form ∀m∃nϕ(m,n), where again ϕ(m,n) has only bounded quantifiers. For each fixed m, there is a Turing machine M_m whose halting is equivalent to ∃nϕ(m,n). Imagine now a scenario where on day m of an infinite future you build and start M_m. Then there surely will be a fact of the matter whether all of these Turing machines will halt, a fact equivalent to ∀m∃nϕ(m,n).

What about a Σ₃ proposition, one expressible in the form ∃r∀m∃nϕ(r,m,n)? Well, we could imagine for each fixed r running the above experiment starting on day r in the future to determine whether the Π₂ proposition ∀m∃nϕ(r,m,n) is true, and then there surely is a fact of the matter whether at least one of these experiments gives a positive answer.

And so on. Thus there is a fact of the matter whether any statement in the arithmetical hierarchy—and hence any statement in the language of arithmetic—is true or false.

This argument presupposes a realism about deterministic idealized machine counterfactuals: if I were to build such and such a sequence of deterministic idealized machines, they would behave in such and such a way.

The argument also presupposes that we have a concept of the finite and of countable infinity: it is essential that our Turing machines be run for a countable sequence of steps in the future and that the tape begin with a finite number of symbols on it. If we have causal finitism, we can get the concept of the finite out of the metaphysics of the world, and a discrete future-directed causal sequence of steps is guaranteed to be countable.

Greek mathematics

I think it is sometimes said that it is anachronistic to attribute to the ancient Greeks the discovery that the square root of two is irrational, because what they discovered was a properly geometrical fact, that the side and diagonal of a square are incommensurable, rather than a fact about real numbers.

It is correct to say that the Greeks discovered an incommensurability fact. But it is, I think, worth noting that this incommensurability fact is not really geometric fact: it is a geometric-cum-arithmetical fact. Here is why. The claim that two line segments are commensurable says that there are positive integers m and n such that m copies of the first segment have the same length as n copies of the second. This claim is essentially arithmetical in that it quantifies over positive integers.

And because pure (Tarskian) geometry is decidable, while the theory of the positive integers is not decidable, the positive integers are not definable in terms of pure geometry, so we cannot eliminate the quantification over positive integers. In fact, it is known that the rational numbers are not definable in terms of pure geometry either, so neither the incommensurability formulation nor theory irrationality formulation is a purely geometric claim.

I think. All this decidability and definability stuff confuses me often.

Σ10 alethic Platonism

Here is an interesting metaphysical thesis about mathematics: Σ₁⁰ alethic Platonism. According to Σ₁⁰ alethic Platonism, every sentence about arithmetic with only one unbounded existential quantifier (i.e., an existential quantifier that ranges over all natural numbers, rather than all the natural numbers up to some bound), i.e., every Σ₁⁰ sentence, has an objective truth value. (And we automatically get Π₁⁰ alethic Platonism, as Π₁⁰ sentences are equivalent to negations of Σ₁⁰ sentences.)

Note that Σ₁⁰ alethic Platonism is sufficient to underwrite a weak logicism that says that mathematics is about what statements (narrowly) logically follow from what recursive axiomatizations. For Σ₁⁰ alethic Platonism is equivalent to the thesis that there is always a fact of the matter about what logically follows from what recursive axiomatization.

Of course, every alethic Platonist is a Σ₁⁰ alethic Platonist. But I think there is something particularly compelling about Σ₁⁰ alethic Platonism. Any Σ₁⁰ sentence, after all, can be rephrased into a sentence saying that a certain abstract Turing machine will halt. And it does seems like it should be possible to embody an abstract Turing machine as a physical Turing machine in some metaphysically possible world with an infinite future and infinite physical resources, and then there should be a fact of the matter whether that machine would in fact halt.

There is a hitch in this line of thought. We need to worry about worlds with “non-standard” embodiments of the Turing machine, embodiments where the “physical Turing machine” is performing an infinite task (a supertask, in fact an infinitely iterated supertask). To rule those worlds out in a non-arbitrary way requires an account of the finite and the infinite, and that account is apt to presuppose Platonism about the natural numbers (since the standard mathematical definition of the finite is that a finite set is one whose cardinality is a natural number). We causal finitists, however, do not need to worry, as we think that it is impossible for Turing machines to perform infinite tasks. This means that causal finitists—as well as anyone else who has a good account of the difference between the finite and the infinite—have good reason to accept Σ₁⁰ alethic Platonism.

I haven't done any surveys, but I suspect that most mathematicians would be correctly identified as at least being Σ₁⁰ alethic Platonists.