Is there infinity in our minds?

Start with this intuition:

1. Every sentence of first order logic with the successor predicate s(x,y) (which says that x is the natural number succeeding y) is determinately true or determinately false.

We learn from Goedel that:

2. No finitely specifiable (in the recursive sense) set of axioms is sufficient to characterize the natural numbers in a way sufficient to determine all of the above sentences.

This creates a serious problem. Given (2), how are our minds able to have a concept of natural number that is sufficiently determinate to make (1) true. It can’t be by us having some kind of a “definition” of natural numbers in terms of a finitely characterizable set of axioms.

Here is one interesting solution:

3. Our minds actually contain infinitely many axioms of natural numbers.

This solution is very difficult to reconcile with naturalism. If nature is analog, there will be a way of encoding infinitely many axioms in terms of the fine detail of our brain states (e.g., further and further decimal places of the distance between two neurons), but it is very implausible that anything mental depends on arbitrarily fine detail.

What could a non-naturalist say? Here is an Aristotelian option. There are infinitely many “axiomatic propositions” about the natural numbers such that it is partly constitutive of the human mind’s flourishing to affirm them.

While this option technically works, it is still weird: there will be norms concerning statements that are arbitrarily long, far beyond human lifetime.

I know of three other options:

4. Platonism with the natural numbers being somehow special in a way that other sets of objects satisfying the Peano axioms are not.

5. Magical theories of reference.

6. The causal finitist characterization of natural numbers in my Infinity book.

Of course, one might also deny (1). But then I will retreat from (1) to:

7. Every sentence of first order logic with the successor predicate s(x,y) and at most one unbounded quantifier is determinately true or determinately false.

I think (7) is hard to deny. If (7) is not true, there will be cases where there is no fact of the matter where a sentence of logic follows from some bunch of axioms. (Cf. this post.) And Goedelian considerations are sufficient to show that one cannot recursively characterize the sentences with one unbounded quantifier.

Materialism and incompleteness

It is sometimes thought that Goedel’s incompleteness theorems yield an argument against materialism, on something like the grounds that we can see that the Goedel sentence for any recursively axiomatizable system of arithmetic is true, and hence our minds cannot operate algorithmically.

In this post, I want to note that materialism is quite compatible with being able to correctly decide the truth value of all sentences of arithmetic. For imagine that we live in an infinite universe which contains infinitely many brass plaques with a sentence of arithmetic followed by the word “true” or “false”, such that every sentence of arithmetic is found on exactly one brass plaque. There is nothing contrary to materialsim in this assumption. Now add the further assumption that the word “true” is found on all and only the plaques containing a true sentence of arithmetic. Again, there is nothing contradicting materialism here. It could happen that way simply by chance movements of atoms! Next, imagine a machine where you type in a sentence of arithmetic, and the machine starts traveling outward in the universe in a spiral pattern until it arrives at a plaque with that sentence, reads whether the sentence is true or false, and comes back to you with the result. This could all be implemented in a materialist system, and yet you could then correctly decide the truth value of every sentence of arithmetic.

Note that we should not think of this as an algorithmic process. So the way that this example challenges the argument at the beginning of this post is by showing that materialism does not imply algorithmism.

Objection 1: The plaques are a part of the mechanism for deciding arithmetic, and so the argument only shows that an infinite materialistic machine could decide arithmetic. But our brains are finite.

Response: While our brains are finite, they are analog devices. An analog system contains an infinite amount of information. For instance, suppose that my brain particles have completely precise positions (e.g., on a Bohmian quantum mechanics). Then the diameter of my brain expressed in units of Planck length at some specific time t is some decimal number with infinitely many significant figures. It could turn out that this infinitely long decimal number encodes the truth values of all the sentences of arithmetic, and a machine that measures the diameter of my brain to arbitrary precision could then determine the truth value of every arithmetical statement. Of course, this might turn out not to be compatible with the details of our laws of nature—it may be that arbitrary precision is unachievable—but it is not incompatible with materialism as such.

Objection 2: In these kinds of scenarios, we wouldn’t know that the plaques are right.

Response: After verifying a large number of plaques to be correct, and finding none that we could tell are incorrect, it would be reasonable to conclude by induction that they are all right. However, if the plaques are in fact due to random processes, this inductive conclusion wouldn’t constitute knowledge, except on some versions of reliabilism (which seem implausible to me). But it could be a law of nature that the plaques are right—that’s compatible with materialism. In any case, here the discussion gets complicated.

Humean accounts of modality

Humean accounts of modality, like Sider’s, work as follows. We first take some privileged truths, including all the mathematical ones, and an appropriate collection of others (e.g., ones about natural kind membership or the fundamental truths of metaphysics). And then we stipulate that to be necessary is to follow from the collection of privileged truths, and the possible that whose negation isn’t necessary.

Here is a problem. We need to be able to say things like this:

1. Necessarily it’s possible that 2+2=4.

For that to be the case, then:

2. It’s possible that 2+2=4

has to follow from the privileged truths. But on the theory under consideration, (2) means:

3. That 2 + 2 ≠ 4 does not follow from the privileged truths.

So, (3) has to follow from the privileged truths. Now, how could it do that? Suppose first that the privileged truths include only the mathematical ones. Then (3) has to be a mathematical truth: for only mathematical truths follow logically from mathematical truths. But this means that “the privileged truths”, i.e., “the mathematical truths”, has to have a mathematical description. For instance, there has to be a set or proper class of mathematical truths. But that “the mathematical truths” has a mathematical description is a direct violation of Tarski’s Indefinability of Truth theorem, which is a variant of Goedel’s First Incompleteness Theorem.

So we need more truths than the mathematical ones to be among the privileged ones, enough that (3) should follow from them. But it unlikely that any of the privileged truths proposed by the proponents of Humean accounts of modality will do the job with respect to (3). Even the weaker claim:

4. That 2 + 2 ≠ 4 does not follow from the mathematical truths

seems hard to get from the normally proposed privileged truths. (It’s not mathematical, it’s not natural kind membership, it’s not a fundamental truth of metaphysics, etc.)

Consider this. The notion of “follows from” in this context is a formal mathematical notion. (Otherwise, it’s an undefined modal term, rendering the account viciously circular.) So facts about what does or does not follow from some truths seem to be precisely mathematical truths. One natural way to make sense of (4) is to say that there is a privileged truth that says that some set T is the set of mathematical truths, and then suppose there is a mathematical truth that 2 + 2 ≠ 4 does not follow from T. But a set of mathematical truths violates Indefinability of Truth.

Perhaps, though, we can just add to the privileged truths some truths about what does and does not follow from the privileged truths. In particular, the privileged truths will contain, or it will easily follow from them, the truth that they are mutually consistent. But now the privileged truths become self-referential in a way that leads to contradiction. For instance:

5. No x such that F(x) follows from the privileged truths.

will make sense for any F, and we can choose a predicate F such that it is provable that (5) is the only thing that satisfies F (cf. Goedel’s diagonal lemma). Now, if (5) follows from the privileged truths, then it also follows from the privileged truths that (5) doesn’t follow from the privileged truths, and hence that the privileged truths are inconsistent. Thus, from the fact that the privileged truths are consistent, which itself is a privileged truth or a consequence thereof, one can prove (5) doesn’t follow from the privileged truths, and hence that (5) is true, which is absurd.

Anti-S5

Suppose narrowly logical necessity L_L is provability from some recursive consistent set of axioms and narrowly logical possibility M_L is consistency with that set of axioms. Then Goedel’s Second Incompleteness Theorem implies the following weird anti-S5 axiom:

• ∼L_LM_Lp for every statement p.

In particular, the S5 axiom M_Lp → L_LM_Lp holds only in the trivial case where M_Lp is false.

For suppose we have L_LM_Lp. Then M_Lp has a proof. But M_Lp is equivalent to ∼L_Lp. However, we can show that ∼L_Lp implies the consistency of the axioms: for if the axioms are not consistent, then by explosion they prove p and hence L_Lp holds. Thus, if L_L∼L_Lp, then ∼L_Lp can be proved, and hence consistency can be proved, contrary to Second Incompleteness.

The anti-S5 axiom is equivalent to the axiom:

• M_LL_Lp.

In particular, every absurdity—even 0≠0—could be necessary.

I wonder if there is any other modality satisfying anti-S5.

Logicism and Goedel

Famously, Goedel’s incompleteness theorems refuted (naive) logicism, the view that mathematical truth is just provability.

But one doesn’t need all of the technical machinery of the incompleteness theorems to refute that. All one needs is Goedel’s simple but powerful insight that proofs are themselves mathematical objects—sequence of symbols (an insight emphasized by Goedel numbering). For once we see that, then the logicist view is that what makes a mathematical proposition true is that a certain kind of mathematical object—a proof—exists. But the latter claim is itself a mathematical claim, and so we are off on a vicious regress.

Provability from finite and infinite theories

Let #s be the Goedel number of s. The following fact is useful for thinking about the foundations of mathematics:

Proposition. There is a finite fragment A of Peano Arithmetic such that if T is a recursively axiomatizable theory, then there is an arithmetical formula P_T(n) such that for all arithmetical sentences s, A → P_T(#s) is a theorem of FOL if and only if T proves s.

The Proposition allows us to replace the provability of a sentence from an infinite recursive theory by the provability of a sentence from a finite theory.

Sketch of Proof of Proposition. Let M be a Turing machine that given a sentence as an input goes through all possible proofs from T and halts if it arrives at one that is a proof of the given sentence.

We can encode a history of a halting (and hence finite) run of M as a natural number such that there will be a predicate H_M(m, n) and a finite fragment A of Peano Arithmetic independent of M (I expect that Robinson arithmetic will suffice) such that (a) m is a history of a halting run of M with input m if and only if H_M(m, n) and (b) for all m and n, A proves whether H_M(m, n).

Now, let P_T(n) be ∃mH_M(m, n). Then A proves P_T(#s) if and only if there is an m₀ such that A proves H_M(m₀, n). (If A proves P_T(#s), then because A is true, there is an m such that H_M(m, #s), and then A will prove H_M(m₀, #s). Conversely, if A proves H_M(m₀, #s), then it proves ∃mH_M(m, #s).) And so A proves P_T(#s) if and only if T proves s.

A bad idea in the foundations of mathematics

The relativity of FOL-validity is the fact that whether a sentence ϕ of First Order Logic is valid (equivalently, provable from no axioms beyond any axioms of FOL itself) sometimes depends on the axioms of set theory, once we encode validity arithmetically as per Goedel.

More concretely, if Zermelo-Fraenkel-Choice (ZFC) set theory is consistent, then there is an FOL formula ϕ that is FOL-provable according to some but not other models of ZFC. So which model of ZFC should real provability be relativized to?

Here is a putative solution that occurred to me today:

• Say that ϕ is really provable if and only if there is a model M of ZFC such that according to M, ϕ has a proof.

If this solution works, then the relativity of proof is quite innocent: it doesn’t matter in which model of ZFC our proofs live, because proofs in any ZFC model do the job for us.

It follows from incompleteness (cf. the link above) that real provability is strictly weaker than provability, assuming ZFC is true and consistent. Therefore, some really provable ϕ will fail to be valid, and hence there will be models of the falsity of ϕ. The idea that one can really prove a ϕ such that there is a model of the falsity of ϕ seems to me to show that my proposed notion of “really provable” is really confused.

Post-Goedelian mathematics as an empirical inquiry

Once one absorbs the lessons of the Goedel incompleteness theorems, a formalist view of mathematics as just about logical relationships such as provability becomes unsupportable (for me the strongest indication of this is the independence of logical validity). Platonism thereby becomes more plausible (but even Platonism is not unproblematic, because mathematical Platonism tends towards plenitude, and given plenitude it is difficult to identify which natural numbers we mean).

But there is another way to see post-Goedelian mathematics, as an empirical and even experimental inquiry into the question of what can be proved by beings like us. While the abstract notion of provability is subject to Goedelian concerns, the notion of provability by beings like us does not seem to be, because it is not mathematically formalizable.

We can mathematically formalize a necessary condition for something to be proved by us which we can call “stepwise validity”: each non-axiomatic step follows from the preceding steps by such-and-such formal rules. To say that something can be proved by beings like us, then, would be to say that beings like us can produce (in speech or writing or some other relevantly similar medium) a stepwise valid sequence of steps that starts with the axioms and ends with the conclusion. This is a question about our causal powers of linguistic production, and hence can be seen as empirical.

Perhaps the surest way to settle the question of provability by beings like us is for us to actually produce the stepwise valid sequence of steps, and check its stepwise validity. But in practice mathematicians usually don’t: they skip obvious steps in the sequence. In doing so, they are producing a meta-argument that makes it plausible that beings like us could produce the stepwise valid sequence if they really wanted to.

This might seem to lead to a non-realist view of mathematics. Whether it does so depends, however, on our epistemology. If in fact provability by beings like us tracks metaphysical necessity—i.e., if B is provable by beings like us from A₁, ..., A_n, then it is not possible to have A₁, ..., A_n without B—then by means of provability by beings like us we discover metaphysical necessities.

Socratic perfection is impossible

Socrates thought it was important that if you didn't know something, you knew you didn't know it. And he thought that it was important to know what followed from what. Say that an agent is Socratically perfect provided that (a) for every proposition p that she doesn't know, she knows that she doesn't know p, and (b) her knowledge is closed under entailment.

Suppose Sally is Socratically perfect and consider:

1. Sally doesn’t know the proposition expressed by (1).

If Sally knows the proposition expressed by (1), then (1) is true, and so Sally doesn’t know the proposition expressed by (1). Contradiction!

If Sally doesn’t know the proposition expressed by (1), then she knows that she doesn’t know it. But that she doesn’t know the proposition expressed by (1) just is the proposition expressed by (1). So Sally doesn’t know the proposition expressed by (1). So Sally knows the proposition expressed by (1). Contradiction!

So it seems it is impossible to have a Socratically perfect agent.

(Technical note: A careful reader will notice that I never used closure of Sally’s knowledge. That’s because (1) involves dubious self-reference, and to handle that rigorously, one needs to use Goedel’s diagonal lemma, and once one does that, the modified argument will use closure.)

But what about God? After all, God is Socratically perfect, since he knows all truths. Well, in the case of God, knowledge is equivalent to truth, so (1)-type sentences just are liar sentences, and so the problem above just is the liar paradox. Alternately, maybe the above argument works for discursive knowledge, while God’s knowledge is non-discursive.

Epistemic scores and consistency

Scoring rules measure the distance between a credence and the truth value, where true=1 and false=0. You want this distance to be as low as possible.

Here’s a fun paradox. Consider this sentence:

1. At t₁, my credence for (1) is less than 0.1.

(If you want more rigor, use Goedel’s diagonalization lemma to remove the self-reference.) It’s now a moment before t₁, and I am trying to figure out what credence I should assign to (1) at t₁. If I assign a credence less than 0.1, then (1) will be true, and the epistemic distance between 0.1 and 1 will be large on any reasonable scoring rule. So, I should assign a credence greater than or equal to 0.1. In that case, (1) will be false, and I want to minimize the epistemic distance between the credence and 0. I do that by letting the credence be exactly 0.1.

So, I should set my credence to be exactly 0.1 to optimize epistemic score. Suppose, however, that at t₁ I will remember with near-certainty that I was setting my credence to 0.1. Thus, at t₁ I will be in a position to know with near-certainty that my credence for (1) is not less than 0.1, and hence I will have evidence showing with near-certainty that (1) is false. And yet my credence for (1) will be 0.1. Thus, my credential state at t₁ will be probabilistically inconsistent.

Hence, there are times when optimizing epistemic score leads to inconsistency.

There are, of course, theorems on the books that optimizing epistemic score requires consistency. But the theorems do not apply to cases where the truth of the matter depends on your credence, as in (1).

"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.

Logical closure accounts of necessity

A family of views of necessity (e.g., Peacocke, Sider, Swinburne, and maybe Chalmers) identifies a family F of special true statements that get counted as necessary—say, statements giving the facts about the constitution of natural kinds, the axioms of mathematics, etc.—and then says that a statement is necessary if and only if it can be proved from F. Call these “logical closure accounts of necessity”. There are two importantly different variants: on one “F” is a definite description of the family and on the other “F” is a name for the family.

Here is a problem. Consider:

1. Statement (1) cannot be proved from F.

If you are worried about the explicit self-reference in (1), I should be able to get rid of it by a technique similar to the diagonal lemma in Goedel’s incompleteness theorem. Now, either (1) is true or it’s false. If it’s false, then it can be proved from F. Since F is a family of truths, it follows that a falsehood can be proved from truths, and that would be the end of the world. So it’s true. Thus it cannot be proved from F. But if it cannot be proved from F, then it is contingently true.

Thus (1) is true but there is a possible world w where (1) is false. In that world, (1) can be proved from F, and hence in that world (1) is necessary. Hence, (1) is false but possibly necessary, in violation of the Brouwer Axiom of modal logic (and hence of S5). Thus:

2. Logical closure accounts of necessity require the denial of the Brouwer Axiom and S5.

But things get even worse for logical closure accounts. For an account of necessity had better itself not be a contingent truth. Thus, a logical closure account of necessity if true in the actual world will also be true in w. Now in w run the earlier argument showing that (1) is true. Thus, (1) is true in w. But (1) was false in w. Contradiction! So:

3. Logical closure accounts of necessity can at best be contingently true.

Objection: This is basically the Liar Paradox.

Response: This is indeed my main worry about the argument. I am hoping, however, that it is more like Goedel’s Incompleteness Theorems than like the Liar Paradox.

Here's how I think the hope can be satisfied. The Liar Paradox and its relatives arise from unbounded application of semantic predicates like “is (not) true”. By “unbounded”, I mean that one is free to apply the semantic predicates to any sentence one wishes. Now, if F is a name for a family of statements, then it seems that (1) (or its definite description variant akin to that produced by the diagonal lemma) has no semantic vocabulary in it at all. If F is a description of a family of statements, there might be some semantic predicates there. For instance, it could be that F is explicitly said to include “all true mathematical claims” (Chalmers will do that). But then it seems that the semantic predicates are bounded—they need only be applied in the special kinds of cases that come up within F. It is a central feature of logical closure accounts of necessity that the statements in F be a limited class of statements.

Well, not quite. There is still a possible hitch. It may be that there is semantic vocabulary built into “proved”. Perhaps there are rules of proof that involve semantic vocabulary, such as Tarski’s T-schema, and perhaps these rules involve unbounded application of a semantic predicate. But if so, then the notion of “proof” involved in the account is a pretty problematic one and liable to license Liar Paradoxes.

One might also worry that my argument that (1) is true explicitly used semantic vocabulary. Yes: but that argument is in the metalanguage.

A problem for Goedelian ontological arguments

Goedelian ontological arguments (e.g., mine) depend on axioms of positivity. Crucially to the argument, these axioms entail that any two positive properties are compatible (i.e., something can have both).

But I now worry whether it is true that any two positive properties are compatible. Let w₀ be our world—where worlds encompass all contingent reality. Then, plausibly, actualizing w₀ is a positive property that God actually has. But now consider another world, w₁, which is no worse than ours. Then actualizing w₁ is a positive property, albeit one that God does not actually have. But it is impossible that a being actualize both w₀ and w₁, since worlds encompass all contingent reality and hence it is impossible for two of them to be actual. (Of course, God can create two or more universes, but then a universe won’t encompass all contingent reality.) Thus, we have two positive properties that are incompatible.

Another example. Let E be the ovum and S₁ the sperm from which Socrates originated. There is another possible world, w₂, at which E and a different sperm, S₂, results in Kassandra, a philosopher every bit as good and virtuous as Socrates. Plausibly, being friends with Socrates is a positive property. And being friends with Kassandra is a positive property. But also plausibly there is no possible world where both Socrates and Kassandra exist, and you can’t be friends with someone who doesn’t exist (we can make that stipulative). So, being friends with Socrates and being friends with Kassandra are incompatible and yet positive.

I am not completely confident of the counterexamples. But if they do work, then the best fix I know for the Goedelian arguments is to restrict the relevant axioms to strongly positive properties, where a property is strongly positive just in case having the property essentially is positive. (One may need some further tweaks.) Essentially actualizing w₀ limits one from being able to actualize anything else, and hence isn’t positive. Likewise, essentially being friends with Socrates limits one to existing only in worlds where Socrates does, and hence isn’t positive. But, alas, the argument becomes more complicated and hence less plausible with the modification.

Another fix might be to restrict attention to positive non-relational properties, but I am less confident that that will work.

Are there multiple models of the naturals that are "on par"?

Assuming the Peano Axioms of arithmetic are consistent, we know that there are infinitely many sets that satisfy them. Which of these infinitely many sets is the set of natural numbers?

A plausible tempting answer is: “It doesn’t matter—any one of them will do.”

But that’s not right. For the infinitely many sets each of which is a model of the Peano Axioms are not isomorphic. They disagree with each other on arithmetical questions. (Famously, one of the models “claims” that the Peano Axioms are consistent and another “claims” that they are inconsistent, where we know from Goedel that consistency is equivalent to an arithmetical question.)

So it seems that with regard to the Peano Axioms, the models are all on par, and yet they disagree.

Here’s a point, however, that is known to specialists, but not widely recognized (e.g., I only recognized the point recently). When one says that some set M is a model of the Peano Axioms, one isn’t saying quite as much as the non-expert might think. Admittedly, one is saying that for every Peano Axiom A, A is true according to M (i.e., M⊨A). But one is not saying that according to M all the Peano Axioms are true. One must be careful with quantifiers. The statement:

1. For every Peano Axiom A, according to M, A is true.

is different from:

2. According to M, all the Peano Axioms are true.

The main technical reason there is such a difference is that (2) is actually nonsense, because the truth predicate in (2) is ineliminable and cannot be defined in M, while the truth predicate in (1) is eliminable; we are just saying that for any Peano Axiom A, M⊨A.

There is an important philosophical issue here. The Peano Axiomatization includes the Axiom Schema of Induction, which schema has infinitely many formulas as instances. Whether a given sequence of symbols is an instance of the Axiom Schema of Induction is a syntactic matter that can be defined arithmetically in terms of the Goedel encoding of the sequence. Thus, it makes sense to say that some sequence of symbols is a Peano Axiom according to a model M, i.e., that according to M, its Goedel number satisfies a certain arithmetical formula, I(x).

Now, non-standard models of the naturals—i.e., models other than our “normal” model—will contain infinite naturals. Some of these infinite naturals will intuitively correspond, via Goedel encoding, to infinite strings of symbols. In fact, given a non-standard model M of the naturals, there will be infinite strings of symbols that according to M are Peano Axioms—i.e., there will be an infinite string s of symbols such that its Goedel number g_s is such that I(g_s). But then we have no way to make sense of the statement: “s is true according to M” or M⊨s. For truth-in-a-model is defined only for finite strings of symbols.

Thus, there is an intuitive difference between the standard model of the naturals and non-standard models:

1. The standard model N is such that all the numbers that according to N satisfy I(x) correspond to formulas that are true in N.

2. A non-standard model M is not such that all the numbers that according to M satisfy I(x) correspond to formulas that are true in M.

The reason for this difference is that the notion of “true in M” is only defined for finite formulas, where “finite” is understood according to the standard model.

I do not know how exactly to rescue the idea of many inequivalent models of arithmetic that are all on par.

Numerical experimentation and truth in mathematics

Is mathematics about proof or truth?

Sometimes mathematicians perform numerical experiments with computers. Goldbach’s Conjecture says that every even integer n greater than two is the sum of two primes. Numerical experiments have been performed that verified that this is true for every even integer from 4 to 4 × 10¹⁸.

Let G(n) be the statement that n is the sum of two primes, and let’s restrict ourselves to talking about even n greater than two. So, we have evidence that:

1. For an impressive sample of values of n, G(n) is true.

This gives one very good inductive evidence that:

2. For all n, G(n) is true.

And hence:

3. It is true that: for all n, G(n). I.e., Goldbach’s Conjecture is true.

Can we say a similar thing about provability? The numerical experiments do indeed yield a provability analogue of (1):

4. For an impressive sample of values of n, G(n) is provable.

For if G(n) is true, then G(n) is provable. The proof would proceed by exhibiting the two primes that add up to n, checking their primeness and proving that they add up to n, all of which can be done. We can now inductively conclude the analogue of (2):

5. For all n, G(n) is provable.

But here is something interesting. While we can swap the order of the “For all n” and the “is true” operator in (2) and obtain (3), it is logically invalid to swap the order of the “For all n” and the “is provable” operator (5) to obtain:

6. It is provable that: for all n, G(n). I.e., Goldbach’s Conjecture is provable.

It is quite possible to have a statement such that (a) for every individual n it is provable, but (b) it is not provable that it holds for every n. (Take a Goedel sentence g that basically says “I am not provable”. For each positive integer n, let H(n) be the statement that n isn’t the Goedel number of a proof of g. Then if g is in fact true, then for each n, H(n) is provably true, since whether n encodes a proof of g is a matter of simple formal verification, but it is not provable that for all n, H(n) is true, since then g would be provable.)

Now, it is the case that (5) is evidence for (6). For there is a decent chance that if Goldbach’s conjecture is true, then it is provable. But we really don’t have much of a handle on how big that “decent chance” is, so we lose a lot of probability when we go from the inductively verified (5) to (6).

In other words, if we take the numerical experiments to give us lots of confidence in something about Goldbach’s conjecture, then that something is truth, not provability.

Furthermore, even if we are willing to tolerate the loss of probability in going from (5) to (6), the most compelling probabilistic route from (5) to (6) seems to take a detour through truth: if G(n) is provable for each n, then Goldbach’s Conjecture is true, and if it’s true, it’s probably provable.

So the practice of numerical experimentation supports the idea that mathematics is after truth. This is reminiscent to me of some arguments for scientific realism.

Optimalism about necessity

There are many set-theoretic claims that are undecidable from the basic axioms of set theory. Plausibly, the truths of set theory hold of necessity. But it seems to be arbitrary which undecidable set-theoretic claims are true. And if we say that the claims are contingent, then it will be arbitrary which claims are contingent. We don’t want there to be any of the “arbitrary” in the realm of necessity. Or so I say. But can we find a working theory of necessity that eliminates the arbitrary?

Here are two that have a hope. The first is a variant on Leslie-Rescher optimalism. While Leslie and Rescher think that the best (narrowly logically) scenario must obtain, and hence endorse an optimalism about truth, we could instead affirm an optimalism about necessity:

1. Among the collections of propositions, that collection of propositions that would make for the best collection of all the necessary truths is in fact the collection of all the necessary truths.

And just as it arguably follows from Leslie-Rescher optimalism that there is a God, since it is best that there be one, it arguably follows from this optimalism about necessity that there necessarily is a God, since it is best that there necessarily be a God. (By the way, when I once talked with Rescher about free will, he speculatively offered me something that might be close to optimalism about necessity.)

Would that solve the problem? Maybe: maybe the best possible—both practically and aesthetically—set theory is the one that holds of necessary truth.

I am not proposing this theory as a theory of what necessity is, but only of what is in fact necessary. Though, I suppose, one could take the theory to be a theory of what necessity is, too.

Alternately, we could have an optimalist theory about necessity that is theistic from the beginning:

2. A maximally great being is the ground of all necessity.

And among the great-making properties of a maximally great being there are properties like “grounding a beautiful set theory”.

I suspect that (1) and (2) are equivalent.

Musings on mathematics, logical implication and metaphysical entailment

I intuitively find the following picture very plausible. On the one hand, there are mathematical claims, like the Banach-Tarski Theorem or Euclid's Theorem on the Infinitude of the Primes. These are mysterious (especially the former!), and tempt one to some sort of non-realism. On the other hand, there are purely logical claims, like the claim that the ZFC axioms logically entails the Banach-Tarski Claim or that the Peano Axioms logically entail the Infinitude of the Primes. Pushed further, this intuition leads to something like logicism, which we all know has been refuted by Goedel. But I want to note that the whole picture is misleading. What does it mean to say that p logically entails q? Well, there are two stories. One is that every model of p is a model of q. That's a story about models, which are mathematical entities (sets or classes). Claims about models are mathematical claims in their own right, claims in principle just as tied to set-theoretic axioms as the Banach-Tarski Theorem. The other reading is that there is a proof from p to q. But proofs are sequences of symbols, and sequences of symbols are mathematical objects, and facts about the existence or non-existence of proofs are once again mathematical facts, tied to axioms and subject to the foundational worries that other mathematical facts are. So the idea that there is some radical difference between first-order mathematical claims and claims about what logically entails what, such that the latter is innocent of deep philosophy of mathematics issues (like Platonism), is untenable.

Interestingly, however, what I said is no longer true if we replace logical entailment with metaphysical entailment. The claim that the ZFC axioms metaphysically entail the Banach-Tarski Claim is not a claim of mathematics per se. So one could make a distinction between the mysterious claims of mathematics and the unmysterious claims of metaphysical entailment--if the latter are unmysterious. (They are unmysterious if one accepts the causal theory of them.)

This line of thought suggests an interesting thing: the philosophy of mathematics may require metaphysical entailment.

A quick route from mathematics to metaphysical necessity

The Peano Axioms are consistent. If not, mathematics (and the science resting on it) is overthrown. Moreover, it is absurd to suppose that they are merely contingently consistent: that in some other possible world a contradiction follows logically from them, but in the actual world no contradiction follows from them. So the Peano Axioms are necessarily consistent. But they aren't logically necessarily consistent: the consistency of the Peano Axioms cannot be proved (according to Goedel's second incompleteness theorem, not even if one helps oneself to the Peano Axioms in the proof, at least assuming they really are consistent). So we must suppose a necessity that isn't logical necessity, but is nonetheless very, very strong. We call it metaphysical necessity.

System-relativity of proofs

There is a generally familiar way in which the question whether a mathematical statement has a proof is relative to a deductive system: for a proof is a proof in some system L, i.e., the proof starts with the axioms of L and proceeds by the rules of L. Something can be provable in one system—say, Euclidean geometry—but not provable in another—say, Riemannian geometry.

But there is a less familiar way in which the provability of a statement is relative. The question whether a sentence p is provable in a system L is itself a mathematical question. Proofs are themselves mathematical objects—they are directly the objects in a mathematical theory of strings of symbols and indirectly they are the objects of arithmetic when we encode them using something like Goedel numbering. The question whether there exists a proof of p in L is itself a mathematical question, and thus it makes sense to ask this question in different mathematical systems, including L itself.

If we want to make explicit both sorts of relativity, we can say things like:

1. p has (does not have) a proof in a system L according to M.

Here, M might itself be a deductive system, in which case the claim is that the sentence "p has (does not have) a proof in L" can itself be proved in M (or else we can talk of the Goedel number translation of this), or M might be a model in which case the claim is that "p has a proof in L" is true in that model.

This is not just pedantry. Assume Peano Arithmetic (PA) is consistent. Goedel's second incompleteness theorem then tells us that the consistency of PA cannot be proved in PA. Skipping over the distinction between a sentence and its Goedel number, let "Con(PA)" say that PA is consistent. Then what we learn from the second incompleteness theorem is that:

2. Con(PA) has no proof in PA.

Now, statement (2), while true, is itself not provable in PA.[note 1] Hence there are non-standard models of PA according to which (2) is false. But there are also models of PA according to which (2) is true, since (2) is in fact true. Thus, there are models of PA according to which Con(PA) has no proof and there are models of PA according to which Con(PA) has a proof.

This has an important consequence for philosophy of mathematics. Suppose we want to de-metaphysicalize mathematics, move us away from questions about which axioms are and are not actually true. Then we are apt to say something like this: mathematics is not about discovering which mathematical claims are true, but about discovering which mathematical claims can be proved in which systems. However, what we learn from the second incompleteness theorem is that the notion of provability carries the same kind of exposure to mathematical metaphysics, to questions about the truth of axioms, as naively looking for mathematical truths did.

And if one tries to de-metaphysicalize provability by saying that what we are after in the end is not the question whether p is provable in L, but whether p is provable in L according to M, then that simply leads to a regress. For the question whether p is provable in L according to M is in turn a mathematical question, and then it makes sense to ask according which system we are asking it. The only way to arrest the regress seems to be to suppose that at some level that we simply are talking of how things really are, rather than how they are in or according to a system.

Maybe, though, one could say the following to limit one's metaphysical exposure: Mathematics is about discovering proofs rather than about discovering what has a proof. However, this is a false dichotomy, since by discovering a proof of p, one discovers that p has a proof.

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.