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”.)

Snakes and finitude

For years I have thought the finite to be mysterious, and needs something metaphysical like divine illumination or causal finitism to pick it out. Now I am not sure. I think snakes and exact duplicates can help. And if that’s right, then the argument in my other post from today can be fixed.

Here are some definitions, where the first one is supposed to work for snakes that may be in the same or in different worlds:

• Snake a is vertebrally equal to snake b provided that there is a possible world with exact duplicates of a and b such that in that world it would be possible to line up the two snakes vertebra by vertebra, stretching or compressing as necessary but neither destroying nor introducing vertebra.

• Snake a is the vertebral successor of snake b provided there is a possble world with exact duplicates of a and b such that in that world it is possible to line up the two snakes vertebra by vertebra with exactly one vertebra of a outside the lineup, again stretching or compressing as necessary but neither destroying nor introducing vertebra.

• A world w is abundant in snakes provided that w has a snake with no vertebrae (say, an embryonic snake) and every snake in w has a vertebral successor in w.

• A snake a is vertebrally finite provided that in every world in which snakes are abundant there is a snake vertebrally equal to a.

• A plurality is finite provided that it is possible to put it in one-to-one correspondence with the vertebrae of a vertebrally finite snake.

These definitions require, of course, that one take metaphysical possibility seriously.

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.

Megethology as mathematics and a regress of structuralisms

In his famous “Mathematics is Megethology”, Lewis gives a brilliant reduction of set theory to mereology and plural quantification. A central ingredient of the reduction is a singleton function which assigns to each individual a singleton of which the individual is the only member. Lewis shows that assuming some assumptions on the size of reality (namely, that it’s very big) there exists a singleton function, and that different singleton functions will yield the same set theoretic truths. The result is that the theory is supposed to be structuralist: it doesn’t matter which singleton function one chooses, just as on structuralist theories of natural numbers it doesn’t matter if one uses von Neumann ordinals or Zermelo ordinals or anything else with the same structure. The structuralism counters the obvious objection to Lewis that if you pick out a singleton function, it is implausible that mathematics is the study of that one singleton function, given that any singleton function yields the same structure.

It occurs to me that there is one hole in the structuralism. In order to say “there exists a singleton function”, Lewis needs to quantify over functions. He does this in a brilliant way using recently developed technical tools where ordered pairs of atoms are first defined in terms of unordered pairs and an ordering is defined by a plurality of fusions, relations on atoms are defined next, and so on, until finally we get functions. However, this part can also be done in a multiplicity of ways, and it is not plausible that mathematics is the study of singleton functions in that one sense of function, given that there are many sense of function that yield the same structure.

Now, of course, one might try to give a formal account of what it is for a construction to have the structure of functions, what it is to quantify not over functions but over function-notions, one might say. But I expect a formal account of quantification over function-notions will presumably suffer from exactly the same issue: no one function-notion-notion will appear privileged, and a structuralist will need to find a way to quantify over function-notion-notions.

I suspect this is a general feature with structuralist accounts. Structuralist accounts study things with a common structure, but there are going to be many accounts of common structure that by exactly the same considerations that motivate structuralism require moving to structuralism about structure, and so on. One needs to stop somewhere. Perhaps with an informal and vague notion of structure? But that is not very satisfying for mathematics, the Queen of Rigor.

Empirical mathematics

Suppose I want to figure out a good approximation to the eigenvalues of a certain Hamiltonian involving a moderately large number of Coulomb potentials. It could well be the case that the best way to do so is to synthesize a molecule with that Hamiltonian and then measure its spectrum. In other words, there are mathematical problems where our best solution to the problem uses scientific methods rather than mathematical proof.

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.

Provability and truth

The most common argument that mathematical truth is not provability uses Tarski’s indefinability of truth theorem or Goedel’s first incompleteness theorem. But while this is a powerful argument, it won’t convince an intuitionist who rejects the law of excluded middle. Plus it’s interesting to see if a different argument can be constructed.

Here is one. It’s much less conclusive than the Tarski-Goedel approach. But it does seem to have at least a little bit of force. Sometimes we have experimental evidence (at least of the computer-based kind) for a mathematical claim. For instance, perhaps, you have defined some probabilistic setup, and you wonder what the expected value of some quantity Q is. You now set up an apparatus that implements the probabilistic setup, and you calculate the average value of your observations of Q. After a billion runs, the average value is 3.141597. It’s very reasonable to conclude that the last digit is a random deviation, and that the mathematically expected value of Q is actually π.

But is it reasonable to conclude that it’s likely provable that the expected value of Q is π? I don’t see why it would be. Or, at least, we should be much less confident that it’s provable than that the expected value is π. Hence, provability is not truth.

Evolution of my views on mathematics

I have for a long time inclined towards ifthenism in mathematics: the idea that mathematics discovers truths of the form "If these axioms are true, then this thing is true as well."

Two things have weakened my inclination to ifthenism.

The first is that there really seems to be a privileged natural number structure. For any consistent sufficiently rich recursive axiomatization A of the natural numbers, by Goedel’s Second Incompleteness Theorem (plus Completeness) there is a natural number structure satisfying A accordingto which A is inconsistent and there is a natural number structure satisfying A according to which A is consistent. These two structures can’t be on par—one of them needs to be privileged.

The second is an insight I got from Linnebo’s philosophy of mathematics book: humans did mathematics before they did axiomatic mathematics. Babylonian apparently non-axiomatic but sophisticated mathematics came before Greek axiomatic geometry. It is awkward to think that the Babylonians were discovering ifthenist truths, given that they didn’t have a clear idea of the antecedents of the ifthenist conditionals.

I am now toying with the idea that there is a metaphysically privileged natural number structure but we have ifthenism for everything else in mathematics.

How is the natural number structure privileged? I think as follows: the order structure of the natural numbers is a possible order structure for a causal sequence. Causal finitism, by requiring all initial segments under the causal relation to be finite, requires the order type of the natural numbers to be ω. But once we have fixed the order type to be ω, we have fixed the natural number structure to be standard.

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.

A posteriori necessities

The usual examples of a posteriori necessities are identities between kinds and objects under two descriptions, at least one of which involves a contingent mode of presentation, such as water (presented as “the stuff in this pond”, say) and H₂O.

Such a posteriori necessities are certainly interesting. But we should not assume that these exhaust the scope of all a posteriori necessities.

For instance, Thomas Aquinas was committed to the existence of God being an a posteriori necessity: he held that necessarily God existed, but that all a priori arguments for the existence of God failed, while some a posteriori ones, like the Five Ways, succeeded.

For another theistic example, let p be an unprovable mathematical truth. Then p is, presumably, not a priori knowable. But God could reveal the truth of p, in which case we would know it a posteriori, via observation of God’s revelation. And, plausibly, mathematical truths are necessary.

For a third example, we could imagine a world where there is an odd law of nature: if anyone asserts a false mathematical statement, they immediately acquire hideous warts. In that world, all mathematical truths, including the unprovable ones, would be knowable a posteriori.

A superpower

Imagine Alice claimed she could just see, with reliability, which unprovable large cardinal axioms are true. We would be initially sceptical of her claims, but we could imagine ways in which we could come to be convinced of her having such an ability. For instance, we might later be able to prove a lot of logical connections between these axioms (say that axiom A₁₂ implies axiom A₁₄) and then find that Alice’s oracular pronouncements matched these logical connections (she wouldn’t, for instance, affirm A₁₂ while denying A₁₄) to a degree that would be very hard to explain as just luck.

Suppose, then, that we have come to be convinced that Alice has the intuitive ability to just see which large cardinal axioms are true. This would be some sort of uncanny superpower. The existence of such a superpower would sit poorly with naturalism. An intuition like Ramanujan’s about the sums of series could be explained by naturalism—we could simply suppose that his brain unconsciously sketched proofs of various claims. But an intuition about large cardinal axioms wouldn’t be like that, since these axioms are not provable.

Now as far as we know, there is no one exactly like Alice who just has reliable intuitions about large cardinal axioms. But our confidence in the less abstruse axioms of Zermelo-Fraenkel set theory—intuitive axioms like the axiom of replacement—commits us to thinking that either we in general, or those most expert in the matter, are rather like Alice with respect to these less abstruse axioms. The less abstruse axioms are just as unprovable as the more abstruse ones that Alice could see. Therefore, it seems, if Alice’s reliable intuition provided an argument against naturalism, our own (or our experts’) intuition about the more ordinary axioms, an intuition which we take to be reliable, gives us an argument against naturalism. Seeing the axiom of replacement to be true is just as much a superpower as would be Alice’s seeing that, say, measurable cardinals exist (or that they do not exist).

Is our universe of sets minimal?

Our physics is based on the real numbers. Physicists use the real numbers all over the place: quantum mechanics takes place in a complex Hilbert space, and the complex numbers are isomorphic to pairs of real numbers, while relativity theory takes place in a manifold that is locally isomorphic to a Lorentzian four-dimensional real space.

The real numbers are one of an infinite family of mathematical objects known as real closed fields. Other real closed fields than the real numbers could be used in physics instead—for instance, the hyperreals—and I think we would have the same empirical predictions. But the real numbers are simpler and more elegant: for instance, they are the only Dedekind-complete and the minimal Cauchy-complete real closed field.

At the same time, the mathematics behind our physics lives within a set theoretic universe. That set theoretic universe is generally not assumed to be particularly special. For instance, I know of no one who assumes that our set theoretic universe is isomorphic to Shepherdson’s/Cohen’s minimal model of set theory. On the contrary, it is widely assumed that our set theoretic universe has a standard transitive set model, which implies that it is not minimal, and few people seem to believe the Axiom of Constructibility which would hold in a minimal model.

This seems to me be rationally inconsistent. If we are justified in thinking that the mathematics underlying the physical world is based on a particularly elegant real closed field even though other fields fit our empirical data, we would also be justified in thinking it’s based on a particularly elegant universe of sets even though other universes fit our empirical data.

(According to Shipman, the resulting set theory would be one equivalent to ZF + V=L + “There is no standard model”.)

Set theory and physics

Assume the correct physics has precise particle positions (similar questions can be asked in other contexts, but the particle position context is the one I will choose). And suppose we can specify a time t precisely, e.g., in terms of the duration elapsed from the beginning of physical reality, in some precisely defined unit system. Consider two particles, a and b, that exist at t. Let d be the distance between a and b at t in some precisely definable unit system.

Here’s a question that is rarely asked: Is d a real number?

This seems a silly question. How could it not be? What else could it be? A complex number?

Well, there are at least two other things that d could be without any significant change to the equations of physics.

First, d could be a hyperreal number. It could be that particle positions are more fine-grained than the reals.

Second, d could be what I am now calling a “missing number”. A missing number is something that can intuitively be defined by an English (or other meta-language) specification of an approximating “sequence”, but does not correspond to a real number in set theory. For instance, we could suppose for simplicity that d lies between 0 and 1 and imagine a physical measurement procedure that can determine the nth binary digit of d. Then we would have an English predicate M_d(n) which is true just in case that procedure determined the n binary digit to be 1. But it could turn out that in set theory there is no set whose members are the natural numbers n such that M_d(n). For the axioms of set theory only guarantee the existence of a set defined using the predicates of set theory, while M_d is not a predicate of set theory. The idea of such “missing numbers” is coherent, at least if our set theory is coherent.

It seems reasonable to say that d is indeed a real number, and to say similar things about any other quantities that can be similarly physically specified. But what guarantees such a match between set theory and physics? I see four options:

1. Luck: it’s just a coincidence.

2. Our set theory governs physics.

3. Physics governs our set theory.

4. There is a common governor to our set theory and physics.

Option 1 is an unhappy one. Option 4 might be a Cartesian God who freely chooses both mathematics and physics.

Option 2 is interesting. On this story, there is a Platonically true set theory, and then the laws of physics make reference to it. So it’s then a law of physics that distances (say) always correspond to real numbers in the Platonically true set theory.

Option 3 comes in at least two versions. First, one could have an Aristotelian story on which mathematics, including some version of set theory, is an abstraction from the physical world, and any predicates that we can define physically are going to be usable for defining sets. So, physics makes sets. Second, one could have a Platonic multiverse of universes of sets: there are infinitely many universes of sets, and we simply choose to work within those that match our physics. On this view, physics doesn’t make sets, but it chooses between the universes of sets.

Arbitrariness and contingency

I’ve come to be impressed by the idea that where there is apparent arbitrariness, there is probably contingency in the vicinity.

The earth and the moon on average are 384400 km apart. This looks arbitrary. And here the fact itself is contingent.

Humans have two arms and two legs. This looks arbitrary. But it is actually a necessary truth. However there is contingency in the vicinity: it is a contingent fact that humans, rather than eight-armed intelligent animals, exist on earth.

Ethical obligations have apparent arbitrariness, too. For instance, we should prefer mercy to retribution. Here, there are two possibilities. First, perhaps it is contingent that we should prefer mercy to just retribution. The best story I know which makes that work out is Divine Command Theory: God commands us to prefer mercy to just retribution but could have commanded the opposite. Second, perhaps it is necessary that we should prefer mercy to retribution, because our nature requires it, but it is contingent that we rather than beings whose nature carries the opposite obligation exist.

Now here is where I start to get uncomfortable: mathematics. When I think about the vast number of possible combinations of axioms of set theory, far beyond where any intuitions apply, axioms that cannot be proved from the standard ZFC axioms (unless these are inconsistent), it’s all starting to look very arbitrary. This pushes me to one of three uncomfortable positions:

• anti-realism about set theory

• Hamkins’ set-theoretic multiverse

• contingent mathematical truth.

Wilde lectures now online

Videos of my 2019 Wilde Lectures in Natural Theology and Comparative Religion at Oxford’s Oriel College are now available online.

Here are the slides:

• Beauty
• Scepticism
• Forms I
• Forms II
• The finite

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.

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.