Humeanism and knowledge of fundamental laws

On a "Humean" Best System Account (BSA) of laws of nature, the fundamental laws are the axioms of the system of laws that best combines brevity and informativeness.

An interesting consequence of this is that, very likely, no amount of advances in physics will
suffice to tell us what the fundamental laws are: significant advances in mathematics will also be needed. For suppose that after a lot of extra physics, propositions formulated in sentences p₁, ..., p_n are the physicist’s best proposal for the fundamental laws. They are simple, informative and fit the empirical data really well.

But we would still need some very serious mathematics. For we would need to know there isn’t a simpler collection of sentences {q₁, ..., q_m} that is logically equivalent to {p₁, ..., p_n} but simpler. To do that would require us to have a method for solving the following type of mathematical problem:

1. Given a sentence s in some formal language, find a simplest sentence s′ that is logically equivalent to s,

in the case of significantly non-trivial sentences s.

We might be able to solve (1) for some very simple sentences. Maybe there is no simpler way of saying that there is only one thing in existence than ∃x∀y(x=y). But it is very plausible that any serious proposal for the laws of physics will be much more complicated than that.

Here is one reason to think that any credible proposal for fundamental laws is going to be pretty complicated. Past experience gives us good reason to think the proposal will involve arithmetical operations on real numbers. Thus, a full statement of the laws will require including a definition of the arithmetical operations as well as of the real numbers. To give a simplest formulation of such laws will, thus, require us to solve the problem of finding a simplest axiomatization of the portions of arithmetic and real analysis that are needed for the laws. While we have multiple axiomatizations, I doubt we are at all close to solving the problem of finding an optimal such axiomatization.

Perhaps the Humean could more modestly hope that we will at least know a part of the fundamental laws—namely the part that doesn’t include the mathematical axiomatization. But I suspect that even this is going to be very difficult, because different arithmetical formulations are apt to need different portions of arithmetic and real analysis.

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.

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).

The Axiom of Separation

The Axiom of Separation in Zermelo-Fraenkel (ZF) set theory implies that, roughly, for any set A and any unary predicate F(x), there is a subset B of all the x in A such that F(x). But only roughly. Technically the axiom only implies this for predicates definable in the language of set theory. We philosophers tend to forget that technical fact when we use set theory, much as we tend to blithely extend set theory to allow for ur-elements (elements that are not themselves sets). But if we are going to be realists about sets (which I am not saying we should be), we should have a real worry about what predicates can be legitimately used in the Axiom of Separation. (That's one of the lessons of this post.)

Consider the predicate L(x) which holds if and only if someone likes x. This is definitely not formulated in the language of set theory, so ZF set theory gives us no guarantee that there is, say, the set of all real numbers that satisfy L(x). If it turns out that there are only finitely many numbers that are liked, then we have no worries: for any real numbers x₁,...,x_n, there is a set that contains them and only them (this follows from the Axiom of Pairs plus the Axiom of Union). There will be other special cases where things work out, say when all but finitely many numbers are liked. But in general there is no guarantee from the axioms of ZF that there is a set of all liked numbers.

One might use this to try to get out of some paradoxes of infinity, by limiting the applicability of set theory. That's a strategy worth exploring further, but risky. For the above observations also severely limit the physical applicability of set theory. Suppose, for instance, that at each of infinitely many points in spacetime there is a well-defined temperature. It is usual then to suppose that there is a function T from the spacetime manifold to the real numbers such that T(z)=u if and only if the temperature at z (or, more precisely, at the point of spacetime corresponding to the point z in the mathematical manifold that models spacetime) is u. And we need there to be such a function T to be able to make physical predictions.

One solution is to extend the Axiom of Separation to include some or even all predicates not in the language of set theory. This is the solution that is typically implicitly used by philosophers. The Axiom of Separation has a lot of intuitive force thus extended, but we need to be careful since we know that the incoherent Axiom of Comprehension also had a lot of intuitive force.

A second option would be to have physics make set-theoretic claims. Thus, a theory positing that at each point of spacetime there is a temperature would also posit that there exists a corresponding function from the mathematical manifold that models spacetime to the real numbers. I think this would be quite an interesting option: it would mean that physics actually places constraints on what the universe of set theory is like.

Perhaps if we are not Platonists about sets, things are easier. But I am not sure. Things might just be murkier rather than easier.

The Axiom of Choice

For any relation R and world w we can ask the following question: Is it the case that for all x, y and z such that xRy and yRz, we also have xRz?  If the answer is affirmative, we say that R is transitive at w.

Likewise, for any relation R and world w we can ask the following question: Is it the case that for every object x such that

1. for every y if yRx, there is a z such that zRy, and
2. for all u, v and z such that uRy, vRy, zRu and zRv, we have u=v,

there exists an object x* such that for every y such that yRx there is a unique z such that zRx* and zRy?  If the answer is affirmative, we could say that R is choosy at w.

Now, it would be silly to ask: "Is transitivity true?"  Transitivity is not the sort of thing to be true.  Some relations are transitive at a world (and some are transitive at all) and some aren't.  Likewise, it would be silly to ask: "Is choosiness true?"  Choosiness is not the sort of thing to be true.  Some relations are choosy at a world (and some are choosy at all) and some aren't.

As it turns out (not by chance--I rigged it), the Axiom of Choice in a set theory is equivalent to claim that the membership relation in that set theory is choosy.  But just as it is nonsense to ask if transitivity or choosiness is true, I rather like the view that it's nonsense to ask if the Axiom of Choice is true.  We can ask if a particular relation satisfies the Axiom of Choice, i.e., is choosy at some world, or at all worlds, but why think there is a distinguished relation that we can call "the membership relation" and that we can ask about the choosiness of?

I am fairly naively inclined to take this quite far in mathematics, along the lines of ifthenism: Mathematicians simply prove necessary conditionals like that if a relation is Zermelo-Fraenkelish and choosy, then it's Zorny, or--for a much more difficult example--that if a relation is Peanish, it is finally-Fermatish.  This is a thesis about mathematical practice, not mathematical truth.  But I really don't know much philosophy of mathematics (I know a lot more mathematics than philosophy of mathematics) and this version of ifthenism may be untenable.

The deposit of faith

Consider the following objection to the Catholic faith (this is based on something I got by email): Catholicism includes a large number of detailed and substantive doctrines that do not seem to be derivable from God's revelation as completed by around the time of death of the Apostles, even though the Catholic Church herself claims that revelation was completed by around the time of death of the Apostles.

Consider, after all, something like the doctrine that Mary was free of original sin from the first moment of her conception. This is a detailed and substantive doctrine that seems to go far beyond the information given in Scripture and what we know about the faith of the first century Church from non-Scriptural sources. The objection is an incredulous stare at the possibility that such doctrines could be derived from revelation as completed by around the time of death of the Apostles. But:

1. Twenty simple axioms of Euclidean geometry generate an infinity of detailed and substantive theorems. These theorems are such that there is no prima facie way to see that they would follow from the axioms. It can take centuries and centuries for humankind to discover that they can be derived. It should, thus, be no surprise at all that we can derive from a set S of propositions new propositions that are details and substantive, and that seem to go far beyond S. This is particularly true when S is not a list of twenty axioms, but includes about 27,570 verses of the Old Testament, about 7956 verses of the New Testament, as well as decades of Apostolic preaching which Catholics think became embedded in the tradition of the Church, particularly in her liturgy.

2. Furthermore, unlike the development of geometry which is as far as we know is typically done by the unaided human intellect, the development of Catholic doctrine is claimed to be done by the human intellect guided by Holy Spirit.

3. Moreover, the Scriptures and the Tradition of the Church not only contain particular doctrinal axioms from which we can derive further propositions, but contain ways of reasoning or rules of inference that embody an understanding of how God deals with the world. Prominent among these is typology. In the New Testament and the Church's liturgy, we learn that God works through parallels. The people of Israel pass through the sea; Christians pass through baptism. Adam sins and from his sin comes death; Christ conquers sin and from his conquering sin comes life. The New Testament (Luke 24:27) says that all of the Old Testament scriptures tell us about Christ. Thus there may be substantive ways of reasoning embodied in Scripture, liturgy and theological practice, ways of reasoning that include typological reasoning. These ways of reasoning are, plainly, more than just formal rules of logic. They are based, rather, on an understanding of God as acting in certain ways (maybe with certain motives), as producing a certain kind of deeply interconnected history.

And new insights might well come from this. Christ corresponds in an important way to Adam; but Mary in the Church's understanding corresponds in an important way to Eve. Just as Eve was created without sin, so, too, Mary was created without original sin. Now it is true that prima facie one might have tried different typological correspondences--one might, for instance, make Mary's being conceived in sin be parallel-by-contrast to Eve's being sinless (as Christ's raising us is parallel-by-contrast to Adam's bringing death on us). Working out a deep understanding of the typology here, and connecting it with many other aspects of Christian doctrine, is going to be difficult. It may take centuries, thus, for the Church to settle on a particular understanding, e.g., to see that the parallel between the new creation in Christ and the old creation in Adam does in fact call not just for Christ the new Adam to be without original sin, but Mary the new Eve as well, but of course with her freedom from the weight of original sin flowing from Christ's redemption, just as our Church's freedom from the weight of original sin does.

Conclusion: It should be no surprise if from a very large body of axioms, which includes substantive rules of inference, one could derive many doctrines that one is individually surprised by.