Natural law: Between objectivism and subjectivism

Aristotelian natural law approaches provide an attractive middle road between objectivist and subjectivist answers to various normative questions: the answers to the questions are relative to the kind of entity that they concern, but not to the particular particular entity.

For instance, a natural law approach to aesthetics would not make the claim that there is one objective beauty for humans, klingons, vulcans and angels. But it would make the absolutist claim that there is one beauty for Alice, Bob, Carl and Davita, as long as they are all humans. The natural lawyer aestheticist could take a subjectivist’s accounts of beauty in terms, of say, disinterested pleasure, but give it a species relative normative twist: the beautiful to members of kind K (say, humans or klingons) is what should give members of kind K disinterested pleasure. The human who fails to find that pleasure in a Monet painting suffers from a defect, but a klingon might suffer from a defect if she found pleasure in the Monet.

Provability and numerical experiments

A tempting view of mathematics is that mathematicians are discovering not facts about what is true, but about what is provable from what.

But proof is not the only way mathematicians have of getting at truth. Numerical experiment is another. For instance, while we don’t have a proof of Goldbach’s Conjecture (each even number bigger than two is the sum of two primes), it has been checked to hold for numbers up to 4 ⋅ 10¹⁸. This seems to give significant inductive evidence that Goldbach’s Conjecture is true. But it does not seem to give significant evidence that Goldbach’s Conjecture can be proved.

Here’s why. Admittedly, when we learned that that the conjecture holds for some particular number n, say 13, we also learned that the conjecture can be proved for that specific number n (e.g., 13 = 11 + 2 and 11 and 2 are prime, etc.). Inductively, then, this gives us significant evidence that for each particular number n, Goldbach’s conjecture for n is provable (to simplify notation, stipulate Goldbach’s Conjecture to hold trivially for odd n or n < 4). But one cannot move from ∀n Provable(G(n)) to Provable(∀n G(n)) (to abuse notation a little).

The issue is that the inductive evidence we have gathered strongly supports the claim that Goldbach’s Conjecture is true, but gives much less evidence for the further claim that Goldbach’s Conjecture is provable.

The argument above is a parallel to the standard argument in the philosophy of science that the success of the practice of induction is best explained by scientific realism.

Are monads in space?

It is often said that Leibniz’s monads do not literally occupy positions in space. This seems to me to be a mistake, perhaps a mistake Leibniz himself made. Leibnizian space is constituted by the perceptual relations between monads. But if that’s what space is, then the monads do occupy it, because they stand in the perceptual relations that constitute space. And they occupy it literally. There is no other way to occupy space, if Leibniz is right: this is literal occupation of space.

Perhaps the reason it is said that the monads do not literally occupy positions in space is that an account that reduces position to mental properties seems to be a non-realist account of position. This is a bit strange. Suppose we reduce position to gravitational force and mass (“if objects have masses m₁ and m₂ and a gravitational force F between them, then their distance is nothing but (Gm₁m₂/F)^1/2”). That’s a weird theory, but a realist one. Why, then, should a reduction to mental properties not be a realist one?

Maybe that’s just definitional: a reduction of physical properties to mental ones counts as a non-realism about the physical properties. Still, that’s kind of weird. First, a reduction of mental properties to physical ones doesn’t count as a non-realism about the mental properties. Second, a reduction of some mental properties to other mental properties—say, beliefs to credence assignments—does not count as non-realism about the former. Why, then, is a reduction of physical to mental properties count as a non-realism?

Maybe it is this thought. It seems to be non-realist to reduce some properties to our mental properties, where “our” denotes some small subset of the beings we intuitively think exist. Thus, it seems to be non-realist to reduce aesthetic properties to the desires and beliefs of persons, or to reduce stones to the perceptual properties of animals. But suppose we are panpsychist as Leibniz is, and think there are roughly at least as many beings as we intuitively think there are, and are reducing physical properties to the mental properties of all the beings. Then it’s not clear to me that that is any kind of non-realism.

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.

Realism and anti-reductionism

The ordinary sentence "There are four chairs in my office" is true (in its ordinary context). Furthermore, its being true tells us very little about fundamental ontology. Fundamental physical reality could be made out of a single field, a handful of fields, particles in three-dimensional space, particles in ten-dimensional space, a single vector in a Hilbert space, etc., and yet the sentence could be true.

An interesting consequence: Even if in fact physical reality is made out of particles in three-dimensional space, we should not analyze the sentence to mean that there are four disjoint pluralities of particles each arranged chairwise in my office. For if that were what the sentence meant, it would tell us about which of the fundamental physical ontologies is correct. Rather, the sentence is true because of a certain arrangement of particles (or fields or whatever).

If there is such a broad range of fundamental ontologies that "There are four chairs in my office" is compatible with, it seems that the sentence should also be compatible with various sceptical scenarios, such as that I am a brain in a vat being fed data from a computer simulation. In that case, the chair sentence would be true due to facts about the computer simulation, in much the way that "There are four chairs in this Minecraft house" is true. It would be very difficult to be open to a wide variety of fundamental physics stories about the chair sentence without being open to the sentence being true in virtue of facts about a computer simulation.

But now suppose that the same kind of thing is true for other sentences about physical things like tables, dogs, trees, human bodies, etc.: each of these sentences can be made true by a wide array of physical ontologies. Then it seems that nothing we say about physical things rules out sceptical scenarios: yes, I know I have two hands, but my having two hands could be grounded by facts about a computer simulation. At this point the meaningfulness of the sceptical question whether I know I am not a brain in a vat is breaking down. And with it, realism is breaking down.

In order for the sceptical question to make sense, we need the possibility of saying things that cannot simply be made true by a very wide variety of physical theories, since such things will also be made true by computer simulations. This gives us an interesting anti-reductionist argument. If the statement "I have two hands" is to be understood reductively (and I include non-Aristotelian functionalist views as reductive), then it could still be literally true in the brain-in-a-vat scenario. But if anti-reductionism about hands is true, then the statement wouldn't be true in the brain-in-a-vat scenario. And so I can deny that I am in that scenario simply by saying "I have two hands."

But maybe I am moving too fast here. Maybe "I have two hands" could be literally true in a brain-in-a-vat scenario. Suppose that the anti-reductionism consists of there being Aristotelian forms of hands (presumably accidental forms). But if, for all we know, the form of a hand can inform a bunch of particles, a fact about a vector or the region of a field, then the form of a hand can also inform an aspect of a computer simulation. And so, for all we know, I can literally and non-reductively have hands even if I am a brain in a vat. I am not sure, however, that I need to worry about this. What is important is form, not the precise material substrate. If physical reality is the memory of a giant computer but it isn't a mere simulation but is in fact informed by a multiplicity of substantial and accidental forms corresponding to people, trees, hands, hearts, etc., and these forms are real entities, then the scenario does not seem to me to be a sceptical scenario.

If you're going to be a Platonist dualist, why not be an idealist?

Let's try another exercise in philosophical imagination. Suppose Platonism and dualism are true. Then consider a theory on which our souls actually inhabit a purely mathematical universe. All the things we ever observe—dust, brains, bodies, stars and the like—are just mathematical entities. As our souls go through life, they become "attached" to different bits and pieces of the mathematical universe. This may happen according to a deterministic schedule, but it could also happen an indeterministic way: today you're attached to part of a mathematical object A₁, and tomorrow you might be attached to B₂ or C₂, instead. You might even have free will. One model for this is the traveling minds story, but with mathematical reality in the place of physical reality.

This is a realist idealism. The physical reality around us on this story is really real. It's just not intrinsically different from other bits of Platonic mathematical reality. The only difference between our universe and some imaginary 17-dimensional toroidal universe is that the mathematical entities constituting our universe are connected with souls, while those constituting that one are not.

One might wonder if this is really a form of idealism. After all, it really does posit physical reality. But physical reality ends up being nothing but Platonic reality.

The view is akin to Tegmark's ultimate ensemble picture, supplemented with dualism.

Given Platonism and dualism, this story is an attractive consequence of Ockham's Razor. Why have two kinds of things—the physical universe and the mathematical entities that represent the physical universe? Why not suppose they are the same thing? And, look, how neatly we solve the problem of how we have mathematical knowledge—we are acquainted with mathematical objects much as we are with tables and chairs.

"But we can just see that chairs and tables aren't made of mathematical entities?" you may ask. This, I think, confuses not seeing that chairs and tables are made of mathematical entities with seeing that they are not made of them. Likewise, we do not see that chairs and tables are made of fundamental particles, but neither do we see that they are not made of them. The fundamental structure of much of physical reality is hidden from our senses.

So what do we learn from this exercise? The view is, surely, absurd. Yet given Platonism and dualism, Ockham's razor strongly pulls to it. Does this give us reason to reject Platonism or dualism? Quite possibly.

Theism and scientific non-realism

One of the major arguments for scientific realism is that the best explanation for why our best scientific theories are predictively successful is that they are literally true or at least literally approximately true. After all, wouldn't it be incredible if things behaved observationally as if the theories were true, but the theories weren't true?

While this is a pretty good argument, it's worth noting that theists have an alternate explanation: In order that intelligent beings be able to make successful predictions of a sort that lets them exhibit appropriate stewardship over the world, God makes the world exhibit patterns of the sort that human science is capable of finding, patterns that can be subsumed under theories that are sufficiently simple for us to find. And one sort of pattern is of the as-if sort: things behave as if there were photons, which lets us organize the behavior of macroscopic things into patterns by supposing (in a way that need not carry ontological commitment) photons.

That said, there is a value to science over and beyond its helping us exercise stewardship over the world—understanding of the world is valuable for its own sake—so even given theism, a scientifically realist theistic explanation seems better than a scientifically non-realist one.

But even if realism is in general the right policy, maybe theism could provide a tenable Plan B if there turn out to be cases where scientific realism is not tenable. For instance, one might think (incorrectly, I suspect) that there is no metaphysically tenable and scientifically plausible version of quantum mechanics. Then, one might retreat to a theistic explanation of why the world behaves as if the metaphysically untenable theory were true. Or one might think (because of Zeno's paradoxes, say) that it is impossible for spacetime to be adequately modeled by a manifold of the sort that mathematics studies (one locally homeomorphic to a power of the real number line). But why do things behave as if spacetime were such a manifold? Maybe God made them behave so because this lets us organize the world in convenient ways.

Scientific realism

Despite having a pretty good Pittsburgh education in the philosophy of science, I never before read Ernan McMullin's "A Case for Scientific Realism". I was especially struck by one thing that I had never noticed before, which Fr. McMullin briefly notes in one context: things are different, realism-wise, in regard to fundamental physics and other areas of science. The rest of this post is me, not McMullin.

Observe that the pessimistic meta-induction works a lot better for fundamental physics than for the special sciences. The meta-induction says that past theories have tended to be eventually refuted, and hence so will the present ones be. (It's really hard to make the statement precise, but nevermind that for now.) But it is false that the special sciences' theories have tended to be eventually refuted. Some, like the geocentric and heliocentric theories in astronomy and the phlogiston theory of combustion, have indeed been refuted. But many theories have stood for millenia. Here is a sample of these theories: (a) there are seasons that come in a cycle, and the cycle is correlated with various botanical phenomena; (b) tigers eat humans and deer; deer eat neither tigers nor humans; (c) rain comes from clouds; (d) herbivores run from apparent danger; (e) much of the earth's energy comes from the sun. And so on. We do not think of these as scientific theories any more because they are so venerable and well-confirmed. This means that we sometimes mistakenly assent to the inductive premise of the meta-induction because those venerable scientific theories that have not been refuted have often become common-sense and hence we exclude them from the sample.

Nonetheless, the pessimistic meta-induction seems to have some force in regard to fundamental physics: there, the change is much more rapid, and very little remains of past theories. We do sometimes get results like the "classical limit" theorems for Quantum Mechanics where we can show that the earlier theory's predictions approximated the predictions of the newer theory, but this approximation in prediction does not typically yield the approximate truth of the earlier theory. The one kind of exception we sometimes get is that sometimes a part of what used to be a fundamental theory survives, but no longer as fundamental—atoms, for instance.

Non-fundamental concepts—such as cell or season—can survive significant shifts in fundamental theories, but obviously fundamental concepts like force or particle find it much more difficult to do so. There is a kind of multiple realizability in the concepts of the special sciences (not along the metaphysical but the conceptual dimension of a two-dimensional modal semantics) which makes them more resilient.

Van Fraassen proposes we be realists about the observable claims of science and non-realists about the unobservable. This is, I think, really implausible. Van Fraassen would have us believe in ova but not in sperm, just because the ovum is large enough to be seen with the naked eye while a sperm is not. But I think there is a view in the vicinity that is worth taking seriously: that we should be realists about non-fundamental science and at least somewhat skeptical of fundamental science.

Content externalist solutions to sceptical problems

A standard solution to general sceptical problems is to move to an externalist account of content. Grossly oversimplifying, if what makes a thought be about horses is that it has a causal connection with horses, then thoughts about horses can't be completely mistaken. This sort of move might be thought to be anti-realist, though I think that's a poor characterization. If this sort of move works, then we couldn't have thoughts and yet have our whole system of thoughts be completely mistaken. And hence, it seems, scepticism is dead.
But it just occurred to me that there is a hole in this argument. Why couldn't the sceptic who accepts the externalist story about content still say: "So, if I am thinking at all, then global scepticism is false. But am I thinking at all?" This may seem to be a completely absurd position—how could one doubt whether one is thinking? Wouldn't the doubt be a thought? Yes, the doubt would be a thought. Hence, the person who doubts whether she thinks would not be able to believe that she doubts. And, of course, the person who thinks she's not thinking has a contradiction between the content of her thought and the fact of her thought, but it's not so obvious that that's a contradiction in her thought (just as a contradiction between the content of an astronomical belief and an astronomical fact need not be a contradiction in the thinker's thought). Besides, the Churchlands think that they have no thoughts, and have given arguments for this.
If I am right in the above, then the content externalist move does not solve the problem of scepticism—it simply radicalizes it. But it raises the cost of scepticism—it forces the sceptic to stop thinking of herself as thinking. And as such it may be practically useful for curing scepticism if the sceptic isn't a full Pyrrhonian, in the way a rose or some other creature that has no thoughts is. However, if the motivation for the content externalism is to solve the problem of scepticism, rather than cure the sceptic, then the motivation seems to fail. (One difference between solving and curing is this. If a theory T solves a problem, then we have some reason to think T is true by inference to best explanation. But if believing a theory T would cure someone of a problem, inference to best explanation to the truth of T is not available. Though, still, I think the fact that believing T is beneficial would be some evidence for the truth of T in a world created by the good God.)

An anti-sceptical argument schema

1. Concept C is not further analyzable.
2. If C is an actually-had (by a human being) concept that is not further analyzable, then C is possibly satisfied.
3. If C is an actually-had concept that is not further analyzable, then, probably, C is actually satisfied.
4. Someone actually has the concept C.
5. Therefore, both possibly and probably there is something that has C.

Plausible cases: causation, materiality, possibility, goodness, consciousness, numinousness, etc.

Irrealism and Tarski

According to Tarski, Schema (T), of which instances have the form:

1. "..." is true if and only if ...,

where the same text is put for the two instances of "...", is compatible with both realism and irrealism, with correspondence theory and coherentism.

Let's explore this claim. Suppose we are irrealists (nevermind that we might then prefer some other term, like "epistemicist") who have some epistemic notion of truth, e.g, a sophisticated version of the claim that S is true if and only if it would be arrived at in the ideal limit of inquiry. Abbreviate the epistemic definition of the truth of S as E(S). I will at times use the the ideal limit formulation for explicitness, but it should really be considered a stand-in for whatever more sophisticated story is to be given.

If we accept both Schema (T) and the epistemic definition of truth, then we have to accept every instance of:

2. E("...") if and only if ....

But (2) gets us into trouble. First of all, if we accept the Law of Excluded Middle (LEM)—that for all p, p or not p—then we have to accept the implausible claim that for all p, E(p) or E(~p). For many values of p, that is simply implausible for any of the epistemic versions of E. Thus, it is not plausible that in the ideal limit of inquiry we will conclude that Napoleon died with an even number of hairs on his head, and it is not plausible that in the ideal limit of inquiry we will conclude that it wasn't the case that Napoleon died with an even number of hairs on his head.

So, our irrealist who accepts (1) will, it appears, have to deny LEM. This shows that Schema (T) is not neutral between realists and irrealists. For while a realist can accept Schema (T) and either believe or not believe LEM, the irrealist is forced by the acceptance of Schema (T) to deny LEM. And if we see LEM as self-evidently true (though that remark begs the question against the intuitionists), then Schema (T) will in fact be unavailable to our irrealist.

Let us consider the irrealism further. Here is an instance of (2) (with the toy version of ideal-limit irrealism):

3. We would in the ideal limit find out that there are conscious beings in the Andromeda Galaxy if and only if there are conscious beings in the Andromeda Galaxy.

This is a startling claim. Moreover, it is a claim that is part of a large family of equally startling claims relating how things are far away and what we would find out. These claims, furthermore, are not merely accidentally true, since the characterization of truth had better not be an accident.

Let's push on further with instances of (2). For instance:

4. The ideal limit of inquiry is never reached if and only if in the ideal limit of inquiry we would conclude that the ideal limit of inquiry is never reached.

But the right hand side of the biconditional doesn't hold: in the ideal limit of inquiry we would not conclude that the ideal limit of inquiry is reached. So, the left hand side doesn't hold. Consequently, we have an a priori argument that the ideal limit of inquiry is reached. But unless one is a theist (who thinks that God has always already reached that ideal limit), it is absurd to suppose we'd have an a priori argument for that—that would yield give an atheist an a priori argument for the claim that we won't all perish tomorrow. The present example is one that cannot be leveled against irrealists who do not engage in any kind of idealization. But I suspect that non-idealizing irrealist views degenerate into relativism.

If this is all right, then in fact the irrealist cannot afford to accept Schema (T), and Tarski is wrong in thinking Schema (T) is neutral.

But non-acceptance of Schema (T) comes with a price, too. We either have to allow that truth of "There is conscious life in the Andromeda Galaxy" does not suffice to show that there is conscious life in the Andromeda Galaxy, or we have to allow that there could be conscious life in the Andromeda Galaxy, even though it is not true that there is conscious life in the Andromeda Galaxy. That is absurd. Of course, as an argument, this is question-begging.

Let's see if we can do better. If the irrealist's use of the word "truth" does not conform with Schema (T), the word "truth" does not match what seem pretty clearly to be central cases of our use of the word. Thus, when the irrealist says that "truth" depends on inquiry, the irrealist is not actually talking of what we mean by "truth", and is not disagreeing with the realist. And assuming that the irrealist doesn't say crazy things like (3) and (4), it is not clear wherein the irrealist is being an irrealist. (I would be quite happy if it were shown that irrealism is impossible.) But if the realist can give a correspondence theory of the concept of "truth" that conforms with Schema (T), then the conformity with Schema (T) would be evidence that the realist is not using "truth" in a Pickwickian sense.

To put the main points differently, epistemicism can be first and second order. First-order epistemicism affirms all the instances of (2). Second-order epistemicism affirms all the instances of

5. "..." is true if and only if E("...").

Now: (a) first-order epistemicism makes sense but is crazy, (b) second-order epistemicism together with Schema (T) leads to first-order epistemicism, and (c) second-order epistemicism without Schema (T) uses the word "truth" differently from how we use it, since our usage is governed, in part, by Schema T. The challenge for the epistemicist is either to deny that first-order epistemicism is crazy, or to show how second-order epistemicism without Schema (T) is talking about "truth".