Scientific realism about mass

While I’ve grown up as a scientific realist, and been trained as one as a philosophy graduate student, and I suppose I still identify as one, I’ve been finding it more difficult to say what scientific realism claims.

For instance, what does it mean to be a realist about mass in a Newtonian context? A naive thought is that for each physical object, there is a positive real number, the mass of the object, which mathematically enters into the laws of nature such as F = ma and F = Gm₁m₂/r². But that seems to commit one to there being some odd objective facts, such as to which objects have the property that the square of their masses is less than their mass—a property that barely seems to make any sense, since normally in physics, we don’t compare masses with squares of masses, as they are measured in different units.

A more sophisticated thought is that there is a determinable mass, and a family of determinates, with various mathematical relations between them, with the family isomorphic with the positive real numbers with respect to the relations, but without necessarily a single isomorphism being privileged. But this more sophisticated thought is much more philosophy than physics: physicists hypothesize entities like forces and particles and the like, but not such entities like determinables and determinates. Indeed, this approach commits one to the denial of nominalism, and surely realism about mass in a Newtonian context shouldn’t commit one to such a controversial metaphysical thesis.

Is there some alternative? Maybe, but I don’t know.

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.

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.

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.

Evolution and scientific irrealism

Consider the following two statements:

1. We do not have good reason to believe evolutionary theory to be true.
2. Scientific irrealism holds.

Now claim (2) entails that science does not give us good reasons to believe propositions to be true. Moreover, the following claim is uncontroversial:

3. All the good reasons for believing evolutionary theory to be true are scientific in nature.

Thus, (2) together with the uncontroversial (3) entails (1).

But here is an oddity about discourse in our society: There is a lot more outrage against scholars who assert (1) than against scholars who assert (2).

Is there a justification for such a differential attitude?

An explanation for the differential attitude is that those who assert (1) frequently are motivated by religious considerations, while those who assert (2) are rarely motivated by religious considerations (unless they accept occasionalism, like many Muslims, or they are led to (2) by way of (1)). But unless one has a good argument for why it is inappropriate to accept or deny a scientific claim on religious grounds, this explanation of the differential attitude is no justification. Certainly it isn't be a necessary truth that it is inappropriate to affirm or deny scientific claims on religious grounds, unless necessarily God doesn't exist: for if God exists, then he in principle could reveal facts that are of purely scientific interest, or facts of religious interest that entail facts of scientific interest.

Maybe, though, the explanation is like this. If someone asserts (1) by itself, we assume that she doesn't hold (2) (just as someone who says that Elbonians are not human is assumed to think non-Elbonians are). But in fact the only good reason for holding (1) is (2). However, simply the fact that someone believes something for a bad reason surely doesn't justify the kind of outrage that is involved here. After all, one might believe (2) for very bad reasons indeed.

Personally, I deny (2). As for (1), my views are rather complex—I accept common descent and natural selection as a major force, I accept that Behe-Dembski style arguments fail to establish Intelligent Design, but I am also convinced that we do not know that every event in the evolutionary history of every animal was naturalistic.