Non-Empirical Confirmation Of Theories

Difficulties

“Fundamental physics today faces the problem that empirical testing of its core hypotheses is very difficult to achieve and even more difficult to be made conclusive,” says Richard Dawid in his paper “The Significance of Non-Empirical Confirmation in Fundamental Physics“. In olden days “it was plausible to focus on empirical confirmation as the only reliable basis for assessing a theory’s viability.”

That’s so: if a theory says X would happen or was likely, given conditions E, and if E is seen and X was not, the theory is doubted. But that if E is seen and so is X, confidence in the theory is bolstered. If your theory insisted the sun would rise in the east on days ending in ‘y’, you had lots of empirical backing; but if it was west instead of east, you did not.

Sunrises are easy to see. Strings inside quarks are not; nor are multiverses. So if you said every day ending in ‘y’ creates a certain special kind of new universe with slip-knotted strings, orthogonal in every way to our own, there is no way to check your theory. Well, this is the problem with multiverses of any kind. Nobody can see them. Are they real? Which is to ask, is the theory which predicts them true?

String theory has been playing the role of a well established approach towards a universal theory of all interactions for over three decades and is trusted to a high degree by many of its exponents in the absence of either empirical confirmation or even a full understanding of what the theory amounts to…Multiverse scenarios in the eyes of critics raise the question to what degree they can be endorsed as scientific hypotheses at all, given that their core empirical implications to a large extent seem not empirically testable in principle.

Here comes the point. He says “a considerable degree of trust in an empirically unconfirmed theory could be generated based on ‘non-empirical theory confirmation’. Non-empirical confirmation denotes confirmation by evidence that is not of the kind that can be predicted by the theory in question, i.e. that does not lie within the theory’s intended domain.”

About string theory:

In the absence of empirical confirmation, exponents of the theory may rely on different kinds of reasoning. For example, they may argue that the theory is supported by the striking difficulties to come up with promising alternatives. Those difficulties clearly cannot be predicted by string theory itself. The observation that those difficulties exist is a contingent observation about the research process of which the development of string theory is just one part. Therefore, this observation does not constitute evidence within string theory’s intended domain. If one concludes, as I will, that the observation amounts to confirmation of string theory nevertheless, it can only be non-empirical confirmation.

The question is this: can there be non-empirical verification of theories? Which is to ask, can we know a theory is true before witnessing the predictions from the theory? The answer is yes, sometimes.

Theory Truth & Probability

Here’s how I see it. A theory—or model, there is no difference, so I’ll use M as shorthand and not T, which in logic often stands for truth—is a collection of a list of premises or propositions, which are all taken jointly, as one complex proposition, like this:

    Pr( M | P_1 P_2 P_3 … P_m) = 1

The collection is “anded” together, where each component proposition may be very complicated indeed. Some P_i may be observational, others may be mathematical, still more other kinds of assumptions. The model M is deduced from these propositions; thus, given the propositions are true, so is the model.

All models, barring mistakes in calculation or logic, are locally true: true conditional supposing their conditions. Whether a theory is universally true is the real question. A theory is universally (or necessarily) true given all its P_i are themselves universally true, meaning true following a chain of argument to a base which is known to be true based on sense impression. All mathematics and logic follows this rule; so will we.

We can’t tell if M is false or uncertain from the P_i, because we use these to create the model or theory. So if we want to say M is false or uncertain, we have to bring in external evidence E. E itself is usually composed of propositions.

Now we can do things like this

    Pr( P_j | E ) = p_j

We cannot insist each p_j can be quantified, though some can. For instance, if P_j is a known mathematical theorem, the “known” part is in the E somewhere, so that we deduce Pr( P_j | E ) = 1. If some P_j is an observation proposition, such as “X_j = (17, 32, … 50)”, then unless there were measurement error, we usually say Pr( P_j | E ) ~ 1.

If we knew that for some k Pr( P_k | E ) = 0, then

    Pr( P_1 P_2 … P_k … P_m | E ) = 0

And therefore

    Pr ( M | E ) = Pr( P | E) = 0,

since M is deduced from P = “P_1 P_2 … P_m”. We have proven the model is false, even though Pr(M|P) = 1 (notice the absence of E). That Pr(M|E) = 0 is not a proof that

    Pr (Y in s | M ) = q

is false. What this means is that probability statements conditional on assuming M is true are themselves true. Since M has no implicit propositions about human computational error, and indeed has the opposite, any statement we deduce from the model is true, assuming the model is true.

Yet, of course, if the model is false, because we know based on everybody-accepts knowledge E that P is false, then M must be false, too. The point is that

    Pr (E | sense impressions; deeper truths ) = 1.

Meaning that we just accept E as true, based on our intuitions, or via other axiomatic truths. Of course, if some do not accept E, then they are working with something else; thus it will be no surprise they could come to different judgments of M.

In general, assuming no Pr(P_j|E) = 0, which will likely be the case in most theories under consideration, calculating Pr( P | E ) = Pr( P_1 P_2 … P_m | E ) would be hideously difficult. If it could be calculated at all — not all probability can be quantified!

The strategy, then, if we want to have some idea of the non-empirical confirmation of M is to attack the problem in pieces, if possible. For instance, we might be able to peel off a set of P_T = (P_a P _b … P_z) that we know conditional on E are true. These might be the mathematical derivations, some observations, some basics of logic, and the like. Then we’ll be left with

    Pr(M|E) = Pr( P_NT P_T | E ) = Pr(P_NT | P_T E) Pr (P_T | E)

and so

    Pr(M|E) = Pr(P_NT | P_T E)

since Pr (P_T | E) = 1. Interestingly, it does not matter how large P_T is: it could contain a million propositions, or only one. If it’s true, it’s true: and that it is true adds nothing to the truth or falsity of the model as long as there are any P_NT in the model! Since it is likely that the P_NT are logically orthogonal to P_T; any statements in P_NT that are deducible from statements in P_T we can lump in P_T.

The perhaps surprising implication is that piling on true evidence to a model does nothing to help improve its veracity. Neither, then, does removing truths detract from a model’s veracity. This is because

    Pr(M|E) = Pr(P_NT | E).

That means that if P_NT contains just one proposition which gives, say, Pr(P_NT | E) = 0.1, and that Pr(P_T = P_a P _b … P_z … Pr_aa … Pr_z…z)| E ) = 1, so that Pr(M|E) = 0.1!

The joint proposition P_NT will contain all those propositions which we know are not false and we know are not true, conditional on E. So, if we can make a stab at calculating, or approximating, or guesstimating Pr(P_NT | E) we have what we wanted: non-empirical “confirmation”, where “confirmation” is taken in the sense of having less than full proof.

Well, this is just a sketch of how to do non-empirical verification. Real-life efforts will come down to disputes over E. For instance, scholastic versus Humean notions of cause, about potentia and actual versus “potential worlds”. Meaning, I think, lack of agreement on whether theories have been non-empirically verified. After all, how many agreements are there on fundamental “theories” or philosophy, even though we’ve been working at it for thousands of years?