Constitutive principles for physical theories
TL;DR: Constitutive principles are about what science is and provide a stronger foundations for physics
Last week, I attended the Midwest Philosophy of Physics Meeting. Don Howard from the University of Notre Dame gave a historical presentation on Einstein’s distinction between principle and constructive theories. The general idea is that a constructive theory is defined by postulating the existence of certain entities and describing their behavior. For example, it might state that there are particles with charge that interact according to some set of equations. A principle theory, instead, starts with statements that are experimentally well-founded and builds from there. The principle of relativity, the invariance of the speed of light, and the equivalence principle (between gravitational and inertial forces) are examples of such principles. In Einstein’s view, ultimately all physical theories are constructive theories, but principle theories play a more fundamental role as they are needed to guide the constructive theories. If we adopt that view, a natural question to ask is: do we have a good set of principles to guide us to solve problems like quantum gravity?
The key issue is that we are trying to construct a theory for a regime we have no direct experience of. So, how are we going to know whether those evidential principles will apply in those cases? Moreover, not all principles we have are well-formulated in terms of physical ideas. For example, all current physical theories work in a Lagrangian framework, which assumes that the correct way to formulate a theory is through the principle of stationary action. How can we know whether this will still apply or not? In our Reverse Physics approach, we show that, at least in classical particle mechanics, the principle of least action is derived from the assumptions of determinism, reversibility, and the independence of all degrees of freedom. Maybe it is not true that all the information needed for a deterministic and reversible description is physically accessible. Maybe it is not true that degrees of freedom are completely independent. In this case, the principle of least action would not apply and would need to be replaced with something else.
So, if evidential principles are not a good guide, what could we use? In our project, we have found that constitutive principles have a lot more power than one might expect. A constitutive principle is a statement that one has to accept as true, not because of experimental evidence, but because it is something needed to define and proceed with the task at hand. For example, “science is about experimental evidence” is a constitutive principle. We do not find that science is about experimentation…experimentally. It is how we distinguish science from other disciplines. Now, this seems like a very mild assertion, but it is actually very powerful. It means that science does not deal with mere statements, but with statements that are associated with a test. While a statement can simply be true or false, a test may succeed, fail, or may not terminate. Not all statements, then, are experimentally verifiable. For example, right now, we believe there are six quarks. Technically, we know there are “at least” six quarks. We cannot exclude the possibility of there being more. And it will always be the case: if we find two more, then experimentally we will only be able to say that there are “at least” eight. However, we can already exclude the existence of only five. The logic of these statements, then, is slightly different from the logic of standard statements. It turns out that if one studies this logic, he can recover notions of topologies and sigma algebras, which are fundamental mathematical structures at the foundation of all other mathematical structures we use in physics. So, the fact that “science is about experimental evidence” tells us that a set of physically distinguishable cases is a \(\mathsf{T}_0\) second countable topological space. The constitutive principle constrains the math.
Another constitutive principle is that “physical laws are reproducibly testable.” That is, physical laws are not about a particular instance at a particular time, but about setups that can be replicated over and over. Physical laws, then, are really about collections of results: whenever I do this, I get that. Moreover, reproducibility means we can always test the laws “one more time,” which implies that these collections are infinite. In other words, the mere idea of a physical law implies the notion of a statistical ensemble. These statistical ensembles must be experimentally well-defined (i.e. points in a \(\mathsf{T}_0\) second countable topological space), allow for statistical mixtures (i.e. form a convex space), and must have an entropy that represents the variability of the preparations within the ensemble. It turns out that these requirements, by themselves, are enough to recover vector spaces, a generalization of probability measures, geometrical structures… all of these are implied simply by saying that “physical laws are reproducibly testable,” without mentioning whether we have classical or quantum objects, particles or fields.
Focusing on constitutive principles, then, allows us to reach a different type of unification: a general mathematical framework for scientific theories that must apply equally to all of them. It allows us to generalize results and differentiate what “happens to be true” within a set of theories from what “must be true” in all physical theories. If our reasoning is correct, rejecting these results means rejecting that “science is about experimental evidence” or that “physical laws are reproducibly testable.” It is not rejecting the interpretation of a certain class of experimental data collected in some circumstances, which, at least in principle, is open to debate. It is rejecting the very nature of science and physics. Constitutive principles, then, are a much sturdier foundation for both the physics we have and the ones yet to be discovered.