# Metrics of the complexity/richness of theories

Are there methods proposed to "measure" the complexity of a theory? either quantitatively or qualitatively. Let me explain with an example: I'd say that theories of the type of Hooke's law:

F = k.x

in which "the force (F) needed to extend or compress a spring by some distance X scales linearly with respect to that distance" Wiki. would be the simplest in the scale because it only contains a simple causal link.

On the other hand, plate theory with its Partial Differential Equations would be much more complex on the scale, since it has many more causal links and interrelations.

I've checked Rescher's Epistemetrics book, has interesting insights, but couldn't find something that addresses this. Any ideas?

• @PédeLeão thanks for the document. Interesting. I've used this in practical problems, but didn't know it could be used to compare theories. Do you know of any applications for this? Feb 9, 2018 at 19:16
• Do you mean “theory” in the mathematical/logical sense? Or a more informal sense that might apply to your average philosophical theory? One thing you might mean is something like “fewest concepts”. In that case, would you consider a theory with few primitive concepts but many complex concepts constructed from these simpler or more complex than one with a fewer number of concepts but more primitives? In discussion of “theoretic virtues”, Simplicity of a theory is often cited as one such virtue. Does the discussion there seem relevant? Feb 9, 2018 at 20:28
• @Dennis, I was mostly thinking in a physico-mathematical theory. So for instance, we can calculate the displacement of a bolt, subjected to the action of a force, either with (1) Hooke's law which will give a single value, or (2) with a much more sophisticated finite elements method approach which will require many more parameters be fed into the theory, but will give more detailed results as the displacement field of the same bolt. Am I clear? Feb 9, 2018 at 20:32
• @OliverAmundsen Unfortunately I won't be of much help on the physics front. On the mathematical side, you'd be looking at topics in model theory. Increasing the number of parameters would correspond to augmenting the signature of the language with new symbols of the appropriate type. Depending on your interests and mathematical background, Shelah's book on classification theory is a classic in the model theoretic literature. But I'm not sure if that's what you're after. There is a notion of "granularity" of propositions.... Feb 9, 2018 at 20:47
• In your examples "theory" amounts to a collection of computational formulas. Computational complexity can be measured even for more complex algorithms. One can also measure logical complexity of formalized sentences, the simplest case is arithmetical hierarchy, see also descriptive complexity theory . Feb 9, 2018 at 23:21