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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?

Oliver Amundsen
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  • @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? – Oliver Amundsen Feb 09 '18 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](https://plato.stanford.edu/entries/simplicity/) of a theory is often cited as one such virtue. Does the discussion there seem relevant? – Dennis Feb 09 '18 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? – Oliver Amundsen Feb 09 '18 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](http://shelah.logic.at/class/) 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.... – Dennis Feb 09 '18 at 20:47
  • ..."Fine grained" theories will be able to assign distinct meanings to more expressions, but might be thought to overgenerate since they're often sensitive to differences in syntax that don't seem to distinguish the expressions semantically. "Coarse grained" theories, on the other hand, often have "too few" propositions to assign to expressions. A typical issue with the more coarse grained approaches is that every logical truth gets assigned the same proposition and so has the same meaning. I could see your example as describing the same phenomena with different levels of granularity. – Dennis Feb 09 '18 at 20:52
  • @Dennis, I wasn't familiar with this field. It does sounds interesting though. Will see if here's any applications to a simple and complex models to see if it may be applicable. BTW, fine and coarse grained theories seems to express well the difference between a theory with one parameter of one with 1000 parameters... – Oliver Amundsen Feb 09 '18 at 20:52
  • Sorry, I was a bit unclear. The theories I was referring to were theories of propositions and it's the _propositions_ that are fine or coarse grained. (I suppose you could make a derivative distinction among theories based on the kind of propositions used to interpret the theories.) For example, coarse propositions typically assign the same meaning to "a euclidean triangle has 3 sides" and "a euclidean triangle has 3 angles" since they have the same truth-conditions. A fine grained theory of propositions might capture a difference between "side" and "angle" to give them different meanings. – Dennis Feb 09 '18 at 22:19
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    In your examples "theory" amounts to a collection of computational formulas. [Computational complexity](https://en.wikipedia.org/wiki/Computational_complexity_theory) can be measured even for more complex algorithms. One can also measure *logical* complexity of formalized sentences, the simplest case is [arithmetical hierarchy](https://en.wikipedia.org/wiki/Arithmetical_hierarchy), see also [descriptive complexity theory](https://en.wikipedia.org/wiki/Descriptive_complexity_theory) . – Conifold Feb 09 '18 at 23:21

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