Building on the theme of no unnecessary assumptions about the world, @BlackBrane suggested … a position I had not considered before … for entertaining the possibility of a mathematical universe:
[Box’s slogan is] an affirmative statement about Nature that might in fact not be true. Who’s to say that at the end of the day, Nature might not correspond exactly to some mathematical structure? I think the claim is sometimes guilty of exactly what it tries to oppose, namely unjustifiable claims to absolute truth.
I suspect that we won’t learn the answer, at least in my lifetime.
In a sense, the appropriate answer is “who cares?” Whether or not there can in principle be perfect models, the real problem is finding ones that are useful in practice. The slogan isn’t helpful for this. (NIPCC authors seem utterly clueless as well.)
In a related post, AK identifies a 3-part typology of models that suggests a basis for guidance:
- “Insilications – In physics, we are used to mathematical models that correspond closely to reality. All of the unknown or system dependent parameters are related to things we can measure, and the model is then used to compute dynamics, and predict the future value of these parameters. …
- Heuristics – … When George Box wrote that “all models are wrong, but some are useful”, I think this is the type of models he was talking about. It is standard to lie, cheat, and steal when you build these sort of models. The assumptions need not be empirically testable (or even remotely true, at times), and statistics and calculations can be used to varying degree of accuracy or rigor. … A theorist builds up a collection of such models (or fables) that they can use as theoretical case studies, and a way to express their ideas. It also allows for a way to turn verbal theories into more formal ones that can be tested for basic consistency. …
- Abstractions – … These are the models that are most common in mathematics and theoretical computer science. They have some overlap with analytic heuristics, except are done more rigorously and not with the goal of collecting a bouquet of useful analogies or case studies, but of general statements. An abstraction is a model that is set up so that given any valid instantiation of its premises, the conclusions necessarily follow. …”
The social sciences are solidly in the heuristics realm, while a lot of science is in the insilication category. The difficulty is knowing where the boundary lies. Actually, I think it’s a continuum, not a categorical. One can get some hint by looking at the problem context for models. For example:
| Known state variables? | Reality Checks (conservation laws, etc.)? | Data per concept? | Structural information from more granular observations or models? | Experiments? | Computation? | |
| Physics | yes | lots | lots | yes | yes | often easy |
| Climate | yes | some | some | for many things | not at scale | limited |
| Economics | no | some | some – flaky | microfoundations often lacking or unused | not at scale | limited |
(Ironically, I’m implying a model here, which is probably wrong, but hopefully useful.)
A lot of our most interesting problems are currently at the heuristics end of the spectrum. Some may migrate toward better model performance, and others probably won’t – particularly models of decision processes that willfully ignore models.