What if there is no true data-generating process? I've been engaging in a number of forecasting efforts recently, and have rediscovered a well-known truth: That combinations of different forecasts are generally better than the forecasts themselves. In particular, the unweighted mean of forecasts is typically better than any of the averaged forecasts. So far, in my own work, I have not encountered any exceptions to this except when the data is artificially generated from simple models.
I was, and remain, flabbergasted by this. Why should averaging models based on entirely inconsistent assumptions generate anything but nonsense? Why is the unweighted average of the best model with relatively inferior models usually better than the best? Why do we seem to get most of the benefits of sophisticated ensemble methods from the unweighted mean?
I always thought that the modeling process was intended to find the model that most nearly approximated the underlying reality, imperfectly, of course, but still assuming that there would always be the best model given specified constraints of parsimony, data availability, and the like. To me, the fact that the unweighted mean of a more-or-less arbitrary collection of model types (that experience has taught us are pretty good) does not suggest that the true model is roughly the mean of the constituent models---that would be absurd.
Instead, it suggests that there is no true data-generating process that can be approximated by any standard estimating technique, however sophisticated. The data may be generated as some complex summation or composite of many, many agents or sub-processes, each of which or who embodies a unique complex of causal forces, perhaps including multiple layers of non-linear feedback. Perhaps they are influenced or entrained by common exposure to forces that you as a modeler will never see, like the boss's mood or the ionization level in the air or irrational remnants of historical institutional structures that persist and still affect decisions.
You see this in other ways too. For example, sometimes the theory is utterly unambiguous about which models are to be preferred. It is, for example, entirely clear that most macroeconomic variables modeled by VARs or VECMs should be logged or log-differenced for multiple compelling reasons, both statistical (i.e. to avoid heteroskedasticity, to linearize any trend present), and economic. Except when you actually run such models, the opposite is true. I have no idea why.
My question is this. Has anyone found a way of formalizing the belief that processes we strive to understand have no data-generating process that we can capture in a standard mathematical model? Has anyone attempted to describe the foundations of statistics based on such a formalization -- a statistic in which all models are unavoidably misspecified? If so, does it have any known implications for hypothesis testing, and the sort of test-and-redesign process that constitutes normal workflow for a statistician or a data scientist? Should we be multiplying models earlier in the analysis process? If so, how? Should we be choosing which models to aggregate based on some principle other than the quality of fit with a complexity penalty, or some model comparison test like AIC? As things are designed ultimately to be input to ensembles, should we prioritize models that give different predictions, rather than good predictions? Is there a principled way to make such trade-offs?
And if this is the norm, why isn't it in any of the six widely-used introductory statistics texts I went through when composing this post?

Looking at it the other way, if there were no true data generating process, how did the data get generated?
The inability of standard estimating techniques to accurately approximate the true data-generating process doesn't mean that the data generating process doesn't exist, it just means that we don't have enough data to determine the parameters of the model (or more generally the correct form of the model).
However, when we make a model, our goal is not to exactly capture the true data generating process, only to make a simplified representation or abstraction of the important features of the true data generating process (TDGP) that we can use to understand the TDGP or to make predictions/forecast of how it will behave in some situation we have not directly observed.  Our brains are very limited, we can't understand the detail of the TDGP, so we need abstractions and simplified models to maximise what we are able to understand.
Rather than say there is not TDGP, I would say there is no such thing as "randomness" (except perhaps at a quantum level, but even that might not be random either, although the Bell experiment suggests it probably is).  We use the concept of "random" to explain the results of deterministic systems that we can't predict because of a lack of information.  So the purpose of a statistical model is express our limited state of knowledge regarding the deterministic system.  For example, flipping a coin isn't random, whether it comes down heads or tails is just physics, depending on the properties of the coin and the forces applied to it.  It only seems random because we don't have full knowledge of those properties or forces.
At the end of the day, the more data we have, in principle the more information we can extract from it (with diminishing returns), and the better our state of knowledge about the TDGP.
The reason averaging helps is that the error of the model is composed of bias and variance, c.f. @Tim's answer (+1).  If we don't have much data, the variance component will be high, but that variance will not be coherent for models trained on different samples, and so will partially cancel when model predictions are averaged.  This is not telling you anything about the TDGP, it is telling you about the estimation of model parameters (and that you should get more data if you can).
