What is the value of a formal model in science?

In science we often use formal models (by which I mean a mathematical structure composed of assumptions, variables, and equations, which might be solved/simulated to derive analytical insights and/or testable hypotheses). If the objective is to explain a certain phenomenon, one would proceed by (i) looking at data for relevant (to the modeller) stylised facts, (ii) set up a formal model that can capture those facts, and (iii) take the model to the data to test hypotheses derived from the model. This way, is often said, one is providing "support for a model".

Now, it makes perfect sense to build a model based on key insights from the world. But if that is the case, what is the real value of deriving hypotheses in order to "test the model"? I mean, by definition, WYPIIWYGO (what you put in is what you get out). Put good things, and you will get good things. So from an epistemological point of view, what is the actual benefit of a model? Is there real value added in using them?

I know this topic is probably huge in philosophy of science. There is a snippet about it in the Standford Encyclopedia of Philosophy. The answer there is however deeply unsatisfactory (at least to me). In particular, take this paragraph:

We begin by establishing a representation relation (‘denotation’) between the model and the target. Then we investigate the features of the model in order to demonstrate certain theoretical claims about its internal constitution or mechanism; i.e. we learn about the model (‘demonstration’). Finally, these findings have to be converted into claims about the target system; Hughes refers to this step as ‘interpretation’.

More particularly:

Once the model is built, we do not learn about its properties by looking at it; we have to use and manipulate the model in order to elicit its secrets.

This seems to me to be the key. A model has secrets. That's the value they add. But why is this? Is it always the case? And going back to my central question: since WYPIIWYGO, what's the merit of a model anyway? These seem not to be discussed in the link.

PS: references to great articles covering this (without needing to get into a full course of philosophy of science [not yet, at least]), are more than welcome.

• What do you mean by "Put good things, and you will get good things"? It's obviously possible to come up with models that look good at first but then lead to predictions that don't pan out, so if you are suggesting there is something circular or tautological about creating models and testing them, I don't understand what your argument for that would be. Dec 31, 2019 at 13:34
• WYPIIWYGO is false, it is a version of logical omniscience, see What is the difference between depth and surface information?. We put in the axioms of arithmetic, but still do not know if odd perfect numbers exist, even arithmetic has "secrets". It is also false in a more direct sense. A model is a template which is not fully parametrized or even fully structured. Testable consequences are needed to stage measurements that determine parameter values and structures. This is particularly obvious in string theory with its multitude of options. Dec 31, 2019 at 13:57
• @Hypnosifl Well, using "good" is not precise enough, as I would expect an input to be good if it produces a good output (the WYPIIWYGO bit). Yes, my question is about models being tautologies, and thus lacking value. A model is "good" only because it has good inputs. Dec 31, 2019 at 14:08
• @Conifold I guess my point with WYPIIWYGO is that there is complete freedom for the modeler (we can create fake realities/universes in a model). This freedom is what enable a model to be anything. How "good" it will be in explaining something (the output) depends on how it is set up (the input). Dec 31, 2019 at 14:11
• Your point is moot since the modeler does not know at the outset what the model does, and hence how "good" it is. She can not design it to do "good things" because a) they are only partially known, and b) even if they were, reverse engineering a suitable model from that is highly non-trivial. Hence the value of both investigating and testing models "from epistemological viewpoint", it produces new knowledge. Dec 31, 2019 at 15:24

Suppose that M is a formal (mathematical) model and T is a target system. Then interpretation can be thought of as a map I:M->T. You can think of M and I as of "a picture" of T. The picture (as usual) can be to some degree accurate. Nevertheless its accuracy can be verified by comparing M via I with the target system T. Now suppose that you verified your model and you believe that it is quite accurate. Then you can mathematically deduce (this can be highly nontrivial - mathematics is not easy) certain facts about M (this are as far as I understand are its secrets) and since your model is accurate (as you established before), you may hope that translation of this deduced facts via I to T will give you some new insights into how T behaves.

Here are two interesting examples, but history of science gives plenty of them.

1. Quantum Mechanics. The model here is quite complex for layman. It consists of a separable Hilbert space H, self-adjoint operators on H and a one-parameter unitary group of transformations of H. Now it turns out that under certain "interpretation" this model forms a very accurate "picture" of certain phenomena concerning measurements (like for example their probabilistic behavior). So far so good. Physicists and some mathematicians investigate formal properties of the model in order to deduce some facts (this is creative and nontrivial). Then these facts are translated back to the target system and give us some insights into how certain parts of reality work. One such prominent example is Bell's theorem, which under certain philosophical interpretation gives solid argument against local determinism. The other example is Hawking's radiation. This was theoretically deduced and its existence still haven't been tested empirically yet. Suppose that it will be falsified. This would put our physics into a serious crisis, which (as the history shows) would motivate emergence of even more accurate models.

2. Paraconsistent Logics. There is a whole variety of propositional calculi (formal systems) that tolerate contradictions. The motivation here is that ones views are often self-contradicting. This obviously does not mean that ones views are without any value whatsoever. Paraconsistent logics try to encapsulate in a formal manner this intuition i.e. the fact that although some parts of for example my body of knowledge consists of self-contradicting statements nevertheless some of my views are valuable insights into how the world is. Now given a fixed paraconsistent logic you can investigate empirically if its rules of inference adequately describe human's deductions and cognitive customs.

• Thanks, but a model might be accurate in some sense but not in all senses. This is, the dimensions you used to "verify it" is a subset of all dimensions. Just like for analogies, which are always partial. (and I guess that by verification you rather mean non rejection? Isn't it that theories can only be disproved but never confirmed?) Dec 31, 2019 at 14:14
• Yes. I think in the analogy presented above I takes care of the sense in which the model is accurate. Namely the image of I is the part of the target system that the model refers to. Nevertheless the first part of my answer is only an analogy itself :) and as you said it is only partial.
– Slup
Dec 31, 2019 at 14:17
• As for the statement that "theories that can only be disproved and not confirmed" it is hard to say what you mean by the term confirmation at this point. For instance K.Popper, who proposed falsifiability criterion, that you are probably referring to, used the term corroboration, which was supposed to mean that the theory was rigorously tested for a long period of time without any falsification. As you see this is very subjective definition.
– Slup
Dec 31, 2019 at 14:29
• @luchonacho Can theories only be disproved but never confirmed? Depends on your definition of proof, which is itself controversial and relates to the problem of induction. And yet science has led to nuclear cruise missiles and gene editing hinting that proof might be broader than deductive certainty. "Models might be accurate in some sense". No, models are definitely only accurate in some sense, and that is their strength, not their weakness.
– J D
Dec 31, 2019 at 18:52

First a point that may or may not be clear: we care about the 'target system'; the model is in many ways incidental. The target system is some system or process that exists in the real world, outside (as Wittgenstein would put it) the limits of our language. We create models to capture elements of that system in language (including the language of mathematics), and if we craft the model well then it allows us to predict and influence and use the target system. But the target system itself is always outside of language. It does what it does and it is what it is, and we are stuck in the position of having to correct our models as we learn more.

Second — and this is something that most people never really think about — we are constantly, implicitly testing models. Every time we throw a ball, shoot a gun, carry a cup of coffee, or sit up in bed, we are testing the theory of gravity. It's not formal hypothesis testing, perhaps, but it's a test of the principle nonetheless, and while these tests always seem to succeed we have to be open to the possibility that one day they might not. Someday we might find ourselves in a context where we throw a baseball and watch it take a dogleg straight up into the sky and disappear. When and if that happens, we'll have to revise our models.

This is why people do formal hypothesis testing:

• They have some critical task to perform, and they want to make sure that the target system behaves as the model predicts in their particular context.
• They are hoping that the target system behaves somewhat differently than the model predicts in a particular context, because that will allow them to create or do new and interesting things.

And so we get dedicated researchers looking for places where a models do not quite fit the behavior of target systems, because bringing those unanticipated behaviors of the target system into language allows us to use them. Consider the fact that it was hypothesis testing that transformed silicon from an ubiquitous but largely useless element (good for making glass, but that's about it) into the foundation of the modern information age. Researchers kept looking for ways our models of the atom and electrical conductivity failed to conform to the physical system; they kept revising those models and taking advantage of new nuances. And here we are.

What is the value of a formal model in science?

To some extent, this question is the basis of the divide between realist and anti-realist positions in the philosophy of science, so it's a very broad question with a lot of metaphysical territory to cover; however, some gross oversimplifications follow:

First, from Mary Hesse, on page 300 of Blackwell's A Companion to the Philosophy of Science, let's note that models can be seen as analogies that have positive, neutral, and negative aspects and that the function of the model is to work in tandem with experiment to confirm or disconfirm these aspects. In fact:

the better the model, the more of the neutral analogy will eventually be accepted as positive, whereas a poor model will become more and more negatively analagous.

Thus models serve as guides to empirical ongoings. Beyond that, if the emphasis of models is on the degree of their formality, that is their syntactic nature manipulated devoid semantic content, then a formal model in science is essentially a collection of propositions to which logic is applied to generate propositions and hence prognostications and explanations about that which the model models. In science, which is often considered an epistemic tool built from skepticism, rationalism, and empiricism and whose success is based on its record of explanation and facilitation of problem-solving, the tools are essentially defeasible logic as embodied in different variations of a generalized model itself known as the scientific method.

What makes formal models powerful is that they are generally seen to possess a higher degree of ontological committment than natural language versions. This is because the dominant Quinean view regarding ontological commitment is that natural language possesses ambiguity and is not generally ontologically serious. Thus a formal model is more precise regarding which entities exist and can be wedded to various criteria which (purports) to increase the certainty about which we can make claims. Once a formal model is built, it is also manipulated by various logics many of which have deductive certainty. So, in the spirit of Galileo or Newton, when a model is built, say of a system of billiard balls, it is a tool for making predictions and gauging the certainty of predictions. Differential equations to predict behavior and statistical measures of certainty such as confidence intervals are extremely valuable in the natural sciences and have led to such accomplishments such as putting astronauts on the moon and the design of nuclear reactors.

A model has secrets. That's the value they add. But why is this? Is it always the case?

Why does a model have secrets? Because models provide relatively certain answers (explanations and predictions) in the face of combinatorial explosion, and oftentimes, the fruits of working a model arrive at propositions that are counter-intuitive and deductively certain. In addition, certain mathematical branches of theory speak to the nature of what is knowable as true independent of meaning. A prime example of this is Gödel's incompleteness theorems which speaks to the nature of deductive reasoning with arithmetic which is highly relevant in mathematized physics. Relativistic physics, which displaced Newtonian physics, is a classic example of how mathematical models of time-space introduce new ideas that lead to metaphysical implications that would otherwise seem preposterous, but are nonetheless empirically and rationally certain. Formal models led to the discovery that time dilates. This prediction was confirmed by evidence and is now an important scientific fact in the application of satellite technology and GPS, for instance.

Since the questioner mentions the Philosophy of Science, here is a response to 'What role does Philosophy play in the realm of Science?' Your creating models conundrum gives voice to the concern that science performed for its own sake, with no limits or guidelines is problematic. At the moment there does not appear to be any systematic, science based paradigm which could serve as a platform for making informed decisions on what directions scientific research should move into,capably. This is not intended to constrain research, but rather to inform it.

What follows is an excerpt form “The Foundations of Metaphysics in Science” Errol E Harris

“We have long had it dinned in our ears that metaphysicians have no right to pose as superior savants with means of discovering the nature of the real exceeding those of the natural scientist in scope and perspicacity. Nor have metaphysicians any such means. The nature of the world cannot be deduced ‘ex nihilo’, but can be discovered only by patient and careful investigation and research, involving, in its proper context, experiment and observation. Even if such knowledge could be divinely revealed, scientific investigation would still be necessary to test the authenticity of the revelation. It can therefore be only through the natural sciences that a comprehensive knowledge of the world can be approached. But none of the special sciences aspires to a conspectus of the entire field, for each special scientist is inevitably immersed in the interconnected details of his own branch, for the study of which he must acquire special techniques. Interdependence between the sciences, intimate though it is, has not led to a super-science combining all of them, in all their ramifications, into a single discipline, and it is doubtful if any such super-science is even possible. The need accordingly remains for the metaphysician’s effort to see things together, as Plato recommended- not to correct, outdo or modify the pronouncements of science, but to reflect upon them, to develop their implications and mutual interconnections, examine their presuppositions, and to form as complete and systematic a conception of the world as the available evidence permits.” PP. 29

@amalloy- If these were my words then your edits would be welcomed. But this excerpt is from a text by a most respected philosopher of science. He has written dozens of books and hundreds of papers on science, the History of science and the philosophy of science. No offense intended on my part, but I am not even sure what you are trying to accomplish, but thanks for your work. CMS