(Soft Question) Real World Modeling as Understood Through Pure Math

Question

This question is necessarily vague; I'm not looking for an answer so much as I'm checking to see if this is something that has been thought about/discussed before, and if there are any resources out there about it. Apologies if the question is better suited for another forum.

It's common in science/engineering to mathematically model the real world, with mathematical variables standing for some quantity. We might say, "let $a$ be the number of apples in the basket", or "let $h$ be the height of the object falling to the floor". We then discuss relationships between the values of these variables, for example writing $h = 10 - 4.9t^2$ to show how the height of the falling object is a function of time $t$. In the applied-math view, variables come first and functions capture the relationships between variables. This leads to notation, like $dh/dt$ for a derivative, that stresses the variable names and suppresses the function names.

Now I come from a pure math background, where variables and functions are treated much differently. In the language of pure math, variables represent an arbitrary general value, and are interacted with via the symbols $\forall$ and $\exists$. In pure math, the phrase "$x = 4$" is not a single claim (like it would be in a mathematical model that uses $x$ to represent something real) but rather a family of claims indexed by $x$, which is a stand-in for any element of some universal space. In pure math the only things that "exist" are sets, and therefore functions, so when we speak pure-math, the functions come first, and the variables are dummies. This is opposite to the treatment of variables and functions in applied math, and leads to the use of symbols, like $f'$ for a derivative, that stresses the function and suppresses the variables.

This difference in philosophy gives me a headache when I try to understand applied math, so I'm curious if there's a way to generate the applied-math perspective entirely within the pure-math perspective. In general I'm not sure how this would be done, but I've noticed that in statistics the situation is nice: Variables in a statistical model are random variables, which are actually functions (in a pure-math sense) from some sample space. When a statistician says "let $A$ be the number of apples in the basket", we can imagine them saying "there is a set $\Omega$ of all states of reality, and we as observers can agree on a unqiue function $A$, that assigns to each $\omega \in \Omega$ a number $A(\omega)$ that we call 'number of apples in the basket for state $\omega$'". I like this perspective, and wonder if it can be applied in general.

Has this type of thing been discussed before? It's certainly difficult to search.

Answer 1

This is only a tangential comment, but there is a chance that it will be helpful. Terence Tao once made some comments about how mathematical modeling works that I found to be very illuminating (even if it seems obvious in hindsight). Tao posted this long ago on Google Plus (which no longer exists) and I think it deserves to be shared more widely. I'll post here on the off chance that it strikes a chord with you:

Terence Tao - @Pietro: the way mathematical or physical models work, one assumes the existence of a variety of mathematical quantities (e.g. forces, masses, and accelerations associated to each physical object) that obey a number of mathematical equations (such as F=ma), and one also assumes that the result of various physical measurements can be computed in terms of these quantities. For instance, two physical objects A_1, A_2 will be in the same location if and only if their displacements x_1, x_2 are equal.

Initially, the numerical quantities in these models (such as F, m, a) are unknown. However, because of their relationships to each other and to physical observables, one can in many cases derive their values from physical measurement, followed by mathematical computation. Using rulers, one can compute displacements; using clocks, one can compute times; using displacements and times, one can compute velocities and accelerations; by measuring the amount of acceleration caused by the application of a standard amount of force, one can compute masses; and so forth. Note that in many cases one needs to use the equations of the model (such as F=ma) to derive these mathematical quantities. (The use of such equations to compute these quantities however does not necessarily render such equations tautological. If, for instance, one defines a Newton to be the amount of force required to accelerate one kilogram by one meter per second squared, it is a non-tautological fact that the same Newton of force will also accelerate a two-kilogram mass by only one half of a meter per second squared.)

If one has found a standard procedure to compute one of these quantities via a physical measurement, then one can, if one wishes, take this to be the definition of that quantity, but there are multiple definitions available for any given quantity, and which one one chooses is a matter of convention. (For instance, the definition of a metre has changed over time, to make it less susceptible to artefacts.)

In some cases, it is not possible to measure a parameter in the model through physical observation, in which case the parameter is called "unphysical". For instance, in classical mechanics the potential energy of a system is only determined up to an unspecified constant, and is thus unphysical; only the difference in potential energies between two different states of the system is physical. However, unphysical quantities are still useful mathematical conveniences to have in a model, as they can assist in deriving conclusions about other, more physical, parameters in the model. As such, it is not necessary that every quantity in a model come with a physical definition in order for the model to have useful physical predictive power.

So I think of mathematical modeling like this, for example. Let's say we're modeling an object hanging from a spring. Let's introduce or make up the following mathematical quantities:

• A function $x$ that we think of as telling us how stretched out the spring is at time $t$.
• A number $m$ that we think of as telling us how difficult it is to accelerate the object.
• A function $f_{\text{spring}}:\mathbb R \to \mathbb R$ that we think of as telling us how hard the spring is pulling on the object at time $t$.
• A function $f_{\text{gravity}}:\mathbb R \to \mathbb R$ that we think of as telling us how hard gravity is pulling on the object at time $t$.
• A function $f_{\text{drag}}:\mathbb R \to \mathbb R$ that we think of as telling us how hard the air is pushing on the object at time $t$.

And let's make up the following assumptions:

• There exists $k \in \mathbb R$ such that $f_{\text{spring}}(t) = -k x(t)$ for all $t$. (This is Hooke's modeling assumption.)
• There exists $g \in \mathbb R$ such that $f_{\text{gravity}}(t) = -mg$ for all $t$.
• There exists $\gamma \in \mathbb R$ such that $f_{\text{drag}}(t) = -\gamma x'(t)$ for all $t$.
• $m x''(t) = f_{\text{spring}}(t) + f_{\text{drag}}(t) + f_{\text{gravity}}(t)$ for all $t$. (In other words, acceleration is proportional to the sum of all the forces pushing on the object, and $m$ is the constant of proportionality. Thanks to the great mathematical modeler Newton for this idea.)

From the above assumptions (which we made up), it follows that $$ m x''(t) + \gamma x'(t) + k x(t) = -mg. $$ We can solve this differential equation to find a formula for $x(t)$.

(If we were slightly more clever, we could have set things up so as to obtain a homogeneous differential equation. That's what is done in standard treatments of spring-mass systems.)

To get the values of the constants $m, \gamma$, and $g$ (which are parameters in our model) we will have to make some measurements and then choose or tune the values of those parameters so that the predictions of our model agree as well as possible with our measurements. (These days machine learning engineers spend all their time tuning model parameter values, but physicists have been playing this game far longer.)

Of course, we wouldn't expect our model to make perfectly accurate predictions. After all, we just made this up. The miracle of physics is that simple models like this often make extremely accurate predictions. Mathematically modeling the physical world has turned out to be an amazingly fruitful research program, because it works far better than we had any right to expect.

Answer 2

A rough way to understand "variables" in an application to, say, physics, is to imagine that the set of possible states of the system is a smooth structure - a manifold. Then the variables are functions (real or vector valued) whose domain is the manifold. If this is a system of two particles then some useful functions might be the position of the first particle, its velocity, the total energy of the system,...

Then you study relations among those variables/functions - for example, the velocity is the derivative of the position with respect to time, energy is conserved, ...