Mathematical Models vs Comparative Quantitative Reasoning
How we use mathematics in our everyday thinking
One point of contention among practitioners of different “schools” of economic thinking and analysis, is the role and proper application of mathematics. The ultimate way to apply mathematical analysis is through the application of a mathematical model.
What is a Mathematical Model?
A mathematical model is essentially a setting up a problem statement or description of a system, where all the variables and their relationships are explicitly defined in quantitative terms. While this sounds like philosophical determinism, computer science demonstrates how a deterministic and non-deterministic system can both be defined rigorously and completely, and even converted between each other. A “regular expression”, used for searching for matching variations of strings or words in text, is just a non-deterministic finite state machine.
One issue in philosophy that is sometimes overlooked, is that randomness is not the same thing as causality and arbitrariness. Online philosophy personalities like Sam Harris and Alex O’Connor have commented that the universe is either deterministic or random. But here they are combining two ideas of how the word “random” is used. In mathematics a random variable still has a known and well defined relationship or distribution. In more casual language the word “random” means something slightly different, that one thing is seemingly unrelated to another thing or arbitrary. So we can distinguish between randomness and arbitrariness. Randomness is a more specific kind of arbitrariness where we are able to describe a range of possible outcomes.
What Sam and Alex are tacitly assuming, is that the universe in reality is equivalent to some mathematical model, even if that mathematical model is unknown to us at this point in time, or even impossible to determine. A mathematical model is simply a complete description of a system in terms of its variables and quantitative relationships.
A Tangent To (Mathematical) Tangents
While it is not critical to our discussion here, we can do a brief segway into this philosophical issue of free will, determinism, and consciousness.
It is not automatically evident that the universe in terms of its fundamental nature is mathematical, ie clockwork that can be described mathematically. Certainly it is true that there exists clockworks within the universe, however, this does not mean that the universe itself is a clockwork and that all its constituent parts behave in this manner.
The enlightenment was an era when clock making as a process was beginning to be refined and perfected. The simplest clocks are either oscillators, like a pendulum or quartz vibrations, or observers of larger natural oscillators, like a sundial or astrolabe with a sextant.
These cyclical processes behave with such mathematical regularity that we can describe them as “clockworks” a deterministic system that can be totally defined mathematically. As we got better and better at building clockworks, the question then arose whether what appears to be two different kinds of things: animals and inanimate objects, are both just physical clockworks of different kinds.
Panpsychism
As the original original categories of animals and inanimate objects began to be blurred—the more we learned about biology and natural processes, and developed clocks and later computers— one philosophical idea that has begun to gain ground in response to this modern confluence of material categories, is the idea of “panpsychism”. Rather than putting objects into simple mutually exclusive categories of animate or inanimate, panpsychism posits that properties that make living things “willful”, exist within all material, not just living things. The introspective experience of consciousness, therefore, is not unique to living things, it is only more refined within them. The question then arises is how do boundaries of “self” emerge, and one potential response of panpsychism is that self is illusory. The ability of animals to reason and have memory, enhances but does not define the instrospective experience of consciousness.
The simple argument would be that even if you close your eyes, and deprive yourself of all sensory connection to your environment, you are still experiencing consciousness. The metaphor I would use is that even though rocks are not animated like a clock with gears, they both are exhibiting the same kind of mechanical properties: they react to forces from their environment. A clock is just a construction designed organize these mechanical processes to perform a coherent and useful function, and similarly a mind might simply be organized psychological processes that are ubiquitous in all matter.
Is this relevant or necessary to economics?
The development of mathematics through the enlightenment era was hampered by a realization of weight and costs of information and information processing. For a time mathematicians and physicists pursued a “theory of everything”, a set of mathematical equations and relationships that could completely describe universe and how it works.
The Fractal Clock
Imagine a fractal clock. You open up a clock, and inside the clock there are moving parts and gears. But when you zoom in close, then inside the clock is a smaller replica of clock itself. If the larger clock was influenced by the behavior of this micro-clock, then you could conceivably get paradoxes or strange feedback loops. But if these micro-clocks were disconnected from the larger clock, and only “mirrored” its function, then you could avoid any such strangeness.
Enlightenment era mathematicians tried to find equations of everything that would essentially allow us to calculate total operations of universe from within the universe.
But in this search they were frustrated by the difficult of describing even apparently relatively simple systems.
The Three Body Problem
As we zoom in more and more at a refined level of detail on mathematical or physical systems, they start to behave strangely. The simplest physics is perhaps the motion of celestial bodies in gravitational orbits. Just like our fractal clocks, what happens on the surface of these stars or planets, doesn’t really affect much what the planet as a whole does. Such independence may seem to provide hope in discovering our fractal clocks or equations of everything.
But even this simple isolated system of orbital mechanics of point masses that never collide, presented unexpected mathematical complexity and intractability. If you slightly perturb the initial starting positions of the planets or physical parameters like mass, then initially the system appears to evolve along a very similar path. But over time the two systems that were apparently very similar, start to diverge more and more rapidly.
This behavior of rapidly diverging systems, is not exhibited in the orbital mechanics of two planets or bodies, but as soon as you add a third body, then you start to get this rapid divergence.
Imagine that you design two clocks to be as similar as possible. Initially, the clocks are only off by less than a millisecond. The difference is imperceptible. With regular clocks, this error might gradually increase over time. Over the course of a week, the difference becomes one millisecond. Over a year, the clocks might drift apart by 50 milliseconds, or one twentieth of a second.
After 20 years, the clocks are now off by 1 full second, but still, that is a good approximation.
This would be an example of linear drift. The error or discrepancy between the two clocks increases by a constant amount over time. After 1200 years, they would be off by a full minute, and after 72 thousand years they would be off by an hour.
Still, for quite a long time, the two clocks remain very close, and the error or discrepancy between them increases by a very predictable rate.
But imagine that we have second order drift. One of the clocks gets faster and faster over time. The first year each day is 24*60*60=86,400 seconds, or 86 million 400 thousand milliseconds. But by the time the next year has rolled around, its day has shrunk by 1 percent of one second, to 86 million, 359 thousand, and 990 milliseconds.
Instead of linear drift between the two clocks, they begin to drift quadratically. The rate of divergence increases over time.
Well when you have three orbital bodies, the “chaos” they exhibit, means that at some point the error or discrepancy between two very similar simulations, diverges very rapidly, much faster than a quadratic.
How We Compensated For Chaos
The response to this mathematical chaos, described as the “butterfly effect”, was to constantly update information and limit the horizon of useful forecasting or prediction so that simulations and the actual system don’t diverge too much.
Control theory developed tools like Kalman Filters which can actively manage and control systems even with noise and external disruptions.
This is all well and good, but such tools still rely on careful engineering where we can organize or engineer the system in a specific way.
The Economy Is Not Just Chaotic, It is Unorganized and Spontaneous, and Has a Lot of Private Information
The problem with analyzing economies, even with modern mathematical tools like control theory, is that the economy is not only subject to the mathematical system divergence of chaos theory, it is unorganized, spontaneous, and private.
Much of the relevant information to simulate an economy, is completely private information that is not published or shared. One might then go on to posit that game theory would be the ideal mathematical tool, and while it is useful, the problem with game theory is that the parties have clearly defined goals or objectives.
In an economy, not only does every party have private information, but what they even want or value is private as well. Game theory relies on well defined objectives. When I describe the economy as “spontaneous”, it is mostly for this reason, that people's goals and pursuits are constantly changing and unknown, both to others and even in cases their past selves.
So the mathematical tools of dynamical systems, control theory, and game theory, are all insufficient for effective and complete economic analysis. What then can we then do?
The Difficulty of Building Mathematical Models: Combining Partial Insights
The problem with trying to build mathematical models for economic systems, is that the variables and their relationships are not well defined. If we knew absolutely nothing about some variables, that would make the job of economists easier.
But this is not the case!
There are many things that economists can observe and measure which provide only partial insights into what variables are relevant and what state they are in.
What is Comparative Quantitative Reasoning?
Comparative quantitative reasoning is basically like a “shortcut” way to apply mathematical reasoning and skills.
The game “20 questions” is a perfect example of how this works in practice. Instead of clearly defining every detail, 20 questions is best approached by asking broad categorical questions which allow you to gradually eliminate bad answers. When the questions are independent of each other, and equally divide remaining possibilities, that allows you to most rapidly narrow in on good answers, almost like a binary search.
The lesson of comparative quantitative reasoning is simple: too much specificity makes analysis more costly and less effective.
Imagine you played 20 questions by starting with very specific answers. Is it a basketball? Is it the statue of liberty? Is it a giant squid? Is it the constant burden of existential angst in a universe where you don’t even know what or who you are?
By being too specific, the information you get from each question becomes less useful. You eliminate fewer possibilities and gain less information when a guess is too specific.
This is the problem with the approach to mathematical modelling that is rampant in the “mainstream” of the economics discipline. It is too specific, and too well defined. People call all the time for falsifiable testable hypotheses, not realizing that they are putting the cart ahead of the horse.
Premature optimization is said to be the root of all evil in software engineering, and I would argue that premature specificity is the root of all evil in economics discourse.
Certainly the ability to do mathematics and practice and skills in mathematics is essential and useful. But by getting too specific too early, with too many unknowns, then you are doing work that is still useful, but less useful and efficient than it could be.
This is why I spend most of my time talking about broad concepts of accounting. It is why, even though I may not completely agree with the approach, I relate a lot more to MMT’s focus on accounting principles and institutional relationships, than the mainstream focus on price puzzles and exogenous shocks. You are being ambitiously and wastefully specific when you try to construct precise and concrete mathematical models.
These models can still be useful as well as develop analytical skills. But to be effective you need to be able to think in terms of broad categories and general principles. This is one thing that I think my computer science training has helped me to do.
The Basic Premise of Accounting is Persuasion
When we talk about national debts, and market caps, and valuations, and leverage, all these details of accounting, what we are fundamentally doing is trying to persuade and incentivize people to participate in our goals and efforts. However, this happens as much through political processes as it does through the incentive of monetary payments and crediting accounts. In fact, I would argue that faith in monetary credit relies significantly on trust in a political consensus underlying basic rules and resource rights.
The word account has multiple meanings. It involves both counting, or quantitative specifics, as well as narrative, a story or sequence of actions or events. It is about both history, what happened in the past, and expectation, what the impact will be in the future.
To model economics and finance, is a bit like trying to find a clock inside of a clock, because accounting is already the tool we use to describe and understand the world we live in, and the resources available to use. We do not necessarily need to model the financial system, the financial system itself is the model of our resource reality.
Still, it is useful to try, but we also need the big picture. I think unfunded liabilities and political promises are just as important if not more than outstanding treasury bonds. But even so, we should not avoid ambitions, just because we need to be realistic and adapt to unexpected outcomes.
ou to accurately determine information, by dealing with broad open categories, and then slowly refining the information.
In terms