The Democratization of Mathematics
The demystification of mathematical modelling is the first step to raising the level of mathematical literacy in public debate
Under COVID, our lives are dictated by the mandate of a handful of mathematical models. The public understanding of those models has never been more essential. Yet models are often discussed as if they were the exclusive province of a mystical caste of mathematician priests, while at the same time, our news media are full of pundits (and politicians) pontificating about the prognostications of these models with no apparent appreciation either for what they can reasonably predict or what they can not.
In this article, I will argue that there is nothing mystical about mathematics, and that mathematical modelling is just a particular refinement of common, but deliberate (“slow”) thinking. As such, there is no reason why we can’t all become conversant in the language of mathematical modelling.
This is not to say that everyone can build and deploy mathematical models, any more than everyone can draw up a legal contract or diagnose and treat an illness. But just as we expect to be able to converse with our lawyer or our doctor, acquiring the understanding we need to inform decisions and influence outcomes, so should we all be able to engage with the models that are so influential in the decisions that shape our lives.
We will not all become mathematicians (Heaven forfend!), but by demystifying mathematics, we can develop a much broader public engagement with the scope and context of our models, the assumptions that underlie them, and our interpretations of their results. Such enrichment of the discourse around our models would be of immense benefit both to our modelling and to the public whose lives are so influenced by it.
Mathematics is just thinking “slow”
The word mathematics comes from the Ancient Greek máthēma (μάθημα), meaning “that which is learnt”, as distinguished from that which you know by intuition. In modern terms, we might identify this broad, pre-Aristotelian sense of mathematics with Daniel Kahnemann’s concept of “slow” thinking (deliberate, rational, learnt), as distinguished from its limbic forerunner “Fast” thinking (impulsive, emotional, intuitive).
When intuition fails, we must bring to bear what we have learned. This is mathematics in this broad sense. But mathematics in its narrower modern sense is simply a refined form of this slow, deliberate, learnt mode of thought. There is an uninterrupted continuum of experience from the simplest mental models with which we make sense of the world to the most abstruse inaccessible treatise of modern advanced mathematics.
A hierarchy of slow thinking
When we think deliberately, “slowly”, about the world, we try to find patterns in the chaos: repetitions, similarities, commonalities. We corral these patterns into categories of things and we conjecture relationships between these categories.
The most interesting relationships are causal because causal relations allow us to think about interventions. We first describe the world, then we predict how it might unfold, then we try to intervene to make that development better for us.
The next level of refinement — what we might call science, though the broader Wissenschaft is closer — is more carefully to define our categories and to formalize our conjectures about the relationships between them.
This allows us to test and compare paradigms: Do they make sense? Are they internally consistent and consistent with other paradigms? Are they consistent with the available information? Are they insightful?
Mathematical modelling in the modern sense is the final refinement in this hierarchy of slow thinking. It’s the part where we conjecture mathematical relationships between things we can measure and assign quantitative values.
The benefits of this are substantial, but they come at a price: the language of those relationships is now harder to understand and understanding may be limited to a set of practitioners who “speak math”.
But in the same way, as mathematical modelling is just refined, slow, thinking, the language of “math” is just a highly refined and efficient language for describing relationships in a model. It ought always to be possible to unpack that language and explain it to anyone. The process of building, testing, and using mathematical models should be intelligible and accessible to everyone, because everyone does it all the time at some level, whenever they think “slowly”.
The music and the words
“I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of Science, whatever the matter may be.”
William Thomson, 1st Baron Kelvin (1824–1907)
Many people imagine Kelvin’s “knowledge” to be unavailable to them because they imagine it to be a consequence of the mathematical articulation of concepts. In fact, it is a condition for that articulation, and as such, it is entirely accessible even to those who are not fluent in the occult technicalities of the language of mathematics.
Mathematicians are, I think, entirely to blame for this misunderstanding.
First, the words
I advocate an approach to modelling that takes its point of departure in the problems we’re trying to solve. This is the prose, the words. This perspective is continuously refined so that categories and relationships can be expressed mathematically. This process of refinement is immensely valuable, but not in itself mathematical. The last step is mathematical — converting those relationships to equations and solving them. This is the poetry, the music. First the words, then the music.
First, the music
But mathematics is not only a very exact (and exacting) language. It is also an incredibly economic, elegant, and beautiful language. For those of us lucky enough to be fluent in it, it’s incredibly natural to try to shortcut the modelling process and to try to describe relationships mathematically before really being clear what it is you’re describing or what the problem is you’re trying to solve. First the music, then the words.
At best, in the right hands, and with a bit of luck, this can be incredibly powerful. This is Einstein developing the theory of general relativity, Paul Dirac predicting the existence of the positron, and (I suspect) Black and Scholes coming up with their equation for option pricing. But if such an approach shall have any use beyond entertainment for angels (or mathematicians), the connection back to the world, through a model, needs to be made explicitly.
And too often we see this approach as a consequence of laziness or an unwillingness to engage with the real world. We see mathematical methods casting about for something to which to apply themselves; we see mathematical sophistication for the sake of their sophistication, unsupported by available data, furnishing adjustments far beneath the noise floor of the fundamental uncertainties in the model and providing no additional insight.
A robust dialogue between the prose and the poetry of mathematical modelling is critical to ensuring the relevance and fitness for purpose. Such a discourse is also the key critical step to mediating the unique insights of our models to as broad a public as possible.
Conclusion
Mathematical literacy is as crucial to running a business, or a country, as knowledge of the law, finance, and economics.
Our business and political leaders need to be conversant with the relationship of the structures of mathematics to the problems we address with it; how models are (or should be) constructed, conditioned, tested, monitored, and improved, how we engage with data, how models inform decisions and support objectives, what we can reasonably expect from them, and what we can not.
Without this interactional expertise, leaders can not hope to engage with and benefit from any form of quantitative study, economic analysis, data analytics, business intelligence, strategic, portfolio or policy modelling, and they are at the mercy of the political and business acumen of their numerically literate advisors. In the public sphere, we have seen all too clearly what a lack of mathematical literacy in the public debate around COVID modelling has cost the world in lives and money.
None of this requires particularly deep knowledge mathematics, just a level of engagement both on the part of leaders to ask the right questions, and on the part of mathematicians to ensure that their models are relevant and fit-for-purpose and that they are translated into a language everyone can understand.
For further discussion on the difference between contributory expertise (the expertise of the mathematician) and interactional expertise (the expertise I advocate for non-mathematicians), see my article on Expertise.
And for updates and more insights into mathematical modelling (mostly, but not exclusively for business), please connect with me on LinkedIn
