Lexington, VA. In “Paradigms of Math and Non-quantifiable Values,” I argued that our current paradigms of mathematics are not creating the outcomes for which we hope. By thinking of mathematical sciences as an optimizer for anything quantifiable, or as a source of Truth, we ironically end up with sub-optimal results, and truth remains elusive. The problem is not the mathematics itself, as that is only a tool. Math is a good tool, but it is not an end in itself; it is not perfect, and we need to recognize its place in order to avoid idolizing it. If we are to unleash the full potential of mathematics and have it fulfill its proper place in describing some aspects of reality, we must understand its place better than we currently do.
In my early education as a mathematics major, I never dealt with this question. Learning to execute the functions of the trade, I rarely paused or was asked to consider math’s place in the world. I started this line of questioning one evening in Erie, PA while listening to a talk by Guy Consolmagno. During the talk, he posed the question to the audience, “Is mathematics discovered or invented?”
This simple question brought me up short. While studying, I had assumed that all mathematics was discovered from natural reality, but without ever realizing that this was in itself an assumption or that there could be another option. My thinking was rooted in old, still popular, paradigms of math without realizing that paradigms of math existed. Since Guy’s talk, I’ve never stopped considering the question.
By the time I started graduate studies in math modeling, my answer to the question had begun to reverse. The more I tried to use models to describe situations or systems, the more I began to think that we may be inventing mathematics as we need them. Personal study in a variety of subjects, including the history of mathematics, has helped shaped this belief into more of a paradigm, one which seems to fit better with the Porcher mentality.
Before articulating the paradigm further, let us examine an approachable insight from modern thinker Clayton Christensen that helps us understand the gaps in our current models. In his Harvard Business Review article “How Will You Measure Your Life?” he describes himself envisioning two days at work–one in which a subordinate goes home at the end of the day frustrated and unhappy, and another where she goes home happy, engaged, and having accomplished something. He imagines a manager’s ability to produce both kinds of days for employees, and the resulting effect on the employee’s family and personal life. This thought experiment leads him to his conclusion:
More and more MBA students come to school thinking that a career in business means buying, selling, and investing in companies. That’s unfortunate. Doing deals doesn’t yield the deep rewards that come from building up people.
Our paradigms of math tell us that quantifying goals is good, and indeed it has tremendous value. But something that is arguably more rewarding is often overlooked, with Christensen implying that the reason for this is because it is not measured. If we were to set a goal to have our employees go home in a mood that made them better people, how would we judge ourselves against that standard? Most companies would not likely have a metric in place for that. If we were courageous enough to attempt to quantify that goal, we would likely start by understanding the quantifiable elements and knitting them together into some sort of framework. We could essentially try to invent the mathematics or measurement of good management, as defined by Christensen.
Another interesting example comes from Simon Sinek, who turns Christensen’s ideas around. Instead of applying business concepts to personal life, he gives a compelling example of applying personal life to business decisions. While discussing leadership, he talks about Bob Chapman’s reaction to the economic downturn in his business in 2008. Instead of laying off “excess” employees, all employees from CEO to janitor were required to take an equal amount of unpaid vacation. Chapman supposedly said, “Better we should all suffer a little, than some of us suffer a lot.” Sinek goes on to say, “Fundamentally, a leader is like a parent.” Few people try to use mathematics to completely describe their parenting as they would their business management. Why is there a disconnect between those endeavors, even when performed by the same person?
We would struggle to imagine a mathematical model that describes how “parent-like” a manager is, or how much a manager loves their employees. That struggle doesn’t prove that one cannot be created, but given its deeply human context, the underlying principle will have been known for millennia, felt viscerally by people since the beginning of tribal leaders, and articulated long before ever becoming a mathematical formula. Having a mathematical communication of how much one loves or is loved does not tend to add significantly to the overall understanding of that love. Someday we may invent the mathematics of management or love, but these realities can be known without the corresponding mathematical formula.
This line of questioning underlies my previous article about the paradigms of mathematics. How can math optimize everything or be the purest form of truth if we have to invent it to describe situations that the least formally educated people have known to be true for years? What paradigm of mathematics could possibly align to our invention of it in time of need, its rightful place of irrefutable proof, its great power, and its yawning gaps?
As we look for a paradigm of mathematics, as we try to figure out the proper role that mathematics should play in our lives, I offer as a suggestion the paradigm of math as a language. It is important to understand that I offer language as a paradigm, not a definition. I have read many arguments about whether math is or is not a language that center on the definition of language and whether math meets that definition. That is irrelevant. Definitions can be gerrymandered to include or not include what we already want or don’t want.
The paradigm of math as a language, however, means approaching mathematics as nothing more or less than a means of communication. In Pi in the Sky, Barrow discusses the anthropological beginnings of mathematics. Many think that our current and well-known base 10 system developed around fingers and finger analogies that turned into what we see in toddlers–finger counting. Imagine the conversation:
“Do you have many goats?”
“I have as many goats as I have fingers.”
“What about chickens?”
“I have even more chickens than fingers.”
“How many more?”
“What do you mean, how many?”
And so the need for numbers developed to answer questions that naturally arose in early societies. What is perhaps more interesting is that different cultures developed counting very differently. Some societies developed only a base two counting system, with words for one, two, and then many, skipping any other quantification as unnecessary. Base twelve became popular in trading societies, possibly because it is more easily divisible into more parts than base ten, making it more useful for dividing a dozen eggs or donuts. All of these methods developed in relation to something in the real world and as a way to compare and describe ideas and quantities that existed in people’s lives.
Now imagine another scenario: a man with a workshop who likes to tinker and make widgets every day after work. He knows how many widgets he made, whether it took too long or was done quickly. He understands where he had trouble and makes adjustments in his process and equipment to make his hobby easier and better. Perhaps he buys a new tool to help make them better or reorganizes his shop to have items closer at hand.
Imagine the same man, making the same widgets in a factory. He still knows if he is doing well or poorly, and still works to improve his own efforts. But he may not control the money to invest in new tooling or have authority to make other decisions. Perhaps those decisions are made by an executive in an office in a large city far away. The executive does not understand how well the work is going day to day. He doesn’t have a gut feel for the performance of the day or the issues faced by the widget-maker. A conversation may look like this.
“Mr. Executive, we need $200,000 for a new widget-maker. It can make them faster and better and would be great to have.”
“Well, is it worth it?”
“Absolutely. We could work so much better with it.”
“But how do I know? I could spend the money many places or distribute it within the company as bonuses. What makes this a better place for it?”
At this point the conversation can go a couple different directions. One is continued qualitative arguments that the executive does not feel in his gut, having never built a widget, and cannot truly comprehend. The other is to translate the qualitative into something that is easily comparable among various options and that allows better communication. This may come in the form of increased throughput that can bring in more dollars or better-quality widgets that waste less money to rework. In either scenario, a translation of the qualitative benefits into quantitative dollars allows someone without familiarity of the situation to make better, more accurate comparisons.
And in these scenarios lie our paradigm. Math and measurement are not inherently solving the problems. They are not making more widgets or raising more chickens. Math is merely the instrument to more precisely communicate a specific idea. It is a language.
A natural counterpoint in a world where the prevailing paradigms of mathematics are of Truth and complete optimization may be that higher orders of mathematics are so far beyond simple counting that these examples do not apply. Yet these examples were chosen only for their familiarity and ease of assimilation. A longer dive into the subject could well include James Franklin’s book C. This book “provides a history of rational methods of dealing with uncertainty.” In building the history of human dealings with uncertainty up to the development of modern probability theory, Franklin starts in what would be an odd place for a discipline of purely numerical truth–he starts with the law. He argues that the law, trying to build a completely certain case against a suspect, was one of the first places where humans tried to rationally deal with uncertainty in a systematic way.
In many instances, if a person confessed to a crime, this amounted to a “complete” proof, and the person could be punished. In situations where a confession was not forthcoming, what constituent parts could sum to complete proof? If an eyewitness was a half proof, were two eyewitnesses a full proof? If no one witnessed a murder, but instead saw someone running from a building with a bloody knife and then entered to find a body, how far did that take a judge toward confirmation of criminal act?
Did mathematicians like Pascal “discover” probability by formulating ways to measure it, or did they probe the depths of its nuance and hone its precision as they explored ideas that humans had known and worked with for millennia? Franklin follows a fascinating history that starts in real-world problems, discusses the intuition that people used to work on these problems, and then traces it through development and iteration until they were able to build a sufficient language to capture and communicate the depth of that reality.
Another counterpoint to the idea of math as a language could arise in the idea that we do not use it as a sole means of communication but only in conjunction with other existing languages. It is also understood in the same way across all other languages. And this exact question holds the key to understanding math as a language, for if we are to consider it as a language, what exactly does it communicate? How does it differ from existing languages? It is not simply another language, but rather another dimension in a matrix of human language capability.
The examples above have shown how mathematics can take ideas and concepts that people intuit, or situations that exist in a murky reality, and find ways to precisely define all of the content in a way that reduces ambiguity, creates a common understanding, and communicates that certainty succinctly. Put simply, math is the language that reduces uncertainty, especially when people or entities are not experiencing the same reality.
Now imagine another scenario. You are out with your significant other for Valentine’s Day or an anniversary, rekindling feelings of affection and building connection. Over a beautiful Italian secundi, he says “I love you 1.27 times more than I did last year.” The romance of this statement, to many, is severely diminished by the addition of mathematics. “I love you more than the moon and stars,” while trite, holds a grander scale that conveys a magnitude as unimaginable as it is imprecise.
And here is the fit for our lives–for those aspects of reality that require precision and concise communication, the language of mathematics can provide it better than any other. But once we have communicated in those precise ways, we should be mindful that there is a large world beyond what we have just communicated. We should not just accept that there may be more to the story but look actively for the story that was not communicated. Once we have used math to get many people on the same page, we need to continue to develop a deep understanding that comes only by relationship, either with other people or with the world that produced the quantified subject.
Math is a quick way to a certain understanding, but it should never replace completely a long, slow path to deep understanding. This is the benefit that exists in communities and relationships that can never be replaced by a computer, no matter how many factors are allowed to exist in the artificial intelligence model. This is also the benefit that exists in the humanities. Taking days or weeks (or months) to read Moby-Dick will likely reach far deeper into our souls than reviewing a balance sheet from a particular whaling voyage. The truths communicated are deeper, more amorphous, and not in any way concise.
What would the world have lost had Shakespeare recounted the Battle of Agincourt as a Wikipedia article, “7,000 English, mostly archers, defeated 25,000 Frenchmen at 50(o) 27’ 49” N, 2(o) 8’ 30” E.”? It certainly communicates a concise and precise idea, but that account hardly lives up to the emotions of “If we are mark’d to die, we are enough / to do our country loss; and if to live, / the fewer men, the greater share of honour. / God’s will! I pray thee, wish not one man more.”
Math is certainly not the best language for every situation, but it is essential for many situations. And once we understand this, and not merely acknowledge it but shift our paradigms to understand it as a very special method of communication, we can use math without fear. We can employ it effectively without the anxiety that many feel of its perceived complexity. We can also use it without fear that it will usurp the Truth. We can rely on math to communicate some aspects of our reality without surrendering our deeper understandings and relationships to its totalizing modern paradigms.