Do the Math

Applied Math teacher
Throughout history, applied mathematics has been the key to human development. By Elnur/Shutterstock

31 July 2019 – Over the millennia that philosophers have been doing their philosophizing, a recurring theme has been the quest to come up with some simple definition of what sets humans apart from so-called “lower” animals. This is not just idle curiosity. From Aristotle on, folks have realized that understanding what makes us human is key to making the most of our humanity. If we don’t know who we are, how can we figure out how to be better?

In recent decades, however, it’s become clear that this is a fool’s errand. Such a definition of humanity doesn’t exist. Instead, what sets humans apart is a suite of characteristics, such as two eyes in the front of a head that’s set up on a stalk over a main torso, with two legs down below and a couple of arms on each side ending with wiggly fingers and opposable thumbs; a brain able to use sophisticated language; and so forth. Not every human has all of them (for example, I had a friend in Arizona who’d managed to lose his right arm and shoulder without losing his humanity) and a lot of non-humans have some of them (for example, chimpanzees use tools a lot). What marks humans as humans is having most of these characteristics, and what marks non-humans as not human is lacking a lot of them.

On the other hand, there is one thing that most humans are capable of that most non-humans aren’t: humans are capable of doing the math.

Yeah, crows can count past two. I hear that pigeons are good at pattern recognition. But, I’m talking about mathematical reasoning more sophisticated than counting past seven. That’s something most humans can do, and most other animals can’t.

Everybody has their mathematical limitations.Experience indicates that one’s mathematical limitations are mostly an issue of motivation. At some point, just about everybody decides that it’s just not worth putting in the effort needed to learn any more math than they already know.

That’s because learning math is hard. It’s the biggest learning challenge most of us ever face. Most of us give up long before reaching the limits of our innate ability to puzzle it out.

Luckily, there are some who are willing to push the limits, and master mathematical puzzles that no human has solved before. That’s lucky because without people like them, human progress would quickly stop.

Even better, those people are often willing – even anxious – to explain what they’ve puzzled out to the rest of us. For example, we have geometry because a bunch of Egyptians puzzled out how to design pyramids, stone temples and other stuff they wanted to build, then proudly explained to their peers exactly how to do it. We have double-entry accounting because folks in the Near East wanted to keep track of what they had, figured out how to do it, and taught others to help. We’ve got calculus because Sir Isaac Newton and a bunch of his buddies figured out how to predict what the visible planets would do next, then taught it to a bunch of physics students.

It’s what we like to call “Applied Mathematics,” and it’s responsible for most of the progress people have made since the days of stone knives and bear skins. Throughout history, we’ve all stood around scratching our heads about things we couldn’t make sense of until some bright guy (or gal) worked out the right mathematics and applied it to the problem. Then, suddenly what had been unintelligible became understandable.

These days, what I think is the bleeding edge of applied mathematics is nonlinear dynamics and chaos. Maybe there’s some fuzzy logic thrown into the mix, too. Most of the math tools needed to understand (as in “make mathematical models using”) these things is pretty well in hand. What we need to do is apply such tools to the problems that today vex us.

A case in point is the Gini-Simpson Diversity Index I described in this blog two weeks ago. That is a small brick in the wall of a structure that I hope will someday help us avoid making so many dumb choices. Last week I ran across another brick in a paper written by a couple of computer science professors at my old alma mater Rensselaer Polytechnic Institute (aka RPI, or as we used to call it when I was there as a graduate student, “the Tute”). This bit of intellectual flotsam describes a mathematical model the authors use to predict how political polarization evolves in the U.S. Congress.

Polarization is one of four (at my last count) toxic group-dynamics phenomena that make collaborative decision making fail. Basically, the best decisions are made by groups that work together to reach a consensus. We get crappy decisions when the group’s dynamics break down.

The RPI model is a nonlinear differential equation describing an aspect of the dynamics of decision-making teams. Specifically, it quantifies conditions that determine whether decision teams evolve toward consensus or polarization. We see today what happens when Congress evolves toward polarization. The authors’ research shows that prior to about 1980 Congress evolved toward consensus. Seeing this dynamic at work mathematically gives us a leg up on figuring out why, and maybe doing something about it.

I’m not going to go into the mathematical model the RPI paper presents. The study of nonlinear dynamical systems is far outside the editorial focus of this column. At this point, I’m not going to talk about solutions the paper might suggest for toxic U.S. Government polarization, either. The theory is not well enough developed yet to provide meaningful suggestions.

The purpose of this posting is to point out that application of sophisticated mathematics is necessary for solving society’s most intractable problems. As I said above, not everybody is ready and willing to become expert in using such tools. That’s not necessary. What I hope you’ll walk away with today is an appreciation of applied mathematics’ importance for societal development, and a willingness to support STEM (science, technology, engineering and mathematics) education throughout our school system. Finally, I hope you’ll encourage students who show an interest to learn the techniques and follow STEM careers.

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