Seeing the world through mathematical eyes

By Keith Devlin @profkeithdevlin

How does being a mathematician affect the way you approach the world? I was asked this question recently on a live interview, being streamed globally on a multi-national corporate network. The interviewer prefaced the question by noting that on more than one occasion I had gone down in print and on video saying that being a mathematician influenced almost everything I do. “Give me three specific examples,” the host said.

My responses are below. They were not considered carefully in advance; I had no prior knowledge of the interview questions. Rather, they were the ones that came immediately to mind; which likely indicates that they have long been an ingrained part of my cognitive framework, which is exactly what the interviewer was asking for.

If you are a mathematician, before you read on, you might want to do the same, and see how your responses relate to mine. I can think of other examples I could have given, so there are surely a number to choose from.

My first answer was not the one I thought most important, but I gave it first because it connected to an earlier answer I had given to the question “What is number sense and why is it important?”

Number sense does indeed influence many decisions I make every day. This has been particularly stark for me – indeed, life-and-death stark – during the pandemic. Regularly assessing coronavirus risks has been a feature of my daily life since March 2020. I have written about this on “Devlin’s Angle” a number of times, as I noted at the start of last month’s post.

To give one example, I decided long before the vaccine was first made available that I would get my shot at the first opportunity. To be sure, that would mean rolling up my sleeve long before there was a huge volume of research data on the vaccine’s deployment on a large scale, including studies of any long term side-effects.

Since the numerical data available at the time was spotty and being updated frequently, calculating precise odds was not possible. But it wasn’t necessary. We had initial data on the probabilities of infection, of serious illness requiring hospitalization, of death, and of ten-days-of-hell-at-home, and by the time the vaccine was available, there was some emerging data on “long-COVID.” We also had data on the side-effects and risks associated with the vaccines.

None of the numbers were precise, rather they came in ranges. But from a number-sense perspective, there was no issue. The dangers posed by the viral infection were orders of magnitude higher than any dangers associated with the vaccine. The difference was so big, inexactitude of the figures was irrelevant. Getting vaccinated was a no-brainer.

But it requires a good grasp of number sense to be able to reason like that. And it’s clear from observing what is currently going on in society, particularly in the United States, that a great many people do not have that basic, quantitative life skill.

I’ll leave the second example I gave in my interview until last, since it was, as I realized when I was giving it, for me the most important of the three. Let me instead tell you now the example I gave third.

Being a mathematician, I have a professionally developed capacity to look for the key structural or functionally-important features of a situation, and pare away the surface features that immediately grab our attention, but in fact often distract from the parts that matter (in the current circumstance). That, after all, is how mathematics arises, going all the way back to the invention of numbers around 10,000 years ago and the development of geometry a few thousand years later.

In contrast to number sense, which for the most part escews precision (though it can lead to it), the power of this “focusing on the bare essentials” skill is the ability to be more precise, indeed to seek out absolute precision. Knowing that in some cases we can achieve great precision about the key features, and knowing how to do so, plays a role in many things I do in my everyday life.

And it’s not just for numerical precision; knowing that logical precision or structural precision are possible also plays a role in many things I do.

[I wrote about this feature of mathematics in my 1994 book Mathematics: The Science of Patterns, and then later in my 2000 book The Language of Mathematics: Making the Invisible Visible.]

Finally, and as I noted above, most importantly of all for me, here is my third answer. It is a skill of enormous power, that I acquired by way of the most painful lesson of all that mathematics teaches us – painful in large part because we are taught this particular lesson over and over again.

How to cope with being wrong. 

In mathematics, if you are wrong less than 90% of the time, you are engaging in mathematics well below your capacity. Which means you are wasting your time; unless you are being paid to do mathematics, but most of the people I know who do mathematics as part of their job also engage in pet research projects on the side, seeking the pleasurable challenge of being in the “ninety-percent-wrong mathematical activity” as a recreational break from their “routine and usually correct but financially rewarding” day job. But for anyone working at the frontier of their mathematical knowledge, doing mathematics means living with being wrong most of the time.

Admitting (to yourself and others) whenever you are wrong is a critical requirement for being a mathematician. It’s usually not critical in everyday life, but it’s surely a valuable ability

And the thing about being wrong in mathematics, is there is no way out, other than to accept you are wrong, then figure out why you are wrong and how to correct it.

We mathematicians are constantly reminded that what we feel sure is correct is in fact false, and that is sometimes takes considerable effort to “re-wind our mental tape” to see where we went wrong and embark on a modified, or different approach.

Part of correcting our errors is accepting new facts or information, identifying and recognizing false beliefs or assumptions and then fixing them,  or recognizing and then correcting a faulty understanding. In mathematics, we cannot change the facts to match our beliefs. The facts rule.

[Many people around us routinely do seem to try to change the facts to match their beliefs, often with disastrous outcomes, since they are led down a path of following fictions or falsities, not facts. Engineers, on the other hand, can lay claim to sometimes being able to change the facts to match their beliefs, often with dramatic consequences for society.]

To my mind, being able to recognize when we are wrong, and adjust accordingly, always trying to match the facts available to us – often by seeking out new facts – is the single most valuable life skill to come from being a mathematician. Moreover, mathematics is the only discipline where you simply cannot progress without first accepting that reality, and then following it. (So an important corollary of this particular “math lesson” is the importance of making decisions based on the best available information/evidence.)

Well, what do you think? What are your choices for the three most important life skills you have acquired through being a mathematician?