Thinking Outside The Triangle

From a 1970’s Textbook

Normally, I would try to find the best image on the internet for my articles. For this one, I tried to find an image of the expansion of the famous Pascal’s Triangle to negative exponents. I couldn’t. So, I used a picture from an old textbook that I used to teach Finite Mathematics.

The lack of images in my Google Search indicates that this knowledge of extending Pascal’s Triangle beyond the borders of “1” is being shaded into obscurity. What was once fairly common knowledge for many high school teachers and students, is now being relegated to lost knowledge/understanding.

While the expansion is cool, that is not what this article is about. It’s actually going to contain very little mathematics. It’s more about how we think about mathematics, or the flip side, how we don’t think about mathematics.

Namely, that it is human. And, to find out how human it is, we must see mathematicians outside of being more than that. This means a deeper and more exhaustive look into the history of mathematics. So, while we are not going to talk about mathematics, we are going to talk about mathematicians. Specifically, the one that the famous triangle is named after, Blaise Pascal.

For most students and teachers of mathematics, Pascal is primarily known through his triangle. However, as is more known now, the elements of the triangle were known to many mathematicians from all over the world prior to 1600’s. In China it is referred to as the Yang Hui’s Triangle(1238–1298) and in Iran it is referred to as the Khayamm Triangle(1048–1131). And even the Indian Sanskrit poet Pingala knew of the binomial coefficients and the additive formula for generating them back in the 2nd century.

Meru Prastaara 755 A.D.

So, Pascal’s whole being in the realm of math education is defined by presenting a piece of mathematics that is hardly original. Science students might encounter him with Pascal’s Law, an idea about equal distribution of pressure(it is how hydraulics works). So, as often is the case, mathematicians and their contributions are reduced to the numbers, laws, and theorems that define them — often rendering both the mathematician and mathematics inert.

Did you know that Pascal died before his 40th birthday?(39). Did you know that much of his life was consumed by pain, specifically migraines and toothaches? In fact, some of his mathematical insights came from those debilitating periods of pain at night — natural endorphins released to combat the pain might have aided in his mathematical visions. He suffered a great deal, especially towards the end of his life.

It is the end of his life that draws me to Pascal, mathematician, scientist, inventor, philosopher, and theologian. Personally, I am not a religious person(more mathematically spiritual), but humanizing the life of Pascal is symbolic of humanizing mathematics in general.

Pascal believed in suffering and using all his resources and wealth to help the poor. There are even stories that he gave up his house during an outbreak of the plague to the poor, and lived on the streets.

In this uncertain time of plague-like conditions, it is more important than ever to know how much humanity resided in our mathematicians, and that the mathematical work they did was only a subset of their lives. Embracing the whole person gives us more appreciation for the parts that make them. And something like mathematics naturally bleeds into human ideas of grace, kindness, and giving.

We need to rethink the purpose of mathematics for a future that will surely be impacted by global events of 2020.

Fighting for the humanity of mathematics is what awaits us on the other side.

Author of Pi of Life: The Hidden Happiness of Mathematics and Co-Author of Math Recess: Playful Learning in the Age of Disruption. Speaker. Amplify and Mathigon