Why You Should Study the History of Math

 
By Dr. Kim Johnson with The Lukeion Project
(instructor /designer of  "Counting to Computers: Math from the Dawn of Time to the Digital Age")

In the mid 1300s a fad made its way around Italy. Mathematicians would challenge each other to “mathematical duels.” They would post problems for their opponents to solve, sometimes along with their solutions in coded poetry. The winners would get support and funding from rich patrons and the losers would descend into obscurity. One such contest between Fiore and Tartaglia involved a new method for solving the cubic. In order to win Tartaglia worked day and night to find Fiore’s method. Unfortunately, Fiore did not do the same and only knew his own method and no others.

Recall that the formula for solutions to quadratic equations of the form

use the quadratic formula,

 Giorlamo Cardano, a physician, philosopher, astrologer and mathematician, convinced Tartaglia to share his method and promised never to reveal it. Then Cardano figured out a more general method, and wanted to share it, but was blocked by his promises. Fortunately for Cardano, he realized that Fiore’s teacher, del Ferro, had actually figured out Tartaglia’s solution earlier. So Cardano shared del Ferro’s solution (which happened to be the same as the one he had promised not to reveal) along with his more general method. Tartaglia was upset, and eventually lost the duel which would have given him a professorship. He hated Cardano for the rest of his life.

Even though this “mathematical dueling” was exciting, dramatic, and expanded the horizons of what was known mathematically, the practice only lasted a few years. What happened? Why don’t we still engage in mathematical duels today? The change didn’t come because we are so much smarter than the historical mathematicians. Things changed because we have different tools than they did. These days we are solving different problems than they did.

We have better tools.

    Because we have the benefit of tools based on thousands of years of mathematical creativity, things which were difficult to the historical mathematician are literally child’s play to us. The reason that algebra is taught in middle school or early high school rather than being the crowning glory of the life’s work of mathematician has to do with the progression from rhetorical algebra to symbolic algebra. In the beginning of mathematics, every single problem was a word problem. The ancient mathematicians simply lacked symbols. By the 1200s, rhetorical algebra evolved to syncopated algebra, a mix of words with a few symbols thrown in. Finally, in the 16th and 17th centuries Francois Viete (a code-master with the French government) and Rene Descartes were among the mathematicians developed the standard symbology we use today.

    The ancient mathematicians didn’t even have the most useful of symbols, the number zero, until the Hindu mathematicians developed it in the 600s and it was exported around the world by the Arabs (which is why we use Hindu-Arabic numbers). Even then, most European mathematicians used Roman numerals until the 1200s and Liber Abaci by Fibonacci popularized the idea. Although many people,  including a Pope, believed zero and the place value system were useful, they did not become universal until much later. When you try to divide MCMDLXXXII by DCXIV, the ease of the place value system becomes obvious. In most cases, the tools that have been developed make computations much easier for us than they were for our predecessors.

 

We are solving different problems.

    Part of the reason the historical mathematicians didn’t have the same tools as we do is because they were not solving the same types of problems. For example, when algebra was introduced by the Arabs, it was developed for solving specific problems including those relating to inheritance. The mathematicians in India used their work to imagine and build altars of various sizes and shapes. The Romans, whose contribution to mathematics was once summed up as “Cicero finding the tomb of Archimedes,” were interested in solving problems only when they were practical. A problem from one of their texts, for example, talks about “the shortest way across a river when there is an army on the other side.” Babylonians were interested in keeping track of goods and astronomy. Egyptians were interested in the area of plots of land for taxes as well as a calendar for computing the flooding of the Nile. Enlightenment mathematicians were interested in solving problems related to how planets moved.

    There have been dice games and other games of chance available for millennia. Only since the 1300s have these been analyzed mathematically. Cardano, mentioned above, put himself through medical school by gambling since his father wouldn’t pay. He redeemed himself, however, by analyzing the games of chance he played and thus inventing the field of combinatorics and probability. Why had no one developed these methods before? They believed that the results of the roll were the results of favor from the gods. Why look for mathematics in something that was divinely ordained?

    Even though the Babylonians had developed the beginnings of the concept of zero, the Greeks had real trouble with even the idea of zero. The void was meaningless to them, and so in part they never developed zero. The Hindu mathematicians had no such philosophical barrier, and so they could use the powerful tool to solve difficult problems.

We are different people.

    Finally, the mathematicians of the past had unique personalities that led to their discoveries. It helps to understand Cantor and infinity when we understand his mental health. We can understand the development of the calculus in a new way when we know about Newton and how he saw the world vs. Leibnitz and his view of the world---and how Euler brought them all together. Some mathematicians, like Fermat and Mersenne, were extremely interested in communication and collaboration. Some, like Sophie Germain or Carl Gauss, overcame terrible obstacles. Others, like the Bernoulli family and Euler, seemed destined to study math. Still others pursued mathematics as a side passion, like John Napier, a Scottish laird who studied theology and incidentally invented the logarithm. The personalities of the mathematicians themselves work their way into their mathematical discoveries.

    For all these reasons---the tools, problems and personalities of historical mathematics---we should look at the history of mathematics in order to understand the mathematics itself. These historical mathematicians were not stupid, they were merely coming from a different perspective. When we understand their various perspectives, our own perspective broadens and our progress in mathematics becomes richer.