 # The 12 Mathematicians Who Set The Stage For The Modern World

## The Pythagoreans (5th Century BC) Some of the earliest mathematicians were Pythagoras and his followers. Mixing religious mysticism with philosophy, the Pythagoreans' contemplative nature led them to explorations of geometry and numbers. The most famous result attributed to Pythagoras is the Pythagorean theorem: for a right triangle, the sum of the squares of the two shorter legs that join to form the right angle is equal to the square of the long side opposite that angle. This is one of the fundamental results in plane geometry, and it continues to fascinate mathematicians and math enthusiasts to this day. One apocryphal story of the Pythagoreans illustrates the danger of combining religion and math. The Pythagoreans idealized the whole numbers, and viewed them as a cornerstone of the universe. Their studies of geometry and music centered on relating quantities as ratios of whole numbers. As the story goes, a follower of Pythagoras was investigating the ratio of the length of the long side of an isosceles right triangle to the length of one of the two shorter sides, which have the same length as each other. He then discovered that there was no way to express this as the ratio of two whole numbers. In modern terminology, this follower had figured out that the square root of 2 is an irrational number. According to the legend, when the follower who discovered this fact revealed it to his peers, the idea that there could be irrational numbers — numbers that can't be expressed as a ratio of two whole numbers — was so shocking to the Pythagoreans that he was taken out on a boat and murdered by drowning.

## Euclid (c. 300 BC) Euclid was one of the first great Greek mathematicians. In his classic "Elements", Euclid laid the framework for our formal understanding of geometry. While earlier Greek philosophers, like the Pythagoreans, investigated a number of mathematical problems, Euclid introduced the idea of rigorous proof: starting with a handful of assumed axioms about the basic nature of points, lines, circles, and angles, Euclid builds up ever more complicated ideas in geometry by using pure deductive logic to combine insights from previous results to understand new ideas. This process of using rigorous proof to build new results out of existing results introduced in the "Elements" has remained perhaps the most central guiding principle of mathematics for over two millennia.

## Archimedes (c. 287-212 BC) Archimedes was possibly the greatest mathematician of all time. He's best known for his contributions to our early understanding of physics by figuring out how levers work and in the famous legend of his discovery of how water is displaced by a submerged object: while taking a bath, Archimedes watched the water sloshing up to the top of his tub, and in the excitement of his discovery, he ran through the streets naked and shouting "Eureka!" As a mathematician, however, Archimedes was able to outdo even his own accomplishments in physics. He was able to estimate the value of pi to a remarkably precise value and to calculate the area underneath a parabolic curve. What is truly amazing about these accomplishments is that he made these calculations using techniques surprisingly close to those used by Newton, Leibniz and their heirs in the development of calculus about 1,800 years later. He found these values by approximating them with measurements of polygons, adding more and more refined shapes, so that he would get closer and closer to the desired value. This is strongly reminiscent of the modern idea of an infinite limit. As far as his mathematical sophistication was concerned, Archimedes was nearly two millennia ahead of his time.

## Muhammed ibn Musa al-Khwarizmi (c. 780-850) al-Khwarizmi was a ninth century mathematician who created many of the most basic techniques for how we perform calculations. His greatest contributions were in the realm of developing formal, systematic ways of doing arithmetic and solving equations. al-Khwarizmi's writings introduced the Hindu-Arabic decimal number system we use today to Europe, and this system makes it far easier to add, subtract, multiply, and divide quantities of any size than using Roman numerals or other non-positional systems. al-Khwarizmi also came up with systems of rules for solving basic equations, like 4x + 8 = 2, or x2 - 8 = 4. His work marks the beginning of what we today understand as algebra. Indeed, the word "algebra" itself comes from part of the title of his book on solving equations, and the word "algorithm", meaning a systematic set of rules used to solve a problem, descends from al-Khwarizmi's name.

## Johannes Kepler (1571-1630) Johannes Kepler was a gifted geometer who applied his mathematical abilities to solidify our understanding of the solar system. Kepler worked closely with the great empirical astronomer Tycho Brahe, who kept some of the most meticulous records of the movements of the planets up until that time. By analyzing those records, Kepler was able to confirm and refine the Copernican view of the solar system: the planets move around the sun, and the time it takes a planet to move around the sun is described by precisely defined mathematical laws based on the shape of the planet's elliptical orbit. Kepler's laws are impressive because they are a precise and elegant mathematical description of a physical process. The fact that things in the world, like planets orbiting the sun, follow such laws has been referred to quite elegantly by the 20th century physicist Eugene Wigner as "the unreasonable effectiveness of mathematics". Kepler's laws are an early example of that unreasonable effectiveness. Kepler's laws also set the stage for Newton's development of his laws of motion and especially of his theory of gravity. Kepler's contributions to our understanding of planetary mechanics led to him being the namesake of NASA's main probe searching for planets outside our solar system.

## Rene Descartes (1596-1650) Rene Descartes is most widely known for his contributions to philosophy, in particular his development of the idea of the dualism of mind and body, and for his famous saying "I think; therefore I am". However, much of the mathematics we use today owes a great debt to Descartes. Descartes' primary contribution to mathematics was in the development of analytic geometry. Throughout the history of mathematics until Descartes, there was always a divide between algebra and geometry. On the one hand, we had the symbolic and abstract manipulation of numbers and unknown quantities, and on the other hand, we had the investigation of shapes and solids. Descartes' analytical geometry unified these two fields. He pioneered the idea of representing algebraic forms and equations using geometric lines and curves on a coordinate plane. His basic ideas are still taught in high school mathematics today, with students learning how to graph an equation like y = 3x + 5 as a line, or an equation like y = x2 - 4 as a parabola. This combination of geometry and algebra was a significant precursor to the later development of calculus, and is such a central idea of modern mathematics that we take it for granted. Descartes' work was so fundamental that we refer to the coordinate system he invented as the "Cartesian plane".

## Blaise Pascal (1623-1662) The French mathematician Blaise Pascal, like many of the people on this list, contributed to a number of fields of mathematics. Pascal's Triangle provides a remarkably elegant way to calculate binomial coefficients, a set of numbers that are important in algebra and elsewhere. He also developed one of the first mechanical calculators in the world, a distant and primitive relative of modern computers. Pascal was also one of the originators of probability theory, coming from his analysis of games of chance. Pascal's work on the basics of probability represented the beginning of our ability to understand chance and risk in a mathematical way. Pascal's work on probability, and his late in life religious revelations, lead to him coming up with Pascal's Wager, an argument for why one should believe in God rooted in the probabilistic idea of expected value.

## Isaac Newton (1642-1727) No list of great mathematicians could be complete without Newton. With his invention of calculus (an achievement shared with our next entry), mathematics was able for the first time to systematically describe how things change across space and time. Newton developed calculus in the context of developing his theories of physics. The language of calculus is the most natural way to describe motion. A car's speed is the rate at which it is changing position, or the derivative of its position. The acceleration of a ball dropped from a tall building is in turn the rate at which its speed is changing, or the derivative of its speed, and Newton understood that this acceleration was the result of the force of the earth's gravity acting on the mass of the ball. Newton's physics also represented a milestone in our overall view of the world. Earlier physicists and astronomers, like the previously mentioned Johannes Kepler, understood that the behavior and movement of objects followed certain patterns. But Newton and the physicists who would follow him understood, with the help of mathematics, the reasons why objects follow those patterns. Further, Newton's laws were understood to be universal — the same force of gravity that causes a ball to accelerate as it falls is the force that keeps the moon going around the earth. The idea that the same laws of physics apply everywhere in the universe is a core tenet of science, and it is supported by all existing evidence.

## Gottfried Wilhelm Leibniz (1646-1716) Leibniz independently developed calculus in Germany at the same time Newton was developing it in England, an occasional issue of debate among mathematicians. Leibniz, however, came up with much of the notation for calculus that we continue to use up to the present. Leibniz also anticipated in many ways a huge number of later mathematical developments. He had a strong belief in rationalism, with a focus on formal symbolism that would later come to fruition in the late nineteenth and early twentieth centuries with the development of modern logic and set theory. Leibniz also had a hand in the improvement of mechanical calculators like the one developed by Pascal.