Author Archives: u0752301

Isaac Newton and his Contributions to Mathematics

Sir Isaac Newton. Image: Arthur Shuster & Arthur E. Shipley, via Wikimedia Commons..

In class we discussed the Fundamental Theorem of Calculus and how Isaac Newton contributed to it, but what other discoveries did he make?

Sir Isacc Newton was born on January 4, 1643, but in England they used the Julian Calender at that time and his birthday was on Christmas Day 1642. He was born in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. His father had already passed prior to his birth and his mother remarried after his birth and left Isaac to live with her mother. He went to The King’s School, Grantham from the time he was twelve until he was seventeen. His mother removed him from school after the death of her second husband, but later allowed him to return by the encouragement of the school’s headmaster. He rose to be at the top in rankings in his school, mainly motivated to get revenge towards a bully. He began attending Trinity College in Cambridge in 1661. After receiving his degree he developed his theories on calculus over the span of two years during the plague [1].

Newton’s work in calculus intitially started as a way to find the slope at any point on a curve whose slope was constantly varying (the slope of a tangent line to the curve at any point). He calculated the derivative in order to find the slope. He called this the “method of fluxions” rather than differentiation. That is because he termed “fluxion” as the instantaneous rate of change at a point on the curve and “fluents” as the changing values of x and y. He then established that the opposite of differentiation is integration, which he called the “method of fluents”. This allowed him to create the First Fundamental Theorem of Calculus, which states that if a function is integrated and then differentiated the original function can be obtained because differentiation and integration are inverse functions [2].

Controversy later arose over who developed calculus. Newton didn’t publish anything about calculus until 1693, but German mathematician Leibniz published his own version of the theory in 1684. The Royal Society accused Leibniz of plagiarism in 1699 and the dispute caused a scandal to occur in 1711 when the Royal Society claimed Newton was the real discoverer of calculus. The scandal got worse when it was discovered that the accusations against Leibniz were actually written by Newton. The dispute between Newton and Leibniz went on until the death of Leibniz. It is now believed that both developed the theories of Calculus independently, both with very different notations. It should also be noted that Newton actually developed his Fundamental Theory of Calculus between 1665 and 1667, but waited to publish his works due to fear of being criticized and causing controversy [1].

Newton not only discovered calculus but he is also credited for the discovery of the generalised binomial theorem. This theorem describes the algebraic expansion of powers of a binomial. He also contributed to the theory of finite differences, he used fractional exponents and coordinate geometry to get solutions to Diophantine equations, he developed a method for finding better approximation to the zeroes or roots of a function, and he was the first to use infinite power series.

His work and discoveries were not limited to mathematics; he also developed theories in optics and gravitation. He observed that prisms refract different colors at different angles, which led him to conclude that color is a property intrinsic to light. He developed his theory of color by noting that regardless if colored light was reflected, scattered, or transmitted it remained the same color. Therefore color is the result of objects interacting with colored light and objects do not generate their own colors themselves [1].

Sir Isaac Newton was a truly amazing mathematician and scientist. He achieved so much in his lifetime and the amount of discoveries he made can seem almost impossible. He helped make huge advancements in mathematics and created theorems that we still use heavily to this day.

[1] http://en.wikipedia.org/wiki/Isaac_Newton

[2] http://www.storyofmathematics.com/17th_newton.html

History of Women in Mathematics

In class we discussed the famous mathematician, Sophie Germain. Upon discussing her I had thoughts that I’m sure many women in math have also had, “why have I never heard of a female mathematician until now?” This caused me to look into female mathematicians throughout history and what their accomplishments were.

Image: Jules Maurice Gaspard, via Wikimedia commons.

The first known woman involved in mathematics is Hypatia. She was from Alexandria, Egypt, and was born sometime between 351-355 AD and died in 415. Her father, Theon, played a major role in her becoming the mathematician that she was. Theon was a professor at the University of Alexandria and was determined to have a “perfect child”[1]. This led to him teaching Hypatia to be very well rounded in all subjects, including math. Hypatia went on to teach at the University of Alexandria and became very popular. Her lectures often were on Diophantus’ “Arithmetica” and the techniques he used. One of the more interesting things about Hypatia was that she was not afraid to go to a group of men, and that she was highly admired by those men [2].

Elena Cornaro Piscopia was not only a mathematician, but was highly skilled in music, philosophy, and language. She was from Venice, Italy, and lived from 1646-1684. She was the first woman to ever receive a doctorate degree, in philosophy, not math, and was only thirty-two years old [3]. Another female mathematician who lived near the same time period of Piscopia was Émilie du Châtelet. She was a French mathematician and physicist who lived from 1706-1749. Her most famous work was the translation of Newton’s Principia Mathematica [4].

Sophie Germain is probably one of the most well-known female mathematicians. She was born in Paris, France 1776. She started her journey into mathematics around the age of 13 when she came across mathematical texts in her father’s library. She began to study them relentlessly, even though her parents did not support her. She began to have an interest in number theory after the release of Legendre’s Essai sur la théorie des nombres. From this point on she began to make huge progress in the field of number theory. As we all know she helped make advancements in the proof of Fermat’s Last Theorem, but she also went on to win an Academy Prize for her work in elasticity. While all this went on, I still find it interesting how she had support from many famous male mathematicians, such as Legendre, Gauss, and Poisson, yet when it came to publishing their own work or helping her further her own they were not as supportive. Germain was never mentioned in Poisson’s work, yet she helped him often and he had access to all of her work. Legendre began to help Germain in her work that won her the Academy Prize, but as time went on he refused to help her anymore. Women were not accepted as scholars during this time, so much so that Germain couldn’t even attend the Academy Prize sessions because she was a woman and not the wife of a male mathematician. It is truly amazing that through all the suppression she experienced that she was able to overcome it all to become one of the most well known female mathematicians ever [5].

Another female mathematician from a more current time period was Emmy Noether. She was born in Germany in 1882 and died in America in 1935. Noether was an algebraist and is known for her work in topology. She worked on Algebraic Invariant Theory and allowed the study of the relationships among the invariants to be possible. Invariant Theory deals with action of groups on algebraic varieties from the point of view of their effect on functions [7]. She made huge progress on ascending and descending chain conditions. A partially ordered set satisfies the ascending chain condition if every strictly ascending sequence of elements eventually terminates and it satisfies the descending chain condition if every strictly decreasing sequence of elements eventually terminates [9]. Most objects in abstract algebra that satisfy these conditions are called Noetherian after her. Some of these “Noetherians” include Noetherian induction, Noetherian modules, and Noetherian rings. Noether also did work in physics and published Noether’s First Theorem, which states that every differentiable symmetry of the action of a physical system has a corresponding conservative law, which she also proved [8]. She not only solved the problem for general relativity, but also determined the conserved quantities for every system of physical laws that possesses some continuous symmetry [6]. Noether’s advancements in math have lead others to call her the most important woman in the history of mathematics [6].

Lastly I will discuss a more current female mathematician, Maryam Mirzakhani. She was born in 1977 in Tehran, Iran. She has received many awards in her lifetime; she competed in two different International Mathematical Olympiads and received gold medals at both. She is also the first Iranian student to receive a perfect score at one of these Olympiads. In 2014 she was awarded the Fields Medal, an award that is given out every four years, to up to maybe four mathematicians under the age of forty. The award is often compared to the Nobel Prize of mathematics. She is the first female and Iranian to ever win the award [10]. Jordan Ellenberg explained her research when she won her award:

… [Her] work expertly blends dynamics with geometry. Among other things, she studies billiards. But now, in a move very characteristic of modern mathematics, it gets kind of meta: She considers not just one billiard table, but the universe of all possible billiard tables. And the kind of dynamics she studies doesn’t directly concern the motion of the billiards on the table, but instead a transformation of the billiard table itself, which is changing its shape in a rule-governed way; if you like, the table itself moves like a strange planet around the universe of all possible tables … This isn’t the kind of thing you do to win at pool, but it’s the kind of thing you do to win a Fields Medal. And it’s what you need to do in order to expose the dynamics at the heart of geometry; for there’s no question that they’re there [10].

Mirzakhani is a modern day mathematician we can all look up to; she has accomplished things that no mathematician has, not just female mathematicians.

Overall throughout looking at the history of some famous women in math I have realized how amazing these women truly were. They were able to overcome many obstacles and repression to further advance a subject that they love, and I love: math.

[1] http://www.math.wichita.edu/history/Women/hypatia.html

[2] http://en.wikipedia.org/wiki/Hypatia

[3] http://www.agnesscott.edu/lriddle/women/piscopia.htm

[4] http://womenshistory.about.com/od/sciencemath1/ss/Women-in-Mathematics-History_4.htm#step-heading

[5] http://en.wikipedia.org/wiki/Sophie_Germain

[6] http://en.wikipedia.org/wiki/Emmy_Noether

[7] http://en.wikipedia.org/wiki/Invariant_theory

[8] http://en.wikipedia.org/wiki/Noether%27s_theorem

[9] http://en.wikipedia.org/wiki/Ascending_chain_condition

[10] http://en.wikipedia.org/wiki/Maryam_Mirzakhani

Is Math an Invention or a Discovery?

A few months ago I was sitting at home watching one of those shows about the universe, you know where they try to condense everything there is to know about our world into a few short episodes? This particular episode was about Isaac Newton and all of his work. In this episode they were discussing how he invented Calculus and how he forever changed the way that we understand our universe. At this point I was pretty intrigued when my boyfriend raised a question I never really gave much thought to. He said to me “Do you believe math is something we discovered or something we invented?” My immediate reaction was that math was a discovery; there is no way that we just made all of this up! After this conversation occurred I started to notice that this was a question I began to think about often, but I never really could come up with a solid answer. So I will raise the same question again, was Math invented or discovered?

Fibonacci Sequence in a sunflower. Image: Ginette, via Flickr.

Fibonacci Sequence in a sunflower. Image:
Ginette, via Flickr.

Let’s start with the discovery side of things; there are many different mathematicians who believe that math was a discovery, such as Plato and Euclid. Mathematical Platonism is “the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices.”[1] This philosophical viewpoint is stating that our universe is made up entirely of math. When we begin to understand math we are allowing ourselves to understand more about how the world around us works [2]. Have you ever thought about how math occurs in nature, that there are patterns and sequences all around us? Euclid believed that nature was a physical manifestation of math [3]. Examples of mathematics in nature include honeycombs, wings of insects, shells, and flowers. We also find the opposite of patterns in nature, uniqueness. The theory that no two snowflakes are the same is an example of uniqueness occurring in nature. Another more modern theory that supports the notion that math is a discovery is the mathematical universe hypothesis, which was proposed by a cosmologist Max Tegmark. This theory states, “Our external physical reality is a mathematical structure.”[4] Basically he is saying that math is not necessarily used to describe our universe, but rather our universe is one mathematical object. I think this theory is very intriguing and would make perfect sense. It would explain why math can be applied to everything that we know.

On the other side we have the belief that math is an invention. The most common theory is that math is a completely human construct, which we made up in order to help us have a better understanding of the world around us. This theory is called the intuitionist theory. The theory is a rejection to Mathematical Platonism and states that “The truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that construction by intuition.”[5] Opposing the mathematical universe hypothesis is Gödel’s first incompleteness theorem. His theorem states that any theory that it has axioms can’t be consistent and complete at the same time. [6] This theory would show that math itself is like one giant loop. Every time we solve one problem based on assumptions we gain another problem that we must now base on assumptions we made from the last problem. This cycle will continue to repeat itself over and over and is inexhaustible.

Another common observation about math is how we actually carry out the process. If math were a discovery would we always have the same method for each problem. As shown in class the Egyptians had a completely different way to multiply that can be more effective than our current system of multiplication because it involves less memorization. Are our different methods for the same math problem enough to show that math is an invention? Or is it enough that we can get to the same solution, so the process isn’t as important? There is even the possibility that there are more discoveries to be made which could end our need for different methods to get to the same solution. There could be a missing link in our chain that we have to work around in order to get the solutions we need, but if we found that missing link we would only need one method to solve our mathematical problems.

In my own opinion the recurring theme of mathematics in nature is evidence enough for me to believe that math is a discovery and not an invention. With that said there are compelling arguments on both sides and it may take us years, if ever, to really prove whether or not math is a discovery or an invention

[1] http://plato.stanford.edu/entries/platonism-mathematics/

[2] http://science.howstuffworks.com/math-concepts/math4.htm

[3] https://www.youtube.com/watch?v=X_xR5Kes4Rs

[4] http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis

[5] http://en.wikipedia.org/wiki/Intuitionism

[6] http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems