“The best way to predict the future is to invent it.” Catmull, Ed; Wallace, Amy. Creativity, Inc. . Random House Publishing Group. Kindle Edition.
I feel so small With my hands up to the sky I am reaching out tonight 'Cause this is bigger than us I give my all But it's just too much to hope No, I can't do this alone 'Cause this is bigger than us Bigger than us It's like everything changes Still nothing changes We arrive and we begin Running in circles Searching for purpose With an open hand - “Bigger than Us” - Josh Groban
Math is alive and well
We have journeyed through a lot of math so far in Lazarus Math. I have written Lazarus Math to communicate what math means to me in the hope that it will inspire you. To me, math is like music that I “hear” and “play” rather than a subject that I study. This wasn’t always true but is something that has evolved over time. As I think back, I ask myself how this change occurred. What happened in my life to change math from a subject to something I experience and enjoy? In this last section, we’re going to ease up on the heavy math and pivot toward understanding where the math came from and how it inspired me.
One key turning point occurred in February 2018 when my sister gave me a book on the mystery of the zeta function and Bernhard Riemann’s ideas that have changed the world. The book gave the history of how math ideas developed over the centuries. I was fascinated by the story, the people, and the mystery. This is ironic because I never considered myself that interested in history. I’ve always been a forward-thinking person who focuses on the future and the present, not wanting to waste time with the past. As I write this, it is clear how narrow and shallow my thinking was. The stories of how math got to where it is today and, more importantly, the people who have contributed to the math we now use, have captured my attention, making me feel connected to these math pioneers and their ideas. It is difficult for me to articulate this change. Perhaps it is easier to communicate why I enjoy reading these stories of old. There are several reasons, but I will give three main reasons.
One reason I read these stories is I find them incredibly interesting and entertaining. For example, the lives of people like Gauss and Riemann are full of adventure and mystery. Not only are these two lives interesting, but how they connected and impacted each other adds a new twist to the story. Riemann attended school at Göttingen, intending to pursue a degree in theology. Gauss was a faculty member at Göttingen and recognized Riemann’s math talent and convinced him to change majors. We have the benefit of hindsight to recognize the impact of Gauss’s initiative and influence on Riemann. It reminds me of all the math teachers reading Lazarus Math. It is impossible to know the impact you have on future generations as you teach and inspire future mathematicians. However, reading these stories is a friendly reminder of the impact we have when we mentor others. I think of it this way: reading stories of math people from the past is not just a journey through ideas but a taste of humanity. What if these two math heroes never met? Would Riemann have pursued math? Would Einstein have had the math required for his theory of relativity?
A second reason for reading these stories is I have admired these math geniuses of old and have been intrigued by their lives. As I read about them, they became my heroes and their lives inspired me. Reading stories about the pioneers of math ideas helped me understand the ideas themselves better. It gave me context. Suppose I gave you a math problem. If you work on the solution rather than being given the final answer, you gain a deeper understanding of what I was thinking. Similarly, reading about the twists and turns of how math developed into what it is today has given me better insight and understanding of our current math. Like so many things in life, context is critical. I have a much richer understanding of calculus from learning about the history of how calculus arrived at where it is today. Reading its history highlights that math is not so much about formulas as it is about ideas, relationships, and concepts. Reading the ideas, relationships, and concepts the greats wrestled with has helped me understand which concepts are most important. Now when I work with exponential functions and logarithms, for example, I appreciate the richness embedded in these operations.
The third reason I read these stories is they teach me about life. Viewing stories of different times and places about people with different perspectives stretches my empathy. My world becomes larger and richer and not bound by time and place. To read about Euler losing his eyesight and then declaring he had one less distraction to do his math impacts me. My problems got smaller when I heard that story. It gave me courage to face challenges. These people of old who formed timeless math ideas impacted their culture. They had incredible vision for the future. Many of them did their best work in the most difficult circumstances. Some of these people have become role models for me as I navigate my life. We live in a world with so much noise and volume. Even though many of the stories of math are filled with adventure and surprise, they are often quiet stories of people with humility yet brilliance.
Hopefully, at least one of these reasons will motivate you to read a few more stories. In the remaining part of this subsection, I will summarize a few math stories to inspire further learning. To get a richer and fuller perspective, find a book about Euler or Riemann or Euclid or another personality that appeals to you.
We have already touched on the lives of the early pioneers: Archimedes, Euclid, and Pythagoras. Few people have impacted the history of the world as long as these three have. Then, after the dark ages, came the giants, Leibniz, Euler, Gauss, and Riemann, whom we have discussed. These seven are the giants among giants, but there are certainly more in this elite list. In the following subsections, I will introduce six more people to you. I am choosing these six because each has something to contribute that is bigger than just the math they developed. Though you may be acquainted with some of these six names, there will likely be one or two you are unfamiliar with.
Al-Khowarizmi
A name that perhaps you haven’t heard of is Muhammad ibn Musa al-Khowarizmi (780-850). One of the key themes for Lazarus Math is to create a system to solve problems. Al-Khowarizmi is mostly known for his contribution to algebra, and some refer to him as the father of algebra. We take for granted equations that involve unknown values such as and . However, it took the math world a long time to develop this system, and al-Khowarizmi was a main contributor. Even though he used verbal language rather than symbols, his verbal method was a systematic way to solve equations with unknown quantities. He created math recipes, if you will, that generalized solutions rather than just being a specific answer. It is easy to underestimate the work required to develop the algebra that we know today. We can thank al-Khowarizmi for getting us started.
Algebra, though, is not the only contribution we received from al-Khowarizmi. He is also pivotal in developing the concept of an algorithm. Now, we often think of algorithms as logic used by programmers. However, we use algorithms to solve math problems as well. Recall how we proved the Euler product formula. This sieving method is an algorithm. You can think of an algorithm as any math system that produces a solution in a finite number of steps. In other words, an algorithm is a series of logical steps that, once executed, achieve some math result.
We have used the trigonometric functions, sine, cosine, and tangent, in Lazarus Math. Al-Khowarizmi was one of the first to create tables that calculated these functions.
It’s not only the work that al-Khowarizmi did that was important but the time when he did his work. Al-Khowarizmi lived during the Middle Ages when there was not much progress in math. However, his work helped start the growth of math ideas in Europe at the end of the Middle Ages. Medieval Europe was “dropping the math ball” during this period and al-Khowarizmi was there to pick it up and keep it rolling.
You can think of al-Khowarizmi as planting a math seed that bore fruit in Europe during the Renaissance. Like many pioneers, al-Khowarizmi blazed difficult trails during difficult times that paved the way for future progress.
Kepler
There were so many incredible mathematicians during the Renaissance. One key person to transition the European world from the Middle Ages to the Renaissance is Johannes Kepler (1571-1630). However, I’m including Kepler in my list not only because of the timing and importance of his work but the context in which he created his work. We all appreciate when others can change our life from chaos to order. Kepler is someone who converted the world of math and science from chaos to order. He was a pioneer of the scientific method. This system of thought has forever changed how we think. Kepler is best known for his three laws of planetary motion. The first law describes how the planets move in ellipses with the sun at one focus point. The second and third laws explain how fast the planets orbit around the sun. All three laws are foundational to the science we know today. They are building blocks that others used to expand their ideas. For example, Newton used Kepler’s third law to build his law of gravitation.
To be clear, Kepler did not always have the right answers. But Kepler was good at asking the right questions. It is impressive that he discovered the three laws, but that he even thought that there were laws to be discovered is impressive as well. He was asking the right questions. Many of his colleagues were plotting data points and using data to answer questions. Kepler went against the grain and believed that there were universal laws that produced that data. It is easy to underestimate the difference in this perspective now. But at the time, it was revolutionary.
In addition to asking the right questions, Kepler stands out to me because of his courage. His courage showed in how he accomplished his work despite a life of hardship, losing many of his family to untimely deaths. Kepler also displayed courage by solving problems that were controversial and resulted in intense persecution. He had the courage to publish his results even though his views opposed the church at that time. One key issue Kepler wrestled with was the center of the universe. The church believed the earth was the center, and Copernicus stated the sun was the center. Kepler essentially agreed with Copernicus that the sun was near the center and identified that the planets revolve around the sun, not in a circle but an ellipse. Thus, the sun was not quite the center of the known universe but close to the center.
However, Kepler not only solved math and science problems. He also saw how math connected to a bigger picture in life. He expanded the concept of symmetry by analyzing the symmetry in a snowflake. He observed that our perception changes when we view life through two eyes rather than one. Like Pythagoras and other math greats, even though he was studying planetary motion, he realized that the math that explains how planets move is connected to the math of music. At the heart of this connection are patterns of addition and multiplication embedded in both. The most common examples are the arithmetic and geometric means. We already uncovered how circles contain this hidden math. Similar examples occur in ellipses, which Kepler popularized. We will uncover some of the beauty of an ellipse and how it relates to a circle in the next subsection. In the process, we will also see how the math of an ellipse connects to the math of music.
Fermat
Over the ages, math people have presented and solved countless math problems. Few have captured the attention as the one presented by Pierre de Fermat (approximately 1607-1665). We have studied and proven Pythagoras’s theorem that for any right triangle with side lengths and and hypotenuse length . Because there are an infinite number of right triangles, there are an infinite number of solutions to this equation. Consider if the exponent is an integer greater than 2, such as 3. Is there a solution to the equation ? Certainly, there are solutions if and can be any real numbers. But, if we restrict and to be integers, is there a solution to this seemingly simple equation? This doesn’t appear to be anything unusual or obscure, so it seems there should be a solution.
Fermat boldly stated that there are no solutions to the equation for any integer and positive integers and . This became known as Fermat’s Last Theorem. Before I share the rest of the story, take a moment and appreciate how bold a claim Fermat made. For any positive integer greater than 2, we cannot identify 3 integers and that meet this simple criterion. Of course, to prove him wrong, we only need to identify one example that has a solution. There are an infinite number of possibilities, yet Fermat claimed that none exists. Of course, if you have no regard for your reputation, you can easily make bold claims like this. However, the main thing math people have going for them is their reputation, so it is not something they risk devaluing.
Fermat made many conjectures without proving them. After his death, mathematicians were able to prove all of his conjectures except this one. The fact that all the other conjectures were proven only strengthened Fermat’s reputation. Thus, this remaining unsolved conjecture became known as Fermat’s Last Theorem because it was the only theorem not yet proved to be true. Fermat stated he had a proof of the theorem but the margins of the paper he was writing in were too small.
Fermat’s Last Theorem captured the attention of the greatest math minds for centuries after his death. There were several reasons. The problem is extremely simple to understand, and it appears like there should be a solution. It is like an extension to the beloved Pythagorean Theorem. Fermat stated that he had a proof, and all of his other conjectures were proven to be true, but it is a mystery whether Fermat actually had a proof for this conjecture.
As math people tackled this problem, they made progress but did not prove it completely. During the 1700’s, Euler essentially proved that there are no solutions to this equation when , which is the cubic equation. During the 1800’s, there were proofs by other mathematicians that there are no solutions when and several prime numbers up to 100. By 1993, it was known that there was no solution for any greater than 2 and less than 4 million. However, there was no proof for the general case where . It wasn’t until 1995 that Andrew Wiles (1953-) finally proved Fermat’s Last Theorem for the general case. How Wiles proved this theorem is another interesting story to pursue if you are interested. Wiles' solution was a dramatic ending to a problem that challenged the greatest minds of the last few centuries.
Do we know if Fermat actually had a proof? The general consensus is that he did not. If he did, his proof must have used a different method from Wiles'. Wiles' proof benefited from work done after Fermat. Specifically, he relied on the Shimura-Taniyama conjecture, or the modularity theorem, which was first introduced in 1955. Interestingly, the proof is connected to elliptic curves, which appears not to be at all related to the integer problem proposed by Fermat. Elliptic curves are curvy and continuous, while Fermat’s Last Theorem is about integers. These two concepts appear to be worlds apart. But surprisingly, mathematicians were able to prove that the solution to the elliptic curve problem is also a solution to Fermat’s Last Theorem.
Fermat has impacted math in many ways in addition to this theorem. His work with tangent functions helped pave the way for calculus. Ironically, Fermat was not a full-time mathematician. His day job was practicing law. Not only is Fermat’s impact on the development of math remarkable, but many of his insights were on topics that were not even recognized topics at the time. For example, his Last Theorem is a problem in number theory which wasn’t a recognized subject at the time. Fermat truly was a person of vision.
Newton
One of the things I enjoy most about problem solving is when I can find a system to follow. Sure, creating a clever solution to a problem is rewarding, but usually those clever solutions are “one-off” solutions. The most useful solutions are the ones that provide a system for solving general problems. We all experience this in little things in life we take for granted. We have a weekly calendar that organizes our lives in a cycle of 7 days. Can you imagine what life would be like without it? Unlike the seasons that are connected to the earth revolving around the sun, the weekly calendar is an external system. Yet the weekly calendar is a useful system that allows us to organize our lives.
What is even better than a system that works for me is a system that works for everyone and is timeless. This is what Isaac Newton (1643-1727) accomplished when he created a system we now call calculus. Remember Leibniz is also credited for discovering calculus as they essentially worked on it at the same time, but Newton was slightly ahead of Leibniz. Regardless of who gets the credit, two of the main things Newton (and Leibniz) accomplished with calculus was to solve the area problem and to solve the slope problem. We have seen both of these problems occur frequently in Lazarus Math, especially the area problem. Before Newton and calculus, the solution to both of these problems were clever one-off solutions.
Not only is calculus a solution to difficult problems, but it also functions like a “math calendar.” It is a system that math people rely upon so much in the work they do. The system of calculus is so engrained in math that we often take its benefit for granted, much like we take for granted the benefit of a weekly calendar.
Newton not only created systems to solve the area and slope problems, but he identified that solutions to the area and slope problems were inverse operations. We have referred to inverse operations often as something that will “undo” something else. Examples have been subtraction that will “undo” addition and the logs that will “undo” an exponent. It certainly is not obvious, at least not to me, that solving for the area under a curve is the inverse operation to calculating the slope of a curve. Yet, that is the gift that calculus brings to our math world.
We saw this inverse operation at play when we worked on the field goal problem. We identified that if we have the inverse tangent function of , we can write an expression that represents the slope of the inverse tangent at any value . The slope expression is . We also went the other direction to calculate the area under the graph of , and the inverse tangent function of gave us the solution to this area problem. Ironically, this area problem was the solution to our Leibniz problem. At first glance, identifying this inverse operation between area and slope may not appear as important. However, we have identified a few examples where we can convert real world problems into problems that involve the slope of a curve or the area under a graph.
If creating this system for the math world wasn’t enough, Newton lived at a time when scientists were just dreaming that there were laws that could explain the world they lived in. Newton developed his calculus to primarily solve physics problems. He was following and expanding upon the works of early pioneers Galileo Galilei (1564-1642) and Kepler who dreamed the world had laws it obeyed. This idea was a foreign concept to the people in the 17th century.
I enjoy learning about math heroes of the past not only to understand their work but who they were as people. Often, I am inspired by how these math legends overcame obstacles to do their work. For me, Newton is a little bit of an enigma. Newton was an extremely private person with many personality and lifestyle quirks. I don’t necessarily consider him a role model to follow. But that is not to say he is not inspirational. I am inspired by how he accomplished his landmark work on calculus. Newton was attending Cambridge University when the bubonic plague arrived in London, killing many in its path. In the summer of 1665, Cambridge University closed classes, so Newton returned home. During the next two years, he developed his calculus among other important discoveries. This story inspired me to write Lazarus Math in 2020 during the Covid-19 pandemic. As is true of many great mathematicians, Newton did his best work during the most trying of circumstances. I consider many mathematicians as role models because of how they lived their lives. Then there are others, like Newton, who I draw inspiration from because they overcame adversity to do their best work.
Noether
One of the best kept secrets in math is group theory. Group theory is an ocean of amazing concepts with a rich history. I am only touching the short tip of the iceberg here, but I hope to give you enough to encourage you to study this further if you find it interesting. I have shared some basic principles from group theory in Lazarus Math when I talked about symmetry.
My focus on group theory is primarily a quick summary through Emmy Noether (1882-1935). I’m including Noether because her story is inspiring, and her work is amazing.
Part of what makes Noether’s story inspiring are the obstacles she had to overcome in order to do her math. Noether had the benefit of living during the time of Albert Einstein. Unfortunately, she also lived in a time when there was discrimination against female mathematicians. For example, two of the leading mathematicians of her day, David Hilbert (1862-1943) and Felix Klein (1849-1925), invited Noether to join the faculty at Göttingen. However, it took 4 more years before she was allowed to teach because of sex discrimination. Hilbert was not going to waste her talent during this 4-year period, so he secretly allowed her to teach classes under his name.
In addition to sex discrimination, Noether’s parents were both Jewish, which led to more persecution while living in Germany. Thus, from a persecution perspective, she could not have lived in a worse time and place. Eventually, she left Germany and moved to the United States. She is only one of many top mathematicians who left Germany and Europe for the States during this time.
On a side note, it is difficult to quantify how much the US benefitted from the talent that fled Europe to the States because of persecution, but it is safe to say that the US benefitted quite significantly. Göttingen was the mecca of math research at that time, and many of the top faculty fled to the US as a result of Nazi persecution in 1933.
There is a lot more that could be said about the obstacles Noether overcame, but let’s focus now on one of her accomplishments. In 1915, Noether proved what is now known as Noether’s theorem. This theorem has the surprising conclusion that there is a connection between the symmetry we see in math and the conservation laws in physics. From a simplified math perspective, symmetry occurs when things remain the same if we change a perspective or change time. You may not have thought of something that doesn’t change over time as being symmetrical. But Noether was able to prove that something that does not change over time is the same thing as the conservation of energy, that we cannot create nor destroy energy.
Of course, all this is deep and complicated. However, at its core, what this is saying is changing our perspective of time and space is the same as changing our perspective on the identity of particles. Because of Noether’s theorem, physicists have an entirely different perspective on basic concepts, such as what an electron is.
It is amazing that conservation laws from physics are connected to symmetry in math. Even more amazing is that this can be proved mathematically. Part of what drives math people to do their work is the beauty they see in symmetries. Then, Noether appears on the scene and identifies math features which have an aesthetic appeal are connected to the foundation of physics.
Much of Noether’s work has been packaged into group theory. Group theory can seem overwhelming and daunting because it is abstract. However, one of the goals of Lazarus Math is to help you become comfortable with the concept of abstraction. It is easy to consider things that are abstract as things that are impossible to understand. But you are already comfortable with abstractions in math. For example, you understand what “2” represents. This symbol “2” is an abstraction for a number, yet we are quite comfortable with it. We would not want to use two dots, for example, to always communicate the concept of two. Using a symbol is an abstract process that allows us to expand what two represents. It doesn’t have to be two dots. It can be two coins or two babies. This type of abstraction has tremendous benefits and it is at the heart of group theory and what Noether has helped develop.
Einstein was well aware of Noether’s contributions to physics. After her death, Einstein wrote to the New York Times, “Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.” Even though Einstein justifiably receives the credit for his theory of relativity, many math people such as Riemann, Noether, Klein, and Hilbert were behind the scenes either providing the framework for Einstein to build on (Riemann) or taking what he started and expanding it (Klein, Hilbert, and Noether).
Ramanujan
All the math people I’ve included in Lazarus Math are pioneers of some sort. It’s more difficult to find pioneers as we move to the 20th century because so much of math has been developed, or at least started. Emmy Noether certainly was an exception. But she is not the only 20th century mathematician who has made a major impact on math and, surprisingly, physics. Another 20th century pioneer is Srinivasa Ramanujan (1887-1920). Ramanujan is a relatively modern-day mathematician that I find interesting and inspirational. He was poor and lived in India most of his life where he had minimal resources to explore his math. It is unusual for this time period for someone to work independently as Ramanujan did. However, he was not aware of much of the math work before him because of the poverty in which he lived. But as he essentially worked independently of previous mathematicians, he rediscovered some of the great works of previous mathematicians and broke new ground himself.
Ramanujan had almost a childlike innocence in how he played with numbers. Because he was not aware of a lot of other people’s work, he had a lot of truly original ideas. He was fascinated with numbers and was driven to see how numbers connected. It has been said that Ramanujan considered every positive integer to be a personal friend. He was a Brahmin of the Hindu faith and considered his work as only uncovering the works of his gods. His goal was not just to learn math but to understand the connection between the math and the god that created the math. He claimed that his formulas came to him in dreams.
Eventually, Ramanujan came to England in 1914 at the request of another great mathematician, GH Hardy (1887-1947). Although brilliant, Ramanujan was a bit raw in his approach to math. Hardy was able to mentor Ramanujan and gave him an opportunity to develop. However, the change in culture from India to England was difficult for Ramanujan, partly because of the persecution he experienced in England. He died at the age of 33.
There are a lot of reasons to enjoy the story of Ramanujan and to be inspired. First is the pure brilliance he possessed. Many consider Ramanujan as one of the most naturally gifted mathematicians of all time.
However, that raw talent needed to be developed. It is easy to overlook the value of mentoring others. Hardy took a chance on Ramanujan and invested valuable time mentoring him. That paid tremendous dividends as Ramanujan developed math that was truly unique. Ramanujan started from the humblest of beginnings, was self-educated, and worked tirelessly on his math. Despite his hardships, he was an enjoyable man whom others appreciated. Solving math problems requires good intuition, and Ramanujan is a good example to learn from. Ramanujan introduced many original methods and ideas, and many of them are being uncovered and unwrapped yet today.
Ramanujan's story is similar to those of Riemann, Noether, and many other mathematicians. A lot of Ramanujan’s work may have appeared as playful fun at first as he authored seemingly obscure formulas and relationships. But many years later, these formulas had major applications to physics and other areas. Ramanujan’s math has had a major impact on string theory. String theory had its early beginnings with Einstein but didn’t gather much momentum until many years after Ramanujan’s death. It is amazing how often pure mathematics has influenced physics without that purpose or intention.
Physics, group theory, and symmetry
In fact, the current trend in physics is to start with beautiful and symmetrical math, rather than starting with data, as the source for new research and development. In other words, as physicists are trying to explain the world we live in, they have identified that it is often better to start with pure math than to start with data from observations. This beautiful and symmetrical math is not only their starting point, but it also impacts each step of the way. There is a strong belief that nature has a behind-the-scenes architect and one of the key properties used by this architect is symmetry. To be sure, there is a controversy in physics as to the validity of this approach. But the fact that many leading physicists demonstrate their confidence in pure mathematics is an amazing testimony to how connected pure math is to the world we live in.
It’s worth restating how surprising this result is. Pure math is developed by starting with a few definitions and assumptions, such as how to treat parallel lines. Once these few assumptions are set, then the math is out of our hands. Once the math people had set their definitions and assumptions, then it is their job to pursue where these assumptions lead. An interesting example is group theory itself. Mathematicians made the definitions and assumptions for a group. Then, they pursued organizing math into this group theory structure. There are some truly amazing results from that research, and there are some things that aren’t as symmetrical as one would like. For example, much of math fits into beautiful types of groups. However, there are 27 types of groups that can be considered as a one-off. As we’ve discussed, one-offs are the enemy of simplicity and beauty, yet we have identified there are 27 of these “monsters” out there. Thus, this pursuit of “all things beautiful” in group theory ended with some clutter. I like how Grant Sanderson put it: “It’s like the world was created by a committee.” Regardless of the clutter, the beauty we do find is undiminished.
The part I find most interesting is that we find years later that pure and beautiful math that seems only good for theory actually explains reality. I find it inspiring that it does not just explain reality; it explains the parts of the universe that are either too large or too small for most of us to comprehend. It is inspiring that a poor Indian mathematician, who could not afford paper to write on and who developed seemingly obscure formulas, is now making an impact on physicists' attempt to unify relativity and quantum mechanics.
Not the physics of Newton
This world of relativity and quantum mechanics is far different from the world of science developed by Newton. Newton’s science was a science of certainty and precision. Today, the perception is we live in a world where the only thing that is certain from a human perspective is that everything that the world is made of is uncertain. This complete change in perspective resulted from the work of modern-day physicists and the math that they are using from people such as Riemann, Noether, and Ramanujan. Unlike Newton, these math heroes developed pure math ideas that were independent of applications to physics, such as understanding about the very large and very small parts of the universe that we occupy.
Once again, I invite you to find a book on one of these stories that you find interesting. The details you read can enlighten and enrich. If you find stories of symmetry and group theory interesting, then I recommend two books that introduce you to many of the contributors to group theory. Surprisingly, the beginnings of group theory travel through al-Khowarizmi because the root of group theory is algebra. This algebra story involves the familiar quadratic equation, , and the formula for based on the coefficients and . Once the quadratic equation was solved, then the challenge expanded to the cubic equation, where the largest exponent is 3, and the quartic equation, where the largest exponent is 4. Eventually, when these equations were solved, a solution to the quintic equation, where the largest exponent is 5, was sought. This is where the story gets interesting as it carries into the 19th century. In the 19th century, you meet interesting characters such as Évariste Galois (1811-1832) and Niels Henrik Abel (1802-1829). In the 20th century, the story includes other hall-of-fame mathematicians such as Klein and Henri Poincaré (1854-1912). I won’t spoil the story, but this famous quintic problem is what leads to group theory. Then, of course, group theory eventually changes the foundation of physics. The books are filled with drama and surprise.
One surprise is that the role of group theory can be thought of in the context of “which came first — the chicken or the egg?” We have the historical answer, that subjects such as algebra and geometry arrived before group theory. Group theory was developed from algebra, so algebra and geometry clearly came first. However, as group theory developed, it was identified to be the foundation of math. Nearly all of math, and now applied physics, fit into a group. Thus, you could view group theory as coming first and all the other subjects being layered on top of it.
If you want a quick overview of group theory, you can go to 3Blue1Brown’s YouTube channel and search for group theory. There are several videos, about 20 minutes in length, that will get you started.
In summary, as I think of the rich history of math and the people who made it possible, I realize that I am not just a passive observer in these stories. I get to enter into each wonderful story and become an active participant when I engage with math. Regardless of how small my math footprint is, I get the opportunity to participate and contribute to something bigger than myself. This idea of something bigger than myself is something I will expand on in our last subsection. But first, let’s play with one more math idea.
