A Mathematical Journey of Thinking

The single biggest problem regarding mathematics and the sciences is motivating younger students to study them.

While the United States excels at welcoming people from all over the world to travel to the U.S. and study science and math, the number of aspiring mathematicians at universities is decreasing. About only 2% of all students in America that pursue a bachelor’s degree are in the fields of mathematics or other physical sciences. On top of that, roughly 48% of students studying a STEM field in a bachelor’s program, and 69% of students pursuing an associate’s degree, changed majors or exited college before earning their degree [1]. One possible explanation for this is the reputation that math has for being boring, uninteresting, and complicated. However, there could hardly be a better time to become a mathematician.

I was watching a documentary about deserts with my 2 year old. At first, she found the documentary to be boring because the desert appeared to be bland and void of life. Then, the screen started showing the lively and colorful aspects of a desert including its oases, various cacti, and small creatures. She was shocked that the arid desert could possess so much color, life, and intrigue. It reminded me of my feelings towards mathematics. For some people, mathematics seems like a dry and dull subject that makes little sense. However, it too becomes full of life and color once one looks in the right places and from the correct perspective.

Ten years ago, the Wall Street Journal ranked jobs based upon a number of parameters such as salary, freedom, the possibility of promotion, and ease of employment. It may surprise you to find out that being a mathematician was ranked as the number one job in the world! [2] Career Cast, a company that started as a spin-off for this kind of ranking in 1988, also ranked mathematician as the number one job in 2014 [3].

In job advertisements, the definition of a mathematician is somebody who applies mathematical theories and formulas to teach or solve problems in a business, educational, or industrial setting. In other words, a mathematician is someone who insists on solving problems, which is what mathematics was invented for. However, a crucial point is missing from this description. There are mathematicians who not only teach or apply mathematics but do math, generate math, and change math.

If a mathematician is one of the best jobs in the world, how many mathematicians are there around the world? Thankfully, it is in the hundreds of thousands: almost 250,000, and it is growing every year, particularly in India and China [4]. These countries encourage their children to participate in STEM fields from a young age and place a very high emphasis upon education that can help foster a love for STEM careers at a young age.

Although mathematics is the science in which theorems last for a long time, it is also a field that is being renewed constantly. There are not many other subjects in which knowledge can last thousands of years and always remain both correct and relevant. For instance, the theorems of Euclid, a Greek mathematician who lived in the third century BC, are still accurate today. If we look at the modern applications of computer science and mathematics we see that people are still applying the contributions of Laplace about the central limit theorem, using the contributions of Shannon in sampling, and applying Fourier’s ideas about signal analysis and signal processing. Paradoxically, mathematics is ancient and contemporary at the same time.

In 2014, Emmanuel Candes of Stanford University gave a short speech about how he had collaborated with medical doctors and mathematicians, and used Fourier’s ideas in analysis and data processing, to drastically reduce the time needed to develop a scan [5]. His machine was a significant evolution for the medical field. In experiments, it was shown that to reconstruct images with the best accuracy, they need to have only 2% of the Fourier transform. Candes managed to reduce this time by a factor of eight by knowing only a very small portion of the Fourier transform. Netflix also uses this technology for reconstructing missing information and predicting the preferences of users for movies.

However, the life of a mathematician is not always full of success. The field is full of uncertainty, intense problem solving that can often take weeks or even months, and waiting. Most of a mathematician’s time as a researcher is spent in failure. That is an objective fact. A common motto is, “Every day is a failure. This is our life.” But when we look back on the amount of time that is spent solving these problems, everything pays off. Every year, the number of new theorems added in mathematics runs into the thousands. Failures are not actually failures, but merely bumps along the path of progress.

Another common problem that befalls mathematicians is that they will be expected to perfectly predict what will happen should their discovery work as intended. This is especially true of work done under the auspices of government agencies, since they will often not grant money without a very clear path towards a desired outcome. Things do not always go as planned, and thus it can be challenging to argue that a desired result will occur 100% of the time.

Despite these drawbacks, one of the most exciting aspects of mathematics is the potential to discover something revolutionary and groundbreaking. History has shown that the ideas and discoveries of one individual can make a tremendous difference in the world. By encouraging and enabling more and more students to study math and science, we thus increase the odds that more discoveries, be they groundbreaking or not, will continue to be made in the future.

Alan Turing was a mathematician who had an outsized impact on human history. He was instrumental in cracking the secret codes that were used by the Nazis during the Second World War. Some have argued that World War II would have lasted at least two more years without Turing’s intervention. In particular, the Normandy operation would have been impossible. What was Turing’s motivation? It was not patriotism, honor, or a strong duty to his country. It was simple: his main motivation was solving riddles and difficult problems. He lived for the thrill of solving his next big challenge.

Paul Erdos, a Hungarian, was the most productive mathematician of the 20th century. Erdos had no home, no car, no bank account, and no salary. He lived with just one suitcase and his ideas. He worked on theorems his entire life. When he got some money from some reward, he would always use part of it to put a reward on another theorem.

Leo Szilard was another revolutionary mathematician. He was the first person to understand the concept of chain reactions between atoms. His inspiration came from a public lecture, in which famed physicist Ernest Rutherford said, “It would be moonshine talking if you are willing to extract energy from the atom.” Rutherford acknowledged that energy existed within atoms but thought it was impossible to do anything with it. Szilard felt that Rutherford was wrong and decided to prove it. He worked and thought for days. And one day, while he was crossing a street in London, he was struck with an idea about the principle of chain reaction and exponential growth of atomic energy. This lead to him eventually meeting with Albert Einstein and the subsequent development of the Manhattan Project.

Steps in developing ideas

But how do scientists find those perfect ideas? What are the steps? Henri Poincare gives us some hints about how the ideas were coming to him. Poincare, who was always considered a genius, experienced many discoveries after working very hard on the problem and he had a lot of failures. He said: “Disgusted from my failure, I went to spend a few days near the sea thinking anything else. And one day, while walking on the cliff, the idea came to me. And as before, it was very brief, sudden, with immediate certainty, that arithmetic transforms of indefinite ternary quadratic forms are identical to those of non-Euclidean geometry.”

So, under which circumstances does a big idea come? In popular culture, we have those myths like Newton saw an apple falling down and changed the world. But in real life, it’s not the way it happens. For Poincare’s example, the cliff has nothing to do with quadratic forms. Poincare was saying that he worked very hard and then he decided to rest a little bit, and finally, he had the enlightening moment. What is important here is if the brain had not been prepared by hard work, the illuminating idea wouldn’t have struck.

So what is the process of discovery?

Of course, a publication will be the last step for the discovery of an idea. Because publication means that the idea is out and going to be seen and read by the world. But before that, we have many steps or ingredients which make our idea stand up on its feet.

1. Fecundation

The first step is fecundation. In other words, conversations and discussions you have with your colleagues. Your interaction has a potential to bring about a new phase, a new idea out of different projects. It may take months or years to decide what you want to prove.

2. Documentation

Documentation is built upon previous research and insights. We may need to document things from several centuries ago or from recent times. Nowadays, all information is stored in computers and the internet, which makes this step easier than before.

3. Motivation

Motivation is the most important ingredient when we pursue a discovery. Psychologists believe that childhood experiences play a big role. For instance, both Szilard and Turing’s lives were strongly influenced by a book they read before they found their ideas. In the case of Turing, the book Natural Wonders Every Child Should Know was his inspiration. When I was a child, I watched “Donald in Mathmagic Land” and thought it was fascinating. It might have played a big role in my choice to become a mathematician.

4. Ecosystem

A discovery or an idea never arrives on its own. A scientist is never alone and there is a whole ecosystem around them. For instance, at one point in history, Persepolis was the most innovative city in the world. Then it was Paris, and then Budapest. Today, we have the well-known Silicon Valley.

If you are working in a lab, you need to have an atmosphere where people can meet, discuss, and be creative. A good idea comes from teamwork.

5. Constraints

The next ingredient that you need for a good idea is constraints. Without constraints, discovering new ideas is not as likely as when there are. Rigorous findings come about mostly with constraints. One of the most famous problems in mathematics is the Riemann hypothesis, which was tested in thousands of experiments. After those experiments, the proof was only a set of logical rules with constraints. Constraints usually help generate authentic results and artistic quality as in rhyming poems.

6. Intuition

Intuition usually comes together with hard work, yet it is not easy to understand its process.

In addition to these six ingredients, whether one has good fortune or not also plays an important role. It is human condition that things may not come out as we like although we might have done everything necessary.

Henri Poincare says, “Thought is only a flash between two long nights, but this flash is everything.” Yet, a big idea might take years of work. Still, even after working so hard, our knowledge is like a tiny island in an ocean of unknown. We know almost nothing, but this is such a precious nothing, because that that knowledge came out of the richness of the thought of many people and their efforts. There are still so many ideas just waiting for us to discover them.

Mathematics, like many scientific fields, is an infinite universe with an infinite amount of problems waiting to be discovered and solved. Their applications within our world are endless, and the possibilities to use these results to create real change in our world are also endless. One tends to ask, is math a stand-alone, abstract concept while its vast horizons of knowledge explains the intricacies of our universe, or does it imply an infinitely bigger wisdom from which this abstract knowledge rises from and that relates to our existence?  We must continue to motivate students to study math so that they may fall in love with it and continue to develop humanity. 

References

  1. United States, Congress, Chen, Xianglei, and Matthew Soldner. “STEM Attrition: College Students’ Paths Into and Out of STEM Fields.” STEM Attrition: College Students’ Paths Into and Out of STEM Fields, National Center for Education Statistics, Nov. 2013. nces.ed.gov/pubs2014/2014001rev.pdf.
  2. Needleman, Sarah E. “Doing the Math to Find the Good Jobs.” The Wall Street Journal, Dow Jones & Company, 7 Jan. 2009, www.wsj.com/articles/SB123119236117055127.
  3. “Jobs Rated 2014: Ranking 200 Jobs from Best To Worst.” CareerCast.com, CareerCast.com, 9 Mar. 2017, www.careercast.com/jobs-rated/jobs-rated-2014-ranking-200-jobs-best-worst.
  4. “Mathematics Genealogy Project.” Welcome! - The Mathematics Genealogy Project, genealogy.math.ndsu.nodak.edu/
  5. Emmanuel J. Candes. “Mathematics of sparsity (and a few other things), August 14, 2014, Seoul. ICM2014 VideoSeries PL3. https://www.youtube.com/watch?v=W-b4aDGsbJk
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