According to the Career Discovery Encyclopedia, mathematicians solve problems in advanced mathematics, including algebra, geometry, number theory, and logic. Theoretical mathematicians develop new ideas and concepts in mathematics and generally teach at colleges and universities. Applied mathematicians focus on using these theories to solve real-world problems through mathematical modeling and computational techniques. They work in business, industry, or government, and their work addresses issues ranging from rocket stability to the effects of new drugs. Financial analysts are applied mathematicians who forecast which investments will be profitable. Computer applications engineers are applied mathematicians who solve scientific and engineering challenges.
Is this all there is to being a mathematician?
Journey to Discovery
As a young man in the 1960s, I was fascinated by mathematics and enjoyed reading about mathematical oddities, geometric quirks, and numerical coincidences in my local newspaper’s Ripley’s Believe It or Not! As I grew older, I discovered more mathematical ideas that deepened my interest, such as The Seven Bridges of Königsberg, Fibonacci Numbers, and others.
In high school, I began focusing more on learning mathematics, with the goal of attending college to expand my mathematical knowledge. I didn’t know what I wanted to do with mathematics, but I knew I wanted to attend college to learn more. At this point, my preconceived notion was that mathematicians worked in a collegiate setting, solving mathematical problems and delving into the unknowns of mathematics. Naively, I did not associate a degree in mathematics with a career. This raises the following question in my mind: Is anyone who uses mathematics in everyday life a mathematician, e.g., math teachers, engineers, economists, accountants, bookkeepers, or software developers?
After earning my undergraduate degree in mathematics, I realized that what I had learned was a broad overview of applied mathematics, and I wasn’t prepared to fully apply that knowledge in my work. (My understanding is that undergraduate degrees are designed to be broad because universities can’t predict the specific skills needed for a particular job. This broad approach is intentional, aiming to develop adaptability, core skills, and intellectual maturity rather than immediate technical expertise.) During my postgraduate studies, I studied operations research and more statistics. Yet again, I was unable to fully apply what I had learned to my work. Over the years, I discovered that a vast number of careers used mathematics to accomplish many interesting and useful tasks. However, can the individuals in these careers be called mathematicians if they use existing mathematics?
My goal was, and still is, to fill in the gaps not covered in my undergraduate and postgraduate education. I personally need to better understand the mathematics I have learned. Today, I share my increased understanding on my website Mathematical Mysteries. If you visit the site and find it somewhat confusing, that’s my effort to make sense of mathematics and how it all connects. At this point much of what I have learned about mathematicians mostly revolves around a title and a list of tasks that they perform. But there is much more to being a mathematician than just tasks. This is where history and current events (soon to be history) come in and help us to understand what a mathematician is.
My goal was, and still is, to fill in the gaps not covered in my undergraduate and postgraduate education. I personally need to better understand the mathematics I have learned. Today, I share my increased understanding on my website, Mathematical Mysteries. If you visit the site and find it somewhat confusing, that’s my effort to make sense of mathematics and how it all connects.
At this point, much of what I have learned about mathematicians comes from historical articles, their titles, and the tasks they perform. But there is much more to being a mathematician than historical facts, titles, and tasks. For example, was Sir Isaac Newton a mathematician or a scientist? Newton is claimed by both scientists and mathematicians, and for good reasons. He was a Renaissance man. But is he the example or the exception?
History and current events (soon to be history) help us understand what a mathematician is because history is not merely a collection of dates and events. History tells us how we reached the present by reminding us that past actions, decisions, and discoveries shape the present. And history will help us understand and answer the question: What is a mathematician?
Mathematicians
Mathematicians have existed since ancient times, when mathematics was part of a classical curriculum. “Majoring” in a specific degree was rare until the rise of “scientific schools” in the mid-1800s. Undergraduate degrees in mathematics began appearing in the United States and Europe during the mid-to-late 19th century, reflecting a shift from classical “mental training” to specialized scientific and technical fields.
Classical
Hypatia was the first known female mathematician, a renowned teacher, and celebrated for her work in mathematics. Throughout her childhood, her father, a scholar and mathematics professor at the University of Alexandria, raised Hypatia and taught her his knowledge while sharing his passion for uncovering answers to the unknown.
Ancient Indian mathematicians Aryabhata (5th century), Brahmagupta (7th century), and Bhaskara II (12th century) revolutionized mathematics centuries before their European counterparts. Aryabhata likely trained at a major center of learning such as Kusumapura (ancient Pataliputra, modern Patna); Brahmagupta was educated in the astronomical tradition of Ujjain; and Bhaskara II learned mathematics directly from his mathematician father and later trained at Ujjain, India’s premier mathematical observatory.
Medieval mathematicians (c. 500–1450) made notable advances, especially in the Islamic world and later in Europe, linking ancient Greek knowledge to modern arithmetic and algebra. Al-Khwarizmi was trained in the scholarly environment of the House of Wisdom in Baghdad. Fibonacci learned mathematics from Arabic mathematicians in North Africa while accompanying his merchant father. Omar Khayyam was educated in the Persian mathematical–astronomical tradition and later worked in Samarkand and Isfahan.
Mathematicians at this point in history were educated in mathematics through the translation and synthesis of Greek, Indian, and Persian mathematical traditions, and through dialogue with local or foreign mathematicians. A common pathway for many mathematicians was the study of astronomy, which was essential to medieval mathematical education because it presented practical, numerical, and geometric challenges that advanced trigonometry, geometry, and arithmetic. This progress was made possible by Fibonacci’s introduction of Hindu-Arabic numerals. The adoption of Hindu-Arabic numerals marked a significant turning point in mathematics, enabling mathematicians to solve complex problems and further expand their knowledge in fields such as astronomy, physics, mechanics, and the merchant trade.
We see that mathematicians were involved in expanding mathematical knowledge for practical applications and the sciences. The end user of this expanding mathematical knowledge may not have been involved in advancing mathematical knowledge, but they benefited greatly in their profession. Did anything change as we progressed in time when mathematicians acquired degrees from accredited institutions?
Traditional Academic Path
The traditional academic path is the conventional, linear educational sequence: completing primary and secondary school, earning a 4-year bachelor’s degree, and often progressing to graduate studies. Mathematicians Hilda Geiringer and Leonhard Euler made significant contributions during their lifetimes and exemplify those who pursued a degree along this path. Let’s briefly examine how this path led them to become mathematicians.
Hilda Geiringer studied mathematics through unusually rigorous university programs for women of her era. She demonstrated strong mathematical ability in high school, and her parents supported her university studies. She studied mathematics at the University of Vienna, completing her PhD in 1917 under Wilhelm Wirtinger, focusing on Fourier series in two variables. Geiringer immersed herself in applied research environments, where she absorbed methods from mechanics and statistics and worked directly with leading figures of the day.
Leonhard Euler was shaped by unusually strong early formal training and an exceptional innate ability. He received early exposure through the Bernoulli family because his father was friends with Johann Bernoulli, one of Europe’s greatest mathematicians. As a teenager, Euler received private lessons from Johann Bernoulli, who recognized his exceptional talent. At age 13, Euler entered the University of Basel, initially to study theology, but shifted to mathematics under Bernoulli’s influence. He received his Bachelor’s degree (around age 15) and a Master of Philosophy degree (at age 16) from the University of Basel in Switzerland.
What did we learn from these two individuals about pursuing a degree to become a mathematician? Hilda Geiringer’s work on the theory of plasticity (the deformation of materials) is crucial to structural engineering. Leonard Euler pioneered graph theory and complex analysis, studied beam bending and column loads, and developed foundational principles in fluid dynamics and ship navigation.
Mathematics expanded explosively with the advent of calculus, mechanics, and analysis. However, mathematics also became rigorous, axiomatic, and hierarchical. Yet these advancements were not just theoretical but also practical, benefiting science, engineering, and commerce, to name a few.
Not only did mathematics expand into new areas of study, but it also remained modest in scope, allowing more individuals to be educated in mathematics, talent to be nurtured, and some students to receive personal mentorship.
As we conclude our brief tour of how mathematicians have historically learned mathematics, we can now ask: Do you need a mathematics degree to be a mathematician?
Non-traditional Academic Path
Several renowned mathematicians throughout history achieved greatness without formal classical training or a university education, often through self-study and passion, and sometimes with exceptional natural talent.
Srinivasa Ramanujan was almost entirely self‑taught, developing his mathematics from a single book and his own discoveries. He worked alone for years, producing thousands of results without access to modern mathematics. He briefly attended the University of Madras but lost his scholarship for focusing on mathematics to the exclusion of other subjects. In 1913, he began correspondence with G. H. Hardy, who recognized his genius and brought him to Cambridge, where Ramanujan received more formal mathematical training. Overall, Ramanujan made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to problems once considered unsolvable.
As a teenager, Sophie Germain discovered mathematics while reading Jean-Étienne Montucla’s Histoire des Mathématiques. Because women were barred from the École Polytechnique, she obtained lecture notes illegally and submitted her work under the male pseudonym M. Le Blanc. Her work impressed Joseph‑Louis Lagrange, who became her mentor after learning her identity. She also corresponded with Adrien-Marie Legendre and Johann Carl Friedrich Gauss, learning advanced number theory and elasticity through correspondence rather than formal schooling.
Born in 1776, Germain grew up during the tumultuous French Revolution. Confined to her home to avoid the chaos in the streets, she found refuge in her father’s library, where she taught herself Latin and Greek to read the works of Sir Isaac Newton and Leonhard Euler. At the time, women were strictly barred from attending the newly founded École Polytechnique in Paris. Determined to pursue mathematics, Germain obtained lecture notes under the name of a former student, Monsieur Antoine-Auguste Le Blanc. She submitted brilliant mathematical papers under this male pseudonym, astonishing the academy’s professors with her aptitude.
Granted, these mathematicians may be the exception rather than the rule. Both Ramanujan and Germain were almost entirely self‑taught, demonstrating that rigorous mathematical achievement may not require a traditional academic path. Many breakthroughs, some celebrated and others not, have emerged from independent study, self‑taught prodigies, and online collaboration, as shown by the accomplishments of these two mathematicians and others.
Amateurs
“Amateur mathematicians are individuals who engage in mathematical activities outside of formal academic training or professional status. Throughout history, many notable figures have made significant contributions to mathematics without traditional credentials. This raises questions about the nature of mathematical ability and the inclusivity of the mathematical community. While contemporary mathematics often prioritizes professional expertise and rigorous methodologies, there remain opportunities for amateurs to contribute meaningfully. … The evolving landscape of mathematics suggests that there is still room for amateur contributions, particularly as boundaries between professional and non-professional engagements continue to blur.” [Michael Green]
Let’s turn our attention to a few amateur mathematicians—people whose primary vocation did not involve mathematics (or any related discipline) yet made notable, and sometimes important, contributions to the field of mathematics.
Eric Temple Bell (E.T. Bell) called 17th-century French mathematician Pierre de Fermat the “Prince of Amateurs” because Fermat was a lawyer and government official by profession, not a full-time mathematician. Despite his amateur status, he made groundbreaking contributions to number theory, probability, and calculus.
In March 2023, the world of mathematics was excited by an announcement that even reached mainstream news. David Smith, a math hobbyist, discovered a shape that had puzzled mathematicians for some time: an aperiodic monotile, also known as the hat or an Einstein. This aperiodic monotile can tile the plane without repeating.
In 2023, high school seniors Calcea Johnson and Ne’Kiya Jackson from St. Mary’s Academy in New Orleans developed a new, trigonometry-based proof of the 2,000-year-old Pythagorean theorem. Using the Law of Sines, they challenged the long-held belief that such a proof is impossible without circular reasoning.
Another way amateur mathematicians are advancing mathematics is by using artificial intelligence (AI). They are using AI to solve long-standing problems, a move that has taken professionals by surprise. While the problems in question aren’t the most advanced in the mathematical canon, the success of AI models in tackling them shows that their mathematical performance has passed a significant threshold and could fundamentally change the way we do mathematics.
“The integration of AI in mathematics is revolutionizing the field, enabling non-academics to solve complex problems traditionally reserved for experts. While AI accelerates discovery, concerns arise about the lack of intuitive understanding in its solutions. This shift blurs the lines between amateur and professional mathematicians, prompting a reevaluation of what constitutes meaningful mathematical knowledge and collaboration.” [Leslie]
These amateur mathematicians are advancing mathematics not for practical use but for personal curiosity and the need to delve deeper into understanding mathematics. They are doing mathematics for mathematics’ sake.
Can Computers Be Mathematicians?
A few key tools that fundamentally changed how we do mathematics include the Hindu-Arabic numeral system (enabling faster arithmetic), algebra (a shift from concrete arithmetic to abstract symbolic reasoning), book printing (mathematics books for the dissemination and expansion of human knowledge), Napier’s logarithms (simplifying multiplication), calculus (modeling motion), slide rules (enabling quick computation), and computers (enabling simulation).
Each new tool was exciting because it made mathematics easier. Many of the tools sped up mathematical calculations, but a user still had to understand what was being done. For example, a computer makes writing easier to revise, but the user still must come up with the ideas to write. Is AI just another tool to aid mathematicians, or will it eventually perform mathematical research beyond calculations?
AI is a powerful tool that mathematicians currently use for computation, verification, pattern matching, and routine tasks, serving as both a researcher and a collaborator. Even though mathematicians leverage AI for scale and tedious work, they focus on higher-level human strengths. Terence Tao states that ‘current AI models are “very good at scouring big lists of problems for low-hanging fruit. It’s tedious and thankless and not something humans want to do.” He cautioned that models are achieving “scattered successes among a big sea of unreported failures.” But the successes are notable.’ [Konstantin-Kakaes]
Human creativity remains crucial for discovering new mathematical ideas and solving existing mathematical problems. Let’s look at a recent example in which AI was used to solve a problem posed by Paul Erdős. He asked a simple question. You draw n dots on a piece of paper. How many pairs of dots can be exactly one inch apart?
‘It would be easy to file this under “AI beats humans at another game” and move on. That would miss the point entirely. This wasn’t a brute-force search. It wasn’t running simulations faster than humans can. The AI did something qualitatively different: it held a long, intricate chain of reasoning together without losing the thread, and it crossed a conceptual gap that humans, by habit, by training, by the way we organize knowledge into departments and disciplines, simply weren’t prone to cross. That’s the uncomfortable part. Not that AI is faster. It’s that AI might be less constrained by the mental walls we’ve built between fields. … AI explores. AI connects. AI suggests. Humans choose the right problems, interpret the beauty, and decide what to ask next.’ [Mishra-Pradeep]
Based on all the reading I have done on the use of AI in mathematics, I conclude that computers are not mathematicians. We must remember that all AI applications are chatbots, computer programs that process user inputs and respond using rules or artificial intelligence. These AI chatbots are programmed using mathematics and logic, and are developed primarily by mathematicians who have left academia to improve AI’s ability to provide more accurate responses.
What I see in all the articles I have read about AI and mathematics is the following.
- AI did not initiate looking into mathematical problems.
- Humans had to frame the problem and ask additional questions to guide the AI toward a viable solution.
- AI was not constrained by a specific discipline and, in some cases, crossed disciplines to solve a problem. In some cases, humans did not think to look to another discipline for an answer.
- Humans needed to review the results to determine whether they were correct and, in some cases, improve upon the results presented by the AI chatbot.
In mathematics the art of proposing a question must be held of higher value than solving it. ~ Georg Cantor — The Father of Set Theory
What is a Mathematician?
Did I answer the question about what a mathematician is? Not directly. But I would like to share my take on answering that question.
According to the Career Discovery Encyclopedia, mathematicians work in various fields, and share common traits and interests. First, they must love mathematics and the challenge of solving problems. Next, they need to work logically, patiently, and creatively on these problems.
The key phrase from the definition above is “they must love mathematics and the challenge of solving problems.” The user of mathematics who plugs numbers into an equation and gets an answer is not a mathematician. A mathematician displays imagination, curiosity, a love of mathematics, a commitment to research, and collaborates with like-minded individuals. A mathematician doesn’t learn just HOW to solve problems but asks WHY this works. A mathematician must ask many questions to gain a deeper understanding of mathematics.
I leave it up to you to decide, based on the information presented, what a mathematician is.
“Good, he did not have enough imagination to become a mathematician.” [Upon hearing that one of his students had dropped out to study poetry.] ― David Hilbert
References
References can be found on the What is a Mathematician? – References webpage.
Mathematical Careers
Core Mathematical Professions
- Actuary: Analyzes financial risk using mathematics and probability, often for insurance firms.
- Statistician: Collects, analyzes, and interprets data to identify trends and inform business decisions.
- Mathematician: Conducts research in fundamental mathematics or applies techniques to solve problems in fields like engineering, medicine, and technology.
- Operations Research Analyst: Uses math and algorithms to improve decision-making, efficiency, and logistical systems.
- Cryptographer/Cryptanalyst: Develops algorithms and codes to secure data, often for national security agencies.
Data and Technical Jobs
- Data Scientist/Analyst: Interprets complex data sets to help companies make data-driven decisions.
- Quantitative Analyst (“Quant”): Applies advanced mathematical modeling to financial markets, trading, and risk management.
- Software Engineer/Programmer: Develops computer systems, algorithms, and software applications.
- Numerical Analyst: Develops techniques for calculating numerical solutions to complex math problems.
Applied Mathematics Roles
- Biomathematician/Biostatistician: Uses modeling and statistics to solve problems in biology, medicine, and epidemiology.
- Meteorologist: Models atmospheric conditions to predict weather, often utilizing fluid dynamics.
- Economist/Financial Analyst: Analyzes economic trends, financial data, and investment opportunities.
- Academician (Professor/Teacher): Teaches mathematical concepts at school or university levels.
Specialized Opportunities
- Animator/Computer Graphics Specialist: Uses mathematics (linear algebra, geometry) to create graphics for films and games.
- Cryptanalyst: Analyzes and breaks encryption systems.
- Psychometrician: Specializes in measuring mental traits, abilities, and behavioral data.
The featured image on this page was generated using the FLUX AI website.