Theoretical mathematics

Summary: The complement to applied mathematics, theoretical mathematics advances the field without necessarily focusing on potential applications.

Often mathematics, as a discipline, is categorized in two general areas: theoretical mathematics and applied mathematics. When this is done, it is common to consider theoretical mathematics (or “pure” mathematics) as the part of mathematics that is carried out for the sheer pleasure of “doing mathematics,” for the intrinsic beauty that lies in the study of the logical patterns and abstract relations that can be found when organizing space (geometry and topology), structures (algebra), quantities (number theory), approximations (analysis), and the thought behind these actions (logic and foundations). However, in many historical cases, the results found in theoretical mathematics have had a practical or “applied” value, often not foreseen—or even understood—until many years after their discovery. Such is the case, for example, of the non-Euclidean geometries that were seen to be coherent within their axiomatic systems but were not thought of as representing physical reality. Riemannian geometry, named for mathematician Bernhard Riemann who lived in the first part of the nineteenth century, became the mathematical context for Albert Einstein’s General Theory of Relativity and led to other non-Euclidean applications in twentieth-century physics.

In much of the research done in theoretical mathematics, the focus is upon extending the field in which the particular mathematician involved is a specialist. Real world applications are not usually relevant to the activity of the pure mathematician, as these belong to the realm of applied mathematics. However, research in pure mathematics often involves the “application” of results to other mathematical objects. It is also important to emphasize that new knowledge in mathematics does not come about by experimentation but by proof.

Algebra: The Study of Structure

People often associate algebra with their experience in secondary school. Algebra studied at this level is known as “elementary algebra,” and, while it is a big step in abstraction for the young student, it still focuses upon real numbers and arithmetic operations in which unknown variables are substituted for numbers. However, the abstract algebra studied and developed by theoretical mathematicians generalizes the structure of the real number system and its arithmetic operations by means of axioms and works with structures that have little to do with the numbers and operations learned in school. Some of the structures most studied in algebra include groups, rings, modules, vector spaces, and fields. These structures are defined by properties and operations. Theoretical mathematicians study the relations that are established between different representations of the same structure, or even between different structures. Once again, even though the exploration, discovery, and development of all these structures is the motivation and a goal in itself, these abstract structures have found applications in areas as diverse as crystallography, computer science, music, and physics.

Geometry and Topology: The Study of Space

The study of symmetries and rigid transformations, such as rotations, reflections, and translations, is associated with the Euclidean geometry that everyone studies in a secondary school program. Euclidean geometry arose from the need to measure and survey as territorial delimitation began to be registered and ancient civilizations developed sophisticated towns and cities. Euclidean geometry, with its study of flat space (where the shortest distance between two points is a straight line), was a faithful representation of “how the world really is.” The discovery of the “other geometries” in the 1800s, when some theoretical mathematicians removed the parallel postulate from the axioms of Euclidean geometry, opened a world of possibilities (or possible geometrical worlds) for exploration on the level of theoretical mathematics. However, it was to be seen that the universe, both on the macrolevel (as in, for example, astronomy) and the microlevel (as in, for example, particle physics), could be much more faithfully described with non-Euclidean geometrical properties.

In general, geometry studies the properties that change when an object is deformed, while topology studies the properties that do not change when an object is deformed. For the topologist, a circle and a square are essentially the same, because there exists a continuous function that transforms one into the other. This is the reason that topology is often called “rubber band geometry.” Whereas in geometry an object remains the same only under rigid transformations, in topology, as long as adjacent points continue to stay adjacent (which means that the object cannot be cut or twisted), the object is considered the same. The rubber band can be stretched, shaped as a square, ellipse, or circle; but points that are close remain close, and the rubber band itself does not change.

Although the study of topology is very axiomatic and theoretical, its results have had important applications in physics, biology, computer science, and robotics. For example, the study of DNA topology by applied mathematicians, together with biologists and chemists, uses results from the theoretical study of “rubber geometry.”

Number Theory: The Study of Quantities

Number theory, also known as “higher arithmetic,” studies the properties of the natural numbers and the integers as well as the properties of those structures that are a generalization of natural numbers and integers—those structures that maintain certain fundamental properties that these numbers possess. Some of these properties are as familiar as divisibility, prime factorization, or congruence, while others have arisen through conjectures that theoretical mathematicians have made. Some of these conjectures are extraordinarily easy to understand by any nonmathematician or young student, but they are also extraordinarily difficult to prove.

Such is the case, for example, of the now famous “Last Theorem of Fermat.” This theorem states that xⁿ + yⁿ = zⁿ can be true only for n = 1 or 2. For over 350 years, some of the best mathematical minds worked on this problem and could not find a proof. In 1995, a proof was presented, but it used some of the most sophisticated and modern mathematical concepts from other areas of pure mathematics to be found.

Number theory has been considered by some mathematicians as a paradigm of pure mathematics. However, since the appearance of computer science, number theory has been applied in a very practical way, especially in cryptography (the encoding of information) and random number generation for statistical analysis; it has even been applied in quantum mechanics.

Analysis: The Study of Approximation

Mathematical analysis began as the process of formalization and axiomatization of calculus, whose dependence on infinitesimally small quantities that “tend” to zero did not have a rigorous foundation. Today, analysis has branched out into different areas of interest. Real analysis is the study of the properties of sequences and functions of real numbers using notions such as limit, continuity, differentiation, and integration. There is also complex analysis, which studies similar notions in the context of the complex numbers, and functional analysis, which studies these notions and others properties of functions that are seen as objects in a “function space.” Probability theory is also considered an area of analysis. Indeed, probability theory is a very abstract and axiomatic subject, based on set theory and measure theory.

It is worthwhile mentioning that in the 1960s an alternative axiomatization of the infinitesimal, known as “nonstandard analysis,” was developed. There are mathematicians who advocate the use of this formalization as a basis for teaching calculus, given that the concept of “limits” is often difficult to comprehend for the beginning student. As seems to be the rule in theoretical mathematics, although the mathematician does not look for applications and the main goal is to expand the particular field of study, applications of analysis have found their way into science, engineering, and economics.

The Theoretical Mathematician: Training and Workplace

The educational systems in the world are not homogeneous, although once a student is at the level of graduate studies, equivalences are usually recognized. In many countries outside of the United States, an undergraduate program consists of a complete submersion in the field and virtually no courses outside the field are taken. In the United States, the majority of fields, mathematics included, are offered as majors at the undergraduate level; therefore, the number of courses taken in the particular area is less, as there are other general education requirements that need to be fulfilled. It is also common for students to take a minor in another area or even a double major.

However, people with undergraduate degrees in any part of the world will not be formally considered theoretical mathematicians. Theoretical mathematicians will have a graduate degree, almost always a Ph.D., and graduate studies are fairly homogeneous worldwide.

People trained in mathematics will have taken a full calculus sequence (single variable and multivariable), followed by an analysis sequence. As undergraduates, they often will have taken linear algebra, abstract algebra, discrete mathematics, and usually some topology or geometry. Once a student has opted to study pure mathematics, and is in a master’s program, the student will orient electives to an area of interest. At the Ph.D. level, students still have to present doctoral comprehensive exams in the subjects of analysis, algebra, and, often, topology as requisites, independently of his or her area of specialization. Students will also present a comprehensive exam in their field of interest, and then they will do doctoral research, culminating in their doctoral dissertation. There are, of course, variants to this process. Some students will specialize in several fields; some will have done research in their master;s program and have produced master’s theses.

The University of Cambridge established the Sadleirian Chair in pure mathematics and, since 1863, there have been eight professors who have held it. This position is usually considered a landmark in the recognition of pure mathematics as separate from applied mathematics. Universities, in general, do not have a standard approach to the separation of theoretical and applied mathematics. Some universities have a single mathematics department; some have mathematics and statistics departments in which applied mathematics is considered a concentration in mathematics. Sometimes computer science is part of the mathematics department, although this is not common at research universities.

The natural ambience of theoretical mathematicians is academia. In this context, they can transmit their knowledge, which is the product of many years of study, reflection, discovery, and creation, to future generations. Academia is also the place where theoretical mathematicians can have the time and resources to dedicate themselves to research. There are also institutions, albeit few, that support theoretical mathematicians to do research, usually at a stage in which they have already produced results and it is clear that they have a big probability of successfully obtaining new ones.

Employment in government and industry is usually reserved for the applied mathematician. However, there are theoretical mathematicians who also have applied knowledge that makes them attractive for these positions. The theoretical mathematician who ends up in an applied context can often provide insights, because of training, that will bring about novel ways of approaching concrete problems.

It is interesting that there is very little difference in the type of work and perspectives of theoretical mathematicians worldwide. The differences have more to do with the size and extension of the university systems in different countries, but the “culture” and daily life of the pure mathematician is remarkably homogeneous.

The Germ of Theoretical Mathematics in School Mathematics

In many universities in the world, prospective schoolteachers who will be teaching mathematics must take a course, or courses, that analyze elementary mathematical concepts from an advanced point of view. This requirement is because many of the concepts that are present from the very beginning of mathematical instruction are very deep, although not necessary to understand for the young student who begins the procedure of basic mathematical operations. Felix Klein (1849–1925) is known for his work in geometry, where he demonstrated that the Euclidean and non-Euclidean geometries could be considered as special cases of projective geometry, that algebra (group theory) can be basic to the study of geometry, and other achievements in theoretical mathematics. The “Klein Bottle,” a two-dimensional object from topological studies that can only be understood as a whole in a four-dimensional context, is named after him. His book Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis is made up of lectures to future teachers over a 20-year period.

At the elementary school level, for example, the concept of “infinity” is present from the moment that children learn to count with natural numbers. The notion of dimension appears when two- and three-dimensional objects are introduced geometrically, and abstraction is required when these physical objects are represented by formulas, often a first contact with algebra. The notions of number theory are omnipresent, for example, in the concept of “divisibility” and integer numbers. The concepts of “equivalence class” from algebra, and “limit” from analysis are also fundamental to work with both rational numbers and roots and real numbers and approximations. Notions from set theory and logic are implicit in teaching methods and explanations about many of the operations and concepts that are expected to be taught and learned at the school level. For this reason, the schoolteacher who is expected to communicate mathematical ideas should have a basic understanding of many of the concepts of theoretical mathematics. Further, schoolteachers who understand the broader theoretical and applied contexts of the mathematics that they teach can answer student questions and plant the seeds of ideas and connections that will later become important. It is the job of mathematicians at universities’ mathematics departments to transmit these ideas to students who, while not pursuing a degree or career in pure mathematics, must understand some of its fundamental components.

Theoretical Mathematicians: Their Work and Their Views

It is usually agreed upon that until the middle 1800s, there was no clear division between theoretical and applied mathematics. Even though, arguably, Euclid’s Elements could be considered pure mathematics, the majority of mathematicians from ancient times until the 1800s were interested in solving problems. It is also true that some of these problems, such as finding the roots of polynomials of varying degrees (which led to the development of Group Theory), might not seem to have much practical application. However, in general, mathematicians as renowned as Isaac Newton, Gottfried Leibniz, Leonhard Euler, Carl Friedrich Gauss, brothers Jacob and Johann Bernoulli, Joseph Fourier, Joseph-Louis Lagrange, Evariste Galois, or Niels Abel, in their contributions to the ideas now considered part of theoretical mathematics, usually were also involved in research in which direct applications were the central objective.

In the 1800s, the axiomatization of calculus, with its convenient but mysterious infinitesimals, was carried out by Augustin-Louis Cauchy (1789–1857) and Karl Weierstrass (1815–1897). George Boole (1815–1864) tried to formalize the laws of thought using algebra and initiated the algebra of logic, called Boolean algebra, in which algebraic symbols represent logical forms. It is interesting that this theoretical endeavor actually laid the ground for the construction of computers and electric circuits, given that these circuits can represent complex logical operations. These mathematicians would now be considered theoretical mathematicians, as their work was oriented to expanding the mathematical areas in which they worked, not to practical applications.

Although the computer does not play the same role in the work of the theoretical mathematician as it does in that of the applied mathematician, it would be false to think that the theoretical mathematician has remained untouched by the advent of the computer. In number theory, for example, if there is a conjecture about properties of, for example, prime numbers, the numbers can be generated to billons or trillions in a short interval of time, detecting in this way if some counterexample could appear. Before this possibility arose, theoretical mathematicians could sometimes spend a lifetime trying to prove a false conjecture because it would have taken several lifetimes to generate enough numbers to arrive at the counterexample. In purely theoretical areas such as commutative algebra and algebraic geometry, computer programs have been developed that permit the calculation of, for example, Gröbner bases, named for Wolfgang Gröbner, that help to further theoretical results. The proof of the Four Color Theorem, which had been attempted by theoretical mathematicians for over 100 years, was done with the aid of the computer, which carried out the multiple calculations that would not have been possible to do by hand in a lifetime.

Of course, there are those who say this proof (of the Four Color Theorem) does not correspond to pure mathematics. This very interesting debate is a product of the transition at the beginning of the twenty-first century that coexistence with computers has become a reality. A quote from theoretical mathematician David Cox, who has played an important role in bridging this gap, is very illustrative:

My fascination with algebra led me to algebraic geometry, which was then among the most abstract areas of pure mathematics. At the time, I never would have predicted that 25 years later I would be writing papers with computer scientists, where we use algebraic geometry and commutative algebra to solve problems in geometric modeling.

Often, quotes from actual theoretical mathematicians best give an idea of how they themselves conceive their work. These quotes may illustrate a perception of pure theoretical mathematics, perhaps not so well known to a general public, rather than absolute importance of these mathematicians over any others—an idea that will always be debatable and impossible to define:

It is not of the essence of mathematics to be occupied with the ideas of number and quantity.

—George Boole (1815–1864)

No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful.

—George Boole (1815–1864)

Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are co-ordinated with concepts introduced previously by means of precise definitions.

—Georg Cantor (1845–1915)

In mathematics the art of proposing a question must be held of higher value than solving it.

—Georg Cantor (1845–1915)

Often theoretical mathematicians are motivated by the knowledge that their abstract research and discoveries will eventually find their way to applications in technology, medicine, or economics. Theoretical mathematics can very well be conceived of as an art by those who find aesthetic pleasure in its logic and patterns, but there is no doubt, as historically has been seen time and time again, that mathematics is science as well.

Bibliography

Cox, David. “What is the Role of Algebra in Applied Mathematics?” Notices of the American Mathematical Society 52, no. 10 (2005). http://www.ams.org/notices/200510/fea-cox.pdf.

“Famous Mathematics Quotes.” http://www.math.okstate.edu/~wli/teach/fmq.html.

Gowers, Timothy, June Barrow-Green, and Imre Leader. The Princeton Campanion to Mathematics. Princeton, NJ: Princeton University Press, 2008.

Klein, Felix. Elementary Mathematics From an Advanced Point of View. New York: Dover, 2004.

Stewart, Ian. Concepts of Modern Mathematics. New York: Dover, 1995.

U. S. Department of Labor. Bureau of Labor Statistics. “Occupational Outlook Handbook: Mathematicians.” http://www.bls.gov/oco/ocos043.htm.