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# The Significance of Theoretical Mathematics

The discipline of working with pure mathematics to expand the breadth of mathematical knowledge. Theoretical mathematicians work to find solutions to unexplained math problems, including those that do not have practical uses. Theoretical mathematicians can also develop theories that are of use in the science and engineering fields.

By creating new principles, theoretical mathematicians help expand what is currently known about mathematics. Often, this expansion will include connecting known principles with previously unknown mathematical relationships. In addition, a theoretical mathematician may use observations and experiments to compare model inferences.

Working as a theoretical mathematician is typically a full-time job that can require more than 40 hours a week. There are jobs available in a number of industries, including the federal government, education and the private sector. When working in the private sector, the mathematician will often work in the research and development department of a science or engineering company. The jobs for a theoretical mathematician in education often combine research and teaching. The research is usually a collaborative effort with several different mathematicians or scientists.

Students who want to become theoretical mathematicians should take as many math courses as possible in high school. From there, a bachelor’s degree in mathematics will help with some jobs, including those with the federal government. Those looking to work in the science or engineering fields will need a master’s degree or Ph.D. in theoretical mathematics. All major university positions will also require an advanced degree. For those looking to teach at the middle or high school level, a master’s degree will be needed. While going to school, it is important to develop critical-thinking and problem-solving skills in order to be successful in the theoretical mathematics field.

# Pure Mathematics

Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. This was a recognizable category of mathematical activity from the 19th century onwards,^{} at variance with the trend towards meeting the needs of navigation, astronomy, physics, economics, engineering, and so on.

Another view is that pure mathematics is *not necessarily* applied mathematics: it is possible to study abstract entities with respect to their intrinsic nature and not be concerned with how they manifest in the real world.^{}Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians.

To develop accurate models for describing the real world, many applied mathematicians draw on tools and techniques that are often considered to be “pure” mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research.

## Ancient Greece

Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between “arithmetic”, now called number theory, and “logistic”, now called arithmetic. Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who “must learn the art of numbers or [they] will not know how to array [their] troops” and arithmetic (number theory) as appropriate for philosophers “because [they have] to arise out of the sea of change and lay hold of true being.” ^{}Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, “since he must make a gain of what he learns.”^{}The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book IV of *Conics* to which he proudly asserted,

They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.

And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of *Conics* that the subject is one of those that “…seem worthy of study for their own sake.”^{}

## 19th century

The term itself is enshrined in the full title of the Sadleirian Chair, founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of *pure* mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between *pure* and *applied*. In the following years, specialization and professionalization (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent.

## 20th century

At the starts of the twentieth century, mathematicians took up the axiomatic method, strongly influenced by David Hilbert’s example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatized and thus subject to the simple criteria of *rigorous proof*.

In fact, in an axiomatic setting *rigorous* adds nothing to the idea of *proof*. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved. Pure mathematician became a recognized vocation, achievable through training.

The case was made that pure mathematics is useful in engineering education^{}

- There is a training in habits of thought, points of view, and intellectual comprehension of ordinary engineering problems, which only the study of higher mathematics can give.

## Generality and abstraction

One central concept in pure mathematics are the ideas of generality; pure mathematics often exhibits a trend towards increased generality. Uses and advantages of generality include the following:

- Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures
- Generality can simplify the presentation of material, resulting in shorter proofs or arguments that are easier to follow.
- One can use generality to avoid duplication of effort, proving a general result instead of having to prove separate cases independently or using results from other areas of mathematics.
- Generality can facilitate connections between different branches of mathematics. Category theory is one area of mathematics dedicated to exploring this commonality of the structure as it plays out in some areas of math.

Generality’s impact on intuition is both dependent on the subject and a matter of personal preference or learning style. Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition.

As a prime example of generality, the Erlangen program involved an expansion of geometry to accommodate non-Euclidean geometries as well as the field of topology, and other forms of geometry, by viewing geometry as the study of a space together with a group of transformations. The study of numbers, called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level; and the study of functions, called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level. Each of these branches of more *abstract* mathematics has many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. A steep rise in abstraction was seen mid 20th century.

In practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, not enough Poincaré. The point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central.

## Purism

Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the most famous (but perhaps misunderstood) modern examples of this debate can be found in G.H. Hardy’s *A Mathematician’s Apology*.

It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that Hardy preferred pure mathematics, which he often compared to painting and poetry, Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express *physical* truth in a mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. Hardy made a separate distinction in mathematics between what he called “real” mathematics, “which has the permanent aesthetic value”, and “the dull and elementary parts of mathematics” that have a practical use.

Hardy considered some physicists, such as Einstein, and Dirac, to be among the “real” mathematicians, but at the time that he was writing the *Apology* he also considered general relativity and quantum mechanics to be “useless”, which allowed him to hold the opinion that only “dull” mathematics was useful. Moreover, Hardy briefly admitted that—just as the application of matrix theory and group theory to physics had come unexpectedly—the time may come where some kinds of beautiful, “real” mathematics may be useful as well.

Another insightful view is offered by Magid:

I’ve always thought that a good model here could be drawn from ring theory. In that subject, one has the subareas of commutative ring theory and noncommutative ring theory. An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the former: a noncommutative ring is a not necessarily commutative ring. If we use similar conventions, then we could refer to applied mathematics and nonapplied mathematics, where by the latter we

mean not necessarily applied mathematics.^{}

## Subfields

Analysis is concerned with the properties of functions. It deals with concepts such as continuity, limits, differentiation, and integration, thus providing a rigorous foundation for the calculus of infinitesimals introduced by Newton and Leibniz in the 17th century. Real analysis studies functions of real numbers, while complex analysis extends the aforementioned concepts to functions of complex numbers. Functional analysis is a branch of analysis that studies infinite-dimensional vector spaces and views functions as points in these spaces.

Abstract algebra is not to be confused with the manipulation of formulae that is covered in secondary education. It studies sets together with binary operations defined on them. Sets and their binary operations may be classified according to their properties: for instance, if an operation is associative in a set that contains an identity element and inverses for each member of the set, the set and operation are considered to be a group. Other structures include rings, fields, vector spaces and lattices.

Geometry is the study of shapes and space, in particular, groups of transformations that act on spaces. For example, projective geometry is about the group of projective transformations that act on the real projective plane, whereas inversive geometry is concerned with the group of inversive transformations acting on the extended complex plane.

Number theory is the theory of the positive integers. It is based on ideas such as divisibility and congruence. Its fundamental theorem states that each positive integer has a unique prime factorization. In some ways it is the most accessible discipline in pure mathematics for the general public: for instance the Goldbach conjecture is easily stated (but is yet to be proved or disproved). In other ways it is the least accessible discipline; for example, Wiles’ proof that Fermat’s equation has no nontrivial solutions requires understanding automorphic forms, which though intrinsic to nature have not found a place in physics or the general public discourse.

Topology is a modern extension of Geometry. Rather than focusing on the sizes of objects and their precise measurement, topology involves the properties of spaces or objects that are preserved under smooth operations such as bending or twisting (but not, for example, tearing or shearing). Topology’s subfields interact with other branches of pure math: traditional topology uses ideas from the analysis, such as metric spaces, and algebraic topology relies on ideas from combinatorics in addition to those of analysis.