Pure Mathematics Definition

[Lotte Donohoe]

Puremathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.

While pure mathematics has existed as an activity since at least ancient Greece, the concept was elaborated upon around the year 1900,[2] after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.

Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections, geometrical curves that had been studied in antiquity by Apollonius. Another example is the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure internet communications.[3]

It follows that, presently, the distinction between pure and applied mathematics is more a philosophical point of view or a mathematician's preference rather than a rigid subdivision of mathematics.[citation needed]

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."[4] 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 gain of what he learns."[5] 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,[6]

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."[6]

The term itself is enshrined in the full title of the Sadleirian Chair, "Sadleirian Professor of Pure Mathematics", 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, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent.

At the start 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 axiomatised and thus subject to the simple criteria of rigorous proof.

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 have 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 Poincar. 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.

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 1940 essay 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 permanent aesthetic value", and "the dull and elementary parts of mathematics" that have practical use.[8]

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 non-commutative ring theory. An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the former: a non-commutative 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... [emphasis added][9]

While there are no explicit scope limitations to pure or applied math, there are some implicit ones. However, you will have to remember that there are several differences between pure math vs. applied math. But, both branches can cover any topic in mathematics, including:

As you progress in your program, the paths diverge. Pure math students may dive deeper into number theory, theoretical physics and topology, while applied math students may explore more analysis, computing and mathematical modeling.

Notably, applied mathematicians will spend more time meeting with clients and other stakeholders. Their schedules may ultimately be split between working with other departments to understand an issue and actually doing math.

However, from a career perspective, an applied mathematics major can certainly be more appealing. Learning skills that translate to real-world professions can increase your job prospects after graduation.

Generally most students are introduced to the concept of Vector as something that has both a "magnitude and direction" and Scalars as something that only has a "magnitude and no direction". Usually this definition of Vectors and Scalars comes straight out of a high-school Physics class, as most students learn about Vectors and Scalars far earlier in Physics than in Mathematics.

However once you get to first and second year undergrad Mathematics courses at Universities, you realize that Vectors and Scalars are really abstract mathematical objects. Once you study further, you realize that there are higher mathematical objects such as Tensors which are generalizations of Vectors and Scalars.

We've all had the "definition" of a Vector drummed into us that : "A vector is a quantity that has both magnitude and direction", but in a Pure Mathematical sense, that is not the case. A Vector is not a "quantity" it's a mathematical object. Likewise a Scalar is not a "quantity" either, but also a mathematical object (e.g a real number).

I argue that saying a Vector is a "quantity that has both magnitude and direction" takes away the Pure Mathematical intuition behind a Vector. I argue that using this "definition" of a Vector is just a shorthand to help us familiarize ourselves with the concept of Vectors, and to find physical intuition for it. Saying that it has magnitude and direction is just a physical interpretation of it, i.e. what we are doing in this definition is giving "physical context" to an abstract Mathematical object.

How do you define the "direction" of a an abstract Mathematical object, such as a Vector in a Pure Mathematical way? The concept of "direction" is given context by the physical world, and hence not a purely mathematical concept. The closest thing we have to "direction" in pure mathematics is associated with a number or axis being either positive or negative.

(I realize in Vector Calculus that there are concepts like "Directional Derivatives", and the like, but again I argue that calling it that is just a shorthand to work with Vectors without getting too much into their true roots in purely mathematical sense (i.e. in terms of tuples of scalars), as you will see below)

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