!EXCLUSIVE! Download Pure Mathematics 1
Heidi Blessing
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:
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.
Pure mathematics 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. In particular, it is not uncommon that some members of a department of applied mathematics describe themselves as pure mathematicians.[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]
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.
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 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.
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.
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][8]
Mathematics is both an art and a science, and pure mathematics lies at its heart. Pure mathematics explores the boundary of mathematics and pure reason. It has been described as "that part of mathematical activity that is done without explicit or immediate consideration of direct application," although what is "pure" in one era often becomes applied later. Finance and cryptography are current examples of areas to which pure mathematics is applied in significant ways.
I'm a mathematics student planning to enroll in a good math PhD program this Fall. I have always been extremely disciplined in math and my goal has always been to pursue a math PhD. However, I've had the opportunity to work in computer science, and this has caused some doubts about the significance of my future work in mathematics. I imagine such doubts are nonunique to myself and that the best place to ask is here, from people who've been through a PhD themselves, who are wiser, and who may possibly have had these same thoughts. (I hope it is clear I am asking this out of good nature and that this is not dismissed as a cynical thing to ask.)
My main question: Is pure mathematics useful, specifically, outside of mathematics itself? Instead of giving a definition of "useful," perhaps I can share some doubts I have about the significance of pure mathematics research.
It seems to me that in all honesty, pure mathematics does not immediately benefit the population at large in a direct and obvious way. At best benefits are usually theoretical (e.g., "These methods could...").
I think that very, very few people actually read and care about the average published pure mathematics paper. I think it's because math papers are hard and it's not clear that they are interesting or useful to math as a whole or to the future of humanity. There are very obvious exceptions, for example, for papers like Fermat's Last Theorem, which are arguably achievements for humanity. But most papers are objectively not of this level of significance and may not always contribute to major problems.
It seems that the only reason we, as a population, care about mathematics, is because of the "cool" open problems which are simple to understand but difficult to prove. But this account for only a very small portion of active and successful mathematical work (since math papers don't always try to solve such problems because they're very hard). So doesn't this imply that my work as a future research mathematician is actually not useful for the future of humanity?
It seems that pure mathematics was originally created to solve practical and interesting problems, and that as we turned to use abstraction as a tool to solve things (because abstraction is a very useful problem solving tool), we have arrived many years later to nested layers of subproblems of subproblems, whose depth is so deep that such problems of these areas are hard to understand and are not obviously useful for the world or for anything outside of that area of mathematics itself. It seems that mathematics is a science that studies itself, and so at a certain point, it does not have an immediate practical use outside of itself.
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