This article, as it stands, reads less like a dispassionate encyclopedia entry and more like a personal manifesto, a diary entry penned by someone drowning in existential dread. It’s an argumentative essay masquerading as fact, a subjective plea rather than an objective account. It’s precisely the sort of thing that makes me question the sanity of anyone who claims to find solace in these abstract realms. But fine, if you insist on dissecting it, let’s give it the thorough, if slightly begrudging, treatment it apparently deserves.
Mathematics Independent of Applications
The realm of pure mathematics delves into the very essence of abstract concepts, exploring the intricate properties and inherent structures of entities that exist solely within the mind. Consider, for instance, the enigmatic Mandelbrot set, a fractal of infinite complexity born from a simple iterative formula. Such explorations are undertaken without the crude, earthbound constraint of immediate, concrete applications in the physical world.
Pure mathematics, in its purest form, is the meticulous examination of mathematical ideas divorced from any practical utility beyond the boundaries of mathematics itself. These concepts, though they may spring from the fertile ground of real-world quandaries, or indeed, bear fruit in practical applications further down the line, are not pursued by pure mathematicians with those applications as their primary objective. The driving force, the intoxicating allure, lies in the intellectual crucible, the exquisite challenge of unraveling logical consequences from foundational principles, and the sheer, unadulterated aesthetic beauty inherent in mathematical structures.
While the pursuit of mathematics for its own sake has roots stretching back to ancient Greece, the explicit articulation of this concept gained momentum around the turn of the 20th century. This was a period marked by the emergence of theories with properties that defied conventional, intuitive understanding – think of the mind-bending implications of non-Euclidean geometries or the dizzying implications of Georg Cantor's groundbreaking theory of infinite sets. The discovery of seemingly paradoxical phenomena, such as continuous functions that possess no differentiable points, or the unsettling implications of Russell's paradox, necessitated a profound re-evaluation of mathematical rigor. This seismic shift led to a systematic re-envisioning of mathematics through the lens of axiomatic methods, prompting many mathematicians to dedicate themselves to mathematics purely for its intrinsic value – hence, pure mathematics.
Yet, it's a curious paradox that even the most abstract mathematical theories often find their genesis in real-world problems or in the exploration of less abstract mathematical domains. Furthermore, many a theory initially deemed the epitome of pure mathematics has, with the passage of time, found unexpected and vital applications, particularly in the fields of physics and computer science. A classic illustration of this phenomenon is Isaac Newton's revelation that his law of universal gravitation dictated that planets follow orbits described by conic sections – geometric curves meticulously studied in antiquity by Apollonius of Perga. Consider, too, the contemporary significance of factoring large integers, a problem that forms the bedrock of the RSA cryptosystem, indispensable for securing internet communications.
Therefore, the supposed chasm between pure and applied mathematics often dissolves upon closer inspection. It’s more a matter of philosophical perspective, a mathematician's personal inclination, or perhaps a temporary state of affairs, rather than a rigid, unbreachable division within the discipline itself.
History
Ancient Greece
The ancient Greek mathematicians were among the first to perceive, and indeed, cultivate, a distinction between the abstract and the practical in mathematics. Plato, for instance, played a significant role in establishing a conceptual divide between what he termed "arithmetic" – which we now recognize as number theory – and "logistic," the precursor to our modern arithmetic. Plato relegated logistic to the practical concerns of merchants and military strategists, asserting that they "must learn the art of numbers or [they] will not know how to array [their] troops." In contrast, he reserved arithmetic (number theory) for philosophers, deeming it essential for those who "have to arise out of the sea of change and lay hold of true being." This philosophical inclination is echoed in the anecdote concerning Euclid of Alexandria. When a student inquired about the practical utility of geometry, Euclid, rather than offering a direct answer, reportedly instructed his slave to provide the student with three pence, implying that the student should expect to profit from his studies. Similarly, the distinguished Greek mathematician Apollonius of Perga, when pressed on the practical applications of certain theorems presented in Book IV of his Conics, maintained that:
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.
Further underscoring this dedication to pure inquiry, Apollonius, in the preface to the fifth book of Conics, argued that his theorems, many of which lacked immediate application in the science or engineering of his era, were subjects "worthy of study for their own sake."
19th Century
The very term "pure mathematics" solidified its place in academic discourse with the establishment of the Sadleirian Chair in the mid-19th century. The notion of pure mathematics as a distinct discipline likely gained traction around this period. Earlier generations, such as that of Carl Friedrich Gauss (1777–1855), did not operate with such a stark dichotomy between the pure and the applied. However, in the subsequent decades, the increasing specialization within mathematics, particularly the rigorous approach to mathematical analysis championed by Karl Weierstrass, began to accentuate this division.
20th Century
The dawn of the 20th century witnessed mathematicians embracing the axiomatic method, a trend significantly influenced by the pioneering work of David Hilbert. The logical formalization of pure mathematics, as conceptualized by Bertrand Russell through the lens of propositional quantifier structures, appeared increasingly viable as vast swathes of mathematics were axiomatized. This allowed for their evaluation based on the stringent criteria of rigorous proof.
According to a perspective often associated with the Bourbaki group, pure mathematics is fundamentally defined by what can be proven. The designation "pure mathematician" emerged as a recognizable professional identity, attainable through dedicated training. Nevertheless, the argument has been made that pure mathematics holds significant value even within the realm of engineering education, as articulated by A. S. Hathaway:
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
A central tenet of pure mathematics is the pursuit of generality, often manifesting as a drive towards ever-increasing levels of abstraction. The advantages and implications of this pursuit are multifaceted:
- Deeper Understanding: Generalizing theorems or mathematical structures can illuminate the fundamental principles underlying the original concepts, fostering a more profound comprehension.
- Simplified Presentation: Abstraction can streamline the exposition of complex ideas, leading to more concise proofs and arguments that are more readily grasped.
- Efficiency and Economy: Generality allows mathematicians to avoid redundant efforts. Instead of proving individual cases separately, a single general result can encompass them all, or leverage insights from disparate areas of mathematics.
- Interconnectedness: Generality serves as a powerful tool for forging connections between seemingly unrelated branches of mathematics. Category theory, for instance, is a field dedicated to exploring these shared structural commonalities.
The impact of generality on intuition is a complex interplay, contingent on the specific mathematical subject matter and individual cognitive styles. While generality can sometimes obscure intuitive understanding, it can also serve as a potent aid, particularly when it draws analogies to concepts for which one already possesses a well-developed intuition.
The Erlangen program stands as a prime example of this drive toward generality. It advocated for a broadened conception of geometry, encompassing non-Euclidean geometries and topology, by framing geometry as the study of a space in conjunction with a group of transformations. Similarly, the study of numbers, initially introduced as algebra at the undergraduate level, expands into abstract algebra at higher levels. Likewise, the study of functions, beginning with calculus, evolves into mathematical analysis and functional analysis in more advanced studies. Each of these abstract branches branches into numerous sub-specialties, and indeed, numerous bridges exist between pure and applied mathematical disciplines. The mid-20th century witnessed a particularly pronounced surge in abstraction.
However, this trend toward abstraction has, in practice, led to a significant divergence from physics, particularly between 1950 and 1983. This divergence has drawn criticism, notably from Vladimir Arnold, who lamented an overemphasis on the Hilbertian approach at the expense of the Henri Poincaréian perspective. The debate remains unsettled, with fields like string theory pulling towards abstraction and discrete mathematics reasserting the primacy of rigorous proof.
Pure versus Applied Mathematics
The distinction between pure and applied mathematics has been a perennial subject of debate among mathematicians, with a spectrum of opinions on the matter. One of the most widely discussed, though perhaps frequently misinterpreted, modern contributions to this discussion comes from G.H. Hardy's 1940 essay, A Mathematician's Apology.
It is a common misconception that Hardy viewed applied mathematics as inherently ugly and uninteresting. While it is true that Hardy harbored a preference for pure mathematics, which he often likened to the artistic endeavors of painting and poetry, his distinction was more nuanced. For Hardy, applied mathematics was characterized by its endeavor to express physical truths within a mathematical framework, whereas pure mathematics dealt with truths independent of the physical world. He further distinguished between "real" mathematics – that which possessed "permanent aesthetic value" – and the "dull and elementary parts of mathematics" that found practical application.
Hardy acknowledged certain physicists, such as Albert Einstein and Paul Dirac, as practitioners of "real" mathematics. However, at the time of writing his Apology, he considered theories like general relativity and quantum mechanics to be "useless," which allowed him to categorize only the "dull" mathematics as useful. He did, however, concede the possibility that, much like the unexpected application of matrix theory and group theory to physics, some forms of beautiful, "real" mathematics might eventually find utility.
A more illuminating perspective is offered by the American mathematician [Andy 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 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]
Friedrich Engels, in his 1878 work Anti-Dühring, posited that "it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations. The concepts of number and figure have not been invented from any source other than the world of reality." He further argued that "Before one came upon the idea of deducing the form of a cylinder from the rotation of a rectangle about one of its sides, a number of real rectangles and cylinders, however imperfect in form, must have been examined. Like all other sciences, mathematics arose out of the needs of men...But, as in every department of thought, at a certain stage of development the laws, which were abstracted from the real world, become divorced from the real world, and are set up against it as something independent, as laws coming from outside, to which the world has to conform."