Elements of the History of Mathematics

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The history of mathematics is nearly as old as humanity itself. Since antiquity, mathematics has been fundamental to advances in science, engineering, and philosophy. It has evolved from simple counting, measurement and calculation, and the systematic study of the shapes and motions of physical objects, through the application of abstraction, imagination and logic, to the broad, complex and often abstract discipline we know today. From the notched bones of early man to the mathematical advances brought about by settled agriculture in Mesopotamia and Egypt and the revolutionary developments of ancient Greece and its Hellenistic empire, the story of mathematics is a long and impressive one.

The East carried on the baton, particularly China , India and the medieval Islamic empire , before the focus of mathematical innovation moved back to Europe in the late Middle Ages and Renaissance. Then, a whole new series of revolutionary developments occurred in 17th Century and 18th Century Europe, setting the stage for the increasing complexity and abstraction of 19th Century mathematics, and finally the audacious and sometimes devastating discoveries of the 20th Century.

Follow the story as it unfolds in this series of linked sections, like the chapters of a book. Even in modern times, various influential thinkers have been stirred by the magnificence of the work. Hobbes, Einstein, and Russell all praised the work—not only for its mathematical precision of planar geometry, number theory, and ancient algebra, but also for its aesthetic beauty and power to show these eternal, preexistent verities to the eye of the mind.

Already, one may start to realize that the ancients and early moderns saw mathematics quite differently than bona fide moderns like ourselves. Value-laden judgments and ascientific notions like beauty are strikingly absent in popular considerations.

Mathematics has turned increasingly praxis-minded in contemporary teaching. Survey a hackneyed word problem one might find in standardized assessments:. If train A leaves Cincinnati for Chicago at AM, traveling at 75 mph, and train B leaves Chicago for Cincinnati at PM, travelling at 85 mph, what time will the two trains meet assuming Chicago and Cincinnati are miles apart? Rather, my qualm lies in looking at mathematics merely as a series of practical word problems to be solved. I fear that, due to the pedagogical ease of focusing on word problems, teachers fail to reveal the splendor of mathematics found so integral to ancients and early moderns.

Plato, Euclid, and even American heroes like Thomas Jefferson and Abraham Lincoln, saw mathematics as much more than a set of tools for solving practical problems.

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These men believed that the elegance and simplicity of mathematics underscored its primacy and power. For Plato and Euclid, this meant that the study of mathematics moved one beyond the realm of the material world and into the more pure realm of the Forms. To the aforementioned men, the truths of mathematics are unchanging, eternal, ordered, and aesthetically beautiful to the eye of the mind.

The study of mathematics prepares one to engage in the disciplines of philosophy and theology; which in turn, prepare one to gaze upon and contemplate God with the clear eyes of a soul that has been enlightened by a sublime mind. It was understood that the study of the trivium grammar, logic, and rhetoric followed by the quadrivium arithmetic, music, astronomy, and geometry would prepare one for the study of liberal arts par excellence philosophy and theology.


While the exact boundaries that confine these fields today differ from those that would have confined them 2, years ago, the general idea was that people must be well-versed in the seven fields of the liberal arts found in the trivium and quadrivium before they would be ready to study the more comprehensive and meta-level fields of philosophy and theology.

Only the best prepared and most motivated minds were allowed to study these disciplines.

They constituted the highest form of human thought, matching that of the divine mind. Yet, these disciplines could be most heinously perverted and used for ill. Throughout history, many atrocities have been perpetrated by the erudite in philosophy and theology. Philosophy and theology had a more comprehensive definition in the ancient and into the early modern period; it was the philosophers and theologians who were to instruct the people about what it means to be human and progress towards flourishing. Geometry is a rigorous discipline that takes intense study, mental sharpness, and clarity of thought and expression.

There is little that can sharpen the mind more than constructing geometric proofs. The reasoning must account for any and all gaps, and in order for a proof to be considered true, each claim must logically follow from those already definitively demonstrated.

Elements of the History of Mathematics | N. Bourbaki | Springer

Errors are exposed quickly in geometry they cannot be veiled as they can in other disciplines. Geometry directs the mind to think on truths that do not pass away with the whims of human opinion. These truths are eternal. These truths are sure. These truths are foundational in the very fabric of the world we inhabit and to the order we find throughout it.

As a failed politician and doubting lawyer, Lincoln sought refuge in the precision and elegance Euclid exuded. Galvanized by such the rigor and assurance of truth laid out in the propositions, Lincoln successfully navigated this troubled point in his life, to becoming a more effective lawyer and one of the greatest politicians America has ever seen.

Leonhard Euler: Mathematical Genius in the Enlightenment

Lincoln biographer Michael Burlingame puts it this way:. Lincoln in his early forties mastered Euclid, whose works he carried with him on business trips. This suggests a desire to strip away all superfluous mental baggage and get at the heart of the matter, psychologically as well as logically… When he emerged from his political semiretirement in , he had formulated a basic critique of the proslavery and popular sovereignty cases, arguing with Euclidean coherence.

To think these thoughts of pure geometry was, in a very real sense, to think the very thoughts of God. Studying this rigorous discipline will train your mind to think on higher thoughts and prepare it for the spiritual ascent to participation in the divine nature. Taking into account this disconnect between an ancient and popular modern understanding of mathematics, the rest of this essay will briefly sketch how we got from the ancient and pre-modern viewpoint to the modern viewpoint, and evaluate some positives and negative implications for us today.

Table of contents 1.

  • The controversial fifth axiom.
  • Elements of Mathematics.
  • Mastering Nmap Scripting Engine?
  • Definition.
  • Navigation menu.
  • Leonhard Euler: Mathematical Genius in the Enlightenment.

Foundations of Mathematics; Logic; Set Theory. Notation; Combinatorial Analysis. The Evolution of Algebra.

Elements of the History of Mathematics

Linear Algebra and Multilinear Algebra. Polynomials and Commutative Fields. Divisibility; Ordered Fields. Commutative Algebra. Algebraic Number Theory.

Non Commutative Algebra. Quadratic Forms; Elementary Geometry. Topological Spaces. Uniform Spaces. Real Numbers. Exponentials and Logarithms. Complex Numbers; Measurement of Angles.

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