History of Mathematics

Introduction

Mathematics is among the oldest and most enduring intellectual achievements of the human species. From the earliest marks scratched into bone by prehistoric hunters keeping tally of game, to the vast edifices of abstract reasoning constructed by twentieth-century logicians and geometers, the story of mathematics is inseparable from the story of human civilization itself. No other discipline has so quietly and so thoroughly shaped the way people understand the universe, build their cities, navigate their seas, conduct their commerce, and push forward the boundaries of science. The complete history of mathematics from ancient number systems to modern mathematics and its greatest discoveries spans at least fifty thousand years of human effort, ingenuity, failure, and triumph.

What makes the history of mathematics so compelling is that it is not merely a chronicle of formulas discovered and theorems proved. It is the record of human beings wrestling with the deepest mysteries of quantity, space, change, and structure. Why do the same mathematical truths appear independently in cultures separated by vast oceans and centuries? How did the simple act of counting sheep eventually lead to the concept of infinity? How did merchants keeping accounts in ancient Mesopotamia lay the foundations for the algebra students still study in school today? These questions make the history of mathematical thought one of the richest narratives in intellectual history.

The development of mathematics has never been a smooth, linear progression. Periods of explosive creativity have been followed by long centuries of consolidation or apparent stagnation. Mathematical knowledge has been gained, lost, and rediscovered. Ideas that seemed merely abstract curiosities in one era turned out, centuries later, to be precisely the tools needed to describe the physical universe. The invention of non-Euclidean geometry in the nineteenth century was dismissed by many as a purely theoretical exercise, yet it became the mathematical language of Einstein's general relativity. The number theory puzzled over by ancient Greek mathematicians found unexpected application in modern cryptography and secure internet communication.

This article surveys the entire sweep of mathematical history, from the prehistoric origins of counting through the towering achievements of the ancient civilizations of Mesopotamia, Egypt, Greece, India, the Islamic world, and China, through the great revolution of calculus and the extraordinary proliferation of mathematical disciplines in the nineteenth and twentieth centuries, to the profound challenges and open problems that still confront mathematicians today. Along the way it profiles the greatest mathematical minds of history, examines how mathematics has shaped everyday life, and considers the unsolved mysteries that continue to beckon from the frontier of human knowledge.

Mathematics is sometimes described as the queen of the sciences, a phrase attributed to Carl Friedrich Gauss, one of the supreme mathematical geniuses of the nineteenth century. It might equally be called the servant of all sciences, since no branch of natural inquiry has been able to advance far without borrowing from the mathematician's toolkit. Understanding how that toolkit was assembled, refined, and expanded over the millennia is to understand a central thread in the fabric of human civilization.

Prehistoric Mathematics and Counting