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Calculation is a powerful mathematical tool. But for hundreds of years after the invention in the 17th century, it stood on a trembling platform. Its core concepts are derived from intuition and informal arguments, instead of accurate and accurate definitions.
The two schools of thought appeared in response, followed Michael BaranyA historian of math and science at the University of Edinburgh. French mathematicians had great content to continue. They are more interested in applying calculations for physical problems, using it to calculate the orbit of the planets, for example, or study the behavior of the current. But by the 19th century, the German mathematicians began to tear everything. They set out to find the reaction samples that will weaken the long -term assumptions, and eventually use those reactions to put the calculation on the feet more stable and more durable.
One of these mathematicians is Karl Weierstrass. Although he showed his early mathematical talent, his father pressured him to study financial and public management, with an eye towards joining a popular civil service. Bored with its university courses, Weierstrass is said to have spent most of the time to drink and fight; In the late 1830s, after he could not get a degree, he became a high school teacher, giving lessons about everything, from math and physics to pen and gymnastics.
Weierstrass did not start his career as a professional mathematician until he was nearly 40 years old. But he will continue to change this field by introducing a mathematical monster.
The pillars of calculation
In 1872, Weierstrass announced a function of threatening everything to mathematicians thinking that they understood calculations. He encountered indifference, anger and fear, especially from the mathematical giants of the French thought school. Henri Poincaré condemned Weierstrass's function as an indignation against common sense. Charles Hermite called it a scary wicked.
To understand why Weierstrass's results are scary, it will first understand two of the most basic concepts in calculation: continuity and difference.
A continuous function is exactly what it is like a function without space or dance. You can track a link from any point on such a function to any other function without lifting your pencil.
Calculating largely about identifying such continuous functions changes quickly. It works, speaking loose, by approximately a certain function with straight, not confident lines.
Illustration: Mark Belan/Quanta Magazine