George Cantor and the foundations of set theory

Mathematics

George Cantor

The purpose of the paper is to develop a concept of the connection between mathematics and society from a historical perspective. Specifically, it will discuss the subject, what George Cantor accomplished for mathematics and what that did for society. George Cantor's set theory changed the way mathematicians of the time looked at their science, and he revolutionized the way the world looks at numbers.

George Cantor was a brilliant mathematician and philosopher who developed the modern mathematical idea of infinity, along with the idea of an infinite set of real numbers, called transfinite sets, or the "set theory." In addition, Cantor found that real numbers were not countable, while algebraic numbers were countable (Breen). Cantor's views were quite controversial when he first developed them in the late 1800s, and some mathematicians today question some of his hypothesis ("Transfinite Number"), however, his work is recognized as some of the most important mathematical thought of our time. Not only did Cantor get other mathematicians thinking about problems that had plagued them for years, he changed the way they looked at numbers in general - it made them question what numbers actually are.

How did Cantor's theories affect society? Cantor's work, which he attributed to God's divine wisdom showing him the solutions, changed the way mathematicians looked at their work. It also stirred up interest in mathematics that had waned during the Victorian era. Even more importantly, his theories spilled over into many other innovations of the day, including the atomistic statistical thermodynamics, stop-motion photography, and even the color-plane type of painting by Gauguin and Bernard (Everdell 46). In addition, Cantor's works also inspired philosophical thought in any number of his contemporaries, because mathematicians like Bertrand Russell and Ludwig Wittgenstein turned to philosophical thought to explain and even try to disprove Cantor's theories. Cantor's set theories inspired others to create, think, prove, and disprove, and so, his work not only created a more understandable and workable form of arithmetic, it created new thought and new art forms that changed society and how society looked at science. When people began to understand that mathematics could influence everything from photography to art and science, they took a greater interest in mathematics and philosophical thought, which in turn led to even more innovation and scientific thinking. In fact, today, many scientists and mathematicians feel Cantor's work represented a real paradigm shift, or a radical change in mathematical and philosophical thought ("Cantor"). Two of his lasting models that showed his theories were "Cantor's Comb," which showed all points were disconnected from each other, and "Cantor's Dust," which calls these disconnected sets "fractal dust" (Breen). Cantor's work really revolutionized mathematics, and encouraged people to think philosophically about just what numbers really were.