The concept of infinity has been evaluated many times throughout history. Only recently, in the nineteenth century, has major progress evolved in the field. The chapter “Beyond Infinity” answers the questions, “what is mathematics and why should I study it?” by reviewing several mathematician’s theories of infinity.
First, the author mentioned Galileo who theorized that a line which measured 3 inches long contained the same amount of points as a line twice it’s length. The author also referred to Bernhard Bolzano, a mathematician who later on attempted to define infinity as well, but failed to do so. Archamedis was also referred to, for he developed a system for infinity called ‘myriad’. Using this, he was able to estimate the number of grains of sand there are on a beach.
The chapter was focussed on matmatician named Cantor who became well known towards the end of the nineteenth century. Cantor believed that one set of numbers is equivalent to another set if they can be paired together. This was referred to as his “stepping-stone” process. He also used this to define numbers beyond infinity. Cantor assigned the first letter of the Hebrew alphabet, along with a subscript, to represent the number of elements in an “ordinary infinite set”. The first letter (Aleph) with zero for a subscript represented real numbers. Aleph with a numeral one stood for real and irrational numbers.
Cantor recognized infinity as a verb, rather than a noun. This was uncommon, for it contradicted previous Platonic theories. Therefor, many mathematicians’ dismissed his theories at first. Also, he was criticized for not having an “absolute infinity”. Some mathematicians eventually started to accept Cantor’s theories. He then went on to prove that there are an infinite equal amount of fractions and whole numbers, and that the set of irrational numbers is larger than whole numbers and fractions. Cantor applied his theories to Geometry as well, thus demonstrating that there are the same infinite amounts of points in every space, despite dimensions. He also showed how points on a line can be paired with points on a plane, which can be paired with points in a volume, and so on.
Based on the accumulated theories, mathematics is defined as a science that has been evaluated, modified, and added to since the beginning of time. It includes all types of numbers and objects that are, in turn, applied to other concepts in the “physical world.” To conclude, it is important to study mathematics in order to understand concepts and things in our lives, and obtain an overall greater understanding. Cantor’s theories are particularly important for, as the author stated, they take us beyond “ordinary infinity”.