Please note: Yet again, I planned too much content for one blog. Therefore, I will only observe the life of one mathematician. Rest assured that many, many other mathematicians followed a similar life as the one described below. I appreciate your cooperation and apologize if you were expecting more from this blog. I promise that I will make it up to you in later blogs!
“In mathematics the art of proposing a question must be held of higher value than solving it.”– Georg Cantor
1845. The beginning of the Mexican-American War. The beginning of the Irish potato famine. The beginning of new life, of Georg Cantor.
Georg Cantor was born of Danish descent in St. Petersburg, Russia. His father was an honest stock broker. His mother was a talented musician. Logically, the fusion of Cantor’s parents blessed him with a natural affinity in mathematics and the fine arts, becoming both an aspiring mathematician and a concert violinist before his fifteenth birthday. Cantor also had five other siblings, all of whom were very musically inclined.
Overcome with illness, Cantor’s father relocated the entire family to Germany in 1856; he then died in 1863. During this period, Cantor studied engineering to appease his father’s wishes. After his father’s passing, however, Cantor dove into mathematics. By 1867, he became the president of the Berlin Mathematical Society and had successfully published dissertations concerning modern number theory. (Number theory is the study of properties of the set of natural counting numbers, the numbers discussed in the first portion of my first blog.)
Throughout the 1870s and 1880s, Cantor worked in the field of infinite sequences and series, or to put in layman’s terms, he worked with infinite lists of number patterns as well as the sum of all numbers within those infinite patterns. (Over the course of the next few months, I will be going more in depth with these notions of infinite sequences and series because they lead to, arguably, the most elegant proof in all of mathematics.) To get a better sense for his work during this time, I planned a simple activity.
Consider the number line from zero to one, including both of those values. Chop this line segment into thirds. Throw away the middle line segment. Take the remaining two line segments. Chop these segments into thirds. Throw away the middle third line segments in both of these two larger line segments. Continue this pattern as long as you are able. If successful, your construction will look similar to the one below.
Congratulations! You have constructed, albeit geometrically, your first infinite sequence! You developed a pattern and forced it to continue for infinity! Now, to express this as a true sequence, one which is written as a list of numbers, assign the length of the initial segment to one. Then, the length of the resulting two segments after taking away the middle third from the initial segment becomes one-third. Following in suit, the lengths of the four segments after removing the middle third of the two line segments that resulted from the first division become one-ninth. Continuing this pattern forever, the sequence of the lengths of the segments would be expressed as 1, 1/3, 1/9, 1/27, 1/81,…, where the pattern is that each number in the sequence can be found by taking the number proceeding it in the sequence and multiplying that proceeding number by one-third.
Of course, this information was already known during Cantor’s time, for infinite sequences had been known since the time of Euclid. Cantor, however, being the mathematical genius that he was, reinvented the wheel. Rather than looking at distance, he looked at the amount of numbers contained within the line segments; he looked at sets of numbers as opposed to the distance between two numbers. His resulting formula became known as Cantor’s set. (For purposes of math phobias, I will not post the formula here.)
Cantor’s set can be thought of as a stepping stone. Through rigorous mathematical reduction, Cantor came to the conclusion that, no matter how small the interval of numbers became in the construction seen above, each interval had an infinite amount of numbers contained in it. Over the course of the next twenty years, this concept of infinity haunted Cantor. In pursuit of understanding the meaning of his set, he discovered something else entirely: There are different sizes of infinity!
Think about it! The counting numbers—1, 2, 3, etc.—can be counted for as long as we want to count them. Therefore, they have a countable infinity. Integers, which include zero and negative numbers, also have a countable infinity because we can assign each number in the set of integers to a number in the set of counting numbers. Rational numbers (i.e. fractions, terminating decimals, or repeating decimals) can also be counted, to the surprise of many. His diagonal proof below shows a method whereby one can count every rational number.
He even went so far as to prove that the roots (solutions) of polynomial equations with integer coefficients are able to be counted to infinity (which is good news if you want to memorize every possible solution to a polynomial equation with integer coefficients for a math test).
Real numbers, however, cannot be counted according to Cantor. To get a better understanding of why this is so, think of the smallest increment with which you can count the real numbers. Hopefully, you are having some trouble with my question and not because you do not understand it. (For the benefit of the doubt, to illustrate what I am asking of you in a better way, I want you to recall the counting numbers. When we count with our counting numbers, we count in intervals of one. We count one, two, three, etc. The distance, in effect, between the counting numbers is one. Essentially, I am asking you the following question: What is the distance between the real numbers?) Well, one could argue the increment is 0.1, but then, I would respond, “What about 0.01?” Another individual could argue the increment is 0.001, but then, I would respond, “What about 0.0001?” A third individual could argue the increment is 0.00001, but then, I would respond, “What about 0.000001?”
Do you see the pattern? Any increment you use to count the real numbers can be made smaller. Therefore, it is impossible to count real numbers. This is the reason behind Cantor’s set. You make the intervals on the segment from zero to one smaller and smaller. Because every interval, no matter how “small” they become when we chop their range into thirds, contains real numbers, each interval contains an infinite, an uncountable infinite, number of points. Think of it in terms of the exercise I had you do in the first blog. The number sqrt(2) never ends yet we can draw it on paper. Likewise, there are an infinite, an uncountable infinite, number of points inside each line we draw, regardless of how small the lines seem to become. How amazing is that discovery?
Sadly, this man of mathematics developed a bipolar disease in the mid-1880s, and for the rest of his life, he struggled to contain his emotions. In fact, during the turn of the twentieth century, he wrote dozens of essays devoted to proving that William Shakespeare was not the original author of his works. Such blasphemy separated him from his colleagues. This separation then induced years of chronic depression, poverty, and illness. Despite such horrible odds against him, Cantor never gave up mathematics. In the last twenty years of his life, he tried in vain to prove what is now known as the “continuum hypothesis.” Because of his inability to prove the hypothesis, however, Cantor fell into a greater depression, and in 1915, his huge seventieth birthday party bash, which may have helped alleviate some of his depression, was cancelled by World War I efforts. A few years later, Cantor, the father of modern set theory, was dead in a sanatorium. Now, only one question remains: Did mathematics drive him to insanity, or was it something else entirely?
For more information about the works of famous mathematicians, click here.