Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
"It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is forever beaten by the waves."
-Proclus Diadochus (412 - 485) Scholium to Book X of Euclid V
"If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation."
-Niels H. Abel (1802 - 1829)
From an early age, we have an intuitive sense that there can be no biggest number. As soon as we learn how to add two numbers together, we have at our disposal a mechanism by which we can make any number bigger—just add one! We have a sense of both a process and a set—the set of all numbers—that are infinite, larger than anything in our daily experience. We also learn a hierarchy of numbers: a billion conquers a million, a googol beats a billion, and infinity is the sovereign value, untouchable in its perfection.
What exactly is infinity? Does it really exist? It certainly doesn't play any obvious role in our everyday lives. We are finite, and we live in a finite world. Our lives have definite beginnings and endings, and we measure the time between these two points using discrete, finite, units such as years, minutes, and seconds. Similarly, the physical space in which we live our lives and enact our everyday pursuits is bounded and separated into fundamental units, such as miles and millimeters. Our best bet for grasping some sensory experience of infinity is to gaze toward the heavens on a starry night. It remains an open question, however, whether or not the universe actually extends forever.
The process of adding the number one to another number to make it greater does not make the result infinite—it merely makes another, greater, finite number. The Greeks called a quantity or a collection “potentially infinite” if, given any finite example of that quantity, a ”larger” example could always be found. In this respect, a line segment (a “collection” of points) is potentially infinite, because it can always be made longer, and the set of counting numbers is potentially infinite, because from one counting number, we can always construct a greater one. To conceive a quantity that is actually infinite, however, is mind-bending and, in many ways, perplexing. What happens if we add the number one to an actually infinite number? It's already infinite—does adding to it make it greater? How could one possibly have something ”more” than infinity?
Such a concept is often called “actual infinity,” and it is much more problematic than “potential infinity.” It defies intuition, forcing us to rely on logic to explore the defining aspects of the concept. The idea of actual infinity has been disturbing to mathematicians since at least the time of the Greeks. At times, it seems to be more an invention of the human imagination than anything real, and, ideally, mathematics should be the language that describes reality, not fantasy.
Despite its nebulous reality, the concept of infinity has long teased at mathematicians' minds. Around 500 BC it manifested in the form of incommensurable quantities, a concept akin to heresy in the view of many, particularly the followers of Pythagoras. At almost the same time, paradoxes posed by the philosopher Zeno showed that infinity was a difficult concept for the human mind to comprehend. It was at this point that the Greeks reluctantly accepted the use of infinity in mathematics, but they left the challenge of understanding it to philosophers and priests.
This view persisted for centuries. Infinity was a tool that could be used in mathematics, even if it was not well understood. As it turned out, infinity proved to be an indispensable tool for 16th- and 17th-century mathematicians seeking to use mathematical concepts to describe real-world physical phenomena, This is most evident in the field of calculus. In order for calculus to work (and we assume that it does work, because it describes the physical world superbly), we have to believe that actual infinity exists—that is, we have to believe at least that an infinite process can have a finite result. So, the concept of infinity proved useful then in much the same way that a modern cell phone does now; we certainly don't have to know how it works in order to make use of it.
Mathematics, however, is supposed to be based on rock-solid, well-understood principles. As the tower of mathematics grew larger and more intricate, with each new idea depending on the validity of those that came before it, mathematicians began to double-check the foundational principles. They were concerned that it might be a bad idea to base large parts of our understanding of the world on a concept, infinity, that we fundamentally do not understand. Enter Georg Cantor. Cantor sought to understand mathematically the infinite and the consequences of believing in an actual infinity. He did this by creating the language of sets, which are just collections of objects, such as numbers.
In doing so, he had to redefine what a number really is. Through some of the most creative and ingenious mathematics ever done, Cantor showed, contrary to intuition, that there can be different sizes of infinity. His polarizing results generated much controversy that, to this day, is not completely resolved. In this unit we will explore infinity by first looking at rational numbers and some of their properties. We will then see how incommensurable quantities and irrational numbers suggest that infinity is at work in the number system. Through Zeno's paradoxes, we will catch a glimpse of how difficult infinity can be to understand. From there we will look at sets of numbers and re-learn how to count in a way that will enable us to approach the concept of infinity. With these tools in hand, we will get a sense of the universe of infinities that Cantor discovered, culminating in Cantor's Theorem, one of the most counter-intuitive ideas in mathematics.
Next: 3.2 Rational Numbers