Infinite Jest
One of the subjects I miss most from college is the philosophy of mathematics. I didn't actually discover it until my senior year, so I did not get to explore it as much as I would have liked. The one seminar I took on the topic was really fascinating.
What is the philosophy of mathematics? It is a branch of both disciplines that concerns itself with the conceptual underpinnings of math. This includes such things as deriving arithmetic from formal logic, advanced set theory, Godel's incompleteness theorem, and the nature and behavior of the infinite.
This last really caught my attention. I had long planned that my master's thesis was going to be exploring the relationship between infinite sets and infinite values. As a specific example, is the infinite value expressed by 1/0 the same as the number of integers, the same as the number of reals, or neither? I have this feeling that there is a conceptual breakthrough lurking in that question, much like the conceptual breakthrough generated by the discovery of complex numbers.
There are many conundrums regarding infinities. Some of them result from misunderstandings. Some are the result of attempting to apply finite operations to infinite sets. And some, of course, are simply the result of the infinite being this wacky thing that is waaay outside our normal experiences.
Let's start with an easy misconception. People think that infinity is really, really big. An easy mistake. After all, we have been trained to believe that if box B contains everything in bag A, then B must be bigger than A. Most infinite sets contain subgroups that are mind-bogglingly large (e.g., the set of all numbers between 0 and a googol). Therefore, the infinite set must be even MORE mind-bogglingly large, right? Wrong. Saying that infinity is "really, really big" doesn't make any more sense than calling it "really, really blue" or "really, really juicy". The infinite has no size, at least not the way we typically think about it.
Similarly, talking about things like "infinity plus one" or "infinity squared" doesn't make sense. Infinity is not a number. Performing arithmetic on it is no more valid than taking the square root of a sonata or subtracting love. Speaking of such things can be occasionally useful to illustrate a point, but only so long as you understand that you are using these concepts figuratively.
So, here are a few infinity-related brain-benders for you:
Suppose you have a lamp. Flip the switch, and you turn it on. Flip it again, and you turn it off. If you flip it an infinite number of times, is it on or off?
Which are there more of, even numbers or negative integers?
Is .999... (I.e., an infinite number of 9's after the decimal) less than 1?
If space is infinite, can it still have edges?
If there are an infinite number of worlds, is there necessarily a world made entirely of cheese?
If anyone takes a crack at answering these in the comments, I will take a crack at explaining why you are wrong or right.
What is the philosophy of mathematics? It is a branch of both disciplines that concerns itself with the conceptual underpinnings of math. This includes such things as deriving arithmetic from formal logic, advanced set theory, Godel's incompleteness theorem, and the nature and behavior of the infinite.
This last really caught my attention. I had long planned that my master's thesis was going to be exploring the relationship between infinite sets and infinite values. As a specific example, is the infinite value expressed by 1/0 the same as the number of integers, the same as the number of reals, or neither? I have this feeling that there is a conceptual breakthrough lurking in that question, much like the conceptual breakthrough generated by the discovery of complex numbers.
There are many conundrums regarding infinities. Some of them result from misunderstandings. Some are the result of attempting to apply finite operations to infinite sets. And some, of course, are simply the result of the infinite being this wacky thing that is waaay outside our normal experiences.
Let's start with an easy misconception. People think that infinity is really, really big. An easy mistake. After all, we have been trained to believe that if box B contains everything in bag A, then B must be bigger than A. Most infinite sets contain subgroups that are mind-bogglingly large (e.g., the set of all numbers between 0 and a googol). Therefore, the infinite set must be even MORE mind-bogglingly large, right? Wrong. Saying that infinity is "really, really big" doesn't make any more sense than calling it "really, really blue" or "really, really juicy". The infinite has no size, at least not the way we typically think about it.
Similarly, talking about things like "infinity plus one" or "infinity squared" doesn't make sense. Infinity is not a number. Performing arithmetic on it is no more valid than taking the square root of a sonata or subtracting love. Speaking of such things can be occasionally useful to illustrate a point, but only so long as you understand that you are using these concepts figuratively.
So, here are a few infinity-related brain-benders for you:
Suppose you have a lamp. Flip the switch, and you turn it on. Flip it again, and you turn it off. If you flip it an infinite number of times, is it on or off?
Which are there more of, even numbers or negative integers?
Is .999... (I.e., an infinite number of 9's after the decimal) less than 1?
If space is infinite, can it still have edges?
If there are an infinite number of worlds, is there necessarily a world made entirely of cheese?
If anyone takes a crack at answering these in the comments, I will take a crack at explaining why you are wrong or right.