When I was a child, I purposely found something to think about to help me fall asleep. Usually I picked cartoons or super powers, but sometimes things just came into my head, like it or not. What was the worst? Thinking about heaven. At first, heaven seems all right. There is a lot to do, gold everywhere (though no purpose for it), people are nice (it’s a prerequisite), you get to see most of your family, and there is plenty to eat (though no one is ever hungry). Anyway, I start thinking about FOREVER.
At first, it is just a sensation; a weird sensation like tingling and falling and nothingness. It is not a sensation that I can make sense of really because forever doesn’t really make sense, at least not to a 10 year old. I try to get away from forever but forever is a huge part of the definition of heaven. Then, the opening credits of the Twilight Zone, with the music, and starry sky, usually appear. Fade to myself standing, looking at heaven, in the dressing room mirrors of infinity. You know, when dressing rooms have those three mirrors that are angled just perfectly so the images are smaller and smaller replicas of one another, on and on, into infinity. This picture, and thoughts of the foreverness of heaven, kept me up at night as a child.
I am glad to say that forever no longer keeps me up at night. While I still find no comfort in the foreverness of heaven, the lack of a middle ground between forever and my time on earth is what usually keeps me up at night now. However; I still can’t stand it when mirrors are angled that way. It creeps me out, and I can’t help but wonder if there is an end, or if I can find a flaw from one image to the next. In my opinion, we are not meant to look into infinity like that, squarely.
When beginning to pursue mathematics, I thought math might clarify, or in some way define, forever (or as adults call it, infinity). On the contrary, Math has actually made it stranger. Theories in math have shown numerous types of infinity, and infinities within infinities, and sizes of infinities, and calculations of infinity. None of this brings me any comfort, except to say that we obviously don’t have this figured out yet because that is just not possible. Infinity is infinity, and it is very large, incalculable and non-denumerable, and there is only one kind; it is called forever. Heaven can only exist in one, all-encompassing infinity.
When reading A History of Mathematics, I read about Zeno’s paradox. That led to an internet search, and then to Numberphile. I watched the video, accepted the idea, and left it alone. The solution seemed reasonable enough. Later in the semester, I was required to do a research project. By some unknown scheme, we picked Georg Cantor, whom I had never heard of. If you haven’t either, he is the creator of set theory but also perhaps the mathematical or scientific father of infinity. You just can’t shake things off in life. They follow you.
My research for that project led me to question the mathematical view of infinity. Let me start by saying, I know very little of Math’s view of infinity. It seems to be an infinite topic. This is where I am in my understanding – so please comment, post, reply, educate me, and critique my understanding. Calculus one is a prerequisite for the course, and being a rule follower, I have that. So, I had experience computing limits to infinity. That is relatively easy. BUT, those are just numbers. They aren’t real things. Numbers aren’t real. So, of course I could compute the infinity of something that isn’t really real. What numbers represent is real; like Zeno’s paradox. Zeno’s paradox applies numbers to something real – something actually happening in the world (theoretically). In other words, when I take the limit of a sequence that goes to infinity, it has no relation to time or space. It is just numbers. But, if I were taking the limit of Zeno’s paradox to see how far Zeno actually travels, or to find the time it takes to travel, or to see if he can ever catch the turtle, I would have to do so in relation to time and space. When I do that, the exact opposite answer occurs. Zeno will never catch the turtle. That mathematics isn’t computing real infinity or perhaps all of infinity is perhaps echoed by the Numberphile narrator when he asks, “What I want to ask a physicist is, can you divide space and time infinitely many times?” Similarly, Kelly MacCarthur wonders in the Calculus 2 video used for online math courses, “Can I take infinitely many steps?”
However, if all of space and time existed at one instant, forever, then Math has it right. It could calculate the infinite because it occurs all at once. There is no sequence, event after event – in essence, no time or space really because it is all at once, everywhere. Yes, there are scientific theories, philosophies, and religions which believe this is the case. Of course, this idea is contrary to most people’s understanding of infinity. Whenever math instructors talk about infinity, they always say, “Infinity is only a concept. It is not a number.” Yes, it is only a concept but is it also something real? If it is only a concept then why are we computing something real that is a concept? Why would we bother to compute a concept? It seems like Math is walking a funny line here.
Math has worked something out though. I’m just not sure what it is. Math is summing an infinite process (as if infinity happened to end). Obviously, Math’s understanding of infinity has proven useful in mathematical calculations and many practical applications. To paraphrase others before Cantor, “It works. So, no need to define it. It works.” So, Math has worked something out about infinity but what has Math worked out, and is it really infinity?
Mathematicians always like to joke about engineers rounding numbers to 3 or 4 places because it doesn’t really matter to engineering after that, but is mathematics rounding off infinity or at least only capturing some aspect of infinity? After all, how can there be different types of infinity? My preferred illustration for the existence of multiple infinities is from Galileo. Galileo used a thoughtful but intuitive approach to understand infinity. He drew a circle. Then, he drew an infinite number of rays from the center of the circle. These rays filled up the space inside the circle. But then, he drew a larger circle around the smaller one and extended those rays to the larger circle. Though he drew as many rays as possible (an infinite number perhaps), the infinite number of rays did not fill up the larger circle; there were spaces between the rays. This led him to believe that first infinity was not large enough for the second circle; not even close. He would need another size of infinity to fill up the larger circle. [BAM! PHH! Did your mind just explode?] It is important to note that intuitively, his illustration makes sense. However, with today’s current understanding of infinity and better ability to calculate infinity, we now know that the infinity in the smaller circle leaves no space between the rays when extending to a larger circle. But, I liked his intuitive approach. Though intuition seems to be severely lacking when it comes to infinity.
Dangerous Knowledge: http://topdocumentaryfilms.com/dangerous-knowledge/
Georg Cantor His Mathematics and Philosophy of the Infinite by Joseph Warren Dauben
TML: The Infinities In Between (1 of 2): http://www.youtube.com/watch?v=WihXin5Oxq8
TML: The Infinities In Between (2 of 2): http://www.youtube.com/watch?v=KhgNiqI-bt0
Infinite Series: http://stream.utah.edu/m/dp/frame.php?f=f55f900bec01a3106121
Zeno’s Paradox – Numberphile: http://www.youtube.com/watch?v=u7Z9UnWOJNY
My new bumper sticker.