“Have you ever thought about how strange it is that we think about infinity every day, but most people think about it only on the rarest of occasions, if ever?” This is the text message I recently sent two of my close friends, who also happen to be mathematicians in my department. I was deep in the midst of studying for preliminary exams, trying to prove Riemann’s Theorem on removable singularities, when I started to think – really think – about infinity.

As mathematicians, there are myriad concepts we think about every day that most people do not. Sometimes this is because they are concepts not known about by non-mathematicians and sometimes it is because they are not pertinent to them. But infinity? Everyone knows about infinity, but no one really thinks about it. Or at least that was my hunch. And, based upon my small [read: VERY small] sample size, I was right. I started texting my close friends outside of the math department, asking them how often they think about infinity. Some said rarely, some said never. This blew my mind. Not because it surprised me, but simply because I couldn’t imagine what life was like without this thought in the back of my head at all times – that there’s always more.

This led me to ask myself how I think about infinity. As a child, infinity meant that I could name a number after myself. I remember telling my parents “If numbers go on forever, then I can name one ‘Deborah-illion’ and it must exist.” As I thought about my current understanding of infinity, I realized I had not progressed much, despite my mathematical studies, because my first thought was “there’s always one more.” But then I realized that this implied my concept of infinity was a countable infinity. So I started to think about uncountable things, and I realized that I think about that as always being able to “stuff more in between” – like how I can choose two real numbers and there are infinitely many more real numbers between them. And then I was overwhelmed with how inadequate my perception of infinity is.

You see, I believe that we as mathematicians fall into this false sense of security thinking that we understand infinity – and justifiably so. We can prove when certain properties hold for finite-dimensional vector spaces but not for infinite-dimensional vector spaces. We can think about limits as some variable tends to infinity and even understand this well enough to explain it to our students. We talk about it over coffee and write it in our proofs. We have a cute little symbol that we have mastered writing (but let’s be honest, that took some of us a while) and a Latex code that we all know by heart. We have taken something so big (is big even the right word?) and stuffed it into a small, tangible little package so that we can carry it around in our finite brains and feel like we know something.

But I think that’s beautiful. Because it’s yet another example of “the more you know, the more you don’t know.” It reminds me that this whole math thing is way out of my reach, but that if I keep reaching and searching for the little things I can understand – the lemma on page 23 or the definition at the beginning of lecture – there is hope that eventually I will be able to use those to prove another lemma, and maybe that will lead to a theorem. It reminds me that there’s always something left to prove, always something I can understand a little better, and that this thing which I have chosen to dedicate my life to will be there as long as I choose to pursue it. Recently I realized how well this concept applies to other areas of life – for me, it’s my faith and the peace that comes along with knowing that while I’ll never understand all of it, I can cling to what is within my reach and hold on. Maybe for you it’s something else. Either way, I think this is a conversation we need to have. So go to your departmental tea and start a conversation about infinity with your colleagues. Talk to your family and your friends and your barista. Ask them how they think about infinity. I would love to hear what you find out.

Nice story 🙂

My early experiences with the infinity go back to my trying to count even more as a child. “What comes after million, billion, trillion, …?” I would ask my uncle. Later a cousin of mine teased my mind with a paradox: Draw an upright triangle and a segment connecting the middle-points of the two of its sides. Now to every point on this segment corresponds a unique point on the base determined by the line crossing through the tip of the triangle and that point. Geometrically, it is obvious that this map is bijective: connect any point on the base to the tip and you’ll get a point on the segment. So, there are just as many points on the smaller segment as on the larger base! How could that be possible.

Then later I read in a book about the paradox of an arrow never reaching the target because first it will have to travel half of the distance left, then half of the remaining distance… There will always be a nonzero half of the distance left to travel, thus at the very end, ultimately, it BARELY reaches it. So, no question of ever reaching going one more foot ahead.

The first mathematical definition of infinity that I really liked and was stunned by its simplicity was that “a set is infinite (in cardinality) iff it has a bijection with one of its subsets.” Then of course came Cantor’s amazing uncountable binary set.

Two incidences that made me reconsider my understanding of infinity were: 1) you can rearrange the terms of a series and end up with a different sum! What the hell?! 2) It is a trivial task to show that [0,1] and (0,1] have the same cardinal number. Thus, a bijection exists between them. Task: Find one! I had this on my mind for some months until I came up with one 🙂 Once you find it you see how simple it is! Good luck with your own bijection.

Endnote: There is an infinity related to values in real line, as in limits of functions, and there is another one related to cardinality of sets.

When I was an undergraduate I had the chance to go to a community to give a series of talks to 7-9 graders. I talked to them about the “Hilbert Hotel problem” they loved it. That’s where I realized that everybody should know about the infinite hierarchy of infinites (in a suitable way), as everybody know the sun doesn’t follow the earth.

“We have taken something so big (is big even the right word?) and stuffed it into a small, tangible little package so that we can carry it around in our finite brains and feel like we know something.” This part made me think about using forcing to collapse cardinals.

Nice article.

Cheers