Remember math class? You instantly had either a flash of great joy or your soul just died. This is a story about math for the latter group. Trust me on this one.
First, why does the latter group exist? Why are there so many people, and let’s be honest, it is a very great number of people, whose experience learning mathematics was pure misery?
Over the years, I cannot tell you how many students have been in my office and said, “I can’t do math.” I invariably say the same thing: “Of course you can do math. What is 5+3? Sometime in your life you had a really awful math teacher who sucked all the joy out of the subject.” Invariably the student’s face instantly lights up with recognition, remembering that math teacher.
The fundamental problem is that mathematics is taught all wrong. Other subjects often start with amazing things that inspire, arousing wonder and curiosity. Mathematics starts with equations and lots and lots of repetitive tasks. Don’t get me wrong. The only way to truly learn mathematics is with lots and lots of repetitive tasks. But, in order for all that work to seem worth the time, isn’t it first necessary to arouse a sense of wonder? Why doesn’t math education start with weird and amazing things and then work backward to learn the mathematics behind all this weirdness?
It is amazing that something we can understand so instinctively can be so difficult to turn into rigorous mathematics. This can make mathematics seem frustrating to some people, impotent to others, and pointless to yet others. To me it is simply fascinating that our gut instincts can be so strong, and so difficult to understand with our brains. It doesn’t mean we shouldn’t try…
Eugenia Cheng wrote that in Beyond Infinity, an exploration of one of those incredibly weird and curious mathematical things. Infinity is an idea everyone learns about fairly early in life as the answer to the question “What is the biggest number?” Remember the moment you learned the answer to that was a number so big it couldn’t get any bigger? Remember when you learned you can’t even count to infinity because it is so big? Even a computer can’t count to infinity. Even Hermione Granger can’t count to infinity.
Then you learned that infinity is so big that if I have an infinite number of marbles and you have one plus an infinite number of marbles, well, you don’t have more marbles than me. In fact, if you have twice as many marbles as me, you have the same number of marbles as me. In fact, if you had infinitely more marbles than me, you have the same number of marbles as me. Honestly, you were amazed about this when you learned about it. Infinity is not just the biggest number possible, it is weird.
But then instead of capitalizing on all that weirdness and keeping you fascinated with numbers, tedium came along. If you made it to calculus, you learned that an infinite series of infinitely smaller things can add up to a finite number. That should be weird and amazing, but it was probably just a thing you had to figure out in order to pass your test. You learned you can calculate the slope of a curvy line by looking at infinitely smaller distances between the point on the line and the point next to it, which is also really weird, but again, you just memorized how to take a derivative. Math wasn’t weird anymore. It was just exam questions.
Unless you had a really amazing teacher, you never learned this in school: that first thing you learned about infinity, that it is the biggest possible number and you can’t get a bigger number, is wrong. There is a number bigger than infinity. The bigger number is also called infinity. Some infinities are bigger than other infinites.
How can that be? Well, imagine you had an infinite series of infinitely long numbers less than one. For example, start with
0.274539754312667532….
0.375631948857323456….
0.984452387503486328….
0.556633995846321885….
0.412386593658235702….
…..
Imagine that each one of those numbers is infinitely long. Imagine that instead of five numbers like that you have a list of infinitely many numbers like that. So, you have infinitely many numbers which are infinitely long. Right? (Make sure you understand that this is an infinite set of infinitely long numbers—ready to have your mind blown?)
There is a number less than one that is not in that infinite set of infinitely long numbers less than one. How is that possible? How do we know?
Take the first number in your list. Look at the first digit in the first number and add one. We now create the number 0.3. Now take the second digit of the second number and add one. 0.38. Now the third digit of the third number and so on. Using the five numbers above this is 0.38579…. Keep doing this. That new number you just created is different than every single number in your infinitely long series! Think about it—pick any number in your infinitely long series and compare it to the number you just created. Consider the 5789th number in your series; the 5789th digit in the 5789th number is, say, 4; in the number you are creating, make the 5789th digit 5. So, the number you are creating is not the same number as the 5789th number in your infinitely long series. Or any other number you choose. There is no way that number you just created is in the original infinitely long series.
So, the series of numbers with the new number you just created and the series you started with must be larger than the series with which you started. Some infinities are larger than other infinites.
I took a lot of math classes in my life, but I only learned this fact about 15 years ago from a colleague of mine in philosophy. When I think back to my calculus class, endless tedium, it is amazing to think how much more excited I would have been about thinking about all those infinitely small distances if I had understood how truly weird infinity actually is.
Another example. Imagine the number 0.999999… where the nines just keep repeating forever. If you subtract that number from 1, what do you get? You get 0. That number 0.99999 and the number 1 are exactly the same number. How can that be? It’s weird. Take the number 1. Say that a number is smaller than 1 if there is some distance between 1 and that number. So 0.9 is smaller than 1 because it is 0.1 smaller than 1. Similarly, 0.99 is .01 smaller than 1. But now imagine the smallest possible distance between 1 and a smaller number. The distance between 1 and 0.99999… is smaller than the distance you just imagined. So imagine an even smaller distance than the smallest possible difference you just imagined. Same thing. In fact no matter how close to 1 you imagine a number has to be in order to equal 1, 0.9999….. is a smaller distance away than that distance. So if there is no distance between 1 and 0.9999…., then 1 must equal 0.9999…. Right? But that can’t be true. Right? Infinity is weird.
Another example. If you have 5 things and you add infinitely many things to it, then you have (5 + infinity) which equals infinity. You know that. But what if you have an infinite number of things and you add five more things? Well, you can’t really do that. In order to add five more things, you would have to add them to the end to your infinite series, but there is no end to your infinite series. So, while (5 + infinity) is a perfectly reasonable thought experiment, (infinity + 5) is meaningless. Which means (5 + infinity) does not equal (infinity + 5). Yeah.
Cheng’s book is an introduction to these sorts of weird things about infinity. Who is the audience? It is those people who really hated math class or those people who took calculus and thought it was boring. Infinity is many things, but it is not boring.
I would love to highly recommend the book to anyone who thinks math is boring. But, alas, while the book has a great many weird and fascinating things about infinity which you will surely enjoy learning if you don’t know them, you have to take the good with the bad. The bad: you have to put up with a lot of corny stores and examples. I was ready to murder the “evil smarty-pants” by the second mention, but he kept coming back for more.
Is infinity an infinitely interesting subject? Probably not. But it does have enough odd and intriguing features that pondering it will give you a finite amount of amusement. Added bonus: from now on when someone tells you something is infinite or forever or whatever, you can ask: Is that the normal infinity or an infinity that is bigger than the normal infinity?
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