When it comes to math, I think it is best that I assume everyone is really ignorant. So forgive me for going over some really basic ideas about infinity. But I have good reasons for doing this. Some time ago, I wrote, Infinity Is Not a Number. In it, I discuss how the great Christian apologist William Lane Craig used a total misunderstanding of infinity to “prove” that the universe is not eternal. I was pointing out that this brilliant man was ignorant of the fact that infinity is not a number.

I guess I can understand. Mathematicians use infinity (∞) along with a lot of numbers. To understand infinity as a mathematical concept, you really have to study limits. And that isn’t something students get into until college — if then. It’s really when you start to study calculus that you get into limits. And that is done for the opposite reason: to understand the infinitesimal. It’s actually the same thing, but I won’t bore you with it. You either understand what I’m talking about (It’s the reciprocal, stupid!) or you would require a good deal more exposition than I’m willing to expend.

## Countable Infinity

But there are some things about infinity that are intuitive to people of this time and place. For example, if you think of the whole numbers (0, 1, 2, 3…), you know that there are an infinity of them. Regardless of what natural number you can come up with, there is always a number that is one unit larger. Let’s combine the whole numbers with all the negative whole numbers (…, -2, -1, 0, 1, 2, …) that infinity space is twice as large. But that doesn’t mean there are twice as many integers as whole numbers. Both sets contain an infinite number of elements. But they are both “countably infinite.”

So what does it mean that they are countably infinite? It means that you can pick any number in the series and count to it. Consider the whole numbers again (because it is easier). You can count from 0 to 15,000,000 or 15,000,000,000 or whatever. It will just take time. If you have an infinite amount of time, I guess you could count all the whole numbers (and, unintuitively, the integers). (Note: this is one reason immortality is so terrifying: eventually you would get to the point of counting all the whole numbers.)

### Rational Numbers

Now let’s consider the rational numbers from 0 to 1. These are just the fractions. You can tell that they are countably infinite. You just start counting: 0/1, 1/1, 0/2, 1/2, 2/2, 0/3, 1/3, 2/3, 3/3, etc. It doesn’t matter that many of the numbers of duplicates. The denominators go on forever. Thus, the number series is infinite. And we know it is countable because we are counting it. In theory, we can count all of the numbers. The same is true of *all* the rational numbers — not just the ones between 0 and 1.

So a countable infinity of countable infinities is a countable infinity. You would think that all infinities would thus be countable. But no.

Irrational numbers are numbers that cannot be represented as a fraction of two integers. You know: numbers like π and *e*. The set of all these numbers does not represent a countable infinity. Indeed, the irrational numbers between any two numbers are not a countable infinity. Think about it. It’s mind blowing.

*Rick and Morty*

I was thinking about this because I was watching an episode of *Rick and Morty*, “Rick Potion No 9.” In it, Rick turns his human world into Cronenberg-world. So in order to fix things, Rick goes looking for another reality, “There’s an infinite number of realities, Morty. And in a few dozen of those, I got lucky and turned everything back to normal. I just had to find one of those realities in which we also happened to both die around this time.” (Note: if there were an infinite number of realities, there would be an infinite number of realities where Rich “got luck,” but I’ll leave that as something to consider on your own time.)

The implication of this episode (and others) is that at every instant, there are an infinite number of realities, and each one creates an infinite reality at that point — and on and on and on. I’ve long thought this is what Planck time must represent: the time at which every possible quantum reality is spawned. But that doesn’t much matter. It’s just a thought. But I like Rick’s idea of an infinite universes (“realities”) constantly spawning an infinite number of universes.

## Countably Infinite Universes

As outrageous as this notion is, it would still represent a countably infinite number of universes. I find that vaguely comforting. Of course, there’s all that dark matter and dark energy. Maybe that’s where the non-countably infinite universes exist. I don’t really care. I don’t like thinking about things that can’t be counted. Irrational numbers are as strong a proof of the non-existence of God that I can think of.

But let me leave you with a practical thought. As many of you know, I think memorizing multiplication tables is a waste of time. I think children would be better served watching *Rick and Morty*. They’d have a chance of thinking about math. And it wouldn’t make them hate math. Finally, it might get them to watch *Existenz*.

Well, yes and no. Is infinity a number? Not under ZFC axioms on the real line, but it can be. You can extend the real numbers by treating infinity as a number. You need a topology for that, which means you need to define what it means to be “close to” infinity. The answer is that you consider z to be within epsilon of infinity if its distance from zero is more than 1/epsilon. That means negative infinity and positive infinity are the same thing, which means the extended number line is similar (“homeomorphic”) to a circle. There are other ways to do this, though. The point is that saying “infinity isn’t a number” is making an absolute statement about a system we created, but we didn’t have to create it that way. These statements are provable, but only because they follow from axioms we choose to accept. And we don’t accept them because they’re true, but because they work for certain types of problems. All models are wrong, but some are useful.

“Let’s combine the whole numbers with all the negative whole numbers (…, -2, -1, 0, 1, 2, …) that infinity space is twice as large.”

What’s an “infinity space?” I’m not sure that term means anything. The cardinality is the main thing you’re talking about in this post, which means you can match the sets up one-to-one. So the interval (0,1) has the same cardinality as the real number line. On the other had, there are measures that can distinguish between these sizes in other ways. The most common, Lebesgue Measure, would say that the interval has a measure 1 whereas the real line has infinite measure. Both the positive integers and all integers would have Lebesgue measure zero, as would any countable set. I can’t think of any measure that would show the integers having twice the measure of the positive integers, but there probably is one. But you can’t blithely say “this space is twice as large as that space,” or you’re committing the same mistake you’re criticizing. This is why definitions in mathematics are so damn complicated- they have to be in order to avoid ambiguities.

“Irrational numbers are as strong a proof of the non-existence of God that I can think of.”

…what? I have no idea how you come up with that. If anything, I’d say the opposite- the existence of things that are beyond the ability of humans to conceive is evidence that there is some intelligence behind them. Newton said his discovery of the law of gravitation made him more confident of God’s existence, because it was part of the universe and not something created by man. Euler’s equation e^i*pi + 1 = 0 is a beautiful mathematical poem, and it’s built into the fabric of the universe. The rules and structures of math are man-made, but the truths they expose were waiting to be discovered. I’m not going to attempt to prove the existence of God, as I figure that’s impossible, but I don’t see how the existence of irrational numbers can be said to disprove it. “You don’t like thinking about it-” fine, don’t, but your not thinking about them doesn’t mean anything. It can be bad to get too invested into mind-bending things, like when Georg Cantor proved the existence of irrational numbers and nearly went insane trying to prove the continuum hypothesis, occasionally taking breaks to try to prove that Francis Bacon wrote Shakespeare’s plays.

Don’t forget, Newton searched for secret messages in biblical text, and doinked around with alchemy. We all have bees in our bonnet. Those smarter than myself (and I’m in the upper half, if my experience with bosses is any indication) have more bees a’ buzzin’.

For me there’s no proving or disproving the existence of the supernatural. All limits prove to me is, while our existence is created by this universe, our minds are not quite capable of grasping it. That’s okay. We don’t need to see the whole electromagnetic spectrum to spot a falling tree and duck outta the way.

What I can’t get is young-Earth creationists. “There can’t be a God and evolution.” Why the hell not? IT’S GOD. If a supreme being imagined all time and matter into existence, surely It could do so any damn way It likes. Including the Big Bang, natural selection, and upright primates clever enough to invent whatever gooey salted substance they call “butter” for their movie popcorn. (Anyone who’s worked in a movie theater can tell you: there ain’t nothin’ ’bout that butter what goes “moo.”) It’s God! It can create anything it wants.

I started watching “Rick and Morty,” after your recent mention of the series. I find myself infinitely amused by it. Thanks. Now, I’ve added “Existenz” to my Nflix queue; it’s not streamable like R&M, so it mike take some time before I get to it. But hey, time is infinite, too, right? ;)

So glad you enjoy “R & M.” Hard to pick a favorite episode, but one that consistently makes me laugh hard is where Rick’s car battery is powered by micro-level slaves. That car cracks me up. “Must protect Summer.” You don’t fuck with that car!

Also, Rick uses slaves, but humans can be dicks. We knew this.

Also: Stephen Colbert!

Although if I had to pick one, it would be the one with Unity. I especially like the very end where he’s trying to contact her. I have this secret fear that even at 80, I could fall madly in love. It seems doubtful though. But I do enjoy watching old men acting just like they did when they were 17.

And of course, by trying to free everyone from the tyranny of Unity, Morty ruins the planet … while Rick gets so much crazy sex fun he becomes bored by it. Generally, this level of cynicism would make me hate a show. But the writing’s so damn funny. It’s like “Arrested Development” that way. You hate yourself for laughing. It’s so good, though.

The other one I’d put at the top is the couples-therapy planet. “Oh no — they’re codependent!”

I do not think it is possible to prove/disprove the existence of an all powerful being or potentially anything else using the concept of infinity or any other mathematical concept (by itself).

Mathematics when not applied for practical stuff is an abstraction. An abstraction only makes sense in the real world if there is the real possibility that it even exists.

The same idea applies to our imagination. We can imagine many things that are possible but that at that moment do not exist.

This is a philosophical issue beyond math by itself.

Math is our only entry to God. But geez, why is everyone grabbing onto that flip remark? But math

isphilosophy.“Math is our only entry to God”

Frank – I would put it slightly differently. Math is as close to perfection on earth as mortals can get. Again math is an abstraction (when) without the possibility that which is being described actually exists.

“Math is philosophy”

Math is absolute in its costructs and calculations (even when coming up with the possibility of an irrational number in an equation for example).

Traditionally, Philosophy as a discipline I mean (beginning with the ancient Greeks) argues the indeterminate concepts us humans often cannot easily differentiate between in an obviously transparent fashion (ie. subjective vs objective for example).

Math is much closer to applied sciences than traditional philosphic discourse for that reason I believe.

This is all just my opinion of course…

Is math as close to perfection as we can get? Maybe! I’d say it’s an interpretation of reality. A really beautiful one, but not reality. I don’t think we’re clever enough to grasp reality.

However, with math and science, we’ve done fairly well, given our limitations. I give humanity an A for effort, a C-/D+ for results.

Frank – Just want to add that I do understand your possibly pointing out that philosophy and math were basically one in the same during the time of Plato and Aristotle certainly. I am not so sure that such a pairing is as appropriate today however (as I see it).

You have to take my first statement in the context that I don’t believe in God — at least not in the way that most people do. Math is the primary way to get past our parochial nature. Science is a very poor second. But I’d be very pleased if we reached a point in science where we could explain why the gravitational constant is what it is. But even that wouldn’t really lead toward an understanding of our existence.

I’m still hung up on Godel — on the idea that logical is ultimately illogical. It is actually because math is

notperfect but that I, in my parochialism, think it should be, that makes me think it the key to existence. That’s my limit at this point. That greatly affects the way I look at reality.I think you are confused about what math is. Sciences use mathematical applications. But math itself is completely abstract. Look at people who work on n-dimensional geometry. They are simply building little universes and seeing what they can about them. That is far closer to moral philosophy than it is to quantum mechanics, even though people use math for the latter.

I’ve taken to watching science videos as I try to sleep, and there was a wonderful one I saw recently. You know how fundamentalists like to say, “how come science can’t create life in the lab?” Well, actually, they’re making progress on that. This video had a wonderful metaphor for the challenge of it. Imagine a nice, fluffy pastry, like a crescent roll. Now go and make one without a recipe. You’ve got the ingredients — flour, water, egg. It’s how to combine them and cook them in just the right order, in just the right way, which gives you a fluffy pastry instead of a flat, burnt bit of crap.

I guess biochemists have made real strides in the kitchen! They’ve come up with about half of RNA. That’s pretty wild, given it’s all trial-and-error and the enormous number of variables. Still, they’ve essentially demonstrated — you mess around enough with the basic ingredients for living cells, you’ll get there. Astronomers have shown us just how many gazillions of stars there are; enough random kitchen ingredient combos. Math gives us a way of feebly grasping what a “gazillion” is.

Now, where did those stars come from? WTF knows. Actually, I do, but you have to pay extra for that additional feature.

“why is everyone grabbing onto that flip remark?”

Probably because it made no damn sense.

Jurgan,

Ah but do not forget; it may not make sense to you while making sense to someone else afterall…

Jon:

Of course, but it’s the author’s responsibility to be clear. If it’s not clear to me, then I will say so.

Quite right! Constructive criticism is better. Suggest how you would have put the point differently, or, if the point is completely unclear, suggest what you think it might be and why you imagine the author was fuzzy about it. You write like someone who gets how tough it is to write. Help other writers get better! Frank would do the same for you. He’d also steal your girlfriend, dog, and farm, but that’s why it is unwise to trust him with such things.

I wrote a pretty detailed response upthread explaining why that remark didn’t make sense to me and how, from my perspective as a practicing mathematician, it is contrary to my experience of the universe. I’m not expecting everyone to have to same perspective, but saying something so weird and not explaining it at all comes off as just trying to stir up shit rather than stimulate thinking.

Fair enough. I’m afraid I lost what little math skills I once had. It’s a real “use it or lose it” pursuit, like foreign languages or basketball jump shots. Believe it or not, I used to be pretty solid from about 15 feet out! Now an eight-year old could whip my butt at HORSE.