Divinity and Infinity

Larry’s talk starts at 6:56.

Thank you so much, Rabbis Mitelman and Zeidman, and good evening everyone. Thank you for joining me tonight on this journey to infinity and beyond. Even a young child thinks about infinity. I mean like, are the stars infinite in lifespan or number? And this past Sunday was Pi Day, which celebrates a number whose decimal never ends.

But sadly, I know that not everyone has been taught math as something beautiful, and the only math-religion link some people might have, or imagine, is wanting prayer in schools when taking a math test. (laughs) So let me offer some some reassurances to anyone who’s nervous about math or about religion. First, there’ll be no grades. Second, I have worked very carefully to minimize jargon. So it’s fine if you don’t know calculus or Hebrew, I’m keeping this self-contained. And third, this is a general-audiences academic talk. It doesn’t teach religion, it teaches about religion, and so we respect and welcome people of any faith, or no faith. And finally, I wanted to go big, so the topic today is infinity, but the talk is finite. And this whole session should should end well before 8:00, which on its side, of course, is a common symbol for infinity!

In case you’re wondering, here is the history of that symbol, of what it dates back to. And it’s only fitting that it was introduced by someone who did math and religion, so that’s fitting with the theme of our sponsoring organization. Someone else who did both math and religion is Pascal. And I just really find this particular quote of his very, very moving:

“When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I’m frightened and astonished to see myself here instead of there, now instead of then.”

And there are many other mathematicians who thought a lot about religion – these are not just exceptions. It’s really not so strange. And even in my own department I’m not alone, I’m not the only one. For example, Kien Lim has a math paper on karma and Granville Sewell has a book on intelligent design. And there’s people in other departments in my University who have also made connections to their respective faiths in their work as well, so we’re not as unusual as you might think.

And so I don’t forget the thank yous – I want to go ahead and do them now. So thank you again to our funders and our sponsors, to Rabbi Zeidman for agreeing to come along on this crazy ride with me, and to all those who gave me feedback on all or part of my slides. But of course, any errors that remain are my responsibility.

We’re also grateful that we’ve managed, in a short period of time, to get print, radio and TV media coverage. This is a screenshot from an interview that Rabbi Zeidman and I did last week, although I  did not know at the time of recording the interview what rotating ad banner they were going to have next to my name, so that’s kind of funny. (laughs)

Okay, so let’s dive in. So what word comes to mind when you think of infinity? Just think about it. If you want to go ahead and type a word in the chat window, that’s fine. I’ll give you 10 seconds to think about it or type. There’s no right answer, it’s just whatever comes up, all right. So I’m seeing some interesting answers here. Oh wow, nice. When I’ve asked this in the past, some of the answers I get are things like “Endlessness, boundlessness, indefiniteness, inconceivability,” and sometimes the answers are more emotional, like “Awe, futility, fear.” So it’s interesting. And so one of the things that we’ve been encouraging all the speakers in this series to do is to include a slide that kind of says what started us on this journey in the first place. And so here’s my slide for that.

So my spark was two decades ago when I was teaching at Emery High School in Houston, Texas, a pluralistic community Jewish day school. And you know this will be hard to believe, because we all know math is cool, but there were some students who really didn’t try to hide that they were only in class because they were required to, and they were just learning enough to do well on the SAT. But it just didn’t captivate them inherently at all.

And so I tried a lot of things to try to counter that, to try to engage them. And one of the things I tried was looking for ways to connect the math content to the things they were learning in their Judaics classes. And it just took on a life of its own. And you know, it made it more fun for me, too, because I was learning things that I hadn’t been taught either – you know, those connections. And so some of the topics that I gathered material on, either creating it or finding it, that other people have done, included – I mean, Gematria is sort of the obvious example that a lot of people think “Math and Judaism” would be. That’s like the Hebrew equivalent, the numerical equivalent, of a Hebrew word. But that’s not what this talk, where this whole speaker series, is focusing on, so have no fear, we’re not going to dive into that.

But there’s other things, too, plenty of things besides Gematria, like mathematical modeling and geometry and logic and counting, and of course the topic today: infinity. And I wrote up some of these connections and published them in journals, like Journal of Mathematics and Culture, and some other venues as well. And so I ended up taking something that was really a deep part of my personal identity and work experience, and finding a way to add it to my professional vitae or resume. So that was neat.

But it’s not just something that you see in technical papers. We see the word “infinity” a lot in popular culture. And so here’s four examples of lyrics from the pop world, or the folk world, where you see the word “infinity” used. And I’m sure if you think there’s a great example I should add for next time, you can let me know through the chat window as well. Here’s a kids’ album that has the title “InFINity” with a picture that I like a lot. That’s kind of fun.

And then the concepts of infinity have also appeared in math songs. As Rabbi Zeidman mentioned, I do write math songs also, and two of the songs I’ve written, “Cantor’s Coat” and “Hotel (Called) Infinity,” to the tune of Hotel California. You can find those at the webpage larrylesser.com/greater-lesser-hits, if you’re interested in hearing a recording of that, but we’re not going to fit that into today.

Okay, so in the chat window, type the English name of the biggest finite number you know. An actual number, not just something like jillion, gazillion, brazilian, or whatever. I’ll give you 10 seconds to do that. Wow, we got some ringers in the crowd, some people have definitely done their homework. Okay, great.

So there are prefix patterns. For every three more digits of a number, you get to the next prefix, and if you know some Latin root stuff, you can see that there are patterns to them. It’s hard to have intuition when the numbers get really big, right. I mean, it’s one thing to talk about millions and billions, but we don’t have a lot of experience beyond billions. Well, I guess actually, this month, we do. This month we had a stimulus pass that’s in the trillions. And also this month we have March Madness starting, and the number of possible brackets for March Madness is in the quintillions, believe it or not. So that’s why no one’s ever had a perfect bracket, it’s really hard to do that.

So it turns out that a really big named number, that a number of you mentioned in the chat window, is a googol, and notice it’s spelled slightly differently from the search engine Google. It is 10 to the power of 100, which means it is a 1 followed by 100 zeros. I’ve written it out there. And just to give you context for just how big that is, it’s more than the number of atoms in the universe, so it’s pretty big! However, their number is even bigger than that. I see at least one person in the chat window has already anticipated – namely, a googolplex, which is the the numeral one followed by a googol of zeros. And there’s no time even to write that number out, even if you started when the universe was born, so just to give you some context there (And I find it really fun that the company Google named its headquarters Googleplex, so very fitting, very fun).

All right, so as I was preparing for this talk, I found myself asking myself and wondering, “So what’s the biggest named number in Jewish text?” And if you go into the book of Daniel, you see that talks about “myriads upon myriads.” And a myriad is like 10,000. So you’ve got ten thousands of ten thousands. And if you do 10,0000 times 10,000, you get 100 million. So yeah, well, that’s pretty big.

And if you’re willing to go beyond Tanakh into Talmud, which is a commentary on the Torah, then you can construct a number that’s a lot bigger than that, namely 10 to the power of 18, which is one quintillion. See, we did reach the quintillions. And this refers to the number of stars in the sky from Tractate Brachot (Brachot 32b). If you multiply all the red numbers together, you get a product that is bigger than one quintillion. So that’s pretty big. I don’t know, but maybe there’s an astronomer in the crowd who can tell us how close that number actually is.

Okay, so you’re not going to be shocked by this big news that there is no biggest number. If you have the set of what we call the counting numbers – 1, 2, 3 and so on, there is no largest element in that set, because you can always add 1 to whatever you think the largest number is, and get something even bigger, right.

So let’s start, simple as one, two, three. Do you think the quantity 1 + 2, added to 3, is equal to 1 + plus the quantity 2 + 3? So in the chat window, you can type yes or no or if at your webcam you can give me a thumbs up or thumbs down for yes or no, respectively. Let’s see what you all think. And you probably remember, order of operations says that you have to do what’s in the parentheses first, and remember the answers are not graded, okay. Most people seem to be saying “yes,” and that is correct, they are both equal to 6.

And the reason that this works is, you may remember from your schooling, there is a property of addition called the associative property, which says that if you’re adding up a bunch of numbers, that no matter how you group those numbers, the sum is going to be the same. See, how you group them is how you’re associating them together, so that is  a property.

Okay, so let’s try our first anonymous zoom poll that Rabbi Zeidman will activate for us, because now i’m going to ask you a question about an infinite sum. What if you have 1 + -1 + 1 + -1 + 1 + -1, and so on? What is that going to be equal to? And I’ll give you 10 seconds or so to – all right, so it looks like most people are saying zero, but a few people are saying “it depends.” And only one person has voted for #1. So I’ll let you see what your results were. Okay, so here’s the answer. It does depend on grouping, okay, because if you group two at a time, then yes, you would get zero, but if you let the first element stand first and then start grouping two at a time, you would end up with 1. And the person who noticed that was – guess what –  another guy who did math and religion.

Now, by the way, I realized that all of the examples so far are indeed guys, but let me reassure you that in our speaker series, about half the speakers are amazing women, so we’ve addressed that and corrected the historical imbalance.

But the big takeaway here is that when sums or sets are infinite, we may need new ideas. That just because something as intuitive as the associative property seems to be so obvious and so common-sense that like, how could that ever not be true – well, we just saw an example where it wasn’t true. And so we want to sort of be prepared, and have an open mind to not assume that the rules that work in finite mathematics are automatically going to transfer over to the world of infinity. That’s sort of the “lightbulb moment” here, okay.

So ein sof is a term in Hebrew which translates “to without end,” or “infinity.” And it’s actually one of the many names of God that is used in Judaism. It was introduced in the 13th century by a mystic, someone who worked with Jewish mysticism, otherwise known as Kabbalah. And that’s a Kabbalistic diagram in the lower-right corner there – don’t ask me to explain it. And what’s cool is that I asked people, “So if you’re a classroom math teacher in Israel teaching a secular subject, what word would you use to refer to infinity?” And the answer was “ein sof”! So this connection is quite direct here.

There’s also the infinite in Jewish liturgy. And by that we’re not necessarily saying that services feel like they take forever, but two prayers that are pretty common in many denominations of Judaism are Yigdal and Adon Olam. And as you can see, the Yigdal prayer mentions Ein Sof, referring to God being without end. And the Adon Olam long prayer talks about God having no beginning and no end. And so you have that language in your prayer, that maybe you didn’t think about it in this context, but it’s there, it’s in there.

And so in general, religious studies scholars have written about what the theological infinite, or what is sometimes called the absolute infinite – how to describe it. And they have described it, as Jill Le Blanc describes, summarizes it, as “Unbounded, unlimited,” – it’s not just a collection of things, it’s nothing you can construct from the finite, it’s beyond all that,” okay. And so that’s how how religious studies people see it.

And so now the question is “Well, how do math people see infinity?” It turns out there’s two conceptions of mathematical infinity that you will see very commonly. The first is called potential infinity, and that’s usually associated with that infinity symbol, the sideways eight that we saw at the beginning. And it’s a way of thinking about infinity as a process that doesn’t end, but you’re still always thinking about it as a finite number of steps, where at any given point it’s still finite, but you can always have more. Like with the counting numbers, there’s always one that’s bigger, right. So that’s the potential infinity, or the “process” way of looking at it.

But there’s also a way of looking at it that’s called actual infinity, where you look at infinity as an object that exists all at once in totality, already realized or encapsulated, as if you had done the process an infinite number of times. And so if we talk about the set of counting numbers as a thing, as something that is just all there in one object, even though we know there’s an infinite number of numbers within that set, we can talk about the set itself as an object of actual infinity, as if it has been completed.

So again, it’s a subtle distinction, it’s abstract, it takes a while to get your head around it. If throughout this talk you don’t get everything right away – I mean like, I didn’t get everything I knew about infinity in one hour either, trust me – so get what you can, and at least this gives you a point of departure to look up more things.

It turns out that some UTEP people have actually done some mathematics education research into this. Mourat Tchoshanov and Grace Babarinsa found that college students did not always have both of these conceptions. In other words, the idea is that someone has both conceptions at once and it’s all integrated in their understanding, they can see it both ways. But what they found is that that’s rarely the case, that sometimes college students would see it one way, sometimes the other way, and which way they saw it would depend on the task at hand, it would depend on the context. And so that was an interesting finding, so just to let you know about that.

So let’s make this look even more concrete. So consider a cookie. I know everyone just had dinner, so think about dessert now. So you’ve got a cookie here. Now imagine you’ve got your kid who eats half the cookie one day, but leaves half the cookie. And then the next day they come downstairs and they eat half of what’s left. So now there’s a quarter of the cookie left, right. And then the next day they come and they eat half of that. So now we’re down to an eighth of a cookie, and so on.

So using the process interpretation, or the potential infinity, you could try to make the argument that he’ll never finish eating the cookie, because there’s always half of what’s left remaining for the next day. You never quite finish it, right. Of course, we all know that it is possible to eat an entire cookie, and that we want to believe that there’s a way of looking at it that says that this sequence of 1/2 plus 1/4 plus 1/8 plus 1/16th and so on must, in some, sense converge, so that you can actually eat the whole cookie, right. And so that’s sort of, again, the paradox that a lot of infinite paradoxes are kind of based on. But it helps to kind of see it with this cookie thing.

And so here’s a way to put some equations into it. Again, you’ve got this diagram where you’ve got half, a fourth, and so on. And you can see that less and less is available. So under the potential infinity, at any finite point there is still some cookie left, right. But in the limit where you let that process continue infinitely, you really do reach the whole. And if someone tries to say “Well no, you don’t, you reach something less than the whole,” then I can say, “Well, then, I’ll eat half of half of that so that I get even closer to 1 from what you said was the limit.” So I can always get closer to one than any stopping point you might give me. So therefore, that’s an indirect way of saying you really do reach 1. But again it’s – even when you take a calculus class, it’s a mind-blower to wrestle with this.

Okay, so the founder of set theory is Georg Cantor. And he had a really important idea about looking at infinite sets. And he says that to know whether two sets have the same size, you have to see if you can make a one-to-one correspondence between their elements. So here’s an example. Suppose we have a set containing three letters, T, M and S, in honor of Temple Mount Sinai. And suppose we have another set containing the numerals 1, 2, 3. Well, we can make a one-to-one correspondence. So that shows that those two sets have the same number of elements. Notice that elements don’t have to be numbers; they can be letters, they can be pictures, I mean whatever, right. But these two sets have the same size, they have the same number of elements, because they have a one-to-one correspondence. Those of you who have taken a lot of math will know that there’s a technical term called cardinality, but again, I promised to keep this jargon-free, so I’m not going to use that term. But you know, it’s there.

All right, so now since we’re toggling back and forth between math and religion throughout this talk, you may find yourself wondering: Is there a one-to-one correspondence in religion? Glad you asked. And the answer is yes, there is. So by taking a census, you know, counting a population, it’s done by some kind of proxy, as a way to respect the infinite value of human life. And so in the Torah, we see a couple of examples. We see from Exodus, actually – that’s from the Torah portion we read just two weeks ago in synagogue – isn’t it that by every person donating a half shekel coin, then you just count the coins, and that enables you to indirectly count the people, right? Or in the Book of Samuel, you know each person brought a lamb, so you count the lambs, and therefore you counted the people. So again, it’s this indirect way, so that you’re not just reducing people themselves to a number by counting them in a direct way.

Okay, another example is that when I looked into this further, that even the modern census, the census in the modern state of Israel, also uses some techniques of indirectness, in this sort of similar spirit, by recording names on unnumbered blank lines, or having the tabulation done by devices, not people. So I thought that was fascinating.

And another example where it comes up in everyday current religious life, [where] there are certain prayers in a Jewish service that can only be said if there is a prayer quorum, so to speak, or minyan of at least 10 eligible people. And so rather than just sort of bluntly pointing at people and going “One, two, three, four, five,” you say a verse of Torah as a way to indirectly count them, a 10-word verse of Torah. This is the one that’s perhaps the most frequently used, but some people have used others as well (Psalms 28:9). And so this way each person is being associated with something of infinite value, a word of Torah, right.

One cool thing someone pointed out to me is that the last word of this ten-word phrase is “ha’olam,” which in this context is “forever,” but it can also mean “world.” So if you’re missing that 10th person, you’re missing a world. So I always found that was a beautiful thought.

Okay, so now we’re going to toggle back to math. And we’re going to use this idea of one-to-one correspondence to see if two sets are the same size, okay. So we’ll start out simple. Here’s two finite sets, the set of {1, 2, 3, 4, 5, 6, 7, 8} – is that set bigger than {5, 6, 7, 8}? So we’ll have the Zoom poll here, go ahead. All right, I’ll end the poll. It seems like a pretty strong mandate for “yes,” and indeed that is the answer. So you get a smiley face for that, very good. No breaks, but you get a smiley face. And again, the reason is because we can’t find a one-to-one correspondence. If we try to match them up where everyone’s holding a hand, we see there’s four members of the first set that don’t have a hand to hold in the second set. So that’s why we can really say the sets are not the same size.

Okay, maybe that was too easy. Let’s try a little harder one. So that example was two sets that are each finite, so let’s make this a little harder by making both of those sets infinite. I added a “…” so we’ll launch the next polling question now. So is the set of numbers that goes forever, starting from one bigger than the set of numbers that goes forever starting from five – again, the question is, do they have a one-to-one correspondence? So yes or no, that they have the same size. A little more neck-and- neck here, almost, so slightly more for “no,” but but still fairly close. Okay, 56 said “no.” And that is the correct answer.

Let’s see why: because if you do sort of an offset or shift, you can see that you can still do this one-to-one pairing, right, where everyone in the first set has a hand to hold in the second set. You can’t think of any element in the first set that’s not accounted for, or any element in the second set that’s not accounted for. If you wanted to use algebraic notation, you could say every element n in the first set can be matched with the element n + 4 in the second set. So we’ve demonstrated that there is a one-to-one correspondence, all right.

Let’s do one more Zoom poll, Zoom poll #4. So now it’s going to be a little trickier, because these are still both infinite sets, but now there’s an infinite number of numbers – not a finite number of numbers, but an infinite number of numbers in the first set that are not in the second set, right. Because the first set is all the counting numbers, the second set is just the even counting numbers. So in other words, it’s omitting the odd counting numbers. So yes, they’re the same size. I mean sorry, A) yes, the first set is bigger, or B) No, they are the same size. Okay, so let’s end the polling here and share those results. And we see that 3/4 of you said that they are the same size. So y’all are  great students, you’re catching on, awesome, with this. And because, again, you can draw this one-to-one mapping where every number goes to its double. 1 goes to 2, 2 goes to 4, or in general, n goes to 2n. And so we’ve established the one to one correspondence. And so that concludes our four Zoom polling questions – see, the four questions, just like Passover coming up.

And we could have done this with the counting numbers and the odd numbers, and that all that would have changed is that we would have had n going to 2n – 1 to make the mapping work. So that’s how that would work.

And it was really fun to see that there is actually a comic strip that mentions this result, a syndicated national comic strip – it’s called Frazz, where this brilliant third grader, Caulfield, in the first panel says, “In infinity there are just as many odd numbers as there are even and odd numbers.” And then in the last panel, Caulfield says, “In infinity there are just as many lame jokes as there are lame jokes and good jokes.” I just love that. So you just never know when popular culture is going to incorporate something like math, but it’s always fun when it does.

But let’s just kind of like notice something – here, this is like another light bulb moment, right, because what we just said is that we have an infinite set that has a subset, a clear proper subset, but they’re still the same size. I mean, that’s not true for for any pair of finite sets, right – I mean this is definitely a rule, a phenomenon, that only happens with infinite sets. This does not happen to finite sets. So it’s another example where the laws of finite land may not carry over to infinite land, just like the associative property didn’t automatically remain always true. So just to kind of call that to your attention.

And so of course, since we’re toggling back and forth between math and religion, you’re probably wondering “So Professor Lesser,” – hey, that rhymes – “Is there a concept in religion of the part and the whole having the same size or being equivalent?” Glad you asked. And the answer of course is yes. Here’s three examples that I’ve been able to find so far, I’m sure there are even more examples. And so if you know of one, feel free to put in the chat window for us to look at or discuss at the end of the talk. One is there is a Jewish legal principle of miktzat hayom kekulah, that a small part of the day is considered like the whole day. And the most common way you encounter that is if you have to mourn for seven days, God forbid, on the last day, just a small piece of that day counts as the full day, and you can end the restrictions and start to, you know, return to your routine.

Another example comes from the Mishna, comes from Talmud, that “Whoever saves one life, it’s like they saved the whole world” (Mishna Sanhedrin 4:5). So again, that’s another example of where the part is like the whole. And there’s also a quote by the Rabbi Baal Shem Tov, the founder of Hasidic Judaism, who says when you hold a part of God’s essence, you hold all of it. And that in turn traces back even further back than the Baal Shem Tov. And so this, the sensibility of the part equaling the whole somehow, does have precedent in religion.

So we’ve talked about the counting numbers a lot. But do we have a name for the size? I mean, it’s one thing just to say it’s infinite, but is there a name for this – you know, this this infinite number? And it turns out there is, it’s “aleph null” (ℵ_o). And so if you have some familiarity with Hebrew, you recognize that that is the first letter of the Hebrew alphabet. And so we call this a “trans-finite number.” It’s because it’s beyond finite, it’s an infinite number, it’s certainly larger than any finite number.

And so you might be wondering “Well, where do we get that from?” Why aleph? Well it was chosen by that guy we met earlier, Georg Cantor, who while his name might make you assume that he is Jewish, he actually lived as a devout Lutheran – however, he likely had Jewish ancestry on one or both sides. There’s different sources that say slightly different things, so I’m not going to take a definitive stand on that. Aleph turns out to also be the first letter in the Hebrew words for “infinity,” for “one,” and for “God,” and it’s also mentioned in Joseph Dauben’s biography of Cantor, why he thought that was a particularly nice choice. And so it was a very intentional choice, and it’s very cool that he did that.

So it turns out that these transfinite cardinal numbers have their own rules of arithmetic, that again, just because things are true for finite arithmetic does not mean they’re always going to be true here. So for example, ℵ₀ + 1 is still all of null. That’s not true for finite numbers. We don’t say 7 + 1 is still 7, right. No, it’s 8. Or there’s ℵ_o odd numbers and ℵ_o even numbers, but when you add ℵ_o and ℵ_o, you don’t get 2(ℵ_o), you get still ℵ_o, because all the counting numbers is still ℵ_o. Again, you don’t have that being true for finite things. And so these are some rules for the cardinal arithmetic, so to speak.

Okay, so this sets this up for – so I guess I will do a math song. It’s really short – well, it’s really long or short, depending on how you look at it. It’s the world’s longest math song. “ℵ₀ bottles of beer on the wall, ℵ₀ bottles of beer, take one down, pass it around, and there’s still ℵ₀ bottles of beer on the wall…”. And it’s the same song over and over, right, because you can keep adding 1 or subtracting 1, it’s still ℵ₀, right. It doesn’t change. So the song becomes boring after the first few thousand verses.

So let’s start applying this more to Judaism in some further ways. So in the Torah, there are Commandments. And the one way of tabulating them says that there are 613 of them. But the Talmud, the commentary on the Torah, identifies a few specific commandments where that commandment is so special it equals the other 612 combined in value. And one example they give is tzedakah, which is translated either as “righteousness,” or as “charity,” [and it] is as important as all the others combined (Baba Batra 9a). And when I first came across things like that, I thought “Okay there, this is a Rabbi being dramatic. This is like a rhetorical thing, to give an emphasis, you know.” But when I thought about it mathematically – like, “Okay, but does this actually make sense mathematically?”, I realized actually it does. Because if we call m the value of tzedakah, and we let each of the other commandments also have the value m, what we’re saying is that m = 612 m’s, or 612*m. Right now, that equation, technically, it’s true if you stick 0 in for both m’s, but hopefully we all agree that charity does not have a value of zero. But if you put any non-zero finite number in there, the equation is not true, right. 3 does not equal 612*3. However, if we put our friend ℵ_o in, then it is true, because ℵ_o does equal 612*ℵ_o. So it actually is perfectly consistent with the laws of mathematics for handling infinity. So I thought that was kind of neat.

Let’s apply this now to the value of life. I mentioned, that there’s a quote from the Talmud: “Whoever saves one life saves the whole world.” That quote was actually featured in the movie Schindler’s List. So even if you’ve never seen or studied Talmud, you have been exposed to this particular piece of Talmud. And so this comes up in a number of places, so it comes up in why you can’t say one life is worth more than another life, because who’s to say your blood is redder, basically (Sanhedrin 74a). And so this is why in Judaism, usually, you can kill someone in self-defense. If someone’s trying to kill you, you can kill them. But you’re not allowed to murder someone to save your life. That there’s a difference there. So that’s one life versus another.

But what’s a little more not obvious, perhaps, is well, what about one life versus many lives? Because some of us might be like John Stuart Mill’s utilitarianism, and say “Well, aren’t five lives worth five times more than one life?” or something, right. You might think it’s okay to kill someone if you can save lots of people. So again, Judaism addresses this. Rabbi Joseph Telushkin says that “Saving many lives at the expense of taking one innocent life is not permitted, because you can’t say many infinities are worth more than one infinity.” And in the Talmud, which was codified into Jewish law, there’s a place where it says “If a group of Jews are told ‘Hand over one of your group or we’ll kill you all,’” even if all of them would be killed, they still aren’t allowed to say “Okay, here’s Jake, go ahead and kill Jake,” you know, you can’t do that (Jerusalem Talmud Terumot 8:4, codified in Mishneh Torah and Shulchan Aruch)

And so again, this is totally consistent with the way mathematics views infinity. Because to say five lives can’t be worth more than one life, well, you can think of that metaphorically or mathematically as saying that “5*ℵ_o is still ℵ_o, it’s not greater than ℵ_o”. And you know, if you want to get more into ethics, you can look at the trolley problem, which has been discussed with the Jewish lens as well as well as a secular lens.

Okay, so a natural question you may have at this point, among many probably is: do all infinite sets have this same size? Do they all have the size of null? In other words, is it like either it’s infinite or it’s not? Or is there more than one size of infinity, are some infinities is bigger than others? Are some bigger than ℵ_o? That’s sort of a natural question. If that seems strange to you that there could be more than one size of infinity in Judaism, maybe it’s not so strange. Because in Judaism there are levels of infinite spiritual worlds. And again, I’m not a scholar on Jewish mysticism, so don’t ask me to go too deep into this slide, but there are people on this call who can handle Jewish questions that are above my pay grade, so they they can chime in during Q&A on this. But this is just sort of a quick diagram just to kind of show you that there are these infinite levels of soul and levels of the world. Notice the top level of the world, by the way, is our friend Ein Sof. But again, this gets into really abstract stuff that just can’t be covered in two minutes, but there it is.

Okay, so back to math. So what we’re saying we’re asking is: Is there a set so much bigger than the counting numbers 1, 2, 3 and so on that there’s no way to put it into one-to-one correspondence with the counting numbers? That’s what we’re really asking. So it turns out that Cantor proved – and if someone asked during Q&A, I have some slides that outline how if you’re curious – that the set of all fractions, because obviously there seems like there’s more fractions – you know, numbers like 2/3 and 7/5 – than there are just counting numbers. But it turns out there’s a way to arrange fractions so that you can still match them up with the counting numbers with this one-to-one correspondence. And so therefore, then the size of the set of all fractions is still ℵ_o. It’s not any bigger than the counting numbers.

So then you think, “Ah, okay, well, then what’s bigger than the set of fractions? What’s a set of numbers in math that’s even bigger that might be even bigger than fractions that I can try?” Like the real numbers. The real numbers are all points on a continuum. So that includes not just rational numbers, but also irrational numbers like Pi, and the square root of 2, whose decimals never repeat or die. There are more real numbers than fractions in some intuitive sense. And it turns out it’s true even in this one-to-one correspondence sense. It has been proven – also by Cantor, and again, I can also outline it during Q&A if that’s what someone wants to see a taste of – there is no way to match up, with one-to-one correspondence, the counting numbers and the set of all real numbers, so that means it does make sense to say there are more real numbers than fractions, or there are more real numbers than counting numbers, because it would have a size greater than ℵ_o

Here’s an excerpt from a poet, Amy Uyematsu, who actually taught high school for a few decades as well publishing like five volumes of poetry. I love people who do that. And from this poem called “This Thing Called Infinity,” she has this stanza:

“Mathematicians are the real jokesters

when it comes to playing with the mind –

how counting from 1 to forever

is somehow smaller than

the set of all values between 1 and 2.”

I just love that. But again, it’s not just being poetic, that’s actually totally consistent with what we’ve been talking about. But hold on, I mean, this is like another light bulb moment, that because if we’re saying there’s more than one size of infinity – and for people who haven’t been exposed to this before, because unfortunately, even if you’ve gone through algebra, geometry, trigonometry, and calculus, there’s no guarantee that you actually were taught this stuff – you know most people only learn this if they take like a a math history class or a math for liberal arts kind of class or something like that. But I just think it’s too cool not to share with everyone. So thank you for being here. So there is indeed more than one size of infinity.

But wait, there’s more. It turns out there’s an infinity of infinities! And once again, Cantor was the one to show it. He found that if you form the set of all possible subsets of an infinite set, that resulting set is too big to be matched up in one-to-one correspondence  with that original set. In other words, there is an infinite sequence of infinities – ℵ_o, ℵ_1, ℵ_2, and so on. So that is a mind-blower, another light bulb moment.

And so because the speaker series talks about math and religion, it’s important to mention that Cantor really saw his math as being very much in harmony with his faith. These are some excerpts from letters that we have from him, and you can see that he really sees that the beauty of these transfinite numbers as reflecting the beauty of creation and the creator.

And the quote in red to me really hammers it home, that “Instead of diminishing the extent of God’s nature and dominion, the transfinite numbers actually make it all the greater.” The way I look at that concept is: most religions would probably agree that God is ultimately beyond full human understanding, at least that God is beyond my understanding. And maybe there’s someone on the call who feels they do totally understand God, but I don’t. And so if we now have an expanded concept of how big infinity can be, but now we realize that God has to be even bigger than that, in effect, we’ve raised the floor for how big God is, because God has to be even more amazing than we ever imagined, right. So to me and to Cantor, obviously, this is a way of a testimony to God, rather than something that takes away something from God.

There was also support from the Church at the time at the time Cantor was going through this stuff. Pope Leo XIII encouraged investigations into science, including, specifically infinity. There was a German priest who believed that actual infinity – remember, that’s the more subtle kind of infinity, not the potential infinity – that he believed that actual infinity could be contemplated by the human mind and help us get closer to to understanding God. And so there was that kind of harmony with religion at the time.

But there was also resistance. In particular, there was a German mathematician, Kronecker, who at one point was Cantor’s mentor, but he kind of withdrew because he  was kind of opposed to Cantor’s ideas. Kronecker is one of these guys who really believed that we should stick to the finite and not talk about these actual infinities. You see his quote: “God created the natural numbers, and all the rest is stuff that people are making up,” right. And he really tried to make life hard for Cantor. I mean, he at one point blocked him from getting a job, a position, and another time he kept him from publishing some of his work. It was not fun. Cantor really had some tough times in his life. He was not always fully appreciated during his life. But by the end, you know, we now see that you know Cantor is pretty universally recognized as an amazing genius.

Another German mathematician says that, “From the paradise that Cantor created for us, no one shall be able to expel us.” So in the end, he is definitely recognized as having done some really amazing things.

And so the last point I want to make sort of wraps things up full-circle by sort of linking the infinite back to the finite. As Rabbi Zeidman mentioned, I did release last year an album of original non-liturgical Jewish songs called Sparks, but my first recorded Jewish song was like 13 years before that, on this album by the group Sababa. And the song draws upon a piece of Torah that includes a kind of a paradoxical verse: “And they shall make me a sanctuary and I will dwell in their midst.” Wait, how is that possible that the infinite God is going to somehow be able to fit in our finite space to be with us? You know, like, how is that going to work?

So it turns out there is a Midrash, which is another form of commentary, that is a pretty fun one (Exodus Rabbah 34:1). “When the Holy One said to Moses, ‘Make me a tabernacle,’ Moses was dumbfounded, and said ‘The glory of the Holy One fills the upper worlds and the lower (i.e. Isaiah 6:3), and He said to make Him a tabernacle?’” How is that gonna work, right? “And the holy one replied to Moses, ‘Well, I don’t see things the way you do. I will come down and contract my presence within a space of one cubit by one cubit.” And by the way, a cubit is the distance from your elbow to the tip of your longest finger, so it’s going to vary from person to person, but it’s roughly 18 inches plus or minus whatever, right. And by the way, notice that people who have studied Jewish creation stories in Kabbalah – you know, the Big Bang and everything – there’s this term tzimtzum, where God contracts his presence to allow space for the world to exist.

But what’s fascinating is in that context, God is contracting Godself – God is removing Godself from the world to make room for us, whereas here in this quote, God is contracting Godself to be with us. So it’s a really profound concept in religion. But again, we’re going to try to make sense of it with math. Because again, the theme here is that you know when ideas and religion just seem so “out there” in terms of how abstract they are, or how if they seem exaggerated, if there’s a way to understand them with some of our secular tools, why not? Because it’s a way for religious ideas not to seem so strange. And vice-versa, maybe ideas in religion can help us understand math more comfortably too, right.

So Cantor showed that a line or a square or cube that has finite dimensions actually has the same number of points as that same object with larger or even infinite dimensions. And so here’s an example with a circle. So think of this as we’re going back to this one-to-one correspondence idea, but now it’s more geometrical instead of algebraic. So you’ve got an inner circle and an outer circle, and of course you might say “Well, the outer circle has a larger circumference,” right. Well yeah, it does. And yet somehow the two circles have the same number of points on their respective boundaries, how is that true? Well, imagine this sort of radius arm that’s going to sort of rotate around the circle. For every point A on the outer circle it’s passing through a point b on the inner circle. And so every point on the inner circle and outer circle gets hit, there’s no point left out as that radius arm sweeps around. Therefore, no matter how much how big, how much you can make that outer circle even bigger and just keep extending the radius arm to keep up with it, it’s still true. No matter how big that second circle is, it’s going to have the same number of points on the circumference, on the boundary.

Cantor also showed a result that’s even more mind-blowing, because it crosses dimensions. Cantor showed that a one-unit line segment, like from 0 to 1, that’s one dimensional, actually has the same number of points on it as the region inside a 1×1 square, which is a two-dimensional region. And I know this is too quick to fully contemplate it, but it’ll be recorded, you can pause it right now, right here, if you’re listening to it in the future and contemplate it more. But that’s basically how that can work.

And here’s another thing where you can show that a semi-circle, which clearly has a finite length, has the same number of points as an infinite-length horizontal line, because if you draw these sort of slanted lines from the center of the semicircle ,you can see that every point on the semicircle, with the possible exceptions of the two boundary endpoints – and there’s technical ways to account for that, so don’t worry about that – but all the other points, for sure, any point on the semicircle, takes you, if you follow that diagonal line, to a point on the infinite-length horizontal line, and vice versa.

And so again, this is how something can be infinite and finite at the same time, how the infinite and the finite can coexist, how God can be transcendent and beyond us, and yet be immanent at the same time. It’s not quite the contradiction that – it’s still a magical concept don’t get me wrong. I don’t want to be like a magician, who once you show people the trick, it’s like “Oh, that’s not a big deal anymore.” No, it’s still a big deal, it’s still beautiful, but it’s cool to have more understanding of it.

And if these last comments about dimension intrigue you, we will have more attention on dimension – I have to mention! – with our next speaker in this series, Dr. Kim Jongerius, whose talk will be titled “Does God Live in Another Dimension? (How what we Assume Shapes What We Conclude)”. You can see it’s a different day of the week, it’s going to be on Sunday, we’re mixing it up here. And you obviously know how to sign up, because you were able to get here, but if you’re watching the recording, this is the website you’ll go to to sign up. Each talk requires its own individual sign-in.

This is the big picture of all six speakers, just to let you see the big picture of it all. And you can see we have speakers that bring in a Jewish background, a Christian background speakers, that talk about Muslim mathematics, so we have a lot of diversity here. We’ve got males and females, we have people in the United States and beyond the United States. So it’s a really terrific series that we’ve been able to put together. And we’re so appreciative of the people who come together to make this happen, and our sponsors and our funders. So thank you for sharing my affinity for infinity. See, I could have done a triple-run title, I could have done could have “done infinity for divinity and infinity,” but I thought that was too much. And if you are curious about some of the math and Judaism papers I have written, they all happen to be open access, so by going to this website, larrylesser.com/judaism, you will find links to those papers if you’d like to read more.