It has been three years. What is there left to speak about for the yahrzeit of a person like my brother Eli?
There is the fact that almost all of us know somebody with a mental illness; that Judaism has always urged compassion, understanding, inclusion, and humane treatment. This is a topic which is most vital to talk about - but I've spoken about all of this before.
And then - there is the idea of turning to something my brother loved. So that we may find common ground not only with one another as fellow-travellers, but with him as well, though he is no longer here, and was a pretty strange character even when he was.
There was nothing my brother loved more than math. I love math, too, but not in the same way. If math is a language - which, of course, it is - then he was a native speaker, while I am very much an outsider who enjoys the music of it tripping off the tongue.
There are so many ways that math intersects with Judaism that actually the topic seems almost purpose-built for a dvar Torah or shiur of some sort – and, in fact, lots of people have written very eloquently about the topic over the years. We are not the first to notice that, in general, many Jews love math kind of the same way my brother loved math. As a subject to belabour over not because we have to, but because we can.
In all our lives, whether we love it or not, math intersects with Judaism at least once a year. When? (not rhetorical q)
At the Pesach seder, we come to an interesting bit in the middle where we stop and “do math.” Ten makkos? Not quite. How about fifty? (Rabbi Yosi ha Glili) How about two hundred? (Rabbi Eliezer) How about two hundred and fifty? (Rabbi Akiva)
Why do we sit and obsess over these numbers? Indeed, this is not the only place where the great rabbis of the Gemara debated a mathematical concept. And perhaps this was because they knew that numbers are infinite; so is Hashem. We cannot really think about “seeing” Hashem: He has no body, no face, no actual hands. So numbers, which go on forever, are probably a very good way to think about Hashem.
The Haggadah definitely obsesses over numbers, giving us so many, from its beginning, with the Four Questions, to its ending with Echad Mi Yodeya.
The first mitzvah given to the Jewish people had to do with numbers – sorting out our days, weeks, and years according to nice, tidy Jewish months instead of just letting them all pile up in a heap. And how many times does the Torah tell us that Hashem counted bnei Yisrael – that when something is precious, you count it fifteen ways to Sunday. What is counting, if not turning something into numbers?
But what I really want to talk about is pi. Who doesn’t love pi(e)?
One of the earliest references to math in Jewish tradition comes from sefer Melachim Alef, Perek 7, passuk 23:
כג וַיַּעַשׂ אֶת-הַיָּם, מוּצָק: עֶשֶׂר בָּאַמָּה מִשְּׂפָתוֹ עַד-שְׂפָתוֹ עָגֹל סָבִיב, וְחָמֵשׁ בָּאַמָּה קוֹמָתוֹ, וקוה (וְקָו) שְׁלֹשִׁים בָּאַמָּה, יָסֹב אֹתוֹ סָבִיב.
“And he made the molten sea of ten cubits from brim to brim, round in compass, and the height thereof was five cubits; and a line of thirty cubits did compass it round about.”
What are we talking about here? A circle. And what is this passuk coming to tell us?
Going back to high school math, “From brim to brim” is the diameter, the distance from one side to another side. And as hopefully some of us remember, if you know the diameter of a circle, you can also figure out its area or perimeter as long as you have a special number at your disposal: pi.
Today, we know pi to be an infinitely repeating sequence of digits starting with 3.14159. For most of us, that’s long enough, but for those who like a little more specificity, pi has now been mapped down to its two-quadrillionth digit – that’s a two with 15 zeroes following it. And people continue to be fascinated by it, not only as an excuse to celebrate “Pi Day” on March 14th with the dessert of the same name, which they do.
There are sites where you can search pi for your phone number – I found both of mine, but only if you leave off the area code. My brother Eli’s birthday, May 16, 1971, or 051671, first occurs at position 1,416,644. If I do it the Hebrew way, 160571, it comes up a little sooner: position 562,820.
Anyway, everyone loves pi. But what most people today don’t realize is that pi is and has been a gradual discovery over the entire course of human civilization. The Greeks got to name the thing because pi is the first letter of their word for circumference, “peripheria,” and their philosopher Euclid had some good ideas about ratios and getting closer to understanding its actual value.
For the Greeks, math was all about ratios – they didn’t have decimals, since those are a pretty new invention in the scheme of things – so pi, for them, was all about figuring out ratios, or fractions. For instance 22 over 7 (22/7) comes pretty close to being pi, though as we know today, it isn’t pi.
Sadly for the Greeks, we do know today that pi can’t be expressed as a fraction – one number on top of another number. What do we call a number that can’t be shown as a fraction, because it goes on forever? In modern math terms, that’s called an irrational number.
No less a figure than the Rambam also emphasizes the irrationality of pi. He wrote “You need to know that the ratio of the circle’s diameter to its circumference is not known and it is never possible to express it precisely. This is not due to a lack in our knowledge, as the sect called Gahalia [the ignorants] thinks; but it is in its nature that it is unknown, and there is no way [to know it], but it is known approximately.” (Perush Hamishna Eruvin I 5 source)
Since pi is a Greek letter, invented by ancient Greeks, bnei Yisrael in sefer Melachim had never heard of it, so they had a problem with measuring the circumference of the circle on the water tank they were building. Ten cubits in diameter – times 3, which was also one ancient Babylonian estimate for the magic number we now know as pi. It’s not a very good estimate, but for general building purposes, it seems to have done the trick.
The concept of pi appears twice in the Talmud, though not by its Greek name, in Eruvin 76a and Sukkah 7b, in a discussion of how big a square sukkah should be to have the same area as a round one. This was a popular obsession of ancient mathematicians; in 1882, it was proven impossible to solve.
In Eruvin, the Mishna provides the general rule for figuring it out: “Every circle whose circumference is three handbreadths, is one handbreadth wide.” As we’ve already seen, this isn’t true. As was well-known at the time of the Gemara by anyone who had even dipped his toes in the water of math and science. Yet the rabbis dig in their heels over this. And Rabbi Yochanan brings as proof the passuk from sefer Melachim – ten cubits diameter, thirty cubits circumference – even though they knew better.
Now when I say anyone who had even “dipped his toes in the water of math and science” knew pi better by this era, I mean it. So we actually have a problem here. Rabbi Yochanan, for instance, lived around the years 180-279 CE, and was well-versed in math and other contemporary science and philosophy. He had most certainly heard of the Greeks. He absolutely knew that there were better estimates for pi.
One proof for this which is obscure but interesting lies in the krei/ktiv (written / read) form of the passuk from Melachim alef itself. The word for “line of thirty cubits” is read as קו / “kav”, but is written as קוה / “kava.” The gematria of kav is 106, while the ה in kava adds 5 to make it 111. So if you throw in the three, because we know they knew about three, then stack these values up, you get 333/106. (source; Tzaban & Garber, 1998) Since you’re probably wondering, that’s 3.1415094, which has four decimal places identical with today’s estimates of pi. It is a very good guess indeed.
Did the rabbis of the Gemara know this, or is it something that has been read in by contemporary writers? It’s cute, but it probably was unknown before the 20th century. Nonetheless, medieval commentators used the estimate “a little less than 3 1/7”, which is pretty close – 3.1429 – demonstrating that though they lived a bit earlier, the rabbis of the Gemara almost certainly knew that pi was definitely not three.
So, coming back to the discussion in Eruvin, why is Rabbi Yochanan bringing a “proof” that pi is three?
A lot of people these days feel very modern because they run around with the conceit that ours is the first generation to attack Hashem with science. Today, we have scientific proofs of all sorts of things that were once a matter of faith: genetics, astronomy, the secrets of life and death. Today, they argue, we don’t need the Torah because we have something bigger, better, stronger.
Rabbi Yochanan knew this, too. Not only is this nothing new, this idea that science can be used to defeat Hashem, it goes all the way back to Migdal Bavel – the tower of Babel.
The pinnacle of scientific achievement in those post-flood days was the process of making bricks. This advanced technology gave ordinary humans the power to literally create stones for building with; think about how remarkable that would have been at the time. Hashem gave people the brains to figure this out… and people, in turn, turned around and figured out a way to defeat Hashem, by uniting, building a great tower out of bricks, and making a name for themselves.
So we have to assume that the rabbis of the Gemara were very familiar with heresy in all its forms. Which means that what Rabbi Yochanan is telling us here is not as simple as it looks on its face. What he is telling us is that even though math tells us that pi is not three, the Tanach tells us it is, and we posken (rule / decide) like the Tanach even though we are smart people, we are up on the latest scientific journals of the year 200 CE, we know there are better estimates.
The point Rabbi Yochanan is making is one the Rambam also made later on. For halachic purposes, we will stick with the number three, thank you very much, even if it is not the most scientific, the most accurate, the most modern.
The Torah, though it may help foster in us a love of math and science and logic, is not here to teach us about math or science or history or geography or anything else that’s grounded in the finite. It teaches us about another realm entirely. A realm we might well call irrational.
Coming back to the idea of the irrational number... the word “irrational” has taken on a pretty negative connotation in what we like to think of as our modern world of rationality. We like things to make sense, and irrational things, it seems, don’t make sense.
But actually, irrational numbers get their name in a simpler way. For the Greeks, most numbers and mathematical concepts were discovered through ratios. A ratio is a fraction, one number on top of another number. So when they came across a mathematical idea that couldn’t be expressed as a fraction, or ratio, by stacking one number on top of another, then they said it had no ratio: it was ir-ratio-nal. You can see the word ratio in there.
So the irrational numbers are irrational because you can’t find them with a fraction, and not because they make no sense, which is the modern meaning.
You know what else is irrational? Hashem. Judaism. People with mental illness.
Interestingly, in the 18th century, pi – already irrational – joined another newly-discovered class of numbers: the transcendental numbers. These are numbers that you can’t get at with a polynomial equation – a statement using x’s and y’s in the language of algebra.
Is it a coincidence that the words mathematicians choose to talk about math are the same words we use to talk about religion, and about Hashem?
Also, these definitions seem more about what the numbers are not – countable, findable with a ratio, derivable from a polynomial – than what they are. Because what numbers are is very tough to put your finger on. This is at once a beguilingly simple question and a very difficult one.
What is three? (not a rhetorical question)
The issue philosophers have had is whether three is simply the number you get after two, or if it embodies a concept of “threeness” that exists outside of the numeral, outside of the word itself. When you look at a group of three objects, chances are, you actually do think “three” without having to count them. While most of us can’t do this for very big numbers, there are some autistic people who can look at a group of, say, seventy-three objects, and instantly say that there are “seventy-three,” without counting.
This is very similar to questions we ask all the time about Hashem. Do we know Hashem only in our perceptions, or does Hashem exist outside of everything we see and hear and understand? In religious ethics, as well, there is a question of whether certain actions are objectively good (we call these mitzvot), or whether they are only good because Hashem chose them.
A lot of people these days like to say religion is irrational – it doesn’t make any sense. You can’t derive it in a simple way by adding two plus two and arriving at religious faith. Sure, there are people who try, with Torah codes and things, and what they do looks a lot like math, but I’m going to suggest that it isn’t.
There is no logical proof of religion, though as Lawrence Kelemen has put it in the title of one of his books, there is certainly “permission to believe” based on numerous historical facts and details in the Torah. You don’t have to think of yourself as completely nuts for believing in the Torah, even if much of the world does.
Hashem, too, defies definition according to physical parameters. Remember that a rational number is one you can find by stacking up two numbers – something as simple, once you’ve learned long division, as adding two plus two. Whereas an irrational number is infinite – it goes on forever without repeating and without any predictability.
This last point is very important when we’re thinking about Hashem – which we are. You cannot say, “we know the first two-quadrillion digits, therefore, we can predict what the two-quadrillion and first digit will be.” All you can say is, “we know those digits and this in itself is a tremendous accomplishment.”
This in itself is glorious. This in itself has taken us five thousand years to figure out and we know – because it’s infinite – that we’ll never be finished figuring it out.
I’ve often told my children (who yawn when they hear it) that mathematics is the closest we can get to understanding the mind of Hashem. Not just because of numbers, which go on forever, but because of the idea, as in the study of Torah, that if you turn it and turn it, you will never discover all that is within it.
The rabbis of the Gemara did their very best, perhaps most notably through the Haggadah, to pass this message along to us as well. They’re telling us that saying the numbers over and over does matter. That putting important ideas into numbers isn’t just another way of saying something – it’s another way of thinking. It is a way, perhaps, that is closer to the way Hashem views the world.
If the mind of Hashem is difficult to penetrate, the mind of a person with schizophrenia can be even more difficult; even more irrational for its refusal to be reduced to consistency, to logic, to a worldview based not on circumstances as they are but on a broken perception that the world itself is irreconcilably damaged and broken. When the world is not rational, then the irrational can become a refuge.
I like to believe that math was a refuge for my brother Eli, even at the end, when there was very little comfort in the world. Seeing him playing the piano; seeing him scribbling numbers on the back of a Shreddies box, it was sadly not – as Oliver Sacks has documented in many patients playing music – a return to normalcy, but a return, perhaps, to fluency; to a language which kept on making sense when the rest of society had gone haywire.