Mathematics is about proof — ideas such as the Pythagorean Theorem or that pi is irrational are foundational to mathematics, and ultimately, science. And for most modern Jews, we don’t approach the Bible or religion in the same way. So where can we find value in the relationship between eternal and universal mathematical proofs and particular ethical questions we face today? How can the wisdom of the Bible and the Talmud help us utilize mathematical concepts to make our world better? And how can mathematics and Judaism help us gain knowledge from experts and become better independent thinkers?
Dr. Olga Kosheleva and Dr. Vladik Kreinovich are based in El Paso, and are members of Temple Mount Sinai. They are originally from Russia and emigrated in 1989, and at present they both teach at the University of Texas at El Paso. Olga teaches future teachers and co-chairs the Teacher Education Department, while Vladik teaches computer science. You can see full Dr. Kosholeva’s biography here and Dr. Kreinovich’s here.
(This post is part of Sinai and Synapses’ project Scientists in Synagogues, a grass-roots program to offer Jews opportunities to explore the most interesting and pressing questions surrounding Judaism and science. “Mathematics, Computing, Ethics, and Religion: From Naïve ‘Contradictions’ to Deep Agreement,” a talk given at Temple Mount Sinai on August 19, 2021, was the fourth in the series Higher Meanings: Connecting Religion and Mathematics. The next event in this series, “Gersonides and the Limits of Knowledge,” featuring Dr. Snezana Lawrence, will be on October 17, 2021 at 6:30 pm MT / 8:30 pm ET)Read Transcript
Dr. Olga Kosheleva and Dr Vladek Kreinovich are based here in El Paso, members of Temple Mount Sinai. They are originally from Russia and emigrated in 1989. They came to El Paso and joined Temple of Mount Sinai in 1990. At present they both teach at the University of Texas at El Paso. Olga teaches future teachers and co-chairs the Teacher Education Department, while Vladek teaches computer science. That is the extent of their bios that they permitted me to share with you. Suffice it to say that the details about their professional careers are vast, the number of awards that they both have individually, and the number of publications that they’ve done together as well are – it’s a very long list. And so we’re grateful to you, Olga, to you, Vladek, Dr. Kreinovich and Dr. Olga Kosheleva, thank you so much for your time this evening, and we will turn it over to you.
Vladek Kreinovich: Thank you very much. Thank you for the great introduction. So, let me start sharing. Okay, so this is the title of the talk. The title is “Mathematics, Computing, Ethics, and Religion: From Naive Contradictions to Deep Agreement.” So we will say outrageous things in the beginning, don’t get mad, don’t kind of call the police on us, because we will kind of calm down and get to the point later. But we will start with contradictions, yes.
So, first is the Naive View. And in order to explain where this Naive View comes from, let me just tell, very briefly, the history [of it]. Several years ago, when we were in Israel – Israel, in general, is not exactly Germany, it’s not a place where everything goes by railroad schedule. But there is one place which does, which is the Western Wall Tunnel. If you miss one minute, they will not let you in, you have to wait for the next turn.
And that’s exactly what happened to our group. So we were late, and so what our tour guide did is he brought us to the museum. And at that museum, we were shown a model of the Temple, and our tour guide was kind of very nervous, because he was – this museum as it turned out – is run by – well, unusual people, so to say, who believe that the Temple should be restored exactly as it is. And as our tour guide explained to us, one reason that they’re not very much insisting on it is that while the Bible seems to be describing exactly, in very precise mathematical details, the lengths and cubits of this, the width and [circumference] of that, that these details just don’t match. There is no way to fit this temple on this particular place, with all the measurements that are there.
And he explained it very simply, the Bible was written by poets – by prophets, not by engineers. Numbers don’t match. This is not a unique case of [numerical inaccuracies] in the Bible. There are many such inconsistencies, even in terms of how many years people lived. And actually, Spinoza, famous philosopher, who was kind of, first, a big specialist in the Bible, in Talmud and everything, he noticed a lot of this inconsistencies and he publicized them, and this was one of the reasons why he was excommunicated, because “How can there be inconsistencies in the Word of God?”
So, there is an example – Classical example. If you look at the Bible, at some point, the Altar, on which the sacrifices were made – the circumference of this Altar is exactly three times its diameter. Not π, 3.14. No, exactly 3. And it was like – people who believed in that, literally, in the 19th century, in the state of Indiana, legislature seriously considered declaring π = 3 , because this is what God says in the Bible.
So, on the other hand, according to our tour guide, which we confirmed later, if you look at the books by Josephus Flavius – and here is the photo of the statue of Josephus Flavius – it’s the other way around. When he talks politics, he talks through his teeth, it doesn’t match what others say. But when she talks about, like, “They marched for 10 miles (actually “mile” was the original Roman unit of length – it meant 1,000 steps) – and made a camp, you can dig there and find the remainder of these camps.” And that’s exactly what Israeli archaeologists have been doing. So it looks like math and Bibles do not match at all. So, how can a mathematician take the Bible seriously?
If we take the Bible’s numbers seriously, then not only do they not match between themselves, they don’t match any science. For example, “The world was created in six days.” When? 6,000 years ago. Dinosaurs, old rocks – nothing exists. Only 6,000 and that’s it. So, “Gays should be put to death.” “When you conquer the enemy you should kill their women and children.” And people, unfortunately, in history took these instructions very literally. The Spanish Inquisition was doing that, religious terrorist[s were] doing that. And if you think Islam is different, a lot of stories of Quran are basically the same stories that we have in the Bible. So how can a person who is ethical believe in all this?
Well, let’s have another example. Many people believe that the atomic bomb is evil. Well, it was used for killing people, as all the weapons are. So who created the bomb – a physicist? So what does that mean? That physics is evil. How did they create it? Well, they used some mathematics, they used some computation. So, conclusion? Mathematics is evil, computing is evil. Why there are financial crises in the world – well, one reason is people relied on simplified mathematical models. There is a whole book – you see, the skull and bones and everything, – [called] Weapons of Mass Destruction, describing how belief in mathematical models leads to economic crisis. So, what shall we do? Destroy all the computers? Stop teaching mathematics? (Well, some kids will love that part). Close all the churches and synagogues? Well, it kind of sounds a little bit too much.
So, after we have these outrageous conclusions, let’s start digging deeper. Mathematics, computing, ethics and religion — let us dig deeper. So, is it true that the Bible is not so good about numbers? Absolutely. How is it related to mathematics? Well, there is a common misconception that mathematics is the science of numbers – it’s not. Mathematicians are not very good at computing. People who are good in computing – well in the old days they could earn the living by going on stage and multiplying four- digit numbers real fast without using a piece of paper – not mathematicians. I remember at one of the seminars, when one of the speakers got confused with simple computations. Our professor kind of noticed that in general, that’s a sign of a true mathematician – that this person is thinking very abstractly, he is not really able to multiply or add correctly.
What mathematics is about — it’s about proofs. When you compute, it’s not mathematics. Mathematics is when you prove. And here is a classical example. There is a known thing called Pythagoras’ theory, that if you have a right triangle, the square of the hypotenuse is the sum of the squares of the sides. This was known by Babylonians, it was known by Egyptians, and Babylonians knew it, and the Egyptians knew it, from practice. For example, if you have 3 and 4 for two legs of the right triangle, then 3² + 4² = 5². And this is exactly how they got a right angle in their constructions. They took three rods, of length 3, 4 and 5, put them together, and that became a precise right angle. And again, this is – I cannot claim that I understand how is it related to this, but people who read Babylonian scripts, they say that this is exactly what is described here.
But the difference is Pythagoras – why is it called “Pythagoras’ theorem”? Because he was proving it. Why do we need to prove it? Well, what the Egyptians and Babylonians also knew is that π is approximately equal to 3 and 1/7, but it turned out to be only approximately true. But why is it important? “In the first case it’s exact, the second is approximate. Big deal! In practice, we don’t have that”. Well, if you’re building a pyramid, you may use this approximate value of π. If the pyramid is a little bit like “1 meter to the left, 1 meter to the right,” no big deal. But if you are doing something more precise – for example, you’re predicting solar eclipses centuries ahead – and use approximate formulas, errors accumulate. That was, by the way, one of the reasons why Einstein got interested in, modifying, changing, Newton’s physics, because there was a discrepancy between the orbit of Mercury.
And prediction was important, because people were scared of eclipses. Supposedly, the sun goes out, it’s scary. There’s a legend – well, again, it’s a legend about the Aztecs, how they started sacrificing people. That supposedly there was an eclipse, and so the sun went out. And so they were scared. The sun is the source of life and suddenly it’s out. So what did they know? Well, you need to sacrifice something. So they sacrificed a chicken and the sun continued staying. Then they sacrificed a cow and the sun continued staying. And then the bravest member of the tribe says, “God needs supreme sacrifice,” and so he killed himself, and the sun came out. And that explains why they had this tradition that every year they would have a football competition, a championship, and then the champion, the hero, would be officially sacrificed to the gods.
The history of this Pythagoras theorem is a good example of how science works. First, you have some informal ideas. They’re very wrong. Then they get confirmed experimentally, like, “π is equal to approximately three. Okay, well, 3 and 1/7 is better,” and so on. And then finally, at some point, somebody derives them from first principles, and then it becomes either laws of nature if it is physics, or theorems if it is mathematics.
And so, in general, that’s how many scientific discoveries were made. And that’s almost kind of divine inspiration. Like there’s a famous Indian mathematician, Ramanujan, who really believed that he gets inspiration from the gods. And it’s not just in mathematics, many ideas of modern physics actually are based on the ideas that were proposed many, many centuries ago.
For, like – very simple example, atoms. Atoms were experimentally discovered only in the late 19th century and confirmed definitively in the 20th century. But they were known in ancient Greece. Their Relativity Principle, for which Einstein is famous, was actually formulated first not by Einstein, but many, many centuries before that, by Galileo. Galileo had a very simple observation that if you are in a ship’s cabin, and you don’t look outside, the water is very calm, you don’t feel the motion. You don’t know whether you’re moving or not.
The steam engine was invented in the 19th century, but in reality it was invented already in Ancient Egypt, in Alexandria. Here is the picture of this ancient steam engine. Well, why don’t we know that? Well, because it was not used to do anything useful. There was plenty of cheap labor at that time. What it was used for is to open the doors of the temple.
That was an interesting period, because there were many gods, and one problem, when you have many gods, is there is a big competition between different temples. So, you’ll bring some money, you’ll bring some sacrifices. Do you sacrifice to Zeus? Do you sacrifice to Athena, do you sacrifice to – I don’t know, Poseidon, whatever it is. And so, if in one temple, you go in there and then suddenly, magically, the doors open, and nobody’s opening them by themselves, and a lot of – a cloud of steam comes out. What’s your natural reaction? Your natural reaction is, take out your wallet and give some money to that particular temple. That was the only reason for using this steam engine.
How is it related to science? Well again, many ideas can be found in myths and in holy books, not just in the Bible. For example, according to modern physics, our universe had a beginning, which scientists call a Big Bang. And these ideas can be traced to the Bible, where of course the world was created. And there are many other holy books which have similar creation stories. Moreover, in the beginning, what did God say? God said, “Let there be light.” And indeed, that’s exactly what physicists noticed. That at the first, we had the period where the universe was filled with light.
And many discoveries of quantum physics similarly can be traced to the ancient books by the Hindu religion. So, some writers conclude, and there are many of them, some journalists – “Well, there is nothing new in modern science. It already was known to the ancients.” And actually, even some Jewish, kind of Orthodox people, believe that. But this is a naive oversimplification. It’s not exactly the same. There is a big difference. On the one hand, Galileo said, “Yes, when we’re in the ship’s cabin in good weather. We don’t feel any emotion.” But what relativity theory does is not just staying at this point. It has some formulas that enables us to predict exactly how to compute locations based, for example, on GPS signals. As you probably know, with GPS signals, if you don’t take into account relativistic effects, with movements of satellites and everything, you miss by kilometers.
Same thing – the Bible says the world was full of light, and modern physics says exactly the same thing. Fine. But the difference is, it’s modern cosmology. Modern physics doesn’t just predict what is full of light, it predicts exactly what we will observe with a radio telescope, so-called 3K radiation, which of course you cannot predict based only on the Biblical stories.
Some scientists make another mistake. They think, “Well, science is everything. Everything else, all these informal things, is unnecessary. Dismiss all these informal arguments altogether.” This makes a nice history of science. “Nobody knew anything, and suddenly comes this genius, Einstein, genius Newton, whatever it is, and everything becomes known.”
Well, it’s a simplification, and even the great scientists realize that this is a simplification. For example, Einstein, who pioneered the idea that the geometry of the world, of spacetime, is not Euclidean – he himself said several times that he got more from Dostoyevsky, the Russian writer who had a lot of informal ideas about non-Euclidean things. (this is Dostoyevsky — he looks a little crazy, and he was a little crazy, but he was a great writer) – than from Carl Friedrich Gauss, who looks very sane, who was a great mathematician, and who actually discovered, [was] one of the discoverers, one of the promoters, of non-Euclidean geometry.
So, Biblical and other myths inspired many scientific theories. Okay, so maybe we should relocate the Bible and other holy books to history. Great. “In the past, when there was no modern physics, no modern mathematics, people had some primitive ideas. That’s great. Wonderful. Let’s applaud these guys. But nowadays, why do we need them?”
Well, it turns out that we still need these original myths. There are still ideas that can be, and need to be, formalized. And here is an example. An example is: what do you do in the case of bankruptcy? Bankruptcy is not a modern problem, of course it existed in ancient times. And in the ancient times, there was always the question: what is bankruptcy? If you owe a lot of money, and you don’t have enough to pay everybody, how do you divide [it]? Different cultures have different ideas, and the Talmud, of course, touches this because Talmud touches everything.
The problem is, unlike more than textbooks, the Talmud, doesn’t give formulas, it only gives examples.
And in other cases – it’s usually in simpler cases – it’s very clear. For example, if you have to contribute on a 1/10, and they give some examples, and it’s clear that what they mean is to give 1/10, or 1/60, or whatever it is. But unfortunately, in this particular example of a bankruptcy, this idea was completely lost. And here I’m reproducing, in a simplified way, what exactly the Talmud says. There are two examples – actually five. In one of them, it shows how much – so suppose we have $100, whatever units – of course it was not dollars at time, shekels probably. So, if the first person was owed 100 shekels, second 200, third 300 – If all you have is 100, so that’s, what, 100 + 200 + 300 = 600. Totally, it’s 600. If you only have 100, how do you divide them?
According to the Talmud, you divide them equally. 33 and 1/3 to each person. If you have 200, then, the first person who was owed 100 gets 50, the other two get 75. If it’s 30/100, everybody gets half of what they owe. And then there was another case, and it’s kind of weird. This is in two places in the Babylonian Talmud, and it was a big mystery until literally like 10 or 20 years ago, when there was a Nobel Prize winner, Robert Aumann – and here’s is this unusual picture because normally he wears a kippah, he is very Orthodox, but this is one of the few photos where you can see his whole head without a kippah. So he gave a formula that explains all Talmudic examples, and moreover, not only he gave a formula, he showed that one of the dozen game theoretical concepts leads exactly to this Talmudic formula. And with Aumann, why we know that, is because with one of our colleagues from California, we kind of also added a little bit – we showed that the Talmudic solution is in some sense better than the other eleven.
So this makes sense. So the ancient rabbis had intuition, and modern game theory, which is used by Aumann, enables us to transform this intuition in precise terms. And there are many such examples of religion-inspired, holy book-inspired, research results, that make us believe that, hopefully, there are many other examples still to be discovered in the teachings of the Holy Books. In other words, from this viewpoint, religion and science supplement each other. Religion provides some intuitive ideas and science helps how to transform them into something very precise.
But, of course, people don’t go to the temple to describe how the world is built. When people go to the temple, usually, they’re more interested in ethical aspects. So, what about ethics? How is ethics related to mathematics? Again, at first glance, it doesn’t[isn’t]. And if you look at the title of that book, it’s awful. Mathematics is very unethical on it.But our Russian teacher – and here Olga will say a few words – taught us that teaching mathematics – here is the picture of Aleksandr. We again selected a picture where he is not looking very official. This is how we know him.
Olga Kosheleva: He was a great person, and he was always teaching by his example, and he liked contradictions, he liked to surprise people. And I think he never prepared ahead of time for lecture.
Vladek Kreinovich: Well, we don’t follow that account. (laughs)
Olga Kosheleva: Instead he would come in and talk very interestingly.
Vladek Kreinovich: Yeah, but his idea was the main important thing: how to make people ethical is to teach them mathematics. It sounds weird, but he gave an argument, because what is it? Mathematics is not about adding numbers; it is not about solving quadratic equations. It’s about proving. The mathematician knows that if you have a statement, you need to prove it. You cannot just say, “famous researcher says, and therefore it’s true.” “Communist Party says, therefore it’s true.” “The rabbi says” — sorry rabbi, it’s not, we still have to believe in it. “The New York Times wrote it. Of course it’s absolute truth. How can the New York Times say something wrong?”
In mathematics, we also know it’s not just our prejudice, we know that famous mathematicians sometimes are wrong. There was a classical case: David Hilbert, the most famous mathematician. He was well-known, so well-known, and so kind of praised as the top mathematician of the end of the 19th century, that in 1900 he was tasked with formulating problems that the mathematicians of the 19th century should give as challenges to mathematicians of 20th century. And not only he formulated open questions, he tried to guess what would be the answer. And of course, being a great mathematician, in about 20 of these 23 problems, he guessed correctly, but in some of them, he goofed completely. I’ll just mention it, but one of his guesses was that you cannot describe every function as a composition of functions of two variables. It turned out to be wrong. And in some sense, all the modern neural networks, deep learning, and all this success, is based exactly on this fact, that he was wrong in this prediction.
But let’s go back to ethics. So, what mathematics teaches us is that you need to think on your own. You need to be convinced on your own. The proof is only a proof that you are convinced. But you may say, “Wait a second. Religion is the opposite, right? We just follow without thinking.” Well, not really the religion that we believe, because, of course, there is what is called fundamentalism. Fundamentalists don’t think, they read the text, and they follow it. Actually, in Russia, we started [reading] Lenin’s papers and the word Talmud just was a curse, was a bad word. Why? Because there are many guys in his time who studied Talmud, not because they wanted to learn better — they just memorized everything, and instead of arguing, they would say “Well, the Talmud says this, so this is what we will follow. I don’t know why – God knows why. I don’t know what will happen, but since the gods said that, it’s right there in the book.” And of course, other Talmudists cited other parts from Talmud. And that’s exactly what Lenin was comparing people to.
So this form of fundamentalism, they say, “I don’t care about geology, I don’t know about dinosaurs, it’s all a fake. Here is the book, the holy book says, ‘The world was created in six days.’ That’s it.” And if you believe otherwise, well, they don’t have the power to burn you at stake nowadays, but still they…
But this is not what serious religions teach. According to the Bible, why was Abraham selected by God? Not because he was obedient. Abraham was selected, why? According to the Bible, because he argued with God several times and succeeded. Why are we called Israel? Israel literally means “The one who fought with God,” because we fought with God, not because we obeyed God. This is the famous story. Well, again, you can interpret it differently, with an angel, with God, whatever it is, but clearly when you are fighting with some will, you are showing that you are not exactly obedient.
And Talmud, it’s not a list of prescriptions, it is basically a discussion club. For every point you have different rabbis who present absolutely different opinions, and not just present opinions, they present arguments. And they justify them. And the true goal of people who study Talmud seriously is not to memorize, although memorizing – although there’s nothing wrong with that. But what they do is they understand the argument, and they use them to make decisions. That’s the whole point.
This is a typical page from the Talmud, and the rabbi probably will understand better what is going on, but usually it’s, what? There is Mishnah, which is in the middle, which is kind of a comment. Then there is a comment on the comment, and so on and then there are some comments there. And overall, you can basically have whatever you want. From this viewpoint, both mathematics and religion teach basically the same thing. Although you may not expect it from mathematics, you may not expect it from religion. But that’s exactly what they teach: “defy authority.”
Here is the illustration about defying authority. It doesn’t mean anarchy. It doesn’t mean ignoring or burning the books, it doesn’t mean not listening to the elders, as was the popular saying in the 1960s, “Don’t trust anybody over 30.” That, of course, changed when those people who proclaimed that became over 30 themselves.
No. We need to listen attentively, we need to read the books – because yes, a lot of ideas that are there in the books, they are valuable ideas, and many ideas that we think are new can be found there. Yes, as we showed with examples of physics and mathematics and everything there. But after that, after you read those ideas, after you argue, then you have to make your own mind. And again, just like the rabbis, they couldn’t convince – not that they managed to [convince] each other. No, they stayed having different opinions. And it’s healthy and it’s reasonable, as long as you’re not imposing these different opinions, as long as you have reasonable arguments about your opinions, you are able to have your own arguments. Like for example, it’s naive to think that some of them were evil, some of them were good. Actually, according to the stories, all the rabbis were kind of, as the rabbi probably put this quite better, the celestial voice came when they were arguing and then said “They’re both words of the living God.” That’s exactly what we have to keep in mind.
So, what is the final conclusion? In our opinion, what does the history of the 20th century teach us? It teaches that evil happens when? When people follow authority without thinking, whether it’s Mein Kampf by Hitler, whether it is the Das Kapital by Karl Marx. Karl Marx had lots of good economic ideas, economists consider him very seriously. But if you just follow literally everything that is written there without thinking too much, that inevitably leads to evil. We all need to be independent thinkers, and that’s the only way, the only key, to an ethical society. And in this, mathematics and religions are in very good agreement. And moreover, deep study of mathematics, and deep study of religion, lead to exactly the same result, that we do become more independent thinkers, we do understand that you need to follow examples. That’s why the rabbi is not a person who kind of tells us what to do, the rabbi is a person who is helping us come up with our own decisions based on his deeper-than-ours knowledge of holy books, knowledge of religion, knowledge of this tradition. So, from this viewpoint, our conclusion is: we need to follow this great advice to be independent thinkers.
And we are very much thankful to Rabbi Zeidman and to Professor Larry Lesser for their invitation, their inspiration and their valuable advice. Again, it’s not like we came up with this idea and the rabbis and Larry are listening in for the first time. We showed them our draft, they gave us a lot of good advice, and I think, hopefully, that helped make it more understandable.
And those who really, really want to see technical details, we have several papers, we have a paper, a mathematical paper – so if you’re afraid of mathematics, don’t go there, but if you are interested, please do. There is a paper on our motivation for this bankruptcy solution, there are corresponding slides, and there are some other papers that we wrote on these topics.
Like, one of the examples is Balaam, the prophet who is well known from the Bible because it’s part of the story about the talking donkey. He had this famous blessing, Ma Tovu, “how great your tents are.” So in what sense are they great? He came from some kingdom, so this tents clearly could not be the richest. They were probably not the best made, because living in the desert doesn’t make you a great kind of – right.
So in what sense were they great? So what the rabbis concluded is: they were great because they provided privacy. If you stand in the opening of one tent, you see as little as possible about what’s happening in the other tents. And it turned out, this leads to an interesting geometric configuration, which kind of shows what is similar to – kind of helps understand where Solomon’s Seal comes from.
There are some papers related to other religions, for example, Mayan mathematics – they had these ideas of zero and infinity, which also looks like simple, naive interpretations or ideas, that eventually lead to some mathematics. Then there is Maimonides’ so-called double negation theology, where Maimonides claimed that you cannot say anything about God positively, you can only make negative statements. You cannot say “God is,” you can say “God is not.” “God is not finite,” “God is not understandable,” and so on. And there is another term Tertulian, the Christian guy. And there was also — we had a paper by Godel – again, one of the great mathematicians – who had a proof that God exists, which is kind of not exactly a mathematical proof, but still some arguments. So thank you very much and we will be glad to answer our questions.