On October 17, Temple Mount Sinai in El Paso, TX welcomed Dr. Snezana Lawrence, who gave a presentation on Gersonides, the history of mathematics, and the discipline’s connection to (or disconnection from) spirituality. Dr. Lawrence is a mathematical historian focusing on creativity, identity and engagement in the learning of mathematics. She is a Senior Lecturer at the Department of Mathematics and Design Engineering at Middlesex University London and is involved in various national and international initiatives to promote the use of the history of mathematics in mathematics education.
She is also the chair of the History and Pedagogy of Mathematics International Group, which is an affiliate of the International Mathematics Union, for 2020 through 2024, and is a Diversity Champion of the Institute of Mathematics and Its Applications in the UK. Dr. Lawrence was the co-editor with Mark McCartney of the book Mathematicians and Their Gods, and recently published a book titled A New Year’s Present from a Mathematician with Chapman and Hall.Read Transcript
Dr. Snezana Lawrence: I’m very grateful to show up for this occasion, and I’m very, very grateful to Rabbi Zeidman and to Professor Lesser for inviting me to do this talk. It has been very enjoyable for me doing the research. And I need to to mention two people, really, that I’m very grateful to: Shai Simonson from Stonehill College in the US, who gave me some of his translations, which were very, very helpful, and of course to Viktor Katz, who was one of the speakers in this series, who I’ve known for many, many years. And I don’t know whether Viktor remembers, but I think he came to one conference I organized in the UK, just when my daughter was getting married, and she was furious that the conference was happening the day before her wedding. And that all worked out at the end. (laughs) So Viktor, thank you for your fantastic contribution to the book on the history of combinatorics, which was also very helpful.
Okay, so I’m going to talk about Gersonides and the limits of knowledge, and his really fascinating work, which spans many areas of mathematics and philosophy and even music. So I promise to talk about his work and his Judaism, and that will be sort of not necessarily always obvious, so if you save your questions for the end, we can talk about it later on. But I think he is one of the mathematicians, theologians, that is very modern in his thinking, despite the fact that he was actually a medieval man. And we will see how his theology influenced his mathematics and vice-versa, and we will see his contribution to combinatorics and his work on harmonic numbers.
So I just want you to hear something before we get on. Okay, so that was a very short clip, that was the work of the composer, his contemporary and friend, who commissioned him to do some work on his own harmonic numbers. So it is very directly related to Gersonides.
So the overview of this talk is that there will be several topics I want to go through with you. First of all, probably the most important, really, for this series, generally speaking, why connect mathematics and religion, and how I’ve done it in my work –and then a little bit about Gersonides as a person and his work. We’re then going to look at the harmonic numbers in a little bit more detail, and we’re going to look at that question of the limits of knowledge according to his philosophy. And we’re going to look at some of the influences that are still very much alive stemming from his work.
So why connect mathematics and religion? Sometimes it is not obvious to people. For example, when I did this work with Mathematicians and Their Gods with my colleague Mark McCartney from Belfast in Northern Ireland. The book, although it received very good reviews, quite a few mathematicians didn’t really want to just look at it, because there is sometimes an attitude that religion has nothing to contribute to mathematics or modern mathematics – and I respectfully disagree with that viewpoint – and also there is a point where you can start looking at mathematicians from, for example, before the 1800s. And if you look at the fantastic website resource that is constantly updated at the University of Santander in Scotland, which is called MacTutor website — if you look through loads of biographies of mathematicians from before the 1800s, quite a lot of them are theologians as well as mathematicians. So there are questions that people are interested in which are within that intersection. So that book was dedicated to sort of various manifestations of those beliefs that mathematicians have, or will investigate and look at from various viewpoints.
The other book, a New Year’s present for my petitioner I did a couple of years ago, doesn’t draw a direct link between theology and mathematics, but it does draw the link between beliefs that mathematicians have, and also the methods by which they come to certain inventions or conclusions. And quite often these are related to their traditions and their beliefs, religious including. So there are different, obviously, stances to this question, but for me that is one of the important questions of mathematics, that connection with theology. And you can see here a little stamp – how Gersonides was celebrated in Israel with a stamp. [The artwork for the stamp was used for the header of this post].
So we’re going to look at a little bit more about this relationship between mathematics and theology. So the first thing is about the mathematical religious structures. The really central question I want to pose here is that this mystery of order, you know, which all mathematicians deal with, of rediscovery and discovering of order of mathematical objects, and of objects – really, of the universe. So there is a direct connection with metaphysics and physics. And so how does mathematics deal with that search for order? And where can we see instances of that in the work of Gersonides’ theology? So that is really the question of this talk that I will attempt to answer, either in my talk or later in the discussion.
And so, there are some hard questions I think we all agree on about the nature of life in the universe, the nature of that sort of the structure of the universe, and how things work out. And as I said, if you look at the history of mathematics from mainly before the 1800’s, you can find many examples of theologians and mathematicians there. And so we’re going to pay attention to one of them, to Gersonides, and look at how he dealt with with those questions. So, Gersonides: the man and his work. He was a Rabbi Levi Ben Gershon, or Gersonides or Magister Leo Hebraeus. Sometimes he’s also known as RaLBaG. (I see we have a translator of RaLBaG in in the audience — I will say hello). So he was born and lived most of his life — well, he lived all of his life, actually, in Provence and Languedoc region of France, in south of France. He wrote exclusively in Hebrew, but some of his work was translated into Latin. And that is really quite an important topic, because when we talk a little bit later about the influence that he made on modern mathematics and computer science, that is one of the aspects that probably is still problematic. He lived in the town of Orange, he was born in Bagnols-sur-Céze, and he comes from this background, where Jews fled from Spain after the takeover by the Almohads in the middle of the 12th century, and they brought with them the rich Greco-Arabic body of knowledge. Quite a lot of them spoke Arabic or read Arabic, so that they actually were able to make commentaries on lots of texts, and Gersonides was himself one of them. (The knowledge of Arabic language gradually diminished in the Jewish Provencal community, and Hebrew became the language in which the traditional texts and foreign wisdom was studied).
So he wrote several books; we’ll come to them. And this is one of his most famous books, Wars of the Lord, or Wars of God. one of the things is that because he was such an individual and a strong thinker, really, and logician, quite a lot of the things that he wrote, people didn’t agree with, and sometimes they even translated or called this, his major book, “wars with God” instead. So this book itself discusses various questions – the immortality of the soul, the nature of prophecy. If we have some time, also, we’ll have a closer look at that, because it is really an interesting question from a logical point of view – and then Divine omniscience and providence, the nature of heavenly spheres and so on. And there is some astronomy there as well.
So Gersonides grew up in this milieu of learning, which came, as I said, from Greco-Arabic culture at the time. And he obviously had access to the elements of Euclid, because he did commentaries on several books of Euclid’s elements. He did one of the first works on trigonometry, which was only published in the 19th century. He did this work, which is on the harmonic numbers — he wrote it in Hebrew, but it was immediately translated into Latin. This particular work was commissioned by Philip, who was the Bishop of Meaux in France, and it was immediately translated into Latin for him. So that book itself was published by someone in the very early 20th century only. So you can see the sort of issue of how publications were late — five, six centuries — to be translated and known.
However, since then, his work has been taken up by quite a lot of people in different areas. This is one of the copies; this page is from the manuscript. The image is the page of the manuscript from the Bavarian Library. And the manuscript was brought from Constantinople in 1480. I believe there is another copy, also brought from Constantinople, [that] is in Paris.
Okay, so what was he famous for? So the first and probably the most important thing is that he’s given credit for inventing this method, or mathematical proof or technique, by induction. So it consisted of proving things in steps. So you need to prove that something is true for one case, for the base case, then you show that it is — whether it holds for any case, or you show different steps. And then finally you show that it is general-case, so therefore it must hold for all cases. Although he doesn’t actually give a general description of the method, he repeats it again and again in his mathematical work.
So, example of mathematical induction. […] So for example, he says, “If one number multiplies another which in itself is the product of two given numbers, the result is the same as when the product of any two of them, of these three factors, is multiplied by the third.” Okay, so in words, that is very complicated, so let’s write it down. So if a number, let’s say a, multiplies a product, which is itself a product of two given numbers – so let’s say b * c – the result is the same as when the product of any of those three factors is multiplied by the third. So, basically that is going to be equal to b * ac or c * ba or a * b. Okay, so that is really what he is saying.
So after that, he does the same thing for a number, multiplying another number which has three factors. So a * b * c * d, right. And he proves it in the same way. And then finally, he says, “If one number multiplies another,” and so on — he comes up with the same, with the product of any collection of these numbers, and multiplies the product of the remaining one. So finally, he says, “You can do that to infinity, you can have as many numbers as you like which are going to be factors of the number which is being multiplied by another number.”
So that is really what the method of induction is. You build from first case, you show it works on another case, and then you show that it works for any such case. Okay, let’s have a look at another thing. So what is important really in his writing is that he does not dismiss the experience. And that is not really very common in his time. So he does actually talk about in his work, he does speak about the importance of experience, and what intellectually one gains, or, “the intellect acquires the knowledge through something which is different from itself,” he says, “and that is experience.” So he really is quite clear, from all his writing, that all the experience is crucially important in his view of the world and his philosophy, and that wasn’t very common.
One of the things he describes in his very important book Maaseh Hoshev is about permutations and combinations. This is still the work that is quite important. So just to make an introduction into that, I’m going to just explain very briefly what it means. So let’s say you have two examples. One example is I can say my fruit salad has apples, grapes, bananas and pineapple, right. So it doesn’t matter in which order I eat them. They’re in a salad. However, if I say my combinations to the safe are a, b, d, c, I want you to sort of think about two things first: that we say “combination,” but actually that is misleading, because if my combination is abdc, it can be in any order whatsoever, right. It can be a, b, c, d, it can be a, d, c, b, and so on, because the combinations do not take into account the order of things. So when the order doesn’t matter, that is a combination. But when it does matter is called the permutation. So we really should call locks — combination locks — we should call them permutation locks.
So he did some work on that. He wasn’t the first there. And why was that important? How does that connect to his Judaism and his theology? So Shabbethai Donnolo, who lived before him a couple of centuries, was the first to look at these combinations — sorry, permutations — of four letters, in order to make words. And the first appearance of this question seems to appear in Sefer Yetsirah, but it continued to appear until the 16th century. It was quite important.
So why was it important? Well, it was important because if God named things, and He created things that he named, how many creations can you have. If you do all these different permutations, right, you would be able to see how many things He has created or He can create. So his example is for n = 5 — so five letters. Of course, it gets a bit complicated in Hebrew because of the vowels, but let’s forget about that for the moment, and we’re going to sort of just stay there. So there was a very important question, really. So how many things can be in the world? Gersonides became interested in that question. There was another question and the importance of that question was posed by another Rabbi from the 12th century, about how many possible conjunctions there are of the seven planets. So for example, the conjunction of Mars and Jupiter was a very important one for astrology or for astronomy. And you know, you can say why that would be important, because — and I’ll throw out a question —things can depend on that very strongly, so it is important to know what and how things are going to be combined.
And Gersonides — one of his greatest sort of interests was astronomy. Actually, he did some work on astrology as well. So that was something that really interested him. […] So let’s say the seven planets then known were Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. So let’s say we have them like that, 6, so 1-2-3-4-5-6-7. So how many conjunctions? We obviously have these conjunctions. These are conjunctions, these are ways you can combine two planets. And you can count those, okay. We’re not going to count them now, but basically you can do that.
So we’re going to have a few questions now about combinations. So let’s say you want to find out two conjunctions of those two planets. So we have Jupiter and we have Mars. How many different combinations can you have of Jupiter and Mars out of the seven? How many combinations can you have all together of these planets?
So I have some questions for you. […] “If the order of these links between the planets does not matter, is the number of those conjunctions greater than if the order does matter? So when would the number of combinations be greater than the number of permutations?” So that was the question, okay. […] In the case of the conjunction of two planets, the order does not matter, right. So those two planets can have a relationship, but in the case of words – order of letters does matter, for example. So what is greater, the number of combinations of two planets out of seven, and/or the number of all possible permutations of two letters out to seven? We’ll come back to that question a bit later on, and we’ll see exact numbers, hopefully.
Now, what is nice is that Gersonides is worked out both permutations and combinations as purely mathematical questions. So despite his great interest in theology and his, you know, big questions that he grapples with throughout his work, when he does mathematics, he actually wants to know exactly how things work in a mathematical sense. So he doesn’t mix his theology in that respect. So the origin of the question is theological, but the work is mathematical. […]
So, number of permutations. So he showed, really, he gives formulas for permutations of given numbers. He shows the number of permutations of a given number of elements of n elements is n factorial (n!). Factorial means you multiply it by all the numbers left – so for example, 7! is going to be 7 * 6 * 5 * 4 * 3 * 2 * 1, okay, and you stop at 1, because if you multiply by 0, things will go wrong.
How does he do it? Well, he uses mathematical induction again. So he says that, for example, “if a number of permutations of a given number of different elements is equal to a given number” — so then he sort of proves that this, for example, is the case. So permutations of seven is the same as 7 * permutations of 6, okay. P(n+1) = (n+1)*P(n). And he does that in a cyclical kind of manner to show that that is the case every time.
So he gives some really important formulations in combinatorics. So the number of permutations – I would give a number out of a set and a number of combinations of a given number out of a given set. I was going to do this, but actually I’m going to skip it, because I want to show you some other things that he did, particularly his harmonic numbers. But I’ll say to you what we did here. So let’s say if you calculate the exact number of permutations of two-letter words out to 7, you will get a certain number, and that would be much larger than – well, it would be, in case of two-letter words, two letters, permutations of two letters out to 7, it will be exactly the double of the combinations. But generally speaking, if you’re dealing with any number of elements and you are trying to find permutations out of the whole set of that element, permutations is going to be always a much larger number.
Again, the question is: how much was this important at the time, and how much it influenced development of combinatorics. Well, it didn’t actually too much, to be fair, because he wrote in Hebrew and his work wasn’t widely known. However, his manuscripts were brought by a French ambassador to Constantinople. He came across his manuscripts and brought his book on numbers to Paris. The copy went to the Oratorian Order, and perhaps from there, the knowledge somehow actually percolated throughout that sort of collection of mathematicians who were very active in Paris or France at the time. So perhaps Mersenne, who was very interested in that, would have known something about it, or have seen it. And through Mersenne, possibly Blaise Pascal, who did quite a lot of combinatorics, would have been able to see and learn from that. But at that time — so the inventions of the people who invented, who wrote something, weren’t necessarily given all the credit. So we don’t really know the exact route, and whether he actually made a direct influence on those mathematicians.
Okay, so what about the harmonic numbers? So you remember that Philip de Vitry, the Bishop of Meaux, who was his contemporary, he was a composer, theorist, poet, philosopher and so on, and he commissioned Gersonides to write De Numeris Harmonicis, a book which was immediately translated from Hebrew to Latin for him. And why did he have interest in numbers? He did because he was a musical theoritician, he wrote about theory of music as well as composed music. And theory of music has, as you know, strong links with mathematics in that way. Boethius, for example, said that “Music is numbers made audible.”
So de Vitry calls these numbers “harmonic.” I mean, there are other harmonic numbers, but he called these harmonic numbers – 2 to the power of something, multiplying 3 to about a power of something else, okay. And you can see here the table of his harmonic numbers up to thousands, okay. So there’s not that many of them, so they multiply, you know, to all the products of powers of 2 and powers of 3.
And what he wanted to know was: are there any pairs that are different to those four pairs, which differ in only one? And the beauty of it is that those pairs – 1, 2, 2, 3, 3, 4, and 8, 9 – correspond to significant ratios in music that music used. So they correspond to octave, fifth, fourth and the whole tone.
So Gersonides did that. He, again, used his inductive method. And he showed that these were the only four pairs, actually. And he showed that those four pairs have to be the multiples of the power of 2 and 3. So it can’t be the only one in it, you know, it has to be those powers. So these were the only ones, if you can see that, four pairs with a difference of 1.
Okay, now what is interesting here is that this obviously wasn’t only some kind of commission. You know, he got commissioned to do something, and it was a nice interesting question in number theory that he wanted to apply his himself to. It was much more than that for him, because for him the playing of music had a little bit more importance. It actually fit into his view of the world and how this order is manifested in the world. So this is what he says: “The player does not have a pre-knowledge of the movements of the fingers required to produce the tune that he wishes to play; rather, he imagines the tune and his limbs are moved by this image until the tune is achieved. This way it can happen that musician plays with his hand while listening to a tune that he has not heard before, and the tune that he plays agrees with the tune to which he listens. This instance is confirmed with [the fact that] emotions of living beings are more wondrous than the emotions of non-animated beings,” and so on.
So it really is quite an important philosophical statement, and one that he is happy to share, because there are things that he, in his writings, we come across that, where he talks about not actually wanting to share everything for various reasons.
Okay, so we come to the final debate, is his thought on metaphysics and epistemology — so, the limits of knowledge. The most important aspect of his work here relates to logic and the nature of knowledge, and the nature of free will. And the common question that many philosophers of the times grappled with was an argument that goes something like this: “If there is divine presence, does an omniscient God who knows everything, know whether I will make a certain choice? So if He does, it doesn’t matter, actually, what I’m going to do, because everything has to be predetermined, and how can he not know if he is an all-seeing and all-powerful God?”
But because God is unchanging — so God would have known this from all eternity, that is another point that has been discussed up to this point. So this means that this event cannot turn in any other way than that which God knows from time immemorial. Because in time, you know, God is unchanging. So he’s able to see everything at all times. So if that is correct, that, you know, there is no free will.
There are other sort of important concepts here that I will very briefly sort of skim over. One is the concept of contingency and necessity. So, contingency and necessity are affected by events of a temporal nature. So, many future events are contingent on our choices in the future, but they become necessary only after they have passed, right. You can’t change things that have happened. So, before it happens, there’s a contingency, but then it becomes necessary.
But again, you have the problem there with the time. And so if God knows everything at all times, from all times, how does that work? And then Maimonides, who was a very famous Jewish philosopher and theologian, puts at the core of his view of Jewish teaching that God, without qualification, knows every particular and event, but like Averroes, who was a Muslim scholar from that period, argues that this is different to our understanding. Okay, so although he does know that, it is not actually something we can understand entirely. We can never understand it entirely.
So what can mathematics do in this respect, according to Gersonides? Well, first of all, we need to just understand that he puts astronomy and metaphysics as the most important sciences. So understanding of the nature and the structures of this order, as it is manifested in the universe, is the most important for him. And then in the middle, physics is important, because through physics you understand that sort of whole picture. And before physics, mathematics is important, because, well, you learn how to think, and also you come with this process of proving things that are true, and that is important. And what is important for him is – and what he says – is the best thing a person can do within their life is to get as much of learning and understanding of existing things of that order, and the equilibrium between things, and the manner of wisdom that has organized them as they are. So really, understanding the existing things, or that existing order among things, is being able to see God’s wisdom. And that is where his mathematics and his science are important for him.
And he also says that, for example, God knows those things which exist in the world, insofar as they possess a universal nature, i.e., essences. So for him, for Gersonides, God doesn’t really bother with the detail. God knows all the order as it is, and that perfect knowledge consists of knowing the nature of the thing. So if you actually want to have a perfect knowledge, you need to understand that structure of the thing, which is a very mathematical sort of approach, especially to viewing things. And also one of the things that this study says is that reason this has been given to people. And this God-given faculty is there for a reason, and the reason is for us to understand God with this instrument.
So how was this influential to modern times? Well, his work in combinatorics was quite important, and it would be. You know, people have tried, but I’m sure there is a lot left to be found, because only now, quite a lot of correspondence is being looked at as it is being digitized to be made available. So hopefully we will see a little bit more about his influence on the development of combinatorics in the years to come. His work on inductive reasoning is incredibly important, because we still use that type of proof. His work on harmonic numbers was important, and it was more important, in a way, than we’re able to really say here, because it is linked to two very important other theorems. One is Fermat’s Last Theorem, which was only really solved in 1995, I think, by an English mathematician, and ABC conjecture, which is a conjecture in number theory. So it’s really still — you know, there is still connection there. So that is an important contribution.
And then his work in logic and epistemology of knowledge has influenced more modern developments, in particular in temporal logic, and via that, computing and artificial intelligence.
So since his work has been mainly known from the, really, 18th — not 18th, 19th and 20th century — it’s quite an achievement to have all these things, all the influences, centuries after one’s life. And he influenced Spinoza and Leibniz, many would argue that. And this is actually one of the streets where he possibly would have lived in the little town of Bagnols where he lived. And of course, there is a fantastic novel by one of my favorite writers, Iain Pears, The Dream of Scipio, where Gersonides appears in some places. And I highly recommend it, not only because I love to think about Gersonides, but also Iain Pears has written another novel, An Instance of the Fingerprints, which to me is a very great novel as well. So there’s the bibliography — as I said, quite a few of things that I’m very grateful to Shai Simonson for — he’s published a [previously] unpublished translation of Gersonides’ work.
(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. “Gersonides and The Limits of Knowledge with Snezana Lawrence,” a talk given at Temple Mount Sinai on October 17, 2021, was the fifth in the series Higher Meanings: Connecting Religion and Mathematics. The next and final event in this series, “Will Computers Make Religion Obsolete?,” featuring Dr. David Novick and Rabbi Ben Zeidman, will be on December 16, 2021 at 6:30 pm MT / 8:30 pm ET).