Feature Article

Mathematical Stability in Our Personal Lives

Rabbi Professor Daniel Hershkowitz

Minister of Science and Technology of the State of Israel

From B’Or Ha’Torah 20 (5771/ 2010)

A Tourist in G-d's World

Five years ago my friend Professor Aaron Ciechanover together with his mentor Professor Avram Hershko, both of the Technion, won the Nobel Prize for chemistry for discovering the ubiquitin-mediated degradation system. Trained as a medical doctor, Ciechanover later branched out into biochemistry. Because he loves Jewish liturgical music, our congregation held a special Shabbat of cantorial music in his honor. During this Shabbat celebration, someone asked him about his profession. Ciechanover replied, “What is my profession? I’m a tourist. I’m a tourist in G-d’s world. I’m trying to explore the wonderful creation, ‘How great are Thy works,’ the laws of nature that were set up by G-d. If I can understand some of it, then I hope that I can employ it to help humankind.”

The Torah Sages say that G-d looked into the Torah and created the world. In other words, the laws of nature created by G-d reflect the Torah. As such, trying to understand the laws of nature can be seen as a high level of Torah study. Thinking this way can open the mind to very interesting links between Torah and science in both directions. Although we could spend a whole conference on this issue alone, I will limit my discussion to one of the biggest conceptual discoveries of science in the twentieth century, of which not too many people are aware.

Why The Evil Inclination?

Let me start with an ancient philosophical question that bothers both Jewish and non-Jewish philosophers. Why did G-d implant in us an evil inclination?

We grew up knowing that everyone has both a good inclination and an evil inclination—the yetser ha’tov and the yetser ha’ra. The yetser ha’tov pushes us toward a positive spiritual direction while the yetser ha’ra pushes us toward a negative material direction. The yetser ha’ra is the source for most of the evil in our world, but why did G-d create us this way? Doesn’t the existence of the yetser ha’ra contradict Rabbi Akiva’s claim in the Talmud that everything G-d does is for the good?

The Talmud Brakhot 54a says on Deuteronomy 6:5, “You shall love the L-rd your G-d with all your heart,” that the Hebrew word for “your heart” is written lvavkha (לבבך) with one ב instead of leibkha (לבך) because we should love G-d with both our good and evil inclinations. So the yetser ha’ra definitely has an important role. But what exactly is its role? This is a classical question. Usually classical questions have classical answers. But instead, I wish to propose another direction, based on our being tourists in G-d’s world. I hope that trying to understand how G-d created our world can help us also to understand our souls.

The Mathematics of Stability

Let’s go back to the big discovery of the twentieth century. Assume we want to design a fast, long-range moving system, such as a high-performance combat jet or a missile or an anti-missile missile, which has to move very fast and must have a long range. I emphasize these two requirements of speed and long range. If asked what is the most essential feature in designing an airplane or a rocket, most people would say the engine, since it is the engine that provides the speed and affects the range. Of course, the engine is a vital part of our airplane or missile, but we would have to invest very little in planning it.

More than twenty years ago, the State of Israel invested tremendous resources in designing the Lavi combat jet. Due to geopolitical considerations, however, the project was abandoned. While a lot was invested in designing that jet, almost nothing was invested in designing its engine. All the designers of the Lavi had to do was to specify to the outsourcers what they wanted from their engine.

Outsourcing is an important concept nowadays: don’t try to reinvent things that other people know how to do better. There are companies in the world that have expertise in manufacturing jet engines, and there is no reason to expend large amounts of money and time in order to get a result that someone else already has.

What, then, is the most difficult part of designing a jet or missile? All you have to do is to remember the experience of taking a birthday balloon, inflating it and then letting it go, without tying the mouthpiece. What happened to the balloon? It flew very fast.

But now let’s assume that you take many balloons. Blow as much air into them as you wish but don’t tie them, then aim them to hit the center of a target. What are your chances of success? Not very good. Why? Each balloon is going to fly very fast. Why shouldn’t it hit the target? Because the balloon travels in an uncontrolled direction. It’s going to move in various directions very rapidly, but with a very unbalanced movement. In science we have a term for this: unstabilized movement.

The balloon basically flies according to the same principle as a jet engine. The problem is not to produce the movement, but to make certain that the movement is going to be stable. When you design a rocket, you don’t want it to travel uncontrolled.

The Lotka-Volterra Model

The problem of stability of movement has bothered mathematicians for over a hundred years. Until about eighty-five years ago, scientists assumed that generally the faster a device travels, the more stable its movement is going to be. By using the example of the balloon it’s easy to prove that this assumption is wrong. The more air you blow into the balloon, the faster it’s going to fly, yet the movement is not going to be more stable. Your best chance of hitting the center of the target is if you don’t blow any air into the balloon at all. Just take the uninflated piece of rubber and throw it. It may travel much more slowly, but its flight is going to be much more stable.

Some eighty-five years ago, a breakthrough that disproved the concept that increased speed increases stability was obtained by two mathematicians, who, as far as I know, never met. One was of Polish origin and lived in the United States: Alfred J. Lotka. The other one was a Jewish Italian mathematician who lived in Rome: Vito Volterra. Until today, the combination of their work on stability is called the Lotka-Volterra model.

Lotka decided that in order to understand the stability of movement, he had to study stability problems in nature. In other words, Alfred Lotka decided to become a tourist in G-d’s world. He selected a classical stability problem in ecology, the predator-prey system. Assume that you have a population of predators, like wolves, and you have a population of prey, like lambs. We are not yet at the end of days when the wolf will lie down with the lamb. Wolves still feed on lambs.

Now, in our predator-prey system, assume that we have many lambs and a few wolves. The wolves have a lot to eat. They eat well, and they multiply. Gradually, the population of the wolves increases. As their numbers increase, they consume more lambs. Consequently, while the population of wolves increases, the population of the lambs decreases, until we reach the stage where there are fewer lambs and more wolves. Now the wolves have much less to eat. Less food, of course, affects their life expectancy, and the po­pulation of the wolves now decreases. As the wolf population decreases, fewer lambs are consumed, so the number of lambs increases. Once again, you have many lambs and few wolves. It’s like a children’s seesaw, but not exactly. When one child is high up on the seesaw, the other is low down at the bottom, and then they switch. The one who had been at the bottom is now at the top, and the other one is at the bottom. But it is not always so in nature.

Assume that the population of lambs reaches rock bottom, since the wol-ves have eaten up all the lambs. The consequence is that not only will there be no more lambs, but there will be no wolves either, because they no longer have enough food. The whole system collapses. This is called an unstable system. A system that does not collapse is called a stable system. Of course, a stable system is always changing, because nature is dynamic and never freezes.

When Lotka developed his theory, he was influenced by the work of a group of biologists who were studying a very rare type of snake that lives only on a particular island. This snake’s food is mice which live on the island. When the biologists arrived there, they made a survey of the snake and mice populations. According to their simple mathematical calculations, the snakes were going to eat up all of the mice in the island within a few weeks. This meant that after several weeks—when there would be no more mice to feed the snakes—the snakes were going to disappear. Worried about the survival of the snake because it exists only on that island, the biologists continued their study. They discovered to their surprise, after half a year went by, that there were still mice, and there were still snakes. In other words, despite their calculations, somehow these two populations had reached stability.

The biologists wondered what made the system stable, until they realized that it was because of the existence of the spotted owl on that island. The snakes survived because of the owl. The owl is a predator, and one of the favorite foods of the owl is snakes. How could this be? Not only was there not enough food for the snakes, but there was also a predator killing them. But this is exactly what made the system stable. The owls were reducing the population of the snakes. Consequently, now there was enough food for the remaining snakes. When there are fewer snakes, there are enough mice to feed the snakes. So, actually, the owls that kill snakes help the snake population survive. In the predator-prey system, the chances of the survivors are increased.

This helped Lotka understand that the forces that make a system stable are forces that act in the reverse direction. The snake population is stabilized because owls kill snakes. Thus, in order to stabilize movement, we have to produce forces that work in the reverse direction.

That’s exactly the purpose of the small wings that every rocket has. These wings reduce the speed because of their air resistance. Their purpose is not to increase the speed but to produce stabilizing forces. The idea that stabilizing forces should act in the reverse direction has affected so many sciences and so many technologies that we could devote a whole conference to this alone. The mathematics of the Lotka-Volterra model is used to calculate the details in each case.

The Lotka-Volterra model has been applied to the field of economics in the study of price stability in multiple markets. This same model has also been employed to save successful companies from collapsing.

If a company is successful, why should it collapse? The reason is very simple. Assume that you’ve invented a cellular phone that can identify when an incoming call is similar in content to previous phone calls and then it conducts the conversation by itself, imitating your voice. (After being in politics for a year, I can vouch for the usefulness of such a feature.) Your future phone has to be recharged only once a month. Moreover, it will also have a small microwave heater that can reheat a cup of coffee.

After a huge investment to develop this cellular phone, you produce a model that sells for $20,000. It’s only for the very wealthy. According to your business plan, the investment will be returned within five years, but your product is so successful that every rich person in the world is motivated to purchase your cellular phone. It takes only half a year to return the whole investment. The company is growing very fast with 500 employees to service this fantastic phone. Your expenses are up to $100 million.

What a success! The problem is, by the second year the market is saturated. You sell only 2,000 phones and bring in $40 million, while your expenses are $100 million.

Some analysts would advise you to improve your product to make it sell better. They suggest introducing an air conditioning device. Users will be able to escape the personal effects of global warming by turning on this feature of your new model when walking down the street. But this won’t increase the sales. The rich people who have already bought your phone are not going to buy another one just for the air conditioning device.

So, improving the product won’t help. Lowering the price would not be useful either, because selling the same device for $200 rather than $20,000 would cause many people to file suits.

What would help? The answer is the Lotka-Volterra model. If you want to ensure continuity of your business, you have to produce stabilizing forces that operate in a reverse direction. This means you should downgrade the telephone, weaken some of its features, change the design, change the brand name, and sell the new model for $200. Now you’re going to sell millions.

This is a very brief description, but the Lotka-Volterra model has been employed by many companies in the world. There are examples of stabilizing forces that work in a reverse direction in many fields of science and technology.

Applying the Lotka-Volterra Model

Let’s conclude with the question with which we started. Why did G-d create in each human being both a good inclination and an evil inclination?

Many people think that the Garden of Eden and Gehenna are located in opposite directions. That’s not so. The Talmud Sages say that the Garden of Eden and Gehenna are right next to each other. This means that you might aim at Eden but arrive at Gehenna. There’s a saying that “the road to hell is paved with good intentions.” We need not only movement in the right direction; we also need stable movement. Just as a rocket needs stabilizing forces in addition to a strong engine in order to remain stable, human beings need stabilizing forces acting in the reverse direction in order to keep them stable. For this purpose, G-d created the forces of evil to push us in the reverse direction.

We see this principle at work in medicine, too. In both physical and mental illnesses, the desired state is stability. And stability is maintained by forces operating in opposite directions.

By being tourists in G-d’s world, by looking at G-d’s wonderful creation, and by understanding predator-prey systems in nature, scientists have come to understand the stability of movement. When I look into the mathematical model of how to stabilize movement, I can also understand how we should maintain stability in our personal lives. First, though, we have to know the proportions. If you place oversized wings on a rocket, it will hardly be able to move. If the stabilizing forces are too great, just as if the power of the engine is too large, then the result will surely be undesirable.

Our Sages of blessed memory said:

Those who rule their evil inclination…” means let us consider the account of the world: the loss incurred by the fulfilment of a mitsvah [commandment] against the reward secured by its observance, and the gain gotten by a transgression against the loss it involves (Talmud Bava Batra 78b).

In other words, we have to do the mathematics. In order to understand our world, we have to understand mathematics, the language in which it is written. When we are talking about the yetser ha’tov and the yetser ha’ra the mathematics is intuitive. By looking at G-d’s world of nature, we can learn a lot about ourselves. May all of us learn how to maintain stability in our personal lives.

Reference

Lotka, A.J. 1925. Elements of physical biology. Baltimore: Williams & Wilkins Co.

Reprinted as: Lotka, A.J. 1956. Elements of mathematical biology. New York: Dover.