Parashat Bechukotai provides an overview of the blessings that the Jewish People will merit if we keep G-d’s Torah and His mitzvot, followed by the curses we will suffer if we, Heaven forbid, choose otherwise. A common denominator of these blessings and curses is that they all pertain to our material world: If we keep the Torah, we will be wealthy, our crops will thrive, we will have peace, and our favourite football team will win the Super Bowl. If we do not keep the Torah, we will suffer famine, war, and twelve consecutive losing seasons. This seems odd. One would have expected to receive blessings and curses with a more spiritual bent: If we act according to G-d’s will, we will be blessed with a better seat in the World to Come and if we do not, we will burn in the bowels of hell.
The Ramban, a leading rabbi, philosopher, physician, and kabbalist, who lived in thirteenth-century Spain, gives us a way ahead. One of the basic tenets of Judaism is the concept of reward and punishment. G-d gives man freedom of choice: If he chooses wisely, he will be rewarded. If he does not choose wisely, he will be punished. Noting that it is G-d, a metaphysical entity, Who is keeping score, it should be expected that punishment and reward are given in the metaphysical world to come. The Ramban explains that Torah is teaching us something unexpected: G-d bridges the infinite void between the physical and the metaphysical, between the corporeal and the surreal. The fact that He metes out punishment and reward in our physical world is nothing less than miraculous.
Let’s keep this idea in our back pocket for future use. One of the blessings that we will hopefully merit reads as follows [Vayikra 26:7-8]: “You will pursue your enemies and they will fall by the sword before you. Five of you will pursue a hundred and a hundred of you will pursue ten thousand, and your enemies will fall by the sword before you”. The arithmetic seems incorrect – there is a mismatch between the kill-ratios in the first part of the verse and in the last part of the verse. The first part of the verse describes a 100:5 or a 20:1 kill-ratio: One Jew will pursue (and kill) twenty of his enemies. The last part of the verse talks about a 10,000:100 or a 100:1 kill-ratio. One would have expected the Torah to maintain a 20:1 kill-ratio by saying something to the effect of “a hundred of you will pursue twenty thousand”. Rabbeinu Bachyeh ben Asher, a contemporary of the Ramban, tries bending the Torah’s words so that it reads, “Five of you will pursue a hundred, and [five] hundred of you will pursue ten thousand”, maintaining the 20:1 kill-ratio, but his explanation seems forced.
Rashi, the ultimate medieval commentator, answers simply, “[The Torah teaches us that] there is no comparison between a few who fulfil the Torah and many who fulfil the Torah.” That is to say, the amplitude of each person’s blessing is proportional to the amount of people being blessed for fulfilling the Torah. As greater numbers of people are blessed, each individual receives a greater blessing and so the sum total is far greater than expected.
Perhaps unwittingly, Rashi is venturing into the world of nonlinear systems. [Note for the non-mathematically inclined: we’re going to get slightly mathematical here, but only for a bit.] The easiest systems to work with are “linear systems”. Let’s introduce a system described by a function f(x). For the system to be linear, our function must satisfy f(aꞏx)=aꞏf(x). Meaning that if I amplify my input by some amount, then my output will also be amplified by the same amount. Cakes are linear systems. If I double the amount of flour, then I will end up with the double the amount of cake. Scientists love linear systems because they have some nice mathematical characteristics that make them easy to work with. Many systems are linear, nearly linear, or linear at least some of the time. But some systems are not linear. Consider the function f(x)=x2. If x=3, then f(x)=9. But if I double the input and set x=6, the output is now 36, which is quadruple the original output. The system described by this function is nonlinear.
An example of a nonlinear system is an army engaged in combat. Let’s say an army is trying to take out a group of enemy leaders who are meeting in some nondescript apartment building in an unfriendly town. The cheapest way to attack the apartment is with rifles. The problem is that to use their rifles, soldiers must enter the building and a high-casualty firefight is nearly a certainty. If the army is better equipped, the soldiers will have shoulder-launched weapons like a bazooka. This gives them some range, or “standoff”, and it gives them a bigger punch. The problem is that a bazooka is unguided, so the chances of missing the target are high. Kill probability is increased if, instead of a bazooka, a guided missile like Spike-LR is used. These missiles cost a pretty penny, so not every army can afford them. Larger armies with larger budgets can increase standoff by using a longer-range missile, like Spike-NLOS, which also has a much larger warhead, further increasing the probability of mission success. Continuing down this path, standoff and warhead size can be further increased by using an air-launched weapon, such as the Popeye Precision Guided Missile, which can deliver 800 lbs of TNT a distance of more than 100 km, allowing it to be launched far out of harm’s way. The problem is that to carry Popeye, a large aircraft, like an F-15, is required. Most Air Forces do not have this capability and so it is limited to only the best-equipped ones.
This is what we mean when we say that armed combat is a “nonlinear system”. Each of the weapons described above offers a vast improvement over its predecessor. It is incorrect to say that one bazooka is worth five rifles or even ten rifles. A bazooka offers performance that no number of rifles can offer. As a result, larger countries with larger military budgets have military capabilities that cannot be rivalled by countries with smaller military budgets. When the Torah says, “Five of you will pursue a hundred, and a hundred of you will pursue ten thousand”, it means, “Five of you [armed with rifles] will pursue a hundred, and a hundred of you [can procure Precision Guided Missiles that] will pursue ten thousand”. There is no comparison between a few who fulfil the Torah and many who fulfil the Torah.
One of the characteristics of nonlinear systems is that they often exhibit highly complex behaviour. A chaotic system is a special type of nonlinear system. One characteristic of a chaotic system is that it is highly sensitive to initial conditions. This means that if two identical chaotic systems start at two different starting points, even if they are infinitesimally close, the outcomes will eventually diverge. Weather is a classic example of a chaotic system. In order to accurately predict the weather, it is necessary to know the humidity, the temperature, and the barometric pressure at every point in the world with infinite accuracy. Obviously this precludes any long-term weather forecasting. Another example of a chaotic system is warfare. Two battles that begin with nearly identical initial conditions can end very differently because in one battle a soldier stood one metre to the left and was hit by mortar fire. In the words of Benjamin Franklin, “For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the message was lost. For want of a message the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail.”
Chaotic systems theory can shine a new light on the blessings and the curses in Parashat Bechukotai. Even though G-d is a spiritual being, His rewards and His punishments have both spiritual and physical components. They are transferred from the spiritual universe to the physical universe via extraordinary conduits that govern, among other things, our weather and our geopolitical climate. These conduits are exquisitely sensitive to the initial conditions – the choices made by a particular person or nation. The performance of even one mitzvah can change the world in ways we cannot even begin to imagine. There is no comparison between a few who fulfil the Torah and many who fulfil the Torah.
Ari Sacher, Moreshet, 5779
Please daven for a Refu’a Shelema for Yechiel ben Shprintza and Tzvi ben Shoshana.
 Our GM will also inexplicably trade away our star wide receiver for peanuts and then waste the 6th overall pick in the draft.
 The word “kill-ratio” typically defines the ratio of losses on both sides. Here we have redefined the term as the ratio of attackers to defenders.
 This is where asymmetric warfare comes in.