Cautiously addressing a span of almost thirty-four centuries, I am fully aware of the analytic hiatus involved in trying to bridge between documented scholarship on one hand and a mixture of rabbinic texts, inference, historical memory, and imagination on the other. Using a mathematical apparatus for the analysis of such delicate subject is another challenge we had to overcome. We think this is critical to explain the hypothesis of the Devine regulation model, examining the timeline from the Exodus to our days. The Jewish world is analyzed in this model as one system underlying a joint mission, where each individual carries a proper share and weight of this divine task.
Light to the Nations
In the Torah passage (Exodus 19:3-6), G–d addresses Moses immediately prior to His revelation at Sinai. G–d called to Moses from the mountain, saying, “So shall you say to the House of Jacob, and relate to the children of Israel: ‘And now, if you hearken well to Me and observe My covenant, you shall be to Me the most beloved treasure of all peoples, for Mine is the entire world. You shall be to me a kingdom of priests and a holy nation.’ These are the words you shall speak to the Children of Israel.”
“Kingdom of Priests and a Holy Nation” – these words condense the reason G–d “chose” the Jews. The reference here to priests does not refer to the Kohanim (Cohens) but to the “priestly function”.
The foremost function of any Jew as a Priest is to bring G-d to the people and to elevate the people to be closer to G-d. The purpose of the Jews is to bring G-d and the world closer to each other. In other words, to act as a genuine ambassador between the G-d and the world. In this specific association every Jew – man or woman- must fulfill priestly functions with the world around us. The reason that a “kingdom of priests” and a “holy nation” are juxtaposed in this passage, is not accidental. Being “holy” means to be dedicated to Torah and mitzvot in our private life, and in that to impact the outside world. Jewish impact on the outside world is intrinsically related to following the Torah rules and mitzvot. Prophet Isaiah has coined the priestly function as a “light to the nations”.
Every Jew, in the Diaspora or in the Land of Israel, is supposed to remember that he or she represents the Jewish people and therefore is mandated with the task. Even in Galut (diaspora) Jews should not ignore nor withdraw from the task, as it is just the physical body that is exiled, while Jewish soul is free from physical suppression or constraints.
Jewish people have been given special directive and the codex of life (Torah and Mitzvot) to exercise spiritual power. This directive and the Codex provide a capacity to carry out the task in the fullest measure. On many occasions the Jewish people stood up among the nations of the world as true implementers of the priestly function. In the time of King Solomon Jewish people has reached the highest degree of perfection, according to many sages. In fact, Jewish prophets have described that time as being “the moon in its fullness”. And as the moon goes through periodic changes in its appearance, Jewish people go through different phases of their history, reflecting the heavenly light to mankind.
Utility of Light in the World
By defining an overall presence of Light of Torah and Mitzvot in the world by some imaginary utility (UtilityLight) and Jewish people are ambassadors carrying this utility, we assume that every single Jew holds a share of this Divine Light (UtilityJ). The smaller the number of Jews is the higher utility of Divine Light is carried by single Jew. In fact, in the times of King Solomon Jews constituted numerically the “fewest of all the nations”.
As true ambassadors carrying the Light, the individual Jew cannot carry a diminished share of this Light. The Diminished share of the Light defeats the whole task of fulfilling the Priestly task in the world. Therefore, the only way the Jewish people can experience a robust increase in their presence (population) in the world is through a sustainable increase of the Light of Torah and Mitzvot (UtilityLight) they are able to create. It was clearly demonstrated by the whole history of Jewish people and multiple Rabbinic sources that when Jews live in accord with the Torah and Mitzvot, true peace is attained as they serve as a guiding light for the nations. In the state of diminished Light of Torah that Jews have occurred to be on a number of occasions due to internal or external reasons, the reduced Jewish presence (population) manifests a better way to embrace such periods, yet still delivering the Divine Light per Jew (UtilityJ) at an acceptable level.
Misinterpretation of the Egalitarian utility
It is important to avoid a misinterpretation of the contribution to the growth of the egalitarian utility as replacing or tantamount to the growth of the Utility of Divine Light. Egalitarian values do stem from the Mitzvot. In fact, Egalitarian values are launched into being by the Mosaic Law, yet they constitute a very small faction of an immeasurable edifice that the Light of Torah brings upon world. While investment or dedication to the continuous growth of the egalitarian human relations is important and noble deed, in no way it can substitute the growth or even sustain the Light of Torah in the world.
Land of Israel Factor
Rabbi Abraham Yitzchak Kook (Z” L) in his writing Olat Ha’Rayah stresses the deep association between Jewish People and its Holy Land. This association first stressed in Genesis 15:18 is beyond any natural affiliation through which all other nations and peoples are tied to their lands. Tannaitic Literature (1st century C.E. to 200 C.E.), Rabbi Kook, Rabbi Soloveitchik, as well as many other Rabbinic sources consider contribution to the wellbeing of the Land of Israel as one of the greatest Mitzvot. In other words, helping develop the Land of Israel and making it more prosperous is a substantial factor in increase of the Utility of Light in the world. We can take an extra step here by suggesting that an essence of living in the Land of Israel is already a factor contributing to the Utility of Light. One who lives in the Land of Israel, carries financial, civic, and moral responsibilities, thus inevitably advancing the growth of the Utility.
Jewish Population as a Reflection of the Utility of Light
Let us suggest that UtilityJLimit is an acceptable limit for the UtilityJ carried by any Jew regardless of whether he/she lives in Diaspora or in the Land of Israel. Coining the term acceptable is a slippery slope so let us assume that acceptability means sustainably positive impact that every Jew has as a part of the Jewish people (or divine ambassadors as it has been explained above). In other words, let us say that acceptable means that the priestly function is successfully fulfilled.
Assuming inseparability in the joint task of carrying the UtilityLight, we can derive UtilityJ by dividing UtilityLight by current populace of the Jewish people (PopJ).
UtilityJ = UtilityLight/ PopJ and should be always equal or greater than UtilityJLimit for the goal of sustainable contribution to the Light of Torah and Mitzvot to be met.
Jewish people, living in the Land of Israel, advance the Utility of Light due to the fact of their direct contribution to the development and the wellbeing of the Land. It would be fair to assume that even observing the Mitzvot at the same level, Jewish people living in the Land of Israel have higher contribution to an overall Utility of Light, than Jews living in Diaspora. Let us designate this extra contribution as LandIsraelFactor. Now let us include the factor of the Land of Israel in the UtilityJ calculation.
UtilityJ = UtilityJ_avg × (Percent of Jews in Diaspora + LandIsraelFactor × Percent of Jews in Israel), where UtilityJ avg is a ratio of UtilityLight to PopJ.
The equation for the can be thus re-written as:
UtilityJ = UtilityLight/PopJ ×(Percent of Jews in Diaspora + LandIsraelFactor× Percent of Jews in Israel).
Our approach leads to conclusion that greater shift of the Jewish population to the Land of Israel boosts the resultant UtilityJ, hence helping to keep it at a level acceptable for the priestly function. This also implies that investing in the well-being to the Land of Israel from the diaspora has similar effect to the growth of the UtilityLight. Mathematically it is analogous to increasing the Land of Israel population, hence utilizing the LandIsraelFactor to boost the UtilityJ. The shift towards Land of Israel also helps to mitigate an insufficient or even degrading contribution to the UtilityLight coming from the diaspora groups.
The growth of the Jewish population in the world requires a sustainable growth of the UtilityLight. Failure to deliver higher UtilityLight triggers divine regulation. Normal outcome of the divine regulation is in balancing the system by forcing UtilityJ to reach or to exceed the UtilityJLimit level.
Likewise, at a times when Jewish populace is at its dip, like it was on the eve of expulsion from Spain in 1492 when total Jewish population in the world was around 1.1M (Sergio DellaPergola, J. Even.2001), it takes noticeably lower effort to turn the tide by delivering an acceptable level of UtilityJ.
Assimilation reduces the Jewish populace in Diaspora. In the situations when Jews fail to increase or even to sustain the Light Utility, assimilation is a mitigation factor helping to maintain the Utility balance or to increase it. In the environment where Utility of Light is in the state of continuous decline, a phenomenon of decrease of the Jewish population at higher rate, ensures UtilityJ remains unchanged or even grows. In other words, in a society where Torah light and Mitzvot are abandoned or substituted by active pursuit of alternative or only superficially overlapping values (i.e., egalitarianism), assimilation is the factor mitigating the downfall of the UtilityJ. In fact, we suggest that assimilation does help Jewish groups that maintain their contribution to the Utility of Light. These Jewish groups can continue in their normal deeds of carrying the Light of Torah and observing Mitzvot without fear that they need to force themselves to move beyond their normal framework, by making up for the contingent not contributing to the process.
The following equation implies that as long as the rate of the assimilation (represented by the PopLoss) exceeds the decline of the UtilityLight, the UtilityJ keeps growing.
UtilityJ = (UtilityLight-Decline UtilityLight)/(PopJ-PopLoss)
Formalization of the Utility of Light
On a formal level the only robust condition for sustainable maintenance or increase of the UtilityJ is when the first derivative of the UtilityLight is greater or equal to the first derivative of the Jewish populace, as stated by the following generic rule:
On a functional level this means that the first derivative of the UtilityLight(t) needs to be greater or equal to the first derivative of the PopJ(t). We can re-write the constraint for the sustainability of the UtilityJ(t) as:
d(UtilityJ(t)*PopJ(t))/dt ≥ d(PopJ(t))/dt
This can be further re-written as:
UtilityJ(t)’×PopJ(t) + UtilityJ(t)× PopJ(t)’ ≥ PopJ(t)’
UtilityJ(t)’ ≥ PopJ(t)’× (1- UtilityJ(t)) / PopJ(t)
Re-writing the as rate of change of UtilityJ:
Rate-Of-Change-Of UtilityJ ≥ PopJ(t)’× (1- UtilityJ(t)) / PopJ(t)
This inequation combined with the rule that UtilityJ(t) needs to be greater or equal to UtilityJLimit at any given time, formalizes the rule for the sustainability or growth of the UtilityJ(t).
UtilityJ Trend Over Timeline
Sergio DellaPergola, J. Even (2001) assesses the development of the Jewish population from the very origins to the present day. It also compares the development with world’s total population.
Genesis (46:8-27) specifies the names of fewer than 70 Jewish men who migrated to Egypt. Exodus (11:37) mentions 600K male adults who left Egypt 430 years later. The first peak of the Jewish population corresponds to the period of the Kings with interpretation of King David’s censuses. The beginning of the second peak corresponds to the end of Hasmonean period, towards the time of construction of the Second Temple in Jerusalem.
Figure 1 shows a trend in the UtilityLight and UtilityJ over timeline, starting from the exodus to our days. This assessment is based on the juxtaposition of the population trend presented in the analysis of DellaPergola (2001), Baron (1976), and Biraben (1979).
At the time of the revelation at Mount Sinai, 600K Jewish People had been tasked with serving as ambassadors and bringing the Light of Torah to the world. Ten Commandments have become a cornerstone of the Mitzvot facilitating the Light of Torah to fill the world. The Utility of Light was high as well as a share of this light carried by every Jew (UtilityJ).
Immediately following the conquest of the Canaan, the Utility of Light had been in a state of decline. Loss of 10 tribes and the Northern Kingdom, destruction of the First Temple by the Babylonians with successive decrease of the Jewish population, have eventually slowed the decline of the UtilityJ. Return from the Babylonian captivity (538BCE), construction of the Second Temple (515BCE) combined with establishment of the schools for religious studies in Babylon have changed the trend, by increasing the Utility of Light and UtilityJ. Priestly scribe Ezra  (~500BCE) inaugurated public reading of the Torah as a new element in Jewish life. The entire period 515BCE-100BCE (called time of Sopherim and Scribes) is associated with establishment academies of high learning in Jerusalem and training of the high priests for the Temple. However, the growth of the Utility of Light has been counterbalanced by the fact that the training has been restricted to a very small elite. Events preceding the destruction of the Second Temple (70 CE) had an offsetting nature in their impact on the UtilityLight. On one hand schools founded by Hillel  and Shamai  have augmented the Light of Torah. On the other hand, the cost of tuition limited the attendance at these schools. These fledgling steps to increase the Utility of Light have been hindered by the inner struggle of multiple Jewish fractions (Sadducees, Pharisees, Sicarii, Zealots) offsetting the growth. Overall, the time period from ~100BCE to ~73CE sees a combination of events preventing the growth of the UtilityLight or even reducing it, due to devastative effect of inner Jewish struggle and inherent limitations in applicability of the Jewish education rulings to certain groups of Jewish population, exacerbated by the disconnect between Jews living in the Land of Israel and in Diaspora.
In an aftermath of the Bar Kohva Revolt (135CE) the process started by Rabbi Ben Zakkai and solidified by Rabbi Judah HaNasi  has boosted the Utility of Light by imposing a unique obligation to Jewish world -the requirement to read and study the Torah. Jewish people that could not afford or meet such requirement, have become susceptible to inevitable assimilation (Botticini M. and Eckstein Z. 2012). On the eve of the expulsion from Spain (1492CE) the worldwide Jewish population (~1M) has reached one of the highest UtilityJ (Figure 1) primarily due to aforementioned process. Further growth of the Jewish population (3.3M in 1825CE based on DellaPergola 1992, 1997) have seen a growth of the UtilityLight primarily due to huge contribution of the Ashkenazi Rabbinic school and Chasidism. UtilityLight growth has continued through 20th century but at a slower rate than the growth of Jewish population. As a result, the UtilityJ downtrend has continued. The establishment of the State Of Israel had an amplifying effect on the Utility Light as we have discussed earlier, hence increasing the UtilityJ. Today the Utility of Light grows primarily due to development of the Land of Israel that has reclaimed its role as a leading center for Rabbinic studies and have become a light for the world nations. The continuous population growth in the Land of Israel is a guarantee to offset the detrimental effect coming from unproductive diaspora by maintaining the UtilityJ at an acceptable level.
Land of Israel Factor in the UtilityJ Trend
Figure 2 shows the trend of UtilityJ when the Land of Israel factor is accounted for. As explained in previous sections, sustainable development of the Land of Israel and the Jewish presence in the Land, constitute fundamental Mitzva, causing further growth of Utility of Light. With the diaspora Jewry dwindling contribution to the Utility of Light, the development of the Land if Israel is the main factor advancing the UtilityJ in our days. Figure 2 shows that UtilityJ moving average, accounting for the previous periods in a prorated way, is in uptrend primarily due to the Land of Israel factor. The moving average reflects the dependency of the current generations on the actions of the previous ones. Talmud reconciles seemingly contradictory statements in the Deuteronomy (24:16) and in Exodus (20:4) about the way children are responsible or affected by their parents’ deeds, by explaining that these verses are referring to two very different kinds of children. We take Deuteronomy guidance here that refers to children (successive generations) who do not continue in the wayward path of parents. The way we assess the moving average of the UtilityJ reduces the impact of the past generations, hence reflecting the Talmud vision .
Our hypothesis explains the way the Jewish world is deeply tied together as one space carrying the Divine Light. The deep mutual responsibility encompasses the most distant (physically and spiritually) parts of this world in fulfilling the priestly function for the humankind. If one part of this world in in the state of decline, moral confuse and unable to sustain the UtilityLight, the only way for Jewish world to succeed is for another part to step up in multiplying the UtilityLight, therefore advancing the UtilityJ. Our hypothesis shows that contribution to the development of the Land of Israel, by advancing its prosperity in any possible way, is one of the best investments in the UtilityJ sustainability due to the Land of Israel factor. We think that holistic view of the Jewish society as one space that contributes to upholding the UtilityJ at a level where the priestly function is successfully fulfilled, is the only way to avoid the divine regulation.
 Ezra descendant of Seraiah (Ezra 7:1) the last high priest to serve in the first temple (kings 2 25:18).
 Hillel (born 90BCE in Babylon) descended from Benjamin (father) and David (mother), founder of the dynasty of sages, that included Rabbi Gamliel, Rabbi Judah HaNasi and others.
 Shamai (born ~50BCE in Israel) – president of Sanhedrin (after Hillel death)
 Descendant of King David, grandson of Rabbi Gamliel II, Nasi of the Land of Israel, compiled the Mishna around 200CE
 Talmud, Berachot 7a.
The Torah: The Five Books of Moses (Genesis, Exodus, Leviticus, Numbers, Deuteronomy)
Rav Avraham Yitzchak Kook. Olat Reiyah. Composed in (c.1900 – c.1930 CE). Commentary on the Siddur (Jewish prayer book)
Sergio DellaPergola, J. Even (2001). Papers in Jewish Demography 1997, Jerusalem, The Hebrew University, 2001
Baron, S. (1976). A Social and Religious History of the Jews, 1976
De Tudela, B (ca 1170). Sefer Massa’ot [The Itinerary of Benjamin of Tudela. Travels in the Middle Ages. London, 1907; Signer, 1983]
Botticini M. and Eckstein Z. (2012). The Chosen Few: How Education Shaped Jewish History 70-1492