I grew up In Russia and was raised on the metric system based on the decimal arithmetic. When we immigrated to the U.S., I had to learn a new to me (but, actually, very old) Imperial system of measurements having 12 inches to a foot. I still struggle with it (decimal system, apparently, was hardwired in my brain).
There are many number systems or positional notation systems. Ancient Babylonians used base-60, Hindu-Arabic system uses familiar decimal base-10 system, Mayans used base-20 system. There are base-2 binary, base-12 duodecimal, and base-16 hexadecimal systems, to name a few. Aside from the binary system used by computers, we humans mostly use decimal and, sometimes, duodecimal systems.
The all-familiar decimal system is used in the metric system of measurements. In base-10 positional notation, there are 10 decimal digits—0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Say, 4359 mean 4 thousand, 3 hundred, 5 tens and 9 digits. In other words:
4359 = 4×103 + 3×102 + 5×101 + 9×100
If a meter is a basic unit of length in the metric system, a decimeter is one-tenth of a meter, a centimeter is one-hundreds of a meter, a millimeter is one-thousands of a meter, a nanometer is one billionth of a meter, and a kilometer is a thousand meters. A kilogram is one thousand grams. Temperature is measured in centigrade, where each degree is one-hundreds of the difference between the temperature of boiling and freezing water.
In duodecimal or “dozenal” arithmetic, which is base-12 position system, there are 12 duodecimal digits—0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ (i.e., 10) and Ɛ (i.e., 11). 12 is written as 10 (which means 1x121+0x120). Duodecimal 50 is the same number as decimal 60: 5012=5×12 is the same as 6010=6×10. We traditionally measure time in the duodecimal system. There are 12 zodiac constellations, 12 months, there 2×12 hours in a day (12 hours from midnight to noon and from noon to midnight or, in the Talmudic system, 12 hours of daylight and 12 hours of the night). Unless you use military time, your clock is divided into 12 hours. Imperial measures are, in part, based on the duodecimal system: there are 12 inches to a foot. A dozen is the basic measure of quantity in the duodecimal system. Gross is 144=122, great gross is 1728=123. In the UK, 12 pence equals one shilling. The Romans used a fraction system based on 12.
Historians connect decimal and duodecimal systems with finger counting. There are 10 digits (fingers) on two hands—hence the use of the decimal system with ten digits. Human fingers (not counting a thumb) each has three bones—phalanges. Four fingers times thee phalanges in each finger give a convenient base-12 counting system.
12 zodiac constellations and approximately 12 lunar months in a solar year helped make duodecimal system popular. Duodecimal system has many mathematical advantages and has been described as the optimal number system.
On a deeper level, the Kabbalah examines spiritual roots of the decimal and duodecimal systems. According to sages of Kabbalah, the spiritual source of the decimal system (and the ultimate reason why we have ten fingers) are ten Sephirot—Divine Emanations.
The duodecimal system has to do with permutations of the letters in the Divine proper name, Tetragrammaton (“Havaya”)—YHWH. Three unique letters—Yud (Y), Heh (H) and Wav (W)—in four positions can have 12 permutations. This is the mystical reason behind twelve months and twelve signs of Zodiak. It is also the mystical reasons behind two sets of 12 hours: there are two letters Heh in the Tetragrammaton —hence two sets of 12 hours making up 24 hours in a day.
The number 12 is also related to the sephirot. The Partzuf Zeer Anpin (Z”A) (Microprosopus) is a configuration of six sephirot—Chesed, Gevurah, Tiferet, Netzach, Chod, Yisod—six emotional attributes of the soul. The number six can be visualized as six sides of a cube, which is why, in the Kabbalah thought, this configuration of six sephirot representing six extremities—four corners of the world plus up-down directions—responsible for our three-dimensional space. Three-dimensional cube has six sides and 12 edges. Or, in other words, there are 12 lines that connect these six sephirot. This, as we will see later, turns out to be highly significant.
Turning to our weekly portion of the Torah, Vayeshev (Gen. 37:1-40:23), we note that it starts with the struggle between Joseph and his brothers.
“And his brothers saw that their father loved him more than all his brothers, so they hated him, and they could not speak with him peacefully.” (Gen. 37:4)
According to the founder of Lurianic school of Kabbalah, Rabbi Isaac Luria Ashkenazi (a.k.a. the Holy Ari or the Arizal), ten brothers (six sons of Leah and four sons of Concubines, Zilpah and Bilhah) believed that the Jewish nation would be built up through ten sons of Jacob. They knew that the Avot—forefathers of the Jewish nation—each personified a specific sephirah: Abraham personified Chesed, Isaak personified Gevurah and Jacob personified Tiferet. Consequently, they reasonably assumed that it was the system of ten sephirot, according to which the foundation of the Jewish nation would be laid; and they, ten sons of Jacob, were this foundation. They also knew that each of the Patriarchs had a son that had to be rejected in order to purify the Jewish nation. Abraham had Ishmael who was sent away. Isaac had Esau, who sold his birthright to Yaakov. They thought that Joseph was the rejected son of Jacob. What about Benjamin, who was a son of Rachel and was not one of ten brothers conspiring against Joseph? According to the Arizal, the brothers thought that they represented ten sefirot of the Ohr Yasher (direct light), while Benjamin represented Ohr Hoser (reflected light). That is why they conspired to kill Joseph. In other words, the brothers were radical “decimalists”!
Little did they realized that the Jewish nation was destined to be built based on the “dozenal” principal—there would be twelve tribes (shevatim) of Jacob corresponding to twelve permutations of the Tetragrammaton!
Where did this division in the brothers’ affinity to different number systems come from? The leader of the ten brothers, Judah, was destined to be the king. Thus he represented the sephirah of Malchut—Kingship—the tenth sephirah. No wonder that Judah and his brothers espoused the decimalist philosophy of ten sephirot.
Joseph, on the other hand, personified sephirah of Yesod, which, in a human body, corresponds to a male procreative organ or a tongue. Both serve as organs of connection. We connect with others through speech, with the tongue being the primary organ of speech. A husband intimately connects to his wife through the sexual organ. The origin of this connectedness is the sephirah of Yesod, which connects the Z”A (Partzuf Zeer Anpin—the Microprosopus) with its female counterpart, Nukvah d’Z”A (Malchut). The number 12 represents twelve lines that connect six sephirot of the Z”A. It is not coincidental then that Joseph personified the “dozenal” principal—the number 12.
The rift between Judah and Joseph started with the original rift between their mothers—Leah and Rachel. It continued through the history resulting in the war between Maccabees (the followers of Judah’s philosophy of isolationism) against Jewish Hellenists (who perverted the extrovert ideology of Joseph), which we commemorate during Chanukah. It will finally be resolved at the end of days with Mashiach ben Yosef (Messiah, the son of Joseph) and Mashiah ben David (Messiah, the son of David) completing the task of our redemption in unity.
In the meantime, the battle between the decimalists and dozenalists rages on. Dozenal Society of America and Dozenal Society of Great Britain vociferously advocate the switch from decimal to the duodecimal system. Should they prevail, I will finally have to learn the Imperial system of measures. Until then, I will stick to the metric system.