The Symbolic Value of Aleph

“Aleph” (\aleph) has a great significance for set theorists. It is the symbol of infinite numbers, the main subject of set theory, and in fact the emblem of this mathematical field. But we rarely notice to its origin and symbolic value and even to the correct way of writing this symbol! In several occasions I saw that my Iranian and non-Iranian Jewish friends complained about the terrible shape of א symbols written by non-native Hebrew speakers, the א symbols that look like X or N!

Why is this matter such important to them? The answer is hidden in an ancient, complicated and mysterious cultural background that exists behind this particular symbol. This cultural aspect of Hebrew language makes א much more significant than just a letter in the alphabet. Here we are going to explain just a fraction of the symbolic meanings of א. However this is a very vast subject which is connected to many historical, social and religious matters that we should omit them in the benefit of shortening the text.

I would like to thank all of my Jewish friends who introduced good references to me about this subject.

א. Why א is called “Aleph”?

To answer the question in the title, first we should note to the root of the Hebrew language. One can divide the current living languages spoken in the middle East and Caucasus into three major families: Languages with Indo-European root like Persian and Armenian, languages with far Eastern root like Turkish and Turkmanian, and languages with Semitic root like Arabic and Hebrew.

Belonging to the same linguistic family means that some time in the past centuries, people who speak these sister languages, were able to understand each other’s language without any need to translation. This often indicates that despite of their seemingly different cultures, religions and traditions, these people belonged to a single tribe and lived in the same region. This historical fact even influenced the ancient texts like Torah. (See Genesis 11: 6 – 7) However in most cases estimating the precise time and place that the major dividing lines happened among them, is hard and the subject of controversial archaeological theories.

For example even now after years of different historical patterns that Arabs and Hebrew people followed in their journey through the history, and despite of different religions that they practiced for centuries, for almost every Hebrew word there is a similar corresponding word in the Arabic language and vice versa! These sister words often differ in just a single letter or the pronunciation of a few vowels. If you know which Hebrew letter is corresponding to which Arabic letter and how you should transfer each Hebrew vowel to an Arabic vowel and vice versa, you will have no problem to see that Hebrew and Arabic languages are in fact different evolutionary forms of a single ancestor language. In a deeper layer and with a little more effort one can see the same similarity in cultures and even religions between these two seemingly different middle Eastern ethnics. The latter fact is often very surprising for Iranians and Europeans who are of a totally different cultural background and linguistic family because in the first view Hebrew and Arabic languages and cultures seem very different to them.

So in order to discover the root of the name “Aleph” for the symbol א, we should go back to the ancient ancestors of Hebrew language because this ancient language underwent a long history of evolution in interaction with several generations of older extincted languages of the region.

Quoted from Wikipedia entry on Aleph:

The name aleph is derived from the “West Semitic word for ox” (i.e. ʾalp), and the shape of the letter derives from a Proto-Sinaitic glyph that may have been based on a Egyptian hieroglyph.

About Proto-Sinaitic Script and current Hebrew and Arabic alphabets one should note that the names of the letters in these alphabets are meaningful words in the corresponding languages. This indicates to the evolution history of these alphabets from a kind of picture-based script. For example:

  • (Home = ب = ב) \longrightarrow (بیت = בית = Bet)
  • (Palm of the hand = ک = כ) \longrightarrow (کف = כף = Kaf )
  • (Eye = ع = ע) \longrightarrow (عین = עין = A’yin)
  • (Head = ر = ר) \longrightarrow (راس = ראש = Resh)
  • (Sun = ش = ש) \longrightarrow (شمس = שמש = Shin/šimš)

See also the related Wikipedia articles on Canaanite languages and Phoenician alphabet. Furthermore note that the “ox” shape of the Aleph symbol carries a deep historical meaning corresponding to the interactions of ancient religions of middle East in Egypt, Mesopotamia and Iran. In all of these ancient cultures which influenced Judaism in their own way, the “ox” was a sacred animal directly related to the deities.

Just as some examples see Apis in Egyptian mythologyGugalanna in Sumerian religion from Mesopotamia, Gavaevodata in Zoroastrianism and Tauroctony in Mithraic mysticism both originated from Iran and compare these mythical sacred bulls with the story of Golden Calf in Judaism. In the next section we will see how these ancient traditions on considering holy aspects for bulls reformulated in Judaism in the form of associating divine properties to Aleph.

ב. Why א is the first letter of the Hebrew alphabet?

The place of א in the Hebrew alphabet could be just a historical accident but there is an important cultural aspect of Hebrew language that connects it to the religion and divinity. According to a Hebrew Midrash (מדרש = fable) the position of Aleph in the alphabet is determined by God himself:

All of the letters come before the Lord giving reasons why they should be the first letter – all, that is, except for the letter Aleph! When the Lord asked why, Aleph explained that since he was a silent letter, he had nothing to say. But the Lord honored Aleph’s humility and declared him to be the first of all the letters and to be honored as the first letter of the first word of the Ten Commandments. (I am Yahweh your God = אנכי יהוה אלהיך)   

This cultural background possibly explains why the fathers of Set Theory who were religious people and in their personal and philosophical writings considered infinite cardinals as a kind of divine objects, chose this new and very special mathematical symbol for referring to the infinities.

Although the divine aspects of Aleph in Hebrew culture is not limited to this fable and Ten Commandments. For example most of the words in Hebrew language and Bible that are related to God begin with Aleph. (e.g. אל, אלהים, אלוה, אדוני ,אדון עולם , אדיר) Interestingly enough, even in those cases in Tanakh which God is referred by a metaphoric word like “Fire” (אש), “Light” (אור) and “Love” (אהבה), Aleph is the first letter of that word!

Some experts in Jewish theology and mysticism presented even more complicated analysis of the connection between Aleph and God. For example Rabbi Dov Ber of Mezeritch (a disciple of Rabbi Israel ben Eliezer the founder of Hasidic Judaism) in one of his teachings appeared in the book, Or Torah, gives an interesting mystic interpretation of the position of the word את consisting from the first and the last letters of the Hebrew alphabet א and ת, in the first sentence of Torah, בראשית ברא אלוהים את השמים ואת הארץ, before the words for Sky/Heaven (שמים = سماء) and Earth (ארץ = ارض).

In his point of view this means that God created the Hebrew alphabet even before Sky and Earth. Thus א is the first creature that God has created because it is the first letter of the alphabet! From this he concludes that the Hebrew alphabets are the basic elements of every creature in the world, a philosophical point of view which reminds Pythagoreans’ idea about the essential role of numbers in the cosmos. This cultural similarity becomes more serious when we notice that Hebrew letters are corresponded to some numerical values too. (See the next section). Maybe Pythagoras took the idea from Rabbis during his travel to the Eastern countries including Iran and Judah.

Remark: There are other fables in Rabbinic literature that explain why ב is the first letter of Torah instead of א. Also there are traditional descriptions of the relation between א and ב as a kind of father – son relation because the word for “Father” in both Hebrew and Arabic languages is אב = اب. Also the Hebrew words for “son” and “create” are בר and  בריאה respectively. Both begin with ב. The usual mystic interpretation corresponded to these relations is that Aleph is the father and Bet is its son. Father sits silently behind while the son does creation. In the benefit of shortening the text we omit the extra explanations and leave this topic to the reader for further research.

ג. The symbolic value of א in Jewish mysticism.

Hebrew alphabet & Gematria NumbersAs it is explained in the previous section there is a strong connection between alphabet and divinity in Jewish culture. These two are also deeply connected with numbers (and in general with mathematics) through the mysticism. In fact Jewish mysticism (Kabbalah) is a combination of three elements; namely ReligionLanguage and Mathematics. In this section we explain some basic elements of this special mystical tradition, however the subject is much more technical and complicated than what is presented here.

At first one should note that the Hebrew alphabet consists of 22 letters, called Authioth, and in Gematria, each Hebrew letter or Auth (and so each word, sentence and text) has a particular numerical value. (See the table). It is a very special aspect of Hebrew culture that makes the language heavily connected to the numbers and mathematics. Nowadays even in ordinary daily uses sometimes Hebrew letters represent the numbers directly. A good example (of many others) is the way that different districts in Beer Sheva are named. Quoted from Wikipedia article on Beersheba:

Today, Beersheba is divided into seventeen residential neighbourhoods in addition to the Old City and Ramot, an umbrella neighborhood of 4 sub-districts. Many of the neighbourhoods are named after letters of the Hebrew alphabet, which also have numerical value, but descriptive place names have been given to some of the newer neighborhoods.


Remark:
 The idea behind this type of coding the language with numbers is quite familiar to the logicians. Recall Godel’s numbering for the symbols (and formulas) of the first order Kurt Godel (Square)language which seems to be inspired by Jewish mysticism. In fact in his Incompleteness
Theorem
, Godel did first order Gematria! The purpose of Gematria is so similar to what Godel did, namely coding entire language with numbers, then doing some pure arithmetical calculations and finally decoding the numerical result and getting a hidden message that wasn’t immediately clear since the beginning. Certainly unfolding the existence of a valid but non-provable arithmetical sentence could be counted as an example of such a surprising result!

In Gematria the number corresponded to א is 1 but it also could represent several other numbers depending on the context. For example א with a dot below is corresponded to the number 1,000; with a line above, it will represent 1,000,000. With a line below it is 10,000 and with two dots below it is 10,000,000.

Remark: The pronunciation of the word “Aleph” in Hebrew and Arabic is so close to the word “Alph” (אלף = الف) which means one thousand in both Hebrew and Arabic languages.

When א represents 1 it usually symbolizes the uniqueness of the God or simply God himself.

When א represents 1000 it may symbolize a millenniumFor example in a Messianic point of view, Messiah will come after 6000 years because there are 6 Alephs in the first sentence of Torah:

בראשית ברא אלוהים את השמים ואת הארץ

Aleph & Star of DavidAnother numerical valuation for Aleph is 111 because the Gematria number of its name אלף is equal to 1 + 30 + 80. In this sense א symbolizes the picture of three-in-one or Hashilush Hakadosh (השילוש הקדוש). With this interpretation Aleph could be used to symbolize the number 666 (= 6\times111) when it is used in hexagonal symbols like a Star of David made by Alephs. The number 666 is of a great importance in Kabbalah as well as many other mystical traditions around the world.

But possibly the most important number corresponding to א is 26. In order to obtain this number one should note that the א symbol consists of three symbols in Hebrew alphabet namely one diagonal Vav (ו) and two upper and lower Yuds (י) beside it. In older calligraphic Hebrew scripts particularly those which belonged to Iraqi Jews, this composite nature of the letter א is more clearer. In this sense א is י+ו+י and its corresponding numerical value is 26 = 10 + 6 + 10. The interesting point is that the number corresponding to the sacred name of the God (Yahweh = יהוה) is also 26 = 10 + 5 + 6 + 5.

In Jewish mysticism the latter correspondence is the source of many interpretations often related to the God-Man relation. In this sense the diagonal Vav is a dividing line which divides the entire world into two upper and lower realms as well as connecting them to each other like a nail.

Remark: The name of the letter ו in Hebrew language is Vav (וו) which means “nail” or “peg”. Also the name of the letter י is Yud (יד = ید) which in both Hebrew and Arabic languages means “hand”. This should not be confused with another Hebrew word Yad (יד) of Persian root with the same spelling and similar pronunciation that means “memorizing” or “remembering“. For example in Hebrew phrases like Yad Vashem (יד ושם) and Yad La-Shiryon (יד לשריון), the word יד is used in its second meaning. Also in Persian there are many words that are built using this latter meaning of the word “Yad”. (e.g. Yad = Memory, Yad Bud = Memorial, Yad Gerfertan = Learning, Yad Dadan = Teaching, Yad Dashtan = Knowing).

The upper Yud in Aleph represents the hidden aspect of God (Ein Sof = אין סוף), whereas the lower Yud represents the revelation of God to mankind. In this sense Aleph is a symbol of human as a half-earthly and half-divine creature. The interesting point for set theorists is that the word Ein Sof may be translated as “no end”, “unending”, “there is no end”, or “Infinity”! Though, I’m not sure whether the first set theorists who used this symbol for referring to infinite cardinals, had this subtle point of Jewish mysticism in mind or not.

Remark: Considering the role of Vav in Aleph as a conjunction of two realms comes from several sources. First is the Hebrew and Arabic grammar where Vav plays a role as same as “And” in English. Second is the position of Vav as the Belly of Torahnamely the central letter of entire 304805 letters of Torah which divides the book into two different parts of the same length. This special Vav happens in the word Gachon (גחון) in Laviticus 11:42 which means “belly”! According to one of over 4000 rules in Soferut (the laws of writing a Kosher scroll of Torah by scribes), this special Vav in the center of Torah should be written in an over-sized form to emphasize on its special position in the book.

ד. A bit Kabbalah using the name of some set theorists!

Finally let me to conclude this post with a bit cardinal arithmetic!

My name in Hebrew is עלי צאדק דקיקי. Its corresponding Gematria number is 529. On the other hand the Gematria number corresponding to the name of one of my Iraqi Jewish set theorist friends, Asaf Karagila (אסף קרגילה),  is 489. If we add these two numbers, we get 1018. Interestingly enough, in two places in the Hebrew Bible there are words with the same Gematria number which are directly derived from the word “Torah (תּוֹרָה)”! (As the word “Torah” appeared here mysteriously it could be interesting to see this related article about Bible Code).

First is the word בְּתוֹרָתִי in Exodus 16:4,

וַיֹּאמֶר יְהוָה אֶל-מֹשֶׁה, הִנְנִי מַמְטִיר לָכֶם לֶחֶם מִן-הַשָּׁמָיִם; וְיָצָא הָעָם וְלָקְטוּ דְּבַר-יוֹם בְּיוֹמוֹ, לְמַעַן אֲנַסֶּנּוּ הֲיֵלֵךְ בְּתוֹרָתִי אִם-לֹא.

Then said the LORD unto Moses: ‘Behold, I will cause to rain bread from heaven for you; and the people shall go out and gather a day’s portion every day, that I may prove them, whether they will walk in My law, or not.

Second is the word וְתוֹרוֹת in Nehemiah 9:13,

וְעַל הַר-סִינַי יָרַדְתָּ, וְדַבֵּר עִמָּהֶם מִשָּׁמָיִם; וַתִּתֵּן לָהֶם מִשְׁפָּטִים יְשָׁרִים, וְתוֹרוֹת אֱמֶת–חֻקִּים וּמִצְו‍ֹת, טוֹבִים

Thou camest down also upon mount Sinai, and spokest with them from heaven, and gavest them right ordinances and laws of truth, good statutes and commandments;

The story becomes even more interesting when we notice to the content of the books of Ezra and Nehemiah which describes their work in rebuilding Jerusalem and its wall during the Second Temple period (see also this post of mine for more information) when the Hebrew people, often from current Iran and Iraq regions, gained the permission of returning to the Persian Judea under Artaxerxes I of Persia, the heir of Iranian emperor Cyrus, the Great, who saved the Jews from slavery in Mesopotamia and gave them their freedom once again! Isn’t it a really interesting accident? If you are not still surprised so look at this …

Beside the usual Gematria number corresponding to a name in Hebrew language there is another number associated to a ciphered version of that name, called its Atbash (אתבש) form, which is produced by replacing the letters with other letters of inverse order. The Atbash form of my name in Hebrew is זכם התקד קדמדם corresponding to the number 764. The Atbash form of Asaf’s name is תחו דגרמכץ which is equal to the number 771. (Final letters are counted as usual letters). Now if we add these two numbers, we get 1535.

Interestingly 1535 is corresponding to the Gematria number of an entire verse in Daniel 2:12,

כָּל-קֳבֵל דְּנָה–מַלְכָּא, בְּנַס וּקְצַף שַׂגִּיא; וַאֲמַר, לְהוֹבָדָה, לְכֹל, חַכִּימֵי בָבֶל.

For this cause the king was angry and very furious, and commanded to destroy all the wise men of Babylon.

Amazingly it is about Daniel’s interpretation of Nebuchadnezzar‘s Dream about the fall of Babylonian Kingdom by Iranians and upcoming freedom of the Babylonian Jews! It is also interesting to know that Daniel’s tomb is now located in the ancient Iranian city Susa, one of the capitals of Persian Empire in 500 B.C.

Not enough? OK, let’s choose another computation method! The Atbash method that we used in the previous paragraph, was just a part of several methods of changing the words in Hebrew language according to Temurah rules. Now let’s try Avgad which is based on changing each letter in a name with its immediate successor. For example replacing א by ב and ג by ד. In this case the Avgad form of my name is פמכ קבהר הרכרכ corresponding to the number 892. Also the Avgad form of Asaf’s name is בעצ רשדכמו corresponding to 732 (= 66 + 666). Now if we add these two numbers, we get 1624.

1624 is the Gematria number of an entire verse in Isaiah 41:29,

 הֵן כֻּלָּם, אָוֶן אֶפֶס מַעֲשֵׂיהֶם; רוּחַ וָתֹהוּ, נִסְכֵּיהֶם.

Behold, all of them, their works are vanity and nought; their molten images are wind and confusion.

The verse is about the vanity of Babylonian idols and surprisingly the content of these middle chapters of the Book of Isaiah (Deutero-Isaiah) is about God’s promise to captive Jews in Babylon that he will send someone (Cyrus, the Great) from the East (Iran) who attacks from the north to free them for returning to Jerusalem! See Isaiah 41:25,

הַעִירוֹתִי מִצָּפוֹן וַיַּאת, מִמִּזְרַח-שֶׁמֶשׁ יִקְרָא בִשְׁמִי; וְיָבֹא סְגָנִים כְּמוֹ-חֹמֶר, וּכְמוֹ יוֹצֵר יִרְמָס-טִיט.

I have roused up one from the north, and he is come, from the rising of the sun one that calleth upon My name; and he shall come upon rulers as upon mortar, and as the potter treadeth clay.

Finally if we apply the Albam (אלבם) rule on Asaf and my names by replacing each letter with the letter in 11 place after it (mod 22) we get another interesting message completely related to the previous examples which is about domination of foreign nations over Hebrew people!

In Albam method the letter א should be replaced by ל (and vice versa), the letter ב should be exchanged by מ and so on. Thus the Albam form of my name is האש זלסח סחשחש corresponding to the number 1087 and Asaf’s name becomes  לדו חטנשאע with Gematria number 478.

The sum of these two numbers is 1565 which is the Gematria number of an entire verse in Isaiah 26:13,

יְהוָה אֱלֹהֵינוּ, בְּעָלוּנוּ אֲדֹנִים זוּלָתֶךָ; לְבַד-בְּךָ, נַזְכִּיר שְׁמֶךָ.

O LORD our God, other lords beside Thee have had dominion over us; but by Thee only do we make mention of Thy name.

Amazingly it is in the Book of Isaiah again! Precisely in Proto-Isaiah part of the book. This time the content is about this inspiring idea that even under domination of foreign non-monotheist nations like Assyrians, that was only the unique God that Hebrew people believed in. The hard situation that came to an end when another group of mono-theist people at that time, namely Iranians, rose to the power.

Remark: It is interesting to mention that there are two places deemed as burial place of Isaiah. First Nahal Dishon in Palestine and second near Emamzadeh Esmaeil in the city Isfahan of Iran. For more information about the possible burial  places of some Biblical figures which are located in the current borders of Iran see here.

Remark: It is not clear whether Judaic tradition of mono-theism is inspired by the same idea in Zoroastrianism or vice versa. Even it is possible that the abstract idea of the existence of an invisible unique God, be an independent discovery of Iranians and Israelite. Anyway as far as I know Judaism and Zoroastrianism were the only mono-theistic religions in the whole world before Christ and there are some undeniable similarities between their world view and terminology. However I can’t mention them all here.

Open question: Is it true that the the sum of Gematria numbers of Asaf and my names after applying any permutation rule on Hebrew alphabet, is the Gematria number of at least a word or verse in Hebrew Bible that is related to Iran and Iraq (Babylon)?!

I wonder what would happen if we add more Hebrew letters to our current mathematical symbols as well as bringing all those infinite cardinal arithmetic into Kabbalah! Anyway, infinitary Kabbalah seems a really interesting research field! Who knows what is waiting for us to be discovered using these strange techniques?!

 Judea under the Persians

6 thoughts on “The Symbolic Value of Aleph

  1. Is there any relation between Kabbalah and Jafr (Science of the Letters in Islam)?

    In Kabbalah there is a tree that all of mysticism is around of that and I heard this is not irrelevant to the story of Adam and Eve and Satan in Heaven, which is quoted in Bible and Quran.

    Also there are rumors that the story of discovering the gravity rule by Isaac Newton by falling of an apple on his head is inspired by Kabbalah’s tree!

    Can you find a relation between Kabbalah’s tree and Set Theory?

    • You brought up very important questions! Although writing a comprehensive answer here is not possible but let me add a brief explanation and some references for further reading.

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      Kabbalah is possibly the most ancient mystic tradition that uses mathematics as a way of unfolding the secrets of the world. It influenced several other Eastern and Western cultures through cultural interactions often made by those Jewish people who immigrated into these communities during several periods of Jewish Diaspora and when some of them converted to the dominating religions of these regions like Christianity in the Europe and Islam in the middle East.

      As you mentioned “Jafr” in Islam is directly inspired by Kabbalah. However its language is Arabic, the language of Islam. There is also a Greek and Latin version of Kabbalah appeared in Pythagoreans’ mathematical mysticism and mystical traditions of secret societies in medieval Europe. However both Christian and Islamic Kabbalistic traditions never became as complicated as what exists in Judaic context because main stream authorities in Catholicism and Islam strongly opposed this way of dealing with religious texts because they counted them as a kind of “magic” or “dark science” which is forbidden in both of these religions but widely accepted in Judaism.

      Anyway Kabbalah continued to exist in its Christian and Islamic forms even up to these days. A good example of Christian Kabbalah which greatly influenced the Western culture and even politics in 20th century is in Canto No. 33 in Dante Alighieri’s Purgatorio where Beatrice makes a mysterious prophecy about the coming of a savior ruler who reestablishes the Roman Empire. Interestingly she refers to the name of this savior as 500 – 10 – 5 (cinquecento dieci e cinque)!

      Non sarà tutto tempo sanza reda
      l’aguglia che lasci? le penne al carro,
      per che divenne mostro e poscia preda;
      ch’io veggio certamente, e per? il narro,
      a darne tempo già stelle propinque,
      secure d’ogn’intoppo e d’ogni sbarro,
      nel quale un cinquecento diece e cinque,
      messo di Dio, anciderà la fuia
      con quel gigante che con lei delinque. (Purg. 33.37-45)

      The interpretation of this number is the subject of many discussions. As far as I know the widely accepted version among Dantelogists is that the numbers 500, 10, 5 represent the Roman letters D, X, V which form the word DXV. Then they permute this word to DVX and reshape V to U to get the meaningful word DUX which in Latin means “Leader”. Read more about this particular part of Dante’s Divine Comedy here.

      In the history of Italy there were people who claimed to be Dante’s predicted savior of Italy. The last and possibly the most important one was Benito Mussolini whose title was “Il Duce” the Italian version of the word “DUX”. The title was chosen in such a way to remind a connection between himself and Dante’s 500, 10, 5.

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      By Kabbalah tree, you possibly mean the tree of ten Sephirot. As same as any other symbol in Kabbalah it symbolizes several things depending on the context:

      – First is the The Tree of Life which its roots are up and its branches are down. Together with the Tree of Knowledge of Good and Evil, they are two trees of the paradise in Bible. Adam and Eve ate the fruit of the latter and found one of the properties of God, namely the ability of recognizing good from evil, then God hid the Tree of Life because he feared that they may eat its fruit as well and become immortal just like God himself! Compare this type of God’s behavior toward humans with several cases of Greek gods and goddesses’ revenges on humans in a fit of jealousy. In Greek myths those humans often were or claimed to be better than gods in a special ability or a particular branch of arts. For example see the story of Greek god Apollo and Marsyas.

      – [Quoted from Wikipedia] Sephirot are the 10 attributes/emanations in Kabbalah, through which Ein Sof (The Infinite) reveals himself and continuously creates both the physical realm and the chain of higher metaphysical realms (Seder hishtalshelus).

      The latter also suggests a pattern that a novice in Judaic mysticism could follow to reach the highest degree of spirituality and finally meet the God himself. Compare it with the “Soluk” pattern of a “Salik” in the Iranian traditional mysticism.

      Compare also the role of a Rebbe in Hasidism with a Murshid in Sufism. Particularly look at the notion of a Tzadiq HaDor (i.e. صادق دوران).

      “Tree of Life” or Etz Chaim in Hebrew is also the name of a book by a famous Kabbalist, Isaac ben Solomon Luria, inspired by Zohar, one of the main references of Kabbalah studies.

      Note that the tree of Sephirot is a graph with 10 vertexes and 22 edges (each labeled with one of the letters of Hebrew alphabet and so a number). The number 10 is of a great symbolic importance here. It is related to Ten Commandments which is a central concept in Judaism. Also it is deeply connected to the sacred notion of Tetractys in Pythagoreans’ secret mysticism. Tetractys is a triangle built by 10 dots arranged in four rows. It is also frequently used by Kabbalists as a rearrangement of the Kabbalah Tree.

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      About Newton and that famous apple, I doubt whether the story is real! Note that as the fruit of the Tree of Knowledge, apple is the symbol of knowledge. Thus it seems possible that the story of falling an apple on Newton’s head is made after his great discovery to relate it to divinity and Biblical stories.

      However one should not forget that despite of his prestigious scientific and academic position, Newton’s life had a dark and less well-known side connected to alchemy and mysticism! He believed in God and as any other believer, Bible played a great role in his life. Maybe the story of apple is made by Newton himself to pretend that he is chosen by God to make such a discovery! Who knows?!

      You can find more about religious beliefs of Isaac Newton in the following book. In this book some evidences of Kabbalah studies of Newton are presented.

      The religion of Isaac Newton by Frank Edward Manuel.

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      About set theory and Kabbalah, well, roughly speaking Set Theory is nothing but formalized Kabbalah! Maybe it is not that clear nowadays because the main ideas and projects are hidden behind heavy technicalities. But three or four generations before us the religious aspects of set theoretic notions weren’t as unclear as now. As Leopold Kronecker once said:

      I don’t know what predominates in Cantor’s theory — philosophy or theology, but I am sure that there is no mathematics there!

      And recall David Hilbert’s famous comment with a clear reference to the Book of Genesis:

      No one can expel us from the paradise which Cantor has created for us!

      As another example of religious aspects of Set Theory, note that despite of the fact that some of the Axioms of ZFC follow from some others but in texts we usually see them in “10 principles” format which reminds Ten Commandments! Also many of set theoretic concepts are named with religious terminology. Things like “cardinal”, “absolute”, or “reflection principle”.

      Regarding the Reflection Principles which are the base of many large cardinal axioms it is interesting to know that the idea behind it is purely theological! In this sense the “proper class of all sets (V)” is corresponded to God that according to a religious belief is expected to be “indescribable” by any man-made expression. Roughly speaking this means that if you choose an arbitrary (first order or second order) property of the universe (V), say “being a model of ZFC“, it should not determine V uniquely. Compare this approach towards V with Jewish approach towards God in Apophatic theology, particularly notice to the notion of שם המפורש.

      In the other words the reflection principle in set theory says that for any given property of the entire universe V, there should be a level of existence like V_{\alpha} (for some ordinal \alpha) that “reflects” this property. In our example this means that V_{\alpha} should be a model of ZFC. This reflection of the properties of the universe usually determines a large cardinal axiom. For example in the case of considering “being a model of ZFC” as a property of V, the reflection principle leads us to the existence of a strongly inaccessible cardinal \kappa because we have V_{\kappa}\models ZFC.

      There are also some interesting papers addressing the special theological aspects of this “reflection principle philosophy” behind large cardinal axioms and the correspondence between “proper class of all sets” and God which is clear in Russell’s philosophical works particularly in connection with his theorem on non-existence of the set of all sets. For example see the following paper:

      Cantor’s Absolute in Metaphysics and Mathematics by Kai Hauser.

      I can send you some more references via mail if you are interested in investigating more around this topic. There are also nice references regarding the connection between Set Theory, Kabbalah and religion. For example look at these interesting books:

      The Mystery of the Aleph: Mathematics, the Kabbalah, and the Human Mind by Amir Aczel.

      Reality in the name of God, or Divine Insistence: An Essay on Creation, Infinity, and the Ontological Implications of Kabbalah by Noah Horwitz.

      Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity by Loren Graham, Jean Michel Kantor.
      (A Persian translation of this book by Dr. Zare Nahandi from math department of the University of Tehran is also available. It is published by Fatemi publications.)

  2. Dear Ali,

    This is to thank you for your article on the aleph. I must admit that even for a native speaker of Hebrew like myself it is interesting.

    Regards,
    Rami.

    • Dear Prof. Grossberg,

      Thank you for your kind words. Exploring the world of Hebrew language, as a historically significant middle Eastern language, was really interesting and eye opening to me particularly when I learned more about the historical interactions of people living in this ancient area of the world through learning about the interactions between their languages.

      Best Regards

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