U+2234 -- The Story of the Symbol ∴

Origins

We don’t know when the symbol as such appeared. Probably very long ago. The triangle is a popular symbol in itself, and three points naturally present themselves as its symbol / representation. If we look at other triangular symbols, such as the triquetra¹ (from Latin tri – three, and quetrus – equipped with a horn, thus three-horned, originally meaning triangle) or triskelion², the triple spiral, tripod, or cyclic group \(C_3\) (e.g., \(ZZ_3\)), we easily reach Neolithic times.

Modern Usage

U+2234 is the UTF-8 codepoint bearing the name “Therefore”. It was added among mathematical operators (section “Mathematical Operators”) in version 1.1 in 1993. Johann Rahn is considered its creator and first user⁵:

Rahn used it in place of the word therefore. A trivial use in mathematics might look like this:

\[ \begin{aligned} x + 1 = 10 \\ ∴ x = 9 \end{aligned} \]

In his 1659 work, he still freely alternates between ∴ and ∵. However, in its English edition of 1668, he almost exclusively uses ∴ to indicate the derivation of mathematical proof. In the 18th century, the symbol was used quite freely, with cases where it simply denoted an unknown (like today’s \(x\)). In the 19th century, it was standardized to its original use.

Similar Symbols

The symbol ∴ is truly simple graphically, so it’s no wonder that there are several very similar typographic symbols:

• ∵ – used in place of the word because
• ஃ – is a character in the Tamil script denoting a special sound of the language – āytam⁶.
• ⁂ – either a typographic symbol indicating minor changes in text – meant to draw attention. Sometimes also used to separate sections in text. The symbol is also used in meteorology to indicate light snowfall.
• ⛬ – a graphically almost identical symbol used by the Japanese geographic service to mark significant historical objects on maps (such as excavations, burial grounds, etc.).
• ɔ: – a typographically similar symbol used by Norwegian and Danish typographers as an abbreviation for viz.⁷

In text typed on a computer/typewriter, ∴ is often replaced with :. or .·..

∴ and Freemasons

The history of the use of ∴ in Freemasonry is not entirely clear. The first use for abbreviating a word or name appears ⁸ in the list of the La Sincerité lodge in Besançon (France) from 1764. This suggests it was used earlier. The “official” establishment of its use is considered to be a circular from Grand Orient de France⁹ dated August 12, 1774, where the name of the Grand Orient was consistently abbreviated as G∴ O∴ de France and informed other lodges about new premises. Other authors, e.g., F. Chapuis¹⁰ claim that the symbol was used earlier, albeit rotated¹¹.

The period of introducing the symbol overlaps with the French Revolution. And the very motto of Freemasonry liberté, égalité, fraternité¹² intertwined with the revolution itself, changes in social order, and is still the motto of France today. It is therefore possible that during this time there arose a need for a simple symbol that would connect Freemasons of different faiths and origins.

There is also a kreuzer minted in 1773 in Frankfurt am Main, which placed ∴ in a prominent position in the shining Delta¹³.

Common Usage

It is standardly used to abbreviate words, names, titles, etc. For example:

• F∴ – Frère (Brother), in the Czech environment B∴
• FF∴ – Frères (Brothers), in the Czech environment BB∴

Symbolic Meaning

It seems that the general attitude is that there is none. At least according to the Encyclopedia of Freemasonry:

Three points in a triangular form (∴) are placed after letters in a Masonic document to indicate that such letters are the initials of a Masonic title or of a technical word in Freemasonry, as G∴M∴ for Grand Master, or G∴L∴ for Grand Lodge. It is not a symbol, but simply a mark of abbreviation. The attempt, therefore, to trace it to the Hebrew three yods [ייי], a sign of the Tetragrammaton, or any other ancient symbol, is futile. […] it is probable that the idea was suggested by the sacred character of the number three as a Masonic number, and these three dots might refer to the position of the three officers in a French Lodge.

— San Pedro Masons¹⁴

I have not been able to discover information about whether the mathematical meaning of ∴ was known and had any impact on the use of the symbol in Freemasonry. However, I found it a shame to stop here when the symbol offers us three main parts – 3, dots, and triangle.

3

The number 3 in Freemasonry symbolizes many key aspects, including morality, equality, justice. It is the basis for many rituals and symbols that are an integral part of Masonic tradition and philosophy. It symbolizes stability between body, mind, and soul (microcosm); heaven, earth, and underworld (macrocosm); … Divine trinities are also popular – Father-Son-Spirit, Jupiter-Juno-Minerva, Osiris-Isis-Horus, Trimurti (Brahma, Vishnu, Shiva).

Where two represents duality, often conflicting, three brings harmony¹⁵¹⁶. Three often appears in architecture/design, such as the rule of thirds. Most people have trichromatic vision (RGB). There are often three sections in storytelling (setup, confrontation, resolution).

• 3 are the Masonic degrees – Apprentice, Fellow Craft, Master¹⁷. The Apprentice is 3 years old.
• Three Great Lights – the Volume of Sacred Law, the Square, and the Compasses.
• Three Lesser Lights – Wisdom, Strength, and Beauty.
• Three immovable jewels of the lodge – the rough ashlar, the perfect ashlar, the trestle board.
• Three movable jewels of the lodge – the plumb, the level, the square.
• Three steps to Solomon’s Temple.
• Three barred windows of Solomon’s Temple. Sometimes there are also three doors.
• Three principal officers of the lodge / three gavels – the Worshipful Master, the Senior Warden, and the Junior Warden¹⁸.
• Three tools of the Apprentice – hammer, chisel, 24-inch gauge.

(Equilateral) Triangle

The triangle is also a very popular symbol. Just because of the simplicity of the shape. It appears in many Masonic and non-Masonic symbols and properties. For instance:

• The Shining Delta, the symbol of the Great Architect of the Universe
• Solomon’s Seal / Star of David
• Masonic square

We’ve already mentioned Pythagoras in the context of the number three. Pythagoras is also credited with discovering intervals in music and inventing the monochord¹⁹.²⁰ The Pythagoreans, of course, noticed that some numbers can be arranged (i.e., objects in a number corresponding to the number) into an equilateral triangle. We call these triangular numbers (\(T_n\)).

So our ∴ actually corresponds to \(T_2\). Another interesting one is \(T_4\), which has its own name – tetractys (τετρακτύς). The Pythagoreans used the tetractys to build and explain the entire world. Their sum gives the magical ten (\(1+2+3+4\)), which for the Pythagoreans signified divine perfection. From the tetractys, one can also derive the basic harmonic ratios: 1:2 – octave, 2:3 – fifth, and 3:4 – fourth (and if we extended to higher \(n\), other intervals as well)²¹.

Conclusion

Symbols are actually shortcuts²² that, besides their conventional and apparent meaning, also have specific connotations (assigned meanings). They are abstractions, shortcuts, visual forms of thought that carry significant emotional charge. We, humans, are responsible for the meaning and interpretation of symbols (e.g., the change in connotation of the swastika).

Symbols help create and transmit stories. And stories are the basic building blocks of human societies and individual lives. It is therefore natural that psychologists pay great attention to them. Symbols are used in religious systems, social structures, political groupings, in science, in business, in personal life, etc. Symbols can be used to communicate universally and also to evoke emotions – and these can be misused for manipulation. I consider the moment when they begin to represent dogma to be critical. Generally, when they are used to suppress critical thinking. An example might be numerology, which uses (natural) numbers, assigns them its own meaning, and ignores the difference between correlation and causality. Or any other example where an inward look is replaced by an external enemy²³.

Symbols are not an anachronism, just look 🔎 at modern emojis in electronic communication²⁴.

Jung describes that various symbols that recur in human history may refer, among other things, to archetypes and thus evoke in us tendencies toward certain emotional or imaginative contents. Symbols then tend to resonate even with our (collective) unconscious:

I am of the opinion that the unification of rational and irrational truth does not occur so much in art as much more in the symbol. The symbol has in its nature two sides, rational (reasonable) and irrational (unreasonable).

— C.G. Jung (²⁵, p. 73)

For me, a symbol is primarily about an invitation to reflection. It’s not just about their generally accepted meaning. The main thing is to explore, and perhaps sometimes understand, what they mean to me personally.