Mathematics in language — Transcendental algebra

Does mathematics transcend all languages? Mathematical equations seem to be able to communicate quantities, derivations, theorems and proofs across a large number of people, which may make it seem that mathematics is generally universally intelligible. The logic it contains is sort of homologous to what we see in language. The concepts of negation, comparison and association all find their own counterparts represented in mathematics. The question begs, would mathematics be a universal writing system to convey messages across any language?

The idea of universalising languages and writing systems has been around for several centuries, with earlier attempts proposing international auxiliary languages which could be intelligible to specific groups of countries, and can exist naturally as lingua franca. Ultimately, international auxiliary languages are foreign languages within a localised region where inhabitants do not share a common first language. Esperanto, and Volapük were historical attempts at creating such a universal system, and have gained some traction in communities around the world. With publications and thousands of “native” speakers, these constructed languages have been self-professed to be successful in achieving the aims of an international auxiliary language. Critics, however, have slammed Esperanto’s claims, citing that it is highly eurocentric, highly sexist, and that the community is just a “stateless diasporic linguistic minority”. Nonetheless, it is still a prominent attempt at creating a universal second language for communication.

In the 17th century, Gottfried Leibniz wrote on characteristica universalis, a universal and formal language which could be used to express mathematical, scientific, and metaphysical concepts. His goal, was an alphabet of human thought. The use of symbols to represent such concepts instead of sounds and phonemes makes it one of the famous examples of attempts to create a pasigraphy. He thought of Egyptian and Chinese writing, as well as chemical signs as examples of real characteristics, demonstrating his interest in the use of ideographs to represent ideas such that they can be read, translated, and understood by each and every language (Couturat, 1901). In several occasions, he represented his philosophical thought using diagrammatic reasoning, as pictograms like De Arte Combinatoria (On the Art of Combinations), where he attempted to show the Aristotelian theory of the use of combinations of the four ancient elements, water, earth, fire and air, to make up all the materials. However, at the turn of the 18th century, Leibniz lost the will to create the universal character he once imagined and worked so long for. He admitted the difficulty in doing so,  and also sort of implied that the human civilisation was not ready, at that time, for such a universality to take hold.

Leibniz never wrote in great detail about what a universal character should be, though scholars have proposed criteria from analysing his works. Cohen (1954) suggested that such a character would serve as an international auxiliary language, symbolism for the exact and systematic expression of all present knowledge, and an instrument of discovery and demonstration. Leibniz’s works have inspired many attempts throughout the centuries to create languages and writing systems which support his intuitions. One such example is Ithkuil, a constructed language by John Quijada in 2004, which has been discussed earlier in this blog, shown here: Hypothetical Representation of a Language — Ithkuil

These thoughts, projects and works have led up to the creation of transcendental algebra by Estonian linguist, Jacob Linzbach, in 1921. Having written on the Principles of Philosophical Language five years prior, Linzbach set off to create a universal writing system based on mathematics and its inner workings. Here we give a few examples of operations featured in transcendental algebra, and their intended or supposed attached meanings in human languages.


One of the four basic arithmetic operations, addition is signified in mathematics by a plus symbol “+”. Addition is commutative and associative, but linguistically, the order of “operations”, that is, how nouns are ordered, may change the meaning semantically. Is there a reason why one noun is placed before the other, or does the order convey a certain level of “semantic power” when it comes to discourse? Nonetheless, the plus symbol conveys the linguistic equivalent of “and” or “with”. When a series of nouns joined by the plus symbol are placed in parentheses ie (a+b+c+d+…), it can linguistically denote a category, like stars, animals, or tools.

In prepositions, addition can be used to express the concepts of on (+y), in front of (+z), and to the right (+x). It can also express “from”. Temporally, addition can be used to represent the future tense, “will” (+t).

I, you and he/she, represented in transcendental algebra


The operation of removal of objects from a collection, subtraction is represented by the minus symbol “-“. Subtraction is anti-commutative and non-associative, meaning that the order of operations matter. A person without an umbrella does not mean the same as an umbrella without a person.

In transcendental algebra, it not only represents the concept of absence of things, but also negation, “towards” and other spatial descriptions. The concept of “under”, to distinguish it from “not on”, is further represented by a set of parentheses “(-y)”, rather than just “-y”. When attached to “x” and “z”, it gives the meaning of “to the left” and “behind” respectively. 

A person without two eyes, i.e. blind


Equations and inequalities are logical judgements, which assert the equality, or lack thereof, of the two expressions on either side of the equation. In transcendental algebra, these symbols still preserve this meaning and representation, often serving to symbolise the association of the expression on the left hand side with the right hand side.

No, the book is not on the table.

The approximation symbol denotes the similarity of one entity to another, but not exactly equal to that entity.

A monkey is similar to a person

The inequality symbols, “<” and “>”, largely present a way to demonstrate comparisons between entities, preserving the original mathematical meaning as well. “>” can mean larger, or more, while “<” means smaller, or less. When used without the intention to compare, these signs will mean large and small respectively.

A big house and a small house

Final remarks

Transcendental algebra has demonstrated the theoretical representation of human language as a universal writing system in the form of mathematical expressions and equations. Although seemingly abstract, and the lack of any ability to properly denote proper nouns and names, this writing system has presented an interesting take on the use of such mathematical operations in the field of linguistics. Obscure, and really more intended to be a proof of concept, transcendental algebra is one of the few writing systems that took mathematics and logic seriously, and pretty much ran with them. While overshadowed by other projects like Blissymbolics, Linzbach’s creation has gained slight prominence, even featuring as one of the questions in the inaugural International Olympiad in Linguistics in 2003. Nonetheless, transcendental algebra has remained as one of the historical projects which attempted to achieve what Leibniz had envisioned two centuries prior.

Further reading

Additional explanations and examples can be found here:


I stumbled upon this writing system when looking up International Olympiad in Linguistics problems back in 2014. At first glance, it did look like a mishmash of mathematical symbols and pictographic glyphs, and almost nothing beyond that. It was upon further looking at these can the logic of the writing system be deciphered. It was an interesting problem in the Olympiad, showing how mathematics was once considered to be the backbone of a universal writing system. With its obscurity, however, there was not much material found on the web which discusses in detail the concept of transcendental algebra, other than the 1921 publication written by Linzbach.

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