A mathematical view of articulation and translation


In a previous article I asked the question “is language inherently inadequate?” and in another previous article I introduced the analogy comparing languages to vector spaces. In this article I will expand on those two articles to draw out the analogy further, in further detail, and also include additional concepts around translation and articulation.  This article assumes knowledge of linear algebra concepts and builds on them without thorough explanation.


The grand question that we begin with here is “does language span thought?”, in other words, are our languages complex enough to articulate every possible thought that we will ever have? The answer I gave is yes, and no. Yes, because of the expansive nature of language, no because as users we may not have sufficient grasp of the languages we use to implement them so powerfully (to articulate any given thought) 

Spanning in mathematics means a set of given vectors can be used to describe any arbitrary vector in a particular vector space. This is an important element of our discussion because the “vector space” of thought is infinite, but the vectors of words / language are finite. When you have a basis for a vector space (that spans the given space) the basis itself is finite, but the vector space is not. Hence the first “grand analogy” is that human thought is like a vector space.

The grand analogy: Thought is the vector space, words and grammar are the basis. (Articulation)

As far as I know, I am the inventor of this analogy. So here goes!

Thought can be analysed and pictured as being similar to the vector space, where the following properties are analogous:

  1. The number of possible vectors in a vector space is infinite, and
  2. The number of thoughts conceivable by man is infinite.

Words and grammatical principles can be analysed and pictured as being similar to vectors, where the following properties are analogous:

  1. The number of vectors required to span an infinite vector space is finite and
  2. The number of words required to express any arbitrary thought is finite

In this analogy, since words are comparable to vectors, one could picture the dimension of the “thought vector space” to be the number of words and grammatical principles that the perfect language would have (This reveals one of the places where this analogy breaks down a bit; where 10 is 10 x 1 and is merely a scalar (10) applied to a vector (1), words don’t function so rigidly). I use the phrase “the perfect language” because as I explained in the previous article at present our linguistic systems are not sufficiently evolved to articulate every particular thought with absolute specificity and accuracy. Often, approximations are made, in many ways this is analogous to an orthogonal transformation where an inherently more complex thought is reduced to finite words, hence as we expand our vocabulary our capacity to more specifically articulate thought grows.

It is interesting to digress here for a moment to note that many fields (such as physics, astronomy, or different sciences) use the language of mathematics to grant them greater power in specificity and precision, again this confirms what I mentioned in the previous article about the expansive nature of both a language itself and an individual’s mastery of that particular language. As an individual masters the language of mathematics, they are empowered with the ability to more precisely articulate scientific fact.

A final point before an example, in finite dimensional vector spaces not all of the vectors in a spanning set are required to express every particular vector. An example from the previous set would be the vector (2, 4) which is simply (1, 2) multiplied by 2. In this case the other vector in the spanning set is not required to express the new vector. In language, this concept is taken to the extreme, since there are so many grammatical principles and words in a language, it is definitely not possible to have a sentence (or even a book) which implements all of them!

A simple example of what I’m talking about:

“I am hungry” this sentence is probably right next to “daddy!” and “mommy!” in simplicity. In this sentence, a particular thought is being articulated; this thought is then expressed in terms of these three words and a very simply grammatical principle. The words “I”, “am”, and “hungry” and the grammar that governs how a statement is made in English (subject then verb then object), using these four, the thought is expressed. The sentence “Am I hungry?” uses the same words but in a different permutation (verb then subject then object) to make a question. This is an example of how a “linear combination” of “words and grammar” can be used to articulate a single thought.

A note on inflection

Perhaps inflection should be added to the “words and grammar”, since speech inflections could also be thought of as vectors which contribute to the articulation of thought. Inflection allows for greater specificity, as the intonation added to a spoken phrase more distinctly expresses what is thought and even felt by the user. Please see the quote at the end of my article about inflection and intonation.

The grander analogy: surjectivity, injectivity and bijectivity. (Translation)

Now we will change gears a bit away from the simpler “language is like a basis, and thought is the vector space” to a bit deeper of a discussion around the concepts of surjectivity, injectivity and bijectivity, which relate to translation.

As an introduction I will briefly recapitulate on the concepts of surjectivity, injectivity and bijectivity. These two are referred to in mathematics when speaking about a transformation from one vector space to another. Surjective means that a particular mode of transformation (or a “linear transformation”) spans the codomain, I.E for every vector in the codomain, there exists at least one vector in the domain that can be translated to it. Injective means that for every output in the codomain there is at most one (possibly none) input that generates that output. Bijective means there exists only one vector in the domain that can be translated to it.

Now we will expand the precious analogy to be:

  1. Thought is a vector space. It is infinite, containing all possible thoughts that a human can conceive.
  2. Language is a vector space (and words, grammar, tonality etc. are basis elements)
  3. Articulation (taking thought and putting it into words) is a kind of linear transformation, taking a vector in the thought space and translating it into a language space.
  4. Each language is its own vector space. Hence you could have the “English space” and the “Chinese space”
  5. Translation (taking a sentence and translating it into a different language) is a kind of linear transformation, from one language space to another.

(diagram too old, missing now!)

In this diagram, I show how an individual can have “thought”, and (for this example) this thought can be articulated into English or Chinese. I have also included the concept of translation; that is how a “vector” (a thought) which has been articulated in English or Chinese (or of course by implication any language) can be translated into the other.

But can any sentence of any language be neatly translated into any other language? This is the crux of the article.

If they could, then this would mean two things 1) both languages fully span thought (according to the old analogy) this means that for any arbitrary thought regardless of how complex, it could be articulated in either language exactly and 2) The process of translation is surjective always (according to the new analogy), meaning that for any sentence in one language there is at least one sentence in the other language which has exactly identical meaning. This also works for each and every word.

Nevertheless anyone experienced in translation and fluent in multiple languages can attest that exact translations are not always possible. Thus translation between languages is not surjective, thus particular sentences or words in particular languages cannot be translated exactly to other languages. If this is true, then by implication, if there exist some sentences or words in some languages which cannot be translated exactly into some other languages, and vice versa going back between the languages in other examples, then neither of those languages fully span thought.

This is monumental, if my logic is true —  then none of the existing languages can perfectly span thought, furthermore if this is true then it is impossible to draw any absolutely certain conclusion without first learning every word and grammatical principle and every inflection in every language currently existing (and good luck with that to whoever tries).

Let’s take this analogy further with more examples:


Mathematically bijectivity implies both surjectivity and injectivity, which means in our analogy that for a particular input in one language there is one and only one output in another language. (This is a translational one to one mapping, often people who are relatively unskilled in translation think that they can simply directly map each word in one language to a corresponding word in the other language, which of course is ineffective)

Here are two examples of possible bijectivity. These are two exactly identical words (in English and Chinese) right down to the etymological roots

Progress <- ->进步 (jin\ bu\)

Regress <- ->退步 (tui\ bu\)

Progress comes from Latin originally, pro- means “forward”, and “gress” comes from gradi “to step”, regress is the same excepting the re- part which simply is the opposite meaning “backward” and “to step”

进步 is the Chinese word for “progress”, where the part 进 literally means “forward’ and 步 literally means “step”, 退步 is the Chinese word for “regress”, where the part 退 literally means “backward” and 步 again means “step”.

Although you can translate the word to other similar words in English like “advancement”, “advance”, “headway”, etc. there is one and only one exact translation of this word, and that is “progress”, making this an example of a bijective translation.


Surjectivity means that any output can be generated by at least one input, possibly more.

While mapping from Chinese to English, in an attempt to translate the word “can”, we will find that there are several words in Chinese which all mean “can”, but in different senses. The words 能, 可以 and 会 can all be translated to “can” in English, thus several inputs can generate the same output.

Nevertheless this argument grows a little “slippery” when considering the fact that we could take the word “can” in English and split it into three senses, where each maps only to one Chinese word. Furthermore each of these words in Chinese could be reasonably explained using words in English, for example 能 means “can”, or more specifically “able to by circumstances”, however still the argument “sort of” holds that there is no single English word which exactly means 能, meaning that going backwards it is injective.


While mapping from Chinese to English, in an attempt to translate the word “dodgy”, we will find that there is not (to the best of my knowledge) a word which exactly means “dodgy”. This is an example of where there is one or less inputs for a particular output. It is arguments like these which form the basis of the theory that languages do not currently span thought, since Chinese has no word for “dodgy” but the concept of “dodgy” still remains a concept conceivable by the mind. There are examples going the other way around of words in Chinese which don’t exist in English also (one of my favourite ones is 羡慕, a kind of jealousy but without any negative inflection or feeling, in English there is no word which exactly means that)

However, in almost all cases, the word that cannot be directly translated still has explanations in the other languages. Again, the analogy is not perfect.


If this article is written correctly and clearly, the analogy should have been fully unfolded to view. As a final remark I will quote from Professor Steven Pinker, a member of the Department of Brain and Cognitive Sciences at MIT, in response to the question:

“Well, language, of course is more than just words. A language has a cadence. It has certain sounds and pitches and timbres. Don’t you think these things may affect the environment in which we think?”

He states:

“Well, those are certainly what make for great literature and poetry and prose, and artists and writers take advantage of those things to get across a certain emotional effect. And I think that’s why great poetry and great literature is often very hard to translate, because even if you translate the meaning you’re not getting the resonances of the sounds. You might have like a harsh staccato set of sounds in one language, and their exact translation might be something very mellow and smooth, and so you might lose that extra layer of meaning that resonates with the literal meaning. But the fact that you can translate at all, when you think about it, shows that there’s got to be something other than words, because what would it mean for two sentences in different languages to be translations of each other, if not for the fact that both of them have the same meaning, where the meaning isn’t exactly the same as either string of words? When we translate, it’s obviously not like one of those phrase books, where it’s, “How do I get to the train station?” and then you find the equivalent in Hungarian, because if you know two languages, you can translate an unlimited number of sentences. There has got to be something, I think, underneath it, something like a set of propositions that don’t really have sounds, that don’t have any left-to-right linear order the way language does, but that has a web of connections between concepts, and that are also connected with other aspects of experience — with visual images, with body sensations.


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