How Vector Spaces and Languages are alike

Note from the Shawn of the future

Quick side note from me — written 13-08-2014. In migrating my blog I found this old article I wrote as a second year university student (February 2011) while I was studying computer engineering. The writing was not very well formatted and a bit hard to understand — I was in the middle of several courses of mathematics and deeply immersed — a vast portion of my thought and activity was focused on understanding linear algebra, which was my favourite area of mathematics. It’s interesting to review articles I’ve written in the past and contemplate on the changes that have happened in my life.  In hindsight, the article itself feels more like a reverie than an article — and is probably a bit difficult for the layman to understand. It seems stuck halfway between trying to explain the concepts and assuming the reader already knows them! What’s interesting is it shows an immersion into mathematics and an attempt to understand the conceptual relationships between things I already knew, and things I was busy learning. Enjoy!

How Vector Spaces and Languages are alike

In this article I will attempt to unite the concepts of two completely independent worlds, and the concept of a “world” in the process. I will try to unite and intermingle the unique expressions of a “complete configuration”, I.E, a setup in which an entire world of some sort can be understood and expressed in terms of other elements within the same world.

This is a unity of two very different spheres; Mathematics (specifically linear algebra), and Linguistics.

Vector spaces (From your first year university linear algebra courses)

Before one can understand what a vector space is, one needs to understand what a vector is, the word “vector” is used to describe an “action” or a “movement” from the origin to a particular point. For example, in R2 (two dimensional space) you could have a point (17, 2) which represents a movement from an origin or “starting point” 17 units in the direction of the first variable, and 2 units in the direction of the second, thus (17, 2) is a movement and not a position, because the origin, or “starting point”, of this movement is undefined.

For me personally, I find understanding vector spaces easier when visualising vectors; for example, R3. R3 is a vector space in which all three dimensional vectors live. In other words — any vector in the form (x, y, z) for any arbitrary x, y, and z, “belong” in R3, an infinite three dimensional space. R2 is also a vector space, and up to RN. The real definition of a vector space relies heavily on the concept of forming a basis for that vector space. For example, the standard basis of R2 is the set of vectors {(1,0),(0,1)}, meaning that any arbitrary element of R2 can be expressed as a linear combination of the vectors (1, 0) and (0, 1) for example, a vector (17, 2) is 17 of the first (1, 0) and 2 of the second (0, 1) thus (1, 0) and (0, 1) span R2, meaning that any element of R2 can be “expressed” in terms of those two vectors.

By heavy mathematical nomenclature, a vector space conforms to ten (or some say eight, two are implications of the others) axioms, they are beautifully logical; I think some of the most important are that it is closed under addition and closed under scaling. This means that if you add any two vectors together, the resulting vector lies within the same vector space, and if you scale any vector, the result lies within the vector space.

In simple words, a vector space is a “world” of vectors, where every element of that world can be expressed in terms of a basis of that world, for example, in R3, if you have three linearly independent (which simply means they are not directly comparable) vectors, then any arbitrary element of R3 can be expressed as a linear combination of the three, scaled with particular scalars.

There are things called subspaces too, which are also vector spaces and subsets of a larger vector space, or mathematical worlds within worlds, systems within systems. For instance, if you had a plane in R3 which contains the origin, then that plane is a vector space, meaning if you could give me any two linearly independent vectors within that plane, I could express any other vector in that plane as a linear combination of those two vectors. (for a side note, you could solve this by constructing a linear system, with the columns of the matrix A being the two input vectors and the x being the scalar vector (contained in R2) and b being the resulting vector (Ax=b))

Suffice it to say, summarily, a vector space is a mathematical “realm” where elements of that realm can be described and expressed in terms of other elements of that realm, or in other words, a vector space is a world of its own.

Languages

A definition of a language is “A systematic means of communicating by the use of sounds or conventional symbols” or “a system of sounds, symbols or a ‘means’ in which thought is packaged, conveyed and interpreted”, language is a tool, essentially, for communicating thought. The interesting correspondence with this articles’ topic is that a language is also a configuration of elements which can be expressed in terms of one another (A world — just like a vector space)

A simple illustration of this concept is a dictionary, within a dictionary you find definitions (or circumscriptions) where the meaning of one particular word or phrase is explicated by using other words or phrases from the same language.

So when you don’t understand the word “gestalt”, you can look it up in the dictionary and find a definition such as “A configuration or pattern of elements so unified as a whole that it cannot be described merely as a sum of its parts”, the word “gestalt” is English, and the definition of gestalt is also in English, therefore English, like any other language is similar to a vector space, it has elements of it (words, or sentences) which can be explained or described in terms of other elements also within it.

Thus English (or any other language) is a complete and independent entity, where the world around us can be expressed in terms of any combination of words, some concepts take a short sentence to express (like “I am hungry”) others, take entire novels or even a series of books (millions of words) to express (like a full answer to the question “why do we exist?” or “what is love?”).

Actually (I believe) one of the best ways to describe a language is a “web” of nodes, where each node is a word, with a particular meaning, with adjacent nodes being similar words of similar meanings, which stretch on (but not infinitely)(in fact, they seem to “wrap around” back into themselves) and thus they are a system or a configuration. And thus they share something with vector spaces, that is, they are closed and complete. (Although the “commensurate” nature of languages can be reasonably well argued, because there are many concepts which are difficult (or nearly impossible) to describe using them — however — the fact that it is a closed and complete system in and of itself is exemplified by the fact that any arbitrary element of a language (a random, single word) can be expressed in terms of a particular combination of other words. If this were not so, then the dictionary would not exist because there would be some words which cannot be described in terms of the others) however, it is important to note that language has a strong dependence on context, were it not for context natural linguistic acquisition would not occur.

Conclusion

There exist worlds, in different fields of life and reality, which contain elements. All elements contained within those worlds can be expressed in terms of the other elements also within the same world. This applies to linguistics, finance, science, art, and almost any of these distinct “spheres” of existence. This is how concepts grow from simplicity to complexity, through iterative expansion and aggregation.

And, perhaps as a final, more philosophical point — existence itself is the master world, which contains them all.

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