Is a circular reference a type of simultaneous equation?

Question

I've long thought that a dictionary doesn't make sense because there is no 1st word that has been defined. This made me think that language is technically 1 big circular reference that only holds water because we can define things physically.

Points at fire : "Fire"

Points at water : "water"

This physical definition allows us to make sense of a lot more words in between.

Is it possible that a dictionary could be thought of as a complex set of simultaneous equations with the rules of grammar being accounted for by an algebraic logic similar to that of boolean algebra?

Hypothetically: If such a system could be created + solved. Could it potentially function as a translator for ancient languages that remain untranslatable?

Has anyone tried to encode grammatical rules into algebra? A quick search has revealed nothing for me.

Answer 1

I am tempted to downvote your question. It's unclear exactly what's being asked here but I am trying. This is more like a long comment to see if I can get some clarification about what is being asked.

In response to your first paragraph:

When we start to make meaning out of symbols we usually start with basic building blocks that can't quite be defined. Euclid doesn't define "point." These basic building blocks might be called "elements" if you read Plato's Theaetetus or "Heuristics/Axioms/Definitions" when we do mathematics. The conclusion of Plato (if that guy had any conclusions) is that knowledge is hard to get at because we have to assume the basic meanings of these elements. This is also the conclusions of modern math. Godel is the modern nail in coffin: Some truths are impossible to access through countably many symbols.

In response to "physical location." This is a problematic way to define stuff. Is anything located anywhere? Aren't you making certain assumptions about time/space?

In response to "Has anyone tried to encode grammatical rules into algebra? A quick search has revealed nothing for me."

Sure! Encoding a syntax for acceptable strings as members of a language is what formal-language is all about. Check out stuff on Turing machines. On some level whenever you interact with an online chatbot you are dealing with something that has an exclusively symbolic notion of grammar.

Could we use this to figure out what ancient Etruscan strings of symbols mean? I am confident that we use computers in attempting to decode Ancient languages. I am not sure that this problem is ever particularly tractable.

Answer 2

The question is strange and unfocussed but suggestive, so I'll respond to some points.

I think that the starting point for Euclid was in fact something like your "define the basic notions with a physical reference". The Greeks thought they knew what points and lines were, so managed to build geometry with essentially undefined primitives. They also thought they knew what the Euclidean plane was, hence the subsequent controversy over the parallel postulate, which mathematicians felt for centuries ought to be a theorem.

In time Gauss, Bolyai and Lobachevsky (independently) discovered that there were consistent geometries in which the parallel postulate failed. That led over time to the view that mathematics wasn't really about anything. As long as it was contradiction free it was OK. In that sense mathematics is just "one big circular reference". That said, most mathematicians believe in their day to day work that the objects they are proving theorems about are quite real, and don't feel the need to justify that belief (so back to the Greek model, in a way). They don't even worry often about the fact that Godel proved that you couldn't prove everyday mathematics was free of contradictions.

As for the dictionary, think of it as a big directed graph, with nodes the words and edges the assertion that one definition depends on another. Then in some sense the meaning of the language is the whole graph, which is more than the collection of words. You can't think of just the words as the primitives.

Computer programs are used to facilitate translations and to decode messages and ancient languages. The best of those programs deal with the contexts in which they find words, not just with one to one correspondences between words, so graph theory is again in play.