The Algebra Of Ideas

The world is an orchestrated piece of chaos within which there exist several mappings to and from its constituents. Math is the lens through which the world can be observed and understood. Her formalisms are irreducible. One must look through one lens and later find intersections: an art of zooming in and out, favoring abstractions and the relationship between things over the properties intrinsic to them.

What mathematical constructions represents the most fundamental truth? Axiom? Proofs?

There is no truth that’s absolute. There’s only truth that’s relative to context.

What premium should thus be placed upon is not what the fundamental truth should be but the most fundamental construction or tool with which the transitive nature of knowledge can be observed, understood, and how rules that hold in one context can be observed in an entirely different one. Hence arming ourselves with a formalized framework for thinking.

Examining Group Theory

Expound upon how group theory formalizes structural similarities between several mathematical domains.

Probability Monad in Natural Language Translation

One way I help myself quickly grasp a concept that's new but important is by looking for ways it maps directly to something I already know or understand. Such mapping (bias) speeds up assimilation.

Have you tried interpreting a statement word for word from one language to another? It often times does not translate well. This is because the mapping between them has to be context aware. By directly translating each word, you lose meaning. The translation must take along with it contextual properties.

Let's say the work of doing the translation is a 'morphism' or an arrow: there's a structure preserved by such morphism going to and from the respective languages. That structure is a bridge upon which not only meaning is conveyed, but contextual properties. It therefore rather limiting to observe systems in isolation.

In natural language translation, there is uncertainty about which words or phrases map best from one language to another. A probability monad can encapsulate this uncertainty by representing translation as a probabilistic process, allowing you to chain together computations that assign different probabilities to different translations.