Intuitively, a plane is a flat surface that extends infinitely in all directions. How can we interpret this definition in a formal way, and in particular use that discussion to derive the standard equation of a plane?
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Tag: Studies
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Sine Angle Addition Functional Equation
In this post, we investigate further the sine angle addition functional equation that I introduced in a previous post, Trigonometric Identities as Functional Equations. (See the work done there before reading this post.) Specifically, the question is to find all functions 
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Interpretations of Model Theory
A typical definition of model theory is “universal algebra plus relations.” However, the term “model” in natural language can typically be used quite generally, especially if we consider how people typically understand its meaning for designing and developing any kind of system. Thus, we could guess just based off terminology that in mathematics, a “model” could be defined, as generally as possible, basically as an “example” (set plus interpretation function) that satisfies any given set of axioms. Is this equivalent to the traditional formulation of model theory? In other words, does the traditional formulation of model theory capture the most general kinds of axioms that an “implementation” could satisfy?
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Topology as an Algebraic Structure 2
In this post, we continue the exploration started in Topology as an Algebraic Structure.
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Topology as an Algebraic Structure
Can we express the axioms of topology equivalently as the axioms of an algebraic structure?
We know that model theory is at least a non-strict generalization of universal algebra. Indeed, we can consider model theory to be “universal algebra, but with relations instead of operations.” We can actually replicate the main definitions of model theory with an exact replacement of “operation” by “relation” in the universal algebra definitions, which yields more generality since every 
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Sine Angle Product Formula
In this post, we address the following question: does there exist a polynomial identity for 
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Linear Independence in Number Theory
In this article, we discuss some questions I had regarding linear independence as it shows up in subjects like transcendental number theory.
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On Logical Independence
There are multiple definitions of logical independence out there in the literature. In this article, we tease out these definitions and investigate their relationships with each other.
We take some basic trigonometric identities and essentially ask whether trigonometric functions are the only ones that satisfy these identities. In other words, what are all the functions that satisfy a given trigonometric identity? Thus, we take trigonometric identities and use them as functional equation problems.
