Mathematics

If you were to invent mathematics from scratch, how would I do it? A good portion of math, both in modern education and throughout history, is simply revisiting basic concepts over and over again with more rigor. High school polynomial algebra is about the study of rings and calculus is reintroduced both in topology and analysis. But if I were to construct a linear process starting from the most fundamental concepts, I would do it as follows.

In order to prove any claim you need logic, and most of mathematics relies specifically on first-order logic. Once this is settled we can talk about set theory, i.e. why naive set theory is inconsistent and why we have the ZFC axioms that is the foundation of all math. These axioms define what a set is, state the existence of the empty set, and provide the operations that one can do on a set, either as a part of the axioms themselves (e.g. union, power set) or as a direct consequence of a combination of the axioms (e.g. intersection, set difference). Through the existence of the power set we can construct Cartesian products of sets, allowing us to then define correspondences as subsets of this Cartesian product. Two popular such correspondences are functions and relations. The axiomization of the empty set allows us to then construct the natural numbers, which allows us to construct the integers, then the rationals, and finally the reals (plus complex, p-adic, etc.). Therefore, given a set, we can define structures on this set, such as elements (an identity), functions (operation, inversion, metrics, norms, inner products, measures), or relations (order, equivalence relations). Set theory is pretty boring without any structure, and so each subfield of math deals with sets having some combination of structures.

Once we have sets, we can branch off into two ways depending on which structure we endow the set with. In one direction, we can talk about certain collections of its subsets, with one popular collection being the topology. A topology allows us to define the concepts of separability, connectedness, compactness, and countability of a set, along with defining the most general notion of continuity of a function. In the other direction, we can define closed operations on an arbitrary set, which turn these sets into algebraic structures, the primary object of study in abstract algebra. This allows us to talk about the behavior of functions and their decompositions (e.g. generators of groups). One of the objects of study in abstract algebra are vector spaces, which with linear maps, constitute the topic of linear algebra. Vector spaces arise in almost every field of math and is the most universally studied topic across all STEM disciplines. The use of algebra to study topological spaces is called algebraic topology and the study of algebraic structures endowed with topologies is called topological algebra.

Unfortunately, abstract algebra is limited in the way that the operations endowed on sets are finary (take in a finite number of arguments), and so repeating them infinitely does not make sense. Therefore, to define such a behavior, we can take our concepts from topology and talk about the limiting behavior of these infinite compositions. This motivates the study of sequences and series (which are sequences of partial sums). To extend limits to differentiability, we cannot just work with arbitrary algebraic structures since they lack certain arithmetic operations and the property of completeness. It seems that vector spaces, specifically Banach spaces, contain the minimal structure that allows one to define differentiability and Riemann integrability of a function. Depending on the type of Banach space we work with, we have different flavors of analysis. Working with real Euclidean space gives us real analysis, complex Euclidean spaces gives us complex analysis, p-adic numbers give p-adic analysis.

It turns out that Riemann integrables are inherently limited, and so a deeper study of the size of subsets is needed. By defining another collection of subsets, called the sigma-algebra and defining a corresponding measure on it, we can talk about Lebesgue integrals, which is the main topic of measure theory. The study of general Banach spaces, usually equipped with a measure, is the study of functional analysis. At this point we can branch off in various directions.

An application of analysis is the study of dynamical systems, i.e. ordinary and partial differential equations, which are the main topics of study in physics. Usually a physical system existing in a phase space is governed by a set of laws that changes the system, expressed in a differential equation. Solving and analyzing these systems hopefully allows us to solve this equation, i.e. find a function that either finds or approximates the true evolution of the system. Another way to branch off is to see that a direct consequence of measure theory is probability theory, which allows us to study random variables as measurable functions. By taking a sequence of these random variables in a function space, we can combine probability and functional analysis to study stochastic processes. The final way is to study smooth manifolds, which are certain topological spaces that are locally homeomorphic to Euclidean spaces. This leads into differential geometry.

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