As a physics student, it is natural for me to describe and analyze complex phenomena in terms of its fundamental elements. This approach to thinking, done by reducing a complex system into its fundamental parts, is called reductionism.

Reductionism

The idea of reductionism was first introduced by Descartes in his work “Discourse” in 1637^[1]. Descartes argues that nature was a just a giant clock, not a living organism, but a machine. ^[2]

[the machine] could be understood by taking its pieces apart, studying them, and then putting them back together to see the larger picture^[2]

I started adopting this philosophy in my quest to learn almost all the knowledge that is available. I believe that by reducing a domain of knowledge into its fundamental parts, it was easier to learn its details. I realize this when I started learning object-oriented programming where I started to adopt a thinking that “everything is an object”. I was introduced to the idea of reducing a system into its fundamental parts or properties and functions.

For instance, consider the genus of canine, they share the same kind of properties, they have tails, they have four legs - they almost have the same generic form. They also share common function or action, e.g. they bark. From these fundamental properties, you can instantiate a bulldog by setting up its skin, its height, etc. And add some action or function features, e.g. bark or roll.

source: wiki
When you think about it, there must be a way to reduce mathematics into its fundamental parts. From the previous post, the notion of sets, we’ve encountered Category theory on how mathematicians found a relationship between different areas of mathematics. This is a hint of an underlying unifying concepts of mathematics.

Mathematics

Just like pornography, when we see mathematics we know it when we see it. But what is mathematics? A common answer is mathematics is about proof. They have a standard workflow that follows:

a sequence of definitions
introduction of several lemmas
a statement of a theorem
a. a proof that applies the lemmas to the definition

This is the universal mathematics workflow: “definition-theorem-proof”. After doing all those steps in the workflow what you have now is a new knowledge true to the span of the lemmas and definitions.

Today, we have several axioms available, and working on one set determines what “mathematical system” is being worked on. That’s why we have Euclidean and non-Euclidean geometry, Riemannian and semi-Riemannian manifold, Cantorian and non-Cantorian set theory, etc.

This brings us to the Bourbaki.

Root of Mathematics

A set of French mathematics students revolted against the “semi-intuitive” view championed by Poincare and started a “new” formalism of mathematics in 1939. It was known as the Bourbaki movement, under the pen name “Nicholas Bourbaki”. ^[3]

The Bourbaki realized that there are “roots” of mathematics –something that was true of mathematics in general. They have identified three basic “mother structures” upon which all of the mathematics depends. These basic structures are called:

This list has been extended as of now, with additions of measures, metric structures, events, equivalence relations, differential structures, and categories.

Note that a structure is just an additional object on a set ^[4]. These mother structures all require a set. They also required something called a relation.

Fundamental Parts

Our basic notions of the set are just a collection of things, referred to as the “element of a set”, which defines the constituent of a mathematical object. A relation/function, on the other hand, is a transformation that acts on a set to produce another set.

This is not enough, a set plus a relation/function does not provide us enough to say we have a structure. An additional thing we need are rules of how this set and relation/function interact.

For a mathematical structure to exist, we need a set, a relation/function, and their rules of interaction.

Mathematics Fundamental Parts

Learning mathematics is like building a house. You lay the foundation brick by brick. The mathematical structures serves as our brick in building mathematics.