we must know
we will know
-David Hilbert
It all begins here. What is the set upon which every set rests? The empty set
How many numbers are there? Replying 'infinitely many!' would be like asking how many fingers you have, and hearing back 'some number!'
Finite numbers come in different flavors, and infinite numbers do as well. What is a number, anyway? The answer depends on what you choose to believe about the higher infinite
Picture a circle, and choose a single point inside. What's its area? Every point inside a circle has area exactly 0 —zero, nada—, and yet, somehow, the sum of all those 0's makes up for an area that is not 0?
How can this be? It took over two thousand years to give a proper —i.e. rigorous— answer. It turns out summing 0 + 0 + 0 ... is not enough
We need to understand the depths of the Continuum, which leads us to the limits of reasoning itself
There are objects —like a square— that don't change appearance when you rotate them by 90 degrees. This invariance is called a symmetry
Symmetries, in turn, form abstract structures whose study goes to the deepest branches of reason, giving rise to groups, rings, fields, homomorphisms, domains, adeles, ideles...
Even the fundamental nature of reality exhibits such order
Take a cup of Play-Doh and form a square. Now deform it to make a circle. In Topology, these two are equal, i.e. they're homeomorphic to each other
While you're at it, make a sphere. Can you deform it and make it pass through itself without tearing a hole through it? You may not, but there's a way
Prepare to delve into higher dimensions
Geometry dominated the minds of Greeks in antiquity. The universe was harmonic, and numbers embodied such essence
The primitives of existence were special kinds of polyhedra, out of which all beauty took form
But their dreams of acme were not meant to be
Grothendieck championed the study not of objects themselves, but of relationships between objects.
Such thinking is at the heart of category theory, a language powerful enough to formulate all of mathematics (including set theory!), and then some.
By studying categories, and categories of categories, and categories of categories of categories, ad infinitum, one arrives at n-categories and "higher" versions of classical objects like vector spaces, groups, and rings. This is the higher mathematics.