# 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**.