First of all, let's address the tea-and-biscuits-shaped elephant in the room. For our UK readers, we're using "Math," not "Maths." Sorry, not sorry :) We'll make our argument(s) for why that's the case eventually, but not here.

Here, we'll focus on the seemingly simple question:

#### What is Math?

Math is one of our oldest intellectual endeavors, yet it lacks a simple definition. We often see Math defined by giving examples of Math: "Math is the study of numbers, shapes, formulas, patterns, and mathematical structures."

It is odd (and logically circular) to have to say that "Math is the study of math-y things". Yet, we tend to know math-y things when we see them. Even if we were content to give a list of math-y things when defining Math, how can we be sure that our list is and always will be all-encompassing? Many of the math-y things that Grothendieck studied 50 years ago would be unrecognizable to Pythagoras 2,500 years earlier. How can we define Math so that both of these gentlemen can be considered Mathematicians?

#### WHAT’S AT STAKE

There is more at stake here than simply whether or not some dead people should still be considered mathematicians. In fact, the health of the entire field relies on finding a better definition. Practitioners of math know that it is immensely diverse, allowing for the study of knots, play-dough, infinities (yes, plural, there’s more than one infinity!), and so much more. However, our current definitions of math don't reflect any of this diversity or creativity, which propels the popular perception of math as something dry and rules-based.#### OUR DEFINITION

We’d like to propose a definition of Math that

- is simple.
- encompasses all current and future “math-y” things.
- exposes how amazingly diverse this field actually is.

Math is the study of questions whose answers are True or not.

Notice the capital T. It’s important to distinguish between capital-T Truths and other truths.

Questions like “is the sky blue?” or “what’s the best way to govern people?” are not questions whose answers are capital-T True. Even questions in the “hard” sciences like physics and chemistry are not of this nature. They rely on implicit assumptions about the time-independence of the laws of nature, domains of applicability, the accuracy of experiments, and more. Hard sciences give us evidence-based truths, which are fantastic and wildly useful, but these truths are of a different nature than the logic-based, capital-T Truth of the answer to a question like “how many prime numbers are there that are smaller than 10,000?”.

One of the reasons we like this definition is that it explains why people tend to say “numbers, shapes, etc” when defining math — numbers and shapes are great at providing questions whose answers are capital-T True or not. Numbers and shapes also * happen* to have a good amount of practical application to our physical world, so we teach these Truths in almost every school, and so people tend to equate these Truths with all of Math.

However, as any mathematician will tell you, there are many more facets of math out there — some with applications to the real world and many without. And this is why we **really** like the above definition — because of its "converse". Namely, ANY set of questions whose answers are capital-T True or not is, by this definition, Math! So, if you can take a question about untying knots (Knot Theory), or deforming play-dough (Topology), or traveling across bridges (Graph Theory), and phrase it in a way where questions have capital-T True answers, then what you're doing is Math.

The study of capital-T Truth is what Pythagoras concerned himself with, it's what Gauss, Gödel, and Grothendieck concerned themselves with millennia later, and it's what mathematicians 2,000 years from now will be concerned with. Mathematicians do not use some finite list of math-y topics to determine what they'll work on, but rather are uncovering new sources of capital-T Truth every day. The definition above is therefore what we’ll be using henceforth at Cohomologous.