Is It Possible to Build Mathematics from Scratch?

Let’s begin this topic by first explaining what it means to build mathematics from scratch. The concept of "building from scratch" or "discovering" actually starts with a person accepting certain definitions and axioms. Without definitions and axioms, the field we call mathematics cannot exist. Once we determine these in a way that aligns most naturally with human perception, the rest is like stepping into a whole new world.

In this new world, the first thing to do is to try to use what we already know to create new concepts or from a mathematical perspective, to discover new definitions. At some point, by using these definitions, we begin to uncover certain rules. These rules give rise to what we call lemmas and theorems. Yes, the very rules that many people fear or dismiss as mere memorization are, in fact, discovered through the concepts we've already defined.

But how, exactly?

We’ve talked about how theorems or lemmas are formed. But how they are actually discovered is a different matter altogether. In my opinion, such discoveries emerge thanks to a combination of human imagination, environmental conditions, scientific progress, human needs, and similar factors. I would also add curiosity and ambition to that list.

For example, the first rule we learn to calculate the area of a triangle is usually: (base × height) / 2. However, when a mathematician asks themselves, "Is there another way I can calculate this?", they begin to use all the definitions and rules they know in an attempt to find a new method.

Sometimes, combining two different definitions does not necessarily lead to a third one, but combining a definition with a rule can indeed give rise to a new rule. And once imagination and deep thought enter the world of mathematics, we begin building new things in this world at a rapid pace.

What is an Axiom?

An axiom is a fundamental proposition in mathematics and logic that is accepted as true without requiring proof. These are statements considered to be self-evident truths. For example, in geometry, we often study using Euclidean axioms, and when defining natural numbers, we refer to the Peano axioms. These form the foundation upon which mathematical systems are built.

Is Belief Based on Proof?

In many areas related to mathematics, when we encounter a rule or piece of information for the first time, we tend to believe it immediately. However, this is not necessarily the right approach and here's why:

First, if we saw that information online or in a non-verified environment, the risk of misinformation is quite high nowadays. Even if it comes from a written source like a book, simply believing it doesn’t prove it’s true because most of the time, there’s no actual proof presented alongside the claim.

Second, we often forget such information the very next day. The concept might never cross our minds again, even though it could have been valuable enough to challenge or even replace our previous understanding. To avoid such situations, it’s important to attempt to prove or justify the information ourselves.

Even if we fail to prove something, the very act of trying is a step forward. Sometimes, a part you couldn't figure out earlier will unexpectedly pop back into your mind and you’ll start thinking it through again. Even in failure, the process helps anchor the concept in your memory and leads to a deeper understanding of the rule or idea.

Building Mathematics

The subject of our blog—building mathematics from scratch—is, in our view, entirely possible. If a person possesses enough imagination, curiosity, and discipline, they can construct their own mathematical system, complete with unique axioms and definitions.

After all, figures like Euclid, Gauss, Euler, Archimedes, Newton, and Leibniz were all human just like us. What set them apart was not some supernatural gift, but rather a stronger sense of discipline and intellectual curiosity. That same potential lives within anyone willing to explore it deeply.