Topology is the study of geometric properties and spatial relations. These are unaffected despite the continuous change of either shape or size within the figures. They are interrelated or arranged in a certain way. The continuous functions from one topological space to another is referred to as “homotopic,” which means that they are similar. If one can be continuously deformed into the other, then the deformation is referred to as a “homotopy.”

So what is homotopy type theory? It’s an idea that brings something new into the world of mathematics. It suggests that there is an invariant conception within the objects of mathematics, offering the idea that intrinsic homotopical content is present. The goal is to create another step toward having all mathematics have consistent foundations, unifying the language of numbers.

### Why Is a New Language Needed for Mathematics?

A majority of the homotopy type theory involves making sure the formulations are fine-tuned. This will allow it to work in conjunction with the traditional homotopy theory so we can understand both at a deeper level. Yet for those who have been involved with mathematics for some time, there is a complaint that what this new theory is doing is creating a new language for reformulation – and one that is not really necessary.

The reason why this new theory is needed is because it allows for ordinary logic to be able to work through hypothetical systems. Instead of needing to deal with equality or formulations that create specific outcomes that could influence the end result, the homotopy type theory allows for a literal interpretation of the equation elements that are being studies.

It’s a subtle shift. Instead of equality, homotopy type theory promotes equivalence instead.

So what do we learn from this process? That mathematics may still be a formal language, but that it can also be a natural language that includes more people. Instead of making people think a specific way, the homotopy type theory allows people to think in a natural way. This changes the formulation of an equation, but it still creates the exact same result.

### It’s Less About Looks and More About Feelings

If you’re not active in the world of mathematics, then discussing dependent sums, product types, homotopy pullback, and other terms is going to feel like a foreign language. You can look at it all you want, but until someone explains what those terms mean, you’re going to be unable to communicate with someone who does speak that language.

What homotopy type theory provides is a more transparent language that helps to understand the equations, proofs, and other elements that are being evaluated when looking at advanced mathematics. This allows it to be understood on a core level by more individuals, even if they are not putting in the effort to solve the equations involved.

Think about it like this. You know 2 + 2 = 4. But why do you know this? Because you can see two items, understand that “+” means to add the numbers together, and the “=” is your final answer. Under that expression, the actual equation becomes transparent. You know how it was solved, even if you didn’t watch the person solve it.

This equation also gives you the information you need to begin solving other equations that are similar to it. It’s become what would become a foundation file. Now that you know 2 + 2 = 4, you can figure out other expressions. For example: 2 + 2 + 2 = 6.

Yet what if the universe doesn’t work in that way? Maybe the universe says that 2 + 2 = 3 + 1. Using homotopy type theory, it becomes possible to express solutions in such a way.

You would also be able to change how these equations are being expressed in a transparent way that is simpler to understand. It’s like transforming 2 + 2 + 2 = 6 into 2 x 3 = 6, but on a much larger scale. Like having heuristics become a theorem.

For Example: LX={x,y: X | (x=y) (z=y)}.

This gives us a mathematical universe where items are similar, but potentially more interesting, then they were before using the foundations of mathematics. It creates simplicity from complication, added transparency, and hopefully an ability to make unfamiliar things become more familiar because expressions are more accurate.

This is because equivalence becomes the point of evidence instead of equality needing to be required in order for a solution to be discovered.