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If all of the math books in the entire world were burned, and all of the digital information about mathematics was deleted, and all of the mathematicians suddenly died in an apocalyptic event… but you could do just one last thing: if you had to decide which are the 5 most fundamental fields of Pure Mathematics that humanity would have to develop in the years to come first, in order to rebuild all of our knowledge, which fields would you choose?

This is a weird question, but your answer can tell you a lot about the way YOU see the world around yourself and what you value most. Because the way you answer this question will likely reflect the kind of mind you have, and what you choose to prioritize. 

I’d say that some of us like to think visually: in terms of shapes and surfaces, and how they transform in space. So, geometrically. People like that tend to imagine curvature before applying some kind of a systematic method of measuring it. For them the world is made of objects that bend, twist and stretch, and mathematics is a way of communicating these ideas with precision.

Now, there are other people who prefer to think more abstractly, or algebraically, which means that the world isn’t full of objects, but of structures. People with this kind of mind ask themselves: What happens when you combine things? What happens when you permute them? They think in terms of rules and transformations, of homomorphisms and isomorphisms, and of how structure persists (or fails) when mapped from one context to another.

If so far this hasn’t described how you think, maybe you are more drawn to logic. Not necessarily because you love proofs, but because you want to know what it means to know something. For these kinds of people, mathematics tends to be a collection of consequences: What follows from that? What must be true, given the axioms? They are the builders of the foundation, who are always suspicious of leaps which are informal, and they are always really precise and rigorous. Plus, these mathematicians are usually very interested in the boundary between what is true and what can’t ever be proven.

I’m sure there are other ways of thinking about the world, but even though I may have not mentioned all possibilities, I hope you understand the point I’m trying to make. The answer to the initial question isn’t only a technical one, it’s also personal. It shows not only what you find to be important, but how your own mind moves through abstraction. Do you think in terms of pictures or more in logical implication? Do you care more about how things look, or how they behave, or how they relate? 

It’s cool to wonder about this apocalyptic thought experiment because, in the end, we’re all trying to describe the same invisible structures, but we look at them and observe them from completely different perspectives. So even though  mathematics is universal, the way we experience it is not. And plus, let’s face it, it’s a pretty fun question. Now, I’m going to give you guys my own list of 5 math fields. And you can disagree with me, that’s ok. But if you do, please explain why in the comments.

So, here we go:

First I would say Geometry, and I think most of you would probably agree. It’s the most intuitive thing for us I’d say, because it gives us the foundational understanding of shapes, distances, and space.

It’s also the first place where we learn how to move from intuition to formal statements. (“these two sticks look the same length” vs “these two segments are equal by construction”). Without it, we don’t have a way of systematically showing if something is true or not. Because it’s one thing to have a hunch and say that “it looks true”, and it’s something completely different to demonstrate something to be true with logical certainly. This is what we can learn from humanity’s own history with Euclid’s axioms, for example, which were one of the earliest systems where definitions, assumptions, and logical steps built an entire body of knowledge. Not to mention that geometry is pretty useful if we want to start building stuff after an apocalypse.

Next, I would say another “obvious one”: Algebra. Algebra lets us systematize concepts like equations, lets us work with unknowns, and it’s where we first develop notation that is symbolic. Without it, we won’t even know how to express generality, or rules which are true universally.

And, even more importantly, it takes us to abstraction, we can now manipulate ideas instead of just numbers and shapes. As it gets more advanced, it will go into groups, which capture symmetry, into rings and fields, which show us how numbers and polynomials behave, etc. We can go on, but the point is that all of these ideas are about finding patterns that are deeper than the specific examples we started with. So in other words, it lets us search for generalizations, because this way of thinking lets us connect completely different areas of mathematics under the same language. 

Ok, now my next point is where some of you may disagree: Mathematical Logic & Foundations, because starting here would naturally take us to things like set theory, formal logic, and so on)

This here is the true foundation of mathematics. Logic gives us a framework for proofs or logical statements, it is how you define mathematics.

Without it, we don’t really know what it means for something to be true. You don’t know what counts as a valid argument, or how to check that an idea is consistent with the rules you started with. Logic tells us what can be proven and what can never be proven, and it forces us to face the limits of reasoning itself. It might feel abstract, or even too philosophical, but it’s literally the foundation that holds all mathematics together.

My next pick is Analysis (both Real and Complex).

At this point we are ready to start talking about limits, continuity, and infinite processes. This is where we make sense of what it means for something to get closer and closer without ever quite reaching it, or for an infinite sum to have a finite value. Real analysis gives us rigor in dealing with functions, sequences, and series, and complex analysis opens up a completely new world where functions behave with surprising regularity, which leads to very powerful results way beyond what real numbers alone can offer.

Not to mention that analysis gives us the necessary tools to get into probability, dynamical systems, and mathematical physics, and it shows up in differential equations, Fourier analysis, and functional analysis, which is the study of infinite-dimensional spaces. And like I said, these tools are essential in physics, in areas like quantum mechanics, signal processing and statistics. All of which would be very useful since we just had an apocalyptic event!

Before I go to my last one, I’d like to say that if you would like to be the first to know when we release our first books and courses, sign up with your email address on our website!

Okay, last but not least: Number Theory.

With number theory I mean the blend of algebra, analysis (like modular forms), and geometry (or something like elliptic curves). Let me explain. I’m not talking “just about primes” anymore, but what is built on top of them. Number theory begins with very simple things, like counting, but it grows into one of the most abstract and far-reaching areas of all mathematics. It’s where some of the most famous problems in mathematics live, like Fermat’s Last Theorem, the Riemann Hypothesis, and the Langlands Program.

Number theory has historically been called the “Queen of Mathematics”, by Gauss, and it’s often seen as the most ancient, self-contained, and intellectually “pure” domain.

I mean, it’s been a hard choice, but deciding on only 5 out of all the areas of mathematics requires sacrifice… Let us know what you think these 5 areas should be.