When I discovered the spark of ‘getting’ mathematics for myself, I couldn’t get but try and get an overview. Find out what’s within mathematics.
There are a lot of interesting and fascinating subfields!
This is an attempt to share my curiosity and new-found appreciation for the field of mathematics. I hope, that the spark can be shared.
Warning: the following list is a selective transcription of the articles above. This is the part most likely to be wrong-ish. There might be mistakes. Take it with a grain of salt.
On the Shoulders of Giants
Most of the stuff below is a result of clicking through the following Wikipedia articles & sections:
- Mathematics - Areas of mathematics
- Areas of mathematics
- Glossary of areas of mathematics
- Outline of mathematics
- (DE) Mathematik - Inhalten und Teilgebiete
- (DE) Teilgebiete der Mathematik
So much for disclaimers and source attributions. Let’s start with the most unobscure fields and expand from there:
Number Theory
Everybody knows about numbers. But what can be said about them? About Natural numbers, integers or real numbers for example.
This field started out being called “arithmetics”, but by now arithmetics usually is understood as the act of doing basic calculations.
Algebra
How to manipulate equations and formulas.
“algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas”
Wikipedia entries: Mathematics/Algebra, Algebra
Geometry
You might be thinking about triangles and circles now. You’re spot on. But it’s more inclusive than that:
“concerned with properties of space that are related with distance, shape, size, and relative position of figures”
Here’s something which surprised me: tiling and tesselation is something which can be studied as well. Check out this video by Veritasium if you’re curious. How do you even go about proving something like this!?
Analysis
Getting curious about functions.
What can be said about a given function? How will it develop if you put in really large numbers? Is there something special to be said about where its highest or lowest points are?
Being able to derive and compute integrals (calculus) is related to this field.
Mathematical Logic
Can we find out interesting things about the way we state that something is true, or that statements “follow” from other statements?
Going to its extreme, we can go full inception here. Proofs about proofs.
Can we even prove everything, or are there things which can’t be proven, even if they are true? Good stuff. Also, my personal kryptonite for some reason.
Once again, there’s a great Veritasium video about a problem from this field.
Combinatorics
“study of finite or discrete collections of objects that satisfy specified criteria”
The classical question: “how many possible ways to arrange 3 different marbles are there?” is part of this. Also, graph theory seems to be contained in this field as well? I’m surprised and somewhat confused.
Note: the nodes in a graph can be seen as just another kind of “marbles” with different properties (connections to other marbles).
Stochastic
Probabilities and statistics are subfields of Stochastic.
The first one is about random events. It seems like people got mostly interested in this when trying to win at chance & gambling-style games. It probably helped them win.
The second one is about dealing with datasets. You can answer questions like: “could the measured datapoints be useful, or could they be noisy due to randomness?”
Numerics
Very relevant for anyone doing calculations with computers.
What if we need to do computations, but our numbers have limited precision? How can we get good results, instead of being unbelievably imprecise due to accumulated mistakes?
Topology
This field blows my mind, but I’m curious to find out more about it.
“properties of a figure that do not change when the figure is continuously deformed”
In Conclusion
What a long list! It took a while to get it together. Even though I didn’t go into much detail on most of these fields.
Keep in mind, there are lots of ways to classify areas of mathematics! The above is my interpretation, and there could be good reasons to disagree on some points. At least I can imagine that’s the case.
Anyway, I hope you had a good time with it. Thanks for reading, and all the best for your future curious pondering.