What Is Math About?

I haven’t really understood mathematics for a long time. It never occured to me, that there was something to be interested in.

There was plenty of exposure to using math. Thanks computer science! But I managed to get by without a deeper understanding of what math is about or an appreciation for it, besides the benefits of practical applications.

I dealt with applications. Getting very familiar with the leaves of the tree, never caring about why the tree came to be in the first place.

That changed recently. Years after the last involuntary structured exposure session.

Some things become less appealing if there is a need to deal with them. And ever more interesting and motivating once that external pressure is gone. Go figure.

This article is about the humble definition which made it happen.

An Obvious Word of Caution

I would bet that my understanding of most things below is at least somewhat incorrect. Some part can be completely wrong. I wouldn’t know at this point. However, I hope that it’s interesting and useful enough to warrant your attention and time.

With that out of the way, let’s get to…

The Sentence That Made It Click

Have you ever experienced a motivation and interest boost for something you had to do, but don’t have to anymore? That happened to me and math.

Eventually, I realized that I don’t know what the word means. Having a way to sum “math” up made the difference. Thanks Wikipedia:

“Most of mathematical activity consists of discovering and proving (by pure reasoning) properties of abstract objects.”

The corresponding German Wikipedia entry is even more delightful:

“Für Mathematik gibt es keine allgemein anerkannte Definition; heute wird sie üblicherweise als eine Wissenschaft beschrieben, die durch logische Definitionen selbstgeschaffene abstrakte Strukturen mittels der Logik auf ihre Eigenschaften und Muster untersucht.”

Which (according to me) means: “There is no commonly acknowledged definition of Mathematics. Today, Mathematics is described as a science, which uses logic to examine abstract structures, self-created through logical definitions, for their properties and patterns.”

Ka-ching.

The Insight

That’s all there is to math. Really. You come up with (or discover) some abstract thing and then you start looking into what can be said about it. You look for interesting properties (for some value of interesting) and just go on from there.

It is tinkering and mentally challenging entertainment. Sometimes motivated by, or leading to an external application. But mostly due to pure curiosity.

I don’t know why exactly, but being able to think about it this way made all the difference for me.

A Math Onramp

You stumble over an abstract construct. You get curious about it. You can’t help but try and find out what can be said about it with absolute certainty.

That’s all there is to “getting” the fascination of math, and getting into math.

You can do it yourself, if you feel like it and the timing is right.

Take something you have encountered in the wild, and try getting curious about it:

• What can be said about whole numbers?
• What can be done with those things called primes?
• What could an ‘infinity’ be?
• …

Then you either read up what others have found out, or try to find out more yourself.

It’s about finding something abstract which you can get intellectually curious about, and following that curiosity.

Well, apart from having some tools to be able to know if the things you suspect to be true, actually are true…

Proofs.

We haven’t talked about proofs yet. It’s the tool you can use to make sure that you found something true.

Proofs are beyond-comprehension awesome.

Imagine, you can say with 100% certainty, that a certain statement is true if certain preconditions are met. And you can be sure that it is always true. There is no doubt left.

For example, you can be sure that an equation will work for any positive number. Without having to try it for all of them, or relying on luck.

Or you can prove that the solution for a problem is the best possible one. Or that there is no solution to a particular problem (so you can stop looking for one and get on with your day). How cool is that!?

You can have proofs about numbers, about subsets of numbers, geometrical structures, equations, the list goes on and on. Any abstract thing (see above) you can come up with.

If you haven’t experienced the power of proofs, or are on the fence - check out Mathematical Induction.

A word of caution: I needed around 10 years to really understand it, instead of just going through the motions. Being in the right state of mind to really get it, instead of having a vague mechanical understanding of the process is never guaranteed I guess.

What Does Math Contain?

Subfields. There are many categories of “abstract things” to be curious about. I’ve had a great time getting an overview for myself, and trying to write it down. Check it out if you’re curious as well!

Closing Remarks and Resources

I hope that I could share some of the awe and wonder that have finally found their way to me.

If you are curious to learn more, here are some things which are on my own list of “possibly cool resources to check out if that mood strikes again”:

Books

I didn’t get through any of these yet, but what I have seen has been quite enjoyable:

• What is mathematics?
• How to solve it (you can also read the summary on Wikipedia)

Great Videos

The list is non-exhaustive of course :) Most of these are from Veritasium.

• The Infinite Pattern That Never Repeats
• (DE) Irrationale Zahlen | Mathewelten | ARTE
• Math’s Fundamental Flaw
• How Imaginary Numbers Were Invented

Random Notes

Things and thoughts which haven’t found their way into the above text. This is for future-me, but if you’ve read this far, they might be interesting to you as well!

1. There are different ways to approach math:

   • historically
   • pure to less pure (applied)
   • from a single application
   • most basic to more complex
   • starting with obvious structures and going to more advanced ones

2. There are lots of ways to classify areas of mathematics! I looked into one way to go about it if you’re interested.

3. Sometimes words which were used in the beginning got replaced by broader terms. I might mix them up, because my understanding is far from deep. However, I think there’s value in having an overview, even if that overview is slightly wrong.

4. Some areas are interrelated, because problems can be viewed as related to both, or one uses techniques and elements from the other.

5. This video contains a small part about how math was seen in ancient times. It was part of the quadrivium:

   • math - numbers
   • music - numbers in time
   • geometry - numbers in space
   • astronomy - numbers in space and time