How mathematicians build their own castles

Illustrative

What I would like to share with you today is my impression of how Mathematics is built and how it works internally. I believe this is an important thing to know about, because, over the course of my early studies, I have met with far too many people who have expressed their interest in Mathematics at first, but literally ran away later on. Why? Well, because they were terribly unaware of how it works at all! We all know that kind of people – their favorite line is “There are no numbers and too many letters, this is no math!”.

This statement couldn’t be more wrong. And I’m very passionate about pointing this out whenever a conversation involving it occurs. What’s the point of math, then?

Definitely not just crunching numbers

Many people think that letters have nothing to do with mathematics.

Many people think that letters have nothing to do with mathematics.

No, math isn’t just about numbers. It’s about abstraction. It studies many, many different kinds of objects and the relationships between them. It is about reasoning. About structure. And, believe it or not, about beauty.

But this is often not what we are being taught on elementary or even high school level. I find this to be a little flaw responsible for spreading such horrible opinions, like that mathematics is just about numbers. Sure, they are the first concept we are introduced to, and we eventually become very familiar with them. So much so that we forget one thing: The idea behind a number is just as abstract as the one behind circle, the one behind function, or derivative! The symbol “3” usually has the same meaning as roman “III”. It is a name we simply give an abstract object within our mind. Just as we name a function f, or call a radius of a circle r.

Numbers, in this manner, are just one specific kind of objects we can study. There are myriads of others, there are probably no limitations. And the clear and strict structure of Mathematics allows us to work with them, create new concepts and eventually, bring them to use in real life. So what does this structure look like?

The basic building blocks

An important thing to realize is that mathematicians don’t like vague expressing. Everything (!) a mathematician ever talks about has to be clearly well-defined. That’s why, the most basic building block of Mathematics is a definition. What is it?

definition introduces a new mathematical object (term), using only objects already known or elementary.

That doesn’t sound too complicated, except for one thing. What are elementary terms? That is indeed a good question. Surely you have encountered the philosophical problem of asking “Why?” ad infinitum. Little kids like to do that. (I have especially loved to do that, drove my parents crazy. But I guess that’s why I eventually fell in love with mathematics.)

Well, this wouldn’t be too practical, and quite frankly, not too aesthetic either. And so, to retain the rigor Mathematics is known for, mathematicians came up with a little trick.

It’s called axiomatization. Some objects or relationships are so intuitive to our minds, that we understand them to be apriori existent and true. And although the choice of axioms is mostly up to you, there are some guidelines. It is also important to understand the difference between an axiom and a dogma. While dogmas are claimed to be true and attempting to disagree might even cost you your life (just recall Middle-Ages), axioms are only supposed to be true and are the basis for reasoning. So to sum up

An axiom is a statement accepted as true without explaining or proving it by using other statements.

The concept of proof is also very important. Every statement other than axioms has to be proven. These statements are called theorems, lemmas, or simply statements or observations. What differs them is mostly their difficulty and applicability – but for the sake of simplicity it’s okay to call them all theorems.

theorem in mathematics is a true statement, derivable and provable using other statements, definitions and axioms within a mathematical theory.

These are indeed the most used building blocks of mathematics. And most importantly, there are the proofs. A proof is like a glue that holds all the building blocks together. Formally, you could say that

proof is a finite sequence of logically correct steps, leading from assumptions to conclusions, using only known facts.

When you give a mathematician a solid ground to stand on (the axioms and definitions) and enough building blocks (theorems) and glue material (proofs), he can be thought of as a builder. This is a nice allegory I like to remind myself of time to time. In this sense, mathematicians really build their own castles. Or, for that matter, they build anything they like, since there are no limitations!

How about you? Do you have anything to add? A similar allegory of yours? Let me know in the comments section below!


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