How good is it for a layman to rediscover the core idea of a math field?

For some background, I’m in high school and I really love thinking about maths from different angles. Recently I had an idea where I imagined placing numbers on different 2D and 3D shapes, then analyzing how operations between the numbers could change depending on the shape itself. Like the geometry/topology of the shape affecting the relationships and operations.

Later I searched online and found out that something somewhat related already exists, called Cellular Homology. I don’t know if my idea is actually close to the real field or if I just stumbled onto a vague resemblance, but honestly I was really happy that I independently arrived at something even remotely connected to an actual area of mathematics.

So my question to math graduates or people deeply into mathematics is: how significant is it when someone independently rediscovers the core intuition behind an existing math field? Does this kind of thinking actually help in becoming better at mathematics, or is it more common than I think?

I’m not claiming I rediscovered the whole theory or anything huge just that it felt exciting to naturally arrive at a similar kind of idea.

How good is it for a layman to rediscover the core idea of a math field?

For some background, I’m in high school and I really love thinking about maths from different angles. Recently I had an idea where I imagined placing numbers on different 2D and 3D shapes, then analyzing how operations between the numbers could change depending on the shape itself. Like the geometry/topology of the shape affecting the relationships and operations.

Later I searched online and found out that something somewhat related already exists, called Cellular Homology. I don’t know if my idea is actually close to the real field or if I just stumbled onto a vague resemblance, but honestly I was really happy that I independently arrived at something even remotely connected to an actual area of mathematics.

So my question to math graduates or people deeply into mathematics is: how significant is it when someone independently rediscovers the core intuition behind an existing math field? Does this kind of thinking actually help in becoming better at mathematics, or is it more common than I think?

I’m not claiming I rediscovered the whole theory or anything huge just that it felt exciting to naturally arrive at a similar kind of idea.

I saw the Collatz Conjecture and also Erdős’s famous comment that “mathematics is not yet ready” for problems like it. I know he obviously meant that current mathematics doesn’t yet have the tools powerful enough to seriously attack or solve it.

But my question is: what exactly did he mean by “tools”?

. Like, what kinds of ideas, theories, or methods are missing right now? What would a future “tool” for solving something like Collatz even look like?

For example, in the past people couldn’t solve certain problems until entirely new areas of math were invented -calculus, group theory, complex analysis, etc. So for Collatz-type problems, what kind of breakthrough are mathematicians waiting for? A deeper theory of randomness in numbers? Better ways to understand iteration and dynamics on integers? Some bridge between number theory and chaos?

I know nobody actually knows the answer yet, but I’m curious how mathematicians think about these “missing tools” and what they might look like in the future.

How hard is it to come up with those insanely simple but deep conjectures in maths? Like I’m still in high school and I genuinely wonder how people like Lothar Collatz, Christian Goldbach, or Adrien-Marie Legendre came up with conjectures that are so easy to state but somehow survive for centuries.

Things like:

• Every even number is the sum of two primes.
• The Collatz process always reaches 1.
• There’s always a prime between consecutive squares.

These statements are so simple that even school students can understand them, yet some of the best mathematicians in history still can’t fully prove them. That feels almost unreal to me.

What amazes me even more is that these conjectures don’t look “complicated” at all. They look like observations anyone could notice, but somehow nobody can crack them completely. It makes me wonder:

• Is coming up with a deep conjecture actually harder than proving one?
• How do mathematicians even notice patterns that are worth studying?

How hard is it to come up with those insanely simple but deep conjectures in maths? Like I’m still in high school and I genuinely wonder how people like Lothar Collatz, Christian Goldbach, or Adrien-Marie Legendre came up with conjectures that are so easy to state but somehow survive for centuries.

Things like:

• Every even number is the sum of two primes.
• The Collatz process always reaches 1.
• There’s always a prime between consecutive squares.

These statements are so simple that even school students can understand them, yet some of the best mathematicians in history still can’t fully prove them. That feels almost unreal to me.

What amazes me even more is that these conjectures don’t look “complicated” at all. They look like observations anyone could notice, but somehow nobody can crack them completely. It makes me wonder:

• Is coming up with a deep conjecture actually harder than proving one?
• How do mathematicians even notice patterns that are worth studying?

How hard is it to come up with those insanely simple but deep conjectures in maths? Like I’m still in high school and I genuinely wonder how people like Lothar Collatz, Christian Goldbach, or Adrien-Marie Legendre came up with conjectures that are so easy to state but somehow survive for centuries.

Things like:

• Every even number is the sum of two primes.
• The Collatz process always reaches 1.
• There’s always a prime between consecutive squares.

These statements are so simple that even school students can understand them, yet some of the best mathematicians in history still can’t fully prove them. That feels almost unreal to me.

What amazes me even more is that these conjectures don’t look “complicated” at all. They look like observations anyone could notice, but somehow nobody can crack them completely. It makes me wonder:

• Is coming up with a deep conjecture actually harder than proving one?
• How do mathematicians even notice patterns that are worth studying?

Coming from a Barça fan, I honestly don’t care if people hate me for saying this-but Vinícius Jr. fights for Real Madrid way more than Mbappé ever seems to. No matter how childish he can be at times or what he does on the pitch, you can’t deny his passion. He shows up, gives everything, and you can feel that he actually cares about the badge. Whether he’s starting, on the bench, or even just watching from the sidelines, he’s always emotionally invested in the team.

Mbappé, on the other hand, sometimes feels more focused on himself-like he switches on for big tournaments like the World Cup but doesn’t always bring that same hunger at club level. Vinícius might not be perfect, but his commitment and energy make a huge difference.

I’m currently in 12th grade, and this is something I’ve been thinking about for a while. Whenever I try puzzles or problems-whether it’s math, logic, or something else-I almost always get at least part of the idea, even if I don’t fully solve it.

Back in 11th, I even tried teaching myself some rocket science stuffs to myself (like from MIT OCW ). Of course, some parts were difficult bcz of the advanced maths and physics which i am not exposed to , but overall I felt like I was understanding things rather than getting completely stuck.

These days, I can sit with something like prime numbers for hours and keep exploring patterns, ideas, etc. Same with puzzles-most of the time I either figure them out or get very close without feeling like I’m forcing it too much.

The only subject I genuinely struggle with is chemistry (especially organic), but I think that’s more bcz of my personal hatred towards it.

So I’m wondering-is this normal? Or am I missing something? Like, am I just not challenging myself enough, or is this how learning is supposed to feel?

Would appreciate honest opinions.

I’m currently in 12th grade, and this is something I’ve been thinking about for a while. Whenever I try puzzles or problems-whether it’s math, logic, or something else-I almost always get at least part of the idea, even if I don’t fully solve it.

Back in 11th, I even tried teaching myself some rocket science stuffs to myself (like from MIT OCW ). Of course, some parts were difficult bcz of the advanced maths and physics which i am not exposed to , but overall I felt like I was understanding things rather than getting completely stuck.

These days, I can sit with something like prime numbers for hours and keep exploring patterns, ideas, etc. Same with puzzles-most of the time I either figure them out or get very close without feeling like I’m forcing it too much.

The only subject I genuinely struggle with is chemistry (especially organic), but I think that’s more bcz of my personal hatred towards it.

So I’m wondering-is this normal? Or am I missing something? Like, am I just not challenging myself enough, or is this how learning is supposed to feel?

Would appreciate honest opinions.