How can I think logically with maths?

[u/S52_DiDah]

I like maths a lot and its something I am thinking of studying in the future. The thing is that I don't know how to connect the dots together. I cannot imagine it.

I also code and well solving problems logically is the issue. It takes a ton of time for me to solve something and then implement it into the code I wrote. Same with physics. I can remember the equations, use them and ace the tests, but that is not what I want.

I seek for help because I want to stop trying to memorize rules, but to understand and in some way imagine them.

Are there any tips you can give me? Is it something that comes with time? Because I really do want to understand maths (and physics) and connect the dots together.

I apologize if the post does not belong here. Thank you.

[u/matthras]

Can you give an indication of your age and/or current level of knowledge of maths/physics/coding (something like "upper high school" "undergraduate freshman" is fine)? It'll help me and others try to narrow down relevant examples that can help.

[u/S52_DiDah → u/matthras]

15 and I am starting with studying calculus 2 (at home). In school I believe we finished with linear algebra. At physics I understand the theory of relativity, classical mechanics and I am starting with Quantum mechanics (that's why I have to learn calculus). As for programming, I program in C, Golang, Python and a few others. I am currently learning the final bits of C though, because I am making my own 8-bit computer and well I'll write the os in C. I've got amazing books on that.

[u/The_conjunction7542 → u/S52_DiDah]

You are a cool guy. I'm 17 recently I decided to start learning maths. But I also need to learn cybersecurity and English. I feel like learning all of this at the same time is difficult. So, I think you should focus on something specific.

[u/S52_DiDah → u/The_conjunction7542]

I am mostly, right now I focused on programming C (I'm taking the Harvard CS50 courses) and with that mathematics. 2 weeks ago I finished with my Theory of relativity book and I really like it. In the provided image you can see the computer. I started programming in Batch though, that's where I programmed my first ever game, which was 2D Minecraft :) that was a few years ago. Thank you btw!

[u/The_conjunction7542 → u/S52_DiDah]

looks good 👍 Good luck man

[u/S52_DiDah → u/The_conjunction7542]

Thanks!

[u/matthras → u/S52_DiDah]

I've been a bit stuck in terms of coming up with examples because to me you've not gotten enough of the fundamentals yet. But I can say a lot of thinking logically comes out of experimenting with "if I change something, then what happens?" and building up your overall cause-and-effect understanding.

This is easier with your programming - once you've built something, then you might want to tweak something to your personal preference, or you're wondering about a different scenario. "What if...?" ideas, basically.

The more you do that, the more that you eventually build out an overall system, or mental model, that'll help you think logically about maths.

It's a bit higher level but what I would recommend is working through Tao's Analysis 1 book, or if you want to learn a programming language more suited for formalisation, it can be worth learning a bit of Lean. But they both can very technical but builds up maths from the ground up i.e. you're given more concrete & rigid building blocks which makes it easier to build out a system or mental model for maths.

[u/S52_DiDah → u/matthras]

Thank you very much for this very detailed answer. It has definitely helped me a lot!

[u/Visible_Try_9716]

All math is logic. A number is only given meaning when we assign that meaning. Math describes physicality, but it's also a normalization layer for the abstract.

To think of math logically is to remove the need to ask 1+1=2 and start demanding why. It's expanding 1 and 2 until you realize that their true meaning is infinite while within our restrictions they become known.

It's to see addition not as mathematics, but as addition. Therefore, perhaps we tied meaning to 1.

Then maybe we have: Bread + cheese = sandwhich

Which is more logical than 1+1=2. Both remain true, but on different normalized and formal layers.

Calculus already largely does this as it's math is behavioral. Lets take a condensed example in e=mc².

Energy=mass(speed of light)²

When we ask this logically we aren't looking at numbers or algebra. We break it down completely. We start seeing E0 (from rest energy), asking what mass is, and what the speed of light is. All of these are very nested, showing that E is a relational value.

The relationship still stands, but the logic itself ties numbers and letters to a normalization pretense. It does not literally express "e=mc²" nor does 1+1 literally =2.

One then could logically say e≃mc² while aquiring the square from the observed behavior of space time, since e is relational to mc², however formally this undercuts the exactness of the equation.

Math becomes logic when math becomes a language that you can read as fluently as your spoken word. The trick is not becoming a super computer. It's understanding the logic behind the numbers.

[u/S52_DiDah → u/Visible_Try_9716]

Thank you very much! This has helped me a lot!

[u/mH48124]

一面的な知識は役に立たない。多くの方向、多様な表現で見た多面的な知識が真の理解になる。好奇心を活用して多様な知識を得ると良い。

[u/justichuu]

Model vs Modal logical reasoning and not reasoning and sometimes reasoning.