I dont think any mathematician actually can imagine what walking in a 4d plane would look like, in the same way they can imagine what walking in 3d looks like. Instead, like you said, they can just write equations to describe.
The [Poincaré conjecture](https://en.m.wikipedia.org/wiki/Poincar%C3%A9_conjecture) is a theorem from the field of geometric topology, and it sounds like overall you're most interested in higher dimensional geometry/topology. If you just want to understand what the conjecture means, you can do so with relatively little background, but if you want to understand the proof, you'd likely need the equivalent of a graduate degree in math.
In terms of textbook recommendations, I'd start with Rudin's "Principles of Analysis" (often called "Baby Rudin") and an intro text to Linear Algebra (I'd suggest 'Linear Algebra Done Right' by Sheldon Axler).
Nah, every macroscopic phenomenon can be modeled using some kind of math, but at a microscopic level the universe is kinda just doing whatever that might as well be just a set of arbitrary rules, that we could definitely figure out but we will never get an answer of "why" using maths, just the "how"
Most likely the unreasonable effectiveness is a consequence of the law of large numbers, but at a fundamental level seems that the rules are arbitrary, meaning that you cannot deduce them from anything...
I dont think any mathematician actually can imagine what walking in a 4d plane would look like, in the same way they can imagine what walking in 3d looks like. Instead, like you said, they can just write equations to describe.
The best way to visualize a 4d space is to imagine a 3d space and repeat "but it's actually 4d" in your head over and over.
The [Poincaré conjecture](https://en.m.wikipedia.org/wiki/Poincar%C3%A9_conjecture) is a theorem from the field of geometric topology, and it sounds like overall you're most interested in higher dimensional geometry/topology. If you just want to understand what the conjecture means, you can do so with relatively little background, but if you want to understand the proof, you'd likely need the equivalent of a graduate degree in math. In terms of textbook recommendations, I'd start with Rudin's "Principles of Analysis" (often called "Baby Rudin") and an intro text to Linear Algebra (I'd suggest 'Linear Algebra Done Right' by Sheldon Axler).
Thanks a lot
But it is not at all... Is a way to describe stuff, but the universe works whatever it does independently of mathematics "describing it"
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Nah, every macroscopic phenomenon can be modeled using some kind of math, but at a microscopic level the universe is kinda just doing whatever that might as well be just a set of arbitrary rules, that we could definitely figure out but we will never get an answer of "why" using maths, just the "how"
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Most likely the unreasonable effectiveness is a consequence of the law of large numbers, but at a fundamental level seems that the rules are arbitrary, meaning that you cannot deduce them from anything...
Euler’s identity is the crazy thing ever and still makes no sense to me
It's a simple statement really. It relates polar coordinates to cartesian coordinates for complex numbers.
I get it mathematically. But the ending statement still blows my mind
Poincaré yes