We can’t physically see or understand how complex numbers exist or work in our world in a nice way, but we know they do exist. Because we’ve made massive advancements in science and technology off the assumption that they exist and work, and our understanding of many things in the world including stuff as basic as the solutions to quadratic equations would fall apart. By the same token, there are many problems for which vectors and problem spaces of nth degree are used, where n>3, and there’s that whole adage where time is considered a 4th dimension. In that way, we often solve many problems, even rudimentary linear algebra ones, using sets in R⁴, R⁵, etc, and there are many, many invisible forces at work in our world such as gravity. We know how easily our brain can trick us, we still are easily fooled by optical illusions even when we know they’re there and what they are/how they work, despite our visual cortex being the one of the most powerful and most used part of our brain. So the idea of forces and things which we don’t have the capacity to perceive existing in the world is not anything new or foreign. There are frequencies we can’t hear, colors we can’t see, etc which other animals can and do. So why is the concept of n dimensions in the world so widely rejected? There must be a simple reason, I have heard that it has to do with the volume of a gas in a container being proportionate to its dimensionality or something
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Math allows for technically and literally anything. Its only a system of extrapolating rules once you have some base rules set. Currently a large portion of the maths we know (and is popular) has high practical value. That is, it is almost as if we have explored parts of maths that are useful.
Now, just because it’s useful, does not mean the universe obeys any maths rules. Not only are there infinite ways of constructing math, there will be infinite different rules. The universe just is. You are finding rules that seem to work out in the universe.
Complex numbers don’t Exist in physical sense. They are a concept which we can use to simplify certain calculation in a certain framework of math. Numbers don’t exist either. You can still assume they ‘exist’ and play around with the idea and see what things can be explained using that math, but it will eventually be insufficient or inconsistent.
TLDR – math is just a tool. If something can exist based on some math, doesn’t mean it should exist in the universe. Universe doesn’t give a F about our math.
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