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So, I don't really understand this stuff, but someone who does told me: When they hear "dimensions" some people think about time being our 4th dimension, or like Marvel style alternate dimensions, forget all that. Completely unrelated to math dimensions.
Didn't get me to understand what came after that, but it helped me not think about it in a completely wrong way.
I am by no means an expert (closer to the opposite, actually), but I always considered Time as the 4th dimension to be a bit of a fallacy that inhibits further thinking on the subject.
Sure, time can be considered a fourth dimension when looking at it from a perspective of trying to determine possible states of a system, as it adds one more axis in which an object can move. However, when it comes down to mathematics and physics, I consider it more like a property that is applied to a system and objects in it, regardless how many axis of movement is applied to it.
And just to be clear: I don't understand it fully either, and I don't know if it is correct. it's just how I prefer to look at it, and it holds true in most circumstances I stumble across. I have no idea how a tesseract would look to a four dimensional eye.
You're more right than you might have expected, but not because it's a fallacy or misleading. You noticed something important in how it all works: time is a dimension, but it doesn't act like "up" or "forwards".
This doesn't make it less of a dimension or a hindrance to understanding, it's an observation that leads to: there are different types of dimensions.
Typically called time like and space like, they can also be thought of as "one directional" and "two directional", although a physicist somewhere is (correctly) coughing politely and glaring at some of the shit photons get up to at the thought of one directional time.
You're thinking of time as a parameter, which is how it is in classical mechanics. It's a different category of thing, but it technically makes the system 4d.
When you start looking at how light moves and relativity you find that you actually need time to act much more like another direction because it no longer defined an order or sequence, and you get stuff like "time slows down when move faster in space because acceleration shifts your movement vector in space time".
It's even simpler in math, because a dimension is simply a number required to specify a point in a space. If you cared to you could use "left" as your parameter and talk about how a thrown ball changes position in time, up, and forward as a function of left.
Then you could do some real math and use that function as a point in some space and talk about how the different components are different dimensional aspects of the infinite dimensional polynomial function space.