> In higher dimensions, there is no geometric analogy and people who rely on geometry alone find themselves unable to think abstractly in higher dimensions. Of

> In higher dimensions, there is no geometric analogy and people who rely on geometry alone find themselves unable to think abstractly in higher dimensions. Often features in higher dimensions can start to be seen in lower dimensions. For example the common example of the strangeness of higher dimensional spaces is the case that a hypercube with side length 2 can have a smaller volume that a hypersphere touched by hyperspheres of radius 1 at each corner of the hypercube. This is not true in 3 or even 4 dimensional space. But if you use geometric intuition to see the trend from 1d to 2d to 3d, it seems entirely reasonable. (Apologies if I didn't explain the problem well. I wanted to find a video on it, but couldn't find one) > I believe intuitions are good for laying a foundation, but abstractions are necessary for generalization and for building larger theoretical systems. Intuitive constructs can only get you so far. I have a slightly different view on this. I feel that abstractions are fundamentally necessary for communicating rather than conceptualization. For example, you may realize that the type of theorems you have been making about vectors equally applies to polynomials, but in order to convince others you would need to write done the properties they both have (resulting in an abstraction). Its kind of a subtle point because communication is key to mathematics. But normally, before I try to write a proof, I figure out the intuition for why it must be true in my head. Because this is a powerful tool, I try to learn a solid intuition for any abstraction I am introduced to.