A bit of explanation I wrote for someone elsewhere:
Scaling it shows how in principle the lines have no width. It’s easier to convince yourself of this in the black-on-white construction. In principle what we have constructed here is invariant of scale. In mathematics that means we have three different ways to describe ’smoothness’. For example pure white noise is also scale invariant, so it is actually smooth.
For shrinking down to infinitely small size we cross two lines and look at their intersection, which is a point. With the black-on-white scenario we can imagine we have additional ways of describing a point, such as a cusp when we lay two black sheets at an angle on the white sheet.
Folding it is Zeno’s paradox. We can actually get to some interesting properties of our extended lines and points by applying that logic. It’s a bit of a hurdle how to fold something which is scale-invariant but we can come up with our own definitions of multiplication and division that way. We should pay attention if these are actually reversible, if multiplication and division are inverses of each other. Thats the difference between an operator/object group and a semigroup, from group theory. This is where we really get into the heart of mathematics, and we can get there pretty quickly!
In mathematics that means we have three different ways to describe ’smoothness’. For example pure white noise is also scale invariant, so it is actually smooth.
There are various definitions of smoothness, but a standard definition is that for a function to be smooth it must be continuous and either infinitely differentiable and all the derivatives must be continuous or differentiable exactly n times and the nth derivative must be continuous. Notice that has nothing whatsoever to do with scale invariance. Here is a math stack overflow discussion which is more concrete and specific[1] and here is a nice treatment of the subject using animation etc[2]. Note that the sine function for example is not scale invariant but is smooth whereas white noise is an example which is scale invariant but not smooth because it is not continuous or differentiable.
I don't think anything you're doing here is mathematics at all let alone getting to the heart of mathematics.
Mathematics is defining axiomatic systems and proving results in those systems. What you're doing is making a physical analogy and then implying all sorts of things that are not rigorous and therefore are not provable or disprovable from that analogy.
Comments
A bit of explanation I wrote for someone elsewhere:
Scaling it shows how in principle the lines have no width. It’s easier to convince yourself of this in the black-on-white construction. In principle what we have constructed here is invariant of scale. In mathematics that means we have three different ways to describe ’smoothness’. For example pure white noise is also scale invariant, so it is actually smooth.
For shrinking down to infinitely small size we cross two lines and look at their intersection, which is a point. With the black-on-white scenario we can imagine we have additional ways of describing a point, such as a cusp when we lay two black sheets at an angle on the white sheet.
Folding it is Zeno’s paradox. We can actually get to some interesting properties of our extended lines and points by applying that logic. It’s a bit of a hurdle how to fold something which is scale-invariant but we can come up with our own definitions of multiplication and division that way. We should pay attention if these are actually reversible, if multiplication and division are inverses of each other. Thats the difference between an operator/object group and a semigroup, from group theory. This is where we really get into the heart of mathematics, and we can get there pretty quickly!
There are various definitions of smoothness, but a standard definition is that for a function to be smooth it must be continuous and either infinitely differentiable and all the derivatives must be continuous or differentiable exactly n times and the nth derivative must be continuous. Notice that has nothing whatsoever to do with scale invariance. Here is a math stack overflow discussion which is more concrete and specific[1] and here is a nice treatment of the subject using animation etc[2]. Note that the sine function for example is not scale invariant but is smooth whereas white noise is an example which is scale invariant but not smooth because it is not continuous or differentiable.
I don't think anything you're doing here is mathematics at all let alone getting to the heart of mathematics.
Mathematics is defining axiomatic systems and proving results in those systems. What you're doing is making a physical analogy and then implying all sorts of things that are not rigorous and therefore are not provable or disprovable from that analogy.
[1] https://math.stackexchange.com/questions/12026/what-does-smo...
[2] https://www.youtube.com/watch?v=vD5g8aVscUI