An exploration of the foundations of logic and philosophy
10 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!
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.
[1] https://math.stackexchange.com/questions/12026/what-does-smo...
[2] https://www.youtube.com/watch?v=vD5g8aVscUI
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
Wildberger has videos about this on his YouTube channels. He came up with 'universal geometry' to get around the parallel assumption in Euclidean geometry if foundations interest you. He at least will enumerate all the cases needed for foundations to work if you're screwing around with your own for fun. Here's one example https://youtu.be/EvP8VtyhzXs
How come we say drawn line has 0 width and using paper instead of drawn line has non 0 width? I can say a paper has 0 width as well. But if we are to stick to physical world, then neither drawn line nor paper has 0 width.
You can't just treat things differently just because.
If your point is to obstruct view of an object, knowing objects dimensions we can calculate it's position behind the paper wall/line.
You can't just treat things differently just because.
If your point is to obstruct view of an object, knowing objects dimensions we can calculate it's position behind the paper wall/line.
The point is to see what we can still calculate when we change our assumptions.
In the case where we construct a black-on-white boundary with two sheets of paper we could for example say that the white paper on its own has two-dimensions but the combination of two papers has fractal dimensions, like 1.5 dimensions. This eventually leads to the Mandelbrot fractal.
But note that we took another step after that, similar to the first. This third construction let us very quickly start to think about highly abstract things and our minds remain capable of dealing with them. We can 'see' otherwise invisible things with our minds, like reading between the lines.
In the case where we construct a black-on-white boundary with two sheets of paper we could for example say that the white paper on its own has two-dimensions but the combination of two papers has fractal dimensions, like 1.5 dimensions. This eventually leads to the Mandelbrot fractal.
But note that we took another step after that, similar to the first. This third construction let us very quickly start to think about highly abstract things and our minds remain capable of dealing with them. We can 'see' otherwise invisible things with our minds, like reading between the lines.
I'm assuming maximum good faith here but this only seems interesting/contradictory because you haven't defined the system rigorously. All the seeming contradictions are coming from the fact that you are applying Euclidian geometric constructions to set theory via this example involving covering things up with actual pieces of paper.
The parallel postulate doesn't deal with pieces of paper. It deals with lines in Euclidian space. These spaces have infinite dimension, and the lines have infinite length, so the very first deduction you are attempting to draw (about "half the logic" being represented) is literally meaningless. If you take an infinite Euclidian space and partition it into two halves with a line, they are each infinite in size. A Venn diagram is not a Euclidian concept, it is a diagramatic representation of set membership. So you can't say anything about logic purely from the idea you can partition a space in half.
Your next paragraph about the logic of camouflage is also meaningless for the exact same reason.
The section about an object on one side or the other of the line is simply a category error. If we have an object with non-zero area and a line in a Euclidian space then the object can be on either side of the line or the line can intersect with the object. Nothing about this changes the position of the object or the line and your "paper" construction simply adds some confusion to a very simple geometric system. If you're actually dealing not with a Euclidian space but with set membership then it's best to abandon this paper/line construction and just talk about set membership. If you have an object and two sets, the object can be in either set, both or neither. This is not a contradiction and the object has no "position" in this system. It's simply a question of definition. If you want to learn how set theory works I recommend you read Naive Set Theory by Paul Halmos.
The final paragraph is meaningless because all the things before are meaningless. If we're talking about Euclidian spaces these have nothing to say about logic (modified or otherwise) and if you are talking about sets then the parallel postulate is irrelevant because it is a geometric concept. I understand the Dedekind construction etc and really don't follow what you're saying about the parallel postulate being a statement about Cauchy completeness. From the fact that the coordinates of a point fully represent that point, the existance of lines in a Cartesian space (not the parallel postulate) is a statement about Cauchy completeness. That said, nothing you've put here affects the construction of numbers in any way.
If you want any of this to be actual mathematics you need to abandon your "paper" construction and define things properly.
The parallel postulate doesn't deal with pieces of paper. It deals with lines in Euclidian space. These spaces have infinite dimension, and the lines have infinite length, so the very first deduction you are attempting to draw (about "half the logic" being represented) is literally meaningless. If you take an infinite Euclidian space and partition it into two halves with a line, they are each infinite in size. A Venn diagram is not a Euclidian concept, it is a diagramatic representation of set membership. So you can't say anything about logic purely from the idea you can partition a space in half.
Your next paragraph about the logic of camouflage is also meaningless for the exact same reason.
The section about an object on one side or the other of the line is simply a category error. If we have an object with non-zero area and a line in a Euclidian space then the object can be on either side of the line or the line can intersect with the object. Nothing about this changes the position of the object or the line and your "paper" construction simply adds some confusion to a very simple geometric system. If you're actually dealing not with a Euclidian space but with set membership then it's best to abandon this paper/line construction and just talk about set membership. If you have an object and two sets, the object can be in either set, both or neither. This is not a contradiction and the object has no "position" in this system. It's simply a question of definition. If you want to learn how set theory works I recommend you read Naive Set Theory by Paul Halmos.
The final paragraph is meaningless because all the things before are meaningless. If we're talking about Euclidian spaces these have nothing to say about logic (modified or otherwise) and if you are talking about sets then the parallel postulate is irrelevant because it is a geometric concept. I understand the Dedekind construction etc and really don't follow what you're saying about the parallel postulate being a statement about Cauchy completeness. From the fact that the coordinates of a point fully represent that point, the existance of lines in a Cartesian space (not the parallel postulate) is a statement about Cauchy completeness. That said, nothing you've put here affects the construction of numbers in any way.
If you want any of this to be actual mathematics you need to abandon your "paper" construction and define things properly.
You are repeatedly begging the question so what follows is falsehood. The premise is to reconstruct the intuition behind the parallel postulate. You can look at Euclid's Elements to get a better understanding of constructivism.
I am most certainly not begging the question. I have addressed each of your points directly. I have a pretty good idea of Euclid's elements and the parallel postulate. Constructivism in the philosophy of maths is the idea that it is necessary to find a specific example of something to prove it exists. That's an idea that significantly postdates Euclid.
This is simply factually incorrect.
First imagine a black line on a white piece of paper. It separates one half from the other and we say the line has 0 width.
Replace the line with a black paper half-way over the white paper. We still have a sharp line separating one side from the other but it is as if just half of the logic that the previous line if 0 width is represented. We are invited to imagine a variation of Venn diagrams which have different logical implications than we are used to, implying a logical context which isn't classical. Perhaps we can use this clue to reason clearly about things which are otherwise paradoxical?
Then see if this experiment is repeatable. Instead of a black paper we place another white paper over the first. Now we have constructed a line again and we can also tell a little more about what happened in the previous step. We know there is a line there because we constructed it but the contrast has disappeared so we can't see clearly where the line is. This is some kind of logic of camouflage.
Let's imagine we have an object which we compare with itself it as it is represented on each side of our line. In the 0-width example the object is clearly either part of one half of the paper or the other. In the last example with our invisible line the object's position is unknown. But, just as we know there is a boundary by the act of constructing the invisible line we can insert the information that the position of the object has changed, even repeatedly. What can we do with the black-on-white paper construction to extract information from the system? What type of information do we need to insert to produce meaningful observations?
Since the parallel postulate states that there is exactly one line through a point parallel to another line, what do our modified logics reveal about unity and equality? If our 0 width line is a number line and the parallel postulate is a statement about Cauchy completeness, what do our extended logics construct as number-like objects?