Bijection has nothing to do with length. That just indicates that no line segment necessarily has more or less individual points than another. Length is not a count of points. Your fallacious thinking is in treating points in a line segment as a finite number.

1) Slant has no effect on the concept of bijection
2) Points are points, and don't have another shape.
3) whatever
4) Points don't have a size, so no.

I don't see how any of this is related to relativism

It has to do with relativism because something can't be both finite and infinite at the same time with respect to the spatial, yet it is so. Relativism is written into space, geometry, and matter. This has much to do with Hegels philosophy

To get from point A to B, a superstar is required, according to human reason. First there is a half length, but since your half there and not there yet, there is another half, and so the process goes to infinity. Yet the endless is within finite bounds. How odd.

Take that infinite series and apply alternating colors to each half step, say green and blue. After this, we can rationally wonder what color is at the end, for the end\limit is there. At this point all our intuitions fail, and we must adopt some form of relativism or say we know nothing of math whatsoever.

At this point all our intuitions fail, and we must adopt some form of relativism or say we know nothing of math whatsoever. — Gregory

It's not that bad. Really. :roll:

It has to do with relativism because something can't be both finite and infinite at the same time with respect to the spatial, yet it is so. — Gregory

What is both finite and infinite? The length of AB is finite, and is never infinite. The number of points that can be described along AB is not finite, and is never finite, hence the existence of the bijuntion between that segment and any other. Neither is both finite and infinite.

Is this just another version of those threads that treat infinity like a number and then draw contradictions?

Anyway, I still don't know what this has to do with relativism, a view that seems to have nothing to say about the sort of mathematics you're discussing here.

If you can subdivide a segment endlessly, to eternity, it's finite and infinite. Figures are round triangles then basically by definition. I am tired of reading that crazy man "Cantor proved" so and so. We know nothing of infinity. Kant, a more accomplished thinker than Cantor, wrote "Concepts without intuition are empty". As long as mathematicians continue to refuse to accept the Second Antimony as a true contradiction in the Kantian and Hegelian sense, as they both did, they will always have unintuitive "definitions" and unsolved " paradoxes". The odd numbers are eternally half the natural numbers, so there is no way at infinity they can be equal. To say otherwise is to make infinity a number. That is as air tight an argument as anything Cantor wrote. So we have contradictions among intuitions on infinity and so we say we know nothing. It's a bridge though to Eliatic realms however, a secret door though that mathematicians don't know about

The odd numbers are eternally half the natural numbers, so there is no way at infinity they can be equal. — Gregory

I can think of at least 3 things very wrong with this statement. For one, what's the difference between X being half Y and X being eternally half Y? I mean, 5 is half of 10, and no matter how long I wait, 5 will remain half of 10, so I suppose it is eternally half of 10, but saying it that way doesn't make '5 is half 10' more true.

Let's see... There are 100 natural numbers with magnitude less than 100, and there are 100 odd numbers with magnitude less than 100, so that's not what you mean by this statement.

2nd try... Half of the natural number 36 is 18, which is not an odd number, so the odd numbers are not the halves of the natural numbers (whereas the integers would be half the even numbers, interpreted this way). So anyway, I don't think you meant this either, but I'm out of ideas.

And where is 'at infinity'? That's the treating it like a number that leads to such nonsense statements.
Surely Kant didn't make any such statements. Your intuitions lead to contradictions it seems.

If you can subdivide a segment endlessly, to eternity, it's finite and infinite.

What, exactly, is finite and infinite? Subdividing a finite segment doesn't affect its total length, so its finite length is unchanged. Also, what's the difference between 'endlessly' and 'endlessly, to eternity'? That's twice you've used that redundant modifier like it means something different than its absence.

It's a bridge though to Eliatic realms however, a secret door though that mathematicians don't know about — Gregory

Wiki: "The Eleatics maintained that the true explanation of things lies in the conception of a universal unity of being. According to their doctrine, the senses cannot cognize this unity, because their reports are inconsistent; it is by thought alone that we can pass beyond the false appearances of sense and arrive at the knowledge of being, at the fundamental truth that the "All is One""

More like a trap door IMHO. :roll:

What, exactly, is finite and infinite? Subdividing a finite segment doesn't affect its total length, so its finite length is unchanged. Also, what's the difference between 'endlessly' and 'endlessly, to eternity'? That's twice you've used that redundant modifier like it means something different than its absence. — noAxioms

I was using "eternal" twice for emphasis. Do you deny that it is unintuitive to be able to divide something finite a literal infinite amount of times?

Do you deny that it is unintuitive to be able to divide something finite a literal infinite amount of times? — Gregory

For one, 'infinite' is not an amount, and 'literal infinite' is no different than 'infinite'.

I would instead find it unintuitive to suggest that there would be a limit to it, a pair of non-identical points for which there is not a location between them. To assert otherwise is to assert that two non-equal numbers A and B can be found such that (A+B)/2 is equal to either A or B or both. You have weird intuition.

So I would say that a finite segment can be divided without limit, and yes, this is intuitive to me.

This is not true of any discreet system. For instance, in the domain of integers, the finite segment from 20 to 6051 can be divided, but only so many times. There is no integer between 21 and 22, and (21+22)/2 does not define a new integer, nor is the answer necessarily equal to either 21 or 22. It might be, depending on how the divide operator is defined for the discreet set in question.

You have weird intuition. — noAxioms

It's the same intuition as Kant's CPR.

There are two aspects here. The finite that is infinitely divisible, and motion over it (Zeno's paradox). If I take a cube, and divide it in half, and then again, ect to infinity and then line them all up biggest to smallest, what is the smallest? Does the series go off into the physical horizon forever? You might say "there is a limit". But there is no final term, so what is right before the limit?

Now motion must travel the infinite series in order to reach the limit. If we add alternating colors to
each descending fraction like I said, what color is the limit? "The series never ends" you say. But motion can get to the limit, so therefore the color can too. So we have a huge paradox here.

I suspect something like dimensions greater than 3 in addition to non-Euclidean geometry might be able to get a handle on this, but aren't we entering Eleatic realms at that point?

If I take a cube, and divide it in half, and then again, ect to infinity and then line them all up biggest to smallest, what is the smallest? ...
You might say "there is a limit". — Gregory

I might not. You're the one making the suggestion and then driving your own assertions to obvious nonsense conclusions.

So we have a huge paradox here.

Exactly so. So maybe try it without positing a last term in the infinite series.

Does the series go off into the physical horizon forever?

Physical now? A physical cube-shaped object has a finite number of particles in it and can only be divided so many times. There is a smallest part at the end of the sorted line and it has a color if you're going to abstractly assign colors to the even and odd ones.

For me at least, this example causes two intuitions to collide, which is the essence of relativism (that is, that truth has no essence).
True? — Gregory

For you, at least? What does that mean? If that means you have run for yourself a nice warm tubful of ignorance and are now taking a public bath in it, please spare us the exhibition(ism).

The truth has no essence? What does that mean?

I am tired of reading that crazy man "Cantor proved" so and so. We know nothing of infinity. The odd numbers are eternally half the natural numbers, so there is no way at infinity they can be equal. To say otherwise is to make infinity a number. That is as air tight an argument as anything Cantor wrote. — Gregory

Eh? This merely says you don't know your subject - at all!

So what's your point?

A physical cube-shaped object has a finite number of particles in it and can only be divided so many times. — noAxioms

Before you said the object has an infinity of points. Now you say otherwise. If finite figures have an infinity of points, that is paradoxical. Qualities apply to objects, objects have infinite parts with no final term, yet the object solid

The truth has no essence? What does that mean? — tim wood

That's what relativism means. Truth is not truth

So what's your point? — tim wood

Comparing infinities using different measures (density vs cardinality) that contradict each other proves the mathematicians don't have any idea of what infinity is

1) how can something have no spatial final term but be finite

2) why can't you compare two infinities by first combining the cardinality with the density in each? Isn't this how you truly compare infinities?

If finite figures have an infinity of points, that is paradoxical. — Gregory

Finite figures have no actual points, but infinitely many potential points. Dimensionless points are not parts of the figures, they are something that we artificially impose on them for particular purposes, such as marking and measuring. Every part of a one-dimensional line is a one-dimensional line, every part of a two-dimensional surface is a two-dimensional surface, and every part of a three-dimensional solid is a three-dimensional solid. Figures of lesser dimensionality--a point on a line, a point or a line on a surface, and a point or a line or a surface on a solid--are limits that we create by arbitrarily dividing the whole into parts.

That's what relativism means. Truth is not truth — Gregory

I.e., Relativism means that X is not X? Or is it some special quality of truth that makes it not truth - in relativism?

Let's first decide/agree that you're not just dealing in solecisms. In as much as you're the one seeking, first decide whether or not the words "true" and "truth" have any meaning whatsoever. Presumably they do, and the question then becomes one of determining what that meaning is, under what circumstances. Consider true. True is a qualitative statement about a proposition wrt well-defined criteria. Truth is the name of that quality. Can they be narrowed down? Give it a try. I think true is criteria-dependent; that is, there's a variety of criteria. As such, truth, then, must be a general and abstract term referring to particulars, but in itself to nothing particular. Can you improve?

So far as I know, there is no corner of relativism that makes truth untruth, or true untrue. The only differences, then, between are differences in specifications and particulars. But that only means that people who differ, differ in regard to things that are not comparable, and to the degree they differ, that they're talking past each other and don't know it.

It would help you, imo, for you to try to define true/truth as rigorously as you can, especially to distinguish them from fact. The more you do that, the more you will know what you're talking about, at least, and whether your interlocutor and you understand each other and are talking about the same thing.

1) how can something have no spatial final term but be finite

2) why can't you compare two infinities by first combining the cardinality with the density in each? Isn't this how you truly compare infinities? — Gregory

1) Eh? Clarify? My ruler is spatial, finite, and its end can be measured to an arbitrarily precise point. It appears you're confusing reality with mathematical concepts. Spend some time with the concept of the least upper bound - lub.

2) I understand cardinality as "how many." Please define "density." Can't compare until the referents are known and understood.

If by density you mean "how many in a defined space," then you're already building in problems in understanding what you're talking about.

Comparing infinities using different measures (density vs cardinality) that contradict each other proves the mathematicians don't have any idea of what infinity is — Gregory

Who is comparing? Please show how the two compare. How do they contradict each other?

https://www.encyclopediaofmath.org/index.php/Density_of_a_set#Density_of_a_measure

Wiki: In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A; that is, the closure of A is constituting the whole set X.

Wiki: The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all those sets which have the same cardinality as A.

To whom it may concern:

Truth is a movement that is not essential and is not substantial. At least that is how Hegel saw it and that is how I've considered it ever since I read his first book for the second time. If you can keep your mind one step ahead of the "but is THAT true then" game, you're a relativist

Zeno broke as much ground with his paradox as Euclid did with all his propositions. The distance from A to A is infinitely divisible (whether speaking of real space or abstract space), yet you can get across it because it's also finite. You don't just cross the finite, you also cross the infinite. Their combination is a wondrous thing, which you are all missing out on seeing. It's not clear exactly how it works. Motion is determinate AND indeterminate, and is like this because space itself is like this. The endless descending fractions as you halve the distance reaches the limit, but what touches the limit? That is the question. There is a fuzziness in there that you can't deny

As for odd numbers having the same cardinality as the natural numbers, it is as obvious to me that *what is greater in the finite must stlil be greater when transcended to infinity* as it is that 2 is greater than 1. Sure, maybe Cantor had some points. But that doesn't cancel what seems obvious to me. This proves MY point (to myself at least) that infinity is a much greater mystery than mathematicians want to admit. In fact I think most of them implicitly want to think infinity is some type of number.