Does it go up forever into space or does it stop at a limit? — Gregory
When we do arithmetic, any number can have a half, so 1 plus 1 can really equal 4 in that case,[/quote}
How do you conclude that? It's a bit much for me.
— Gregory
I hope to become bolder and use my ambition to solve the paradox of Banach and Tarski — Gregory
The vsauce video was where I first encountered B\T. His supertask video also showed me that I was not alone in thinking about "Zenonian cubes". I know that mathematicians look at Banach-Tarski with many equations in mind, but I've always looked at it from the angle of Zeno's paradox alone. So my series of questions has been — Gregory
1) if space is infinitely divisible than it has infinite parts despite the fact that we experience geometric things as finite — Gregory
2) calculus says that a infinite number can be subsumed by a finite measurement. But in spatial terms how is this possible? — Gregory
3) how can something be spatially finite and infinite is what appears to be "the same respect"? — Gregory
4) if an object has infinite parts we can take infinite parts out and have a new object, hence Banach Tarski. But isn't this entirely counter intuitive? — Gregory
5) this is all paradoxical to because of the way I think of objects as finite. What is the way forward? — Gregory
I wanted to explore the non-Euclidean stuff with more care because it is also counter intuitive and might give me a clue on how to find the fundamental principle of all geometry and space. — Gregory
I'm not trying to prove anything to other people, but trying to find an understanding that satisfies myself. Some are ok with Gabriel 's horn. I don't have peace with it — Gregory
If something is spatially finite it's finite, not infinite. — fishfry
Do you mean how can a finite length, like the unit interval [0,1], contain infinitely many points? That's pretty simple, if nothing else 1/2, 1/4, 1/8, 1/16, ... are infinitely many points contained within the interval. — fishfry
There are a lot of buzzwords in there but the Wiki proof is pretty decent. — fishfry
But to be honest, earlier you claimed that 1 + 1 might be 4, and you didn't respond when I asked for clarification. May I suggest nailing that down first. — fishfry
Do you follow the calculus in Gabriel's horn? The integral of 1/x from 1 to infinity is infinite, and the integral of 1/x^2 is finite. It's just how it is and the proofs are perfectly straightforward. — fishfry
Modern mathematicians seem to have forgotten that Aristotle covered up this problem with a sophistry and that Kant presented this problem in one of his antimonies. If I want to know how many parts an oven has or a loaf of bread baked in it, I simply have to ask how many times I can mentally divide it. And it turns out I can do this infinitely, yet the bread and the oven are finite. Mathematicians now longer see this as a problem or as even strange, and I don't know why — Gregory
Yes, but this is contradicted by infinite divisibility, which all space must have. — Gregory
Presenting the problem in terms of numbers instead of space obscures the issue — Gregory
I am trying to comprehend the first few axoims of all geometry, and i'm not sure the specifics of B/T relate. I only was talking about B/T in terms of taking an infinite of points out of another infinity of points. — Gregory
Sure. If we have two 12 inch rulers, they are equal 1 to 1. However with numbers half of 1 is also a number, so if we apply to this the ruler we have 2 six inches on one side and 2 six inches on the other, hence instead of 1 and 1 being compared, it's 2 and 2. — Gregory
The reason is that in arithmetic you have to have basic numbers that are understood as not divided. In geometry, all space is divisible and its impossible to find the basic unit. — Gregory
Not precisely. I was good at pre-calculus in high school but in college I only did geometry and that was over ten years ago. — Gregory
I am coming at this from a more basic fundamental level and perhaps I can't avoid highwe mathematical ideas but I had wanted to find the first few axioms of geometry and am confused why it's become to problematic — Gregory
nit interval [0,1] contains infinitely many points. Is that correct? If so, how many points do you think it should have? — fishfry
And modern physics does not posit infinite divisibility. In fact in physics, space is divisible down to the Planck length, equal to around 1.616255...×10−351.616255...×10−35 meters. Below this distance, our physics breaks down and we cannot sensibly speak of what goes on or how space is. There's a Planck time as well, a minimum time interval below which our physics breaks down and can't be applied. — fishfry
Oh darn, I sandbagged myself again. People always bring up Banach-Tarski, and I say, "B-T is at heart a simply syntactic phenomenon that I could describe in a page of exposition if anyone was interested," and they invariably have no interest. One of these days someone's going to say, "I'd like to see that" and I'll do it. But I see once again that you name-checked B-T but don't actually have an interest in it. And I got hopeful, only to be disappointed again. I am telling you that the heart of B-T is simple and surprising and perfectly clear, but nobody wants to hear about it. I pointed you at the references but you had no questions. One of these days ... — fishfry
In math, given a line, you can pick any two points, label one point 0 and the other point 1, and that length is your basic unit. — fishfry
That falls directly out of calculus. And as andrewk noted in the Gabriel's horn thread, it's analogous to the fact that the infinite series 1/2 + 1/3 + 1/4 + 1/5 ... sums to infinity, yet the series 1/4 + 1/9 + 1/16 + 1/25 + ... has a finite sum. Just a mathematical fact that takes a bit of getting used to, but is undeniably true. — fishfry
Well Euclid's axioms are a fine set of basic axioms. And if you drop the parallel postulate and replace it with either zero or many parallels through a point parallel to a given line, you get various flavors of non-Euclidean geometry. In modern times, Tarski's axiomitization of Euclidean geometry is of interest.
https://en.wikipedia.org/wiki/Tarski%27s_axioms — fishfry
Something discrete. Yet "discrete space" is impossible if it is to remain space. — Gregory
This is a commonly held sophistry. — Gregory
As i demonstrated on this thread, everything in this world is made of infinite parts and I BELIEVE the conclusion is that everything is finite and infinite in the exact same respect. That last part is what I was trying to explore — Gregory
I don't see how anyone with a brain wouldn't want to know how to get two objects out of one without referring to infinities. — Gregory
Such a theorem is incredible and I hope you do codify it into a thesis that others will read and appreciate. — Gregory
I for one am having trouble with it because it's of such a nature which I do not think I will understand it by READING it, as opposed to having it explained in person where I can cross examine every step. Reading it is just to much for me — Gregory
That is arbitrary, as is the Plank length — Gregory
Something I need to consider more, thanks. — Gregory
It becomes very confusing, which is why I was trying to find something basic about space that I could use as "first principles" in a Cartesian fashion — Gregory
People always bring up Banach-Tarski, and I say, "B-T is at heart a simply syntactic phenomenon that I could describe in a page of exposition if anyone was interested," and they invariably have no interest. One of these days someone's going to say, "I'd like to see that" and I'll do it. — fishfry
The Planck length is a fundamental aspect of modern physics. And by modern I mean since 1899, when Planck came up with the idea. He noted that it's defined only in terms of the speed of light, Newton's gravitational constant, and Planck's constant. His idea was that the Planck length was universal, in the sense that aliens would come up with it.
Here's Sabine Hossenfelder discussing the Planck length.
http://backreaction.blogspot.com/2020/02/does-nature-have-minimal-length.html — fishfry
Since space is continuous, it has infinitely many potential parts, but its only actual parts are those that we create by marking them off.if space is infinitely divisible than it has infinite parts — Gregory
That is because there is no basic unit intrinsic to space itself, only arbitrary constructs that we impose in order to measure length/area/volume.In geometry, all space is divisible and its impossible to find the basic unit. — Gregory
Space and time in standard fundamental theories of physics are continua, just as in classical physics. This you can readily see from any dynamical equation, such as Schrodinger equation. — SophistiCat
Since space is continuous, it has infinitely many potential parts, but its only actual parts are those that we create by marking them off. — aletheist
That is because there is no basic unit intrinsic to space itself, — aletheist
Sure, but space itself is not an object in that sense. It has no actual parts, only potential parts, and it is obviously not spatial in the same sense as a physical body. It is the continuous medium in which discrete objects exist.Objects don't potentially have parts. The parts are actual, and also spatial. — Gregory
Likewise if you have a proof that space is discrete. They are two different mathematical assumptions.If you have a proof that space is continuous that would be a discovery on a par with the revolutions of Newton and Einstein. — fishfry
Yes, but I understand it to be a limitation on measurement, not a discrete unit of space itself.I just pointed out that Max Planck introduced the Planck length as something aliens would discover. It's intrinsic to space itself as I understand it. — fishfry
I disagree, everything that is physical occupies space, but space itself is not physical.And they're physical, not mathematical assumptions. — fishfry
I agree, but again, space is not an object that exists--something that reacts with other like things in the environment. Instead, it is a reality--something that is as it is regardless of what anyone thinks about it.When speaking of something that exists, the phrase "potential parts" is an oxymoron. — Gregory
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