The Infinitesimal - In a 3 d environment.

Quantum Heraclitus:

No, in as much as the volume is irreducible, it cannot have a center, which implies a smaller volume.

That being said, I am going to be contacting a mathematician that I know to explain to me something. So when I get back from that, I'll tell you the whole business of my question and the answer and such.

 

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btw taking our ratio workings to 15 dec places indicates a ratio of 1.666666667
so unfortunately no golden ratio....

 

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So the infinitesimal is an conceptual number, the smallest that can be?

 

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This, of course, is quite wrong. Spheres know nothing about volume. You are thinking of balls, as Alpha explained

No, it isn't. n-Spheres simply do not know about their own interior. Paraphrase, the n-sphere may be thought of as the boundary of the (n + 1)-ball provided only that the (n + 1)-ball is closed.

The n-ball is where you may talk about volume; it's boundary (if it has one), is the n - 1 sphere.

Not every n-ball is closed (i.e. has a boundary), but every n - 1 sphere exists. Sorry, but his is elementary geometry

 

To current understanding, this infinitesimal size is of the Planck Length \(10^{-33}\) and the Planck Time \(10^{-44}\), and in these ''physical units,'' we find infinitesimal pointlike objects.

 

and the big question is:
Can this infnitesimal have a curvature or is it somehow fixed to be straight?

In a 3 d environment using a sphere for example the infinitesimal would need to have a curvature if the infinity was applied equally to the ball or sphere. I would think, but maybe I am incorrect in this.

 

First up, I'd get a handle on what things like infinity, or Planck's constant are.

Not just what, but why they are what they are, like why they must exist. Sort of a necessary condition for reality, as it were.

 

And from an honest quantum physical viewpoint, i answer that its curvature is infinite.

A sphere however, is just shakey terminology to apply to spaces where normally they exhibit things like electrons, which seem to have no structure at all.

 

I agree.

If and only If, spacetime actually had a beginning, then there needs to be an infinitesimal unit of measurement for such an occurance to begin with.

 

umm, maybe im missing something but I contend that the so called "infinitessimal" does not exist. how can you draw a line which is so short that any smaller and it would be a point? You can always zoom in on that line and draw a smaller distance.

If you are talking about infinitessimal in the physical world that is limited by reality, then it would be the planck length. but when you are talking about theoreticals in math, then no infinitessimal can exist.

Give any so called "infinitessimal" number and I can give an uncountable infinity of numbers which are smaller than it and nonzero.

 

Hence the idea of a "MkII" infinitesimal that doesn't bang into that little problem.
Instead of calling it an "infinitesimal" it's called a limit.
The limit of a radius that tends to zero is zero. But it never gets there, it's just always smaller than the "smallest possible" number.

It's like, at that scale, the coin can't be tossed, you accept that there is no way to determine "how close" it is to zero, so it "is" zero, but it's still around. Like the idea of the smallest possible gap to see something through. Except a lot more mathematically rigorous.

You're trying to make the gap as small as possible.
But if you think about how a value is approached as the number of intervals increases, it's like counting, you want to count as many as possible to make sure they're as small as you can get them, or find the smallest interval "size". The only way to get an infinitely small "size" is to count an infinite number of intervals.

That's like throwing a die to get an exactly even spread of each of the faces "counted", i.e. only the largest number of throws - an infinite number - will achieve this, otherwise, the spread only approaches even.

 

Exactly, the infinitessimal can only be expressed by a limit. it seems to me that you can easily shrink a sphere to make its radius approach the infinitessimal, but it is an infinite and never ending shrinking. You can't just stop and say "well, now we have our infinitessimal sphere" that makes no sense you could always shrink it further, infinitely further in fact.

I have no problem with expressing the infinitessimal as a limit, but to actually quantify it and make something that small is impossible.

If you dont call the infinitessimal a limit, then it is not small enough--I can always make a smaller number. If you do call the infinitessimal a limit, then it is zero, since that limit equals zero. You can't say that for the limit it is always just ever so close to zero but never gets there--you can say that about the individual terms--you can say the terms tend to zero but never get there, but if you are talking about the limit itself and not just the individual terms, it equals zero, not approaches zero. You can't say that the last term of that infinite series is the answer--what is the last term? that is impossible to declare some 'last' term of an infinite sequence.

Im ust remembering basic calculus when you did infinite sequences and series it was never the limit (i dunno how to do an 'approximate' equals sign, the squiggly one) some number, it was always =, but perhaps that was oversimplification

lim (x-->inf) 1/x = 0, not approximately equals zero, it equals zero.

So as a limit it equals zero, as a number it can be further subdivided--I contend the infinitessimal does not exist.

 

No, the infinitesimal doesn't exist in the reals, so the limit \(\lim_{x\to \infty}\frac{1}{x} = 0\). In the hyperreals, you can generate the infinitesimals from a basis quantity e, just as you generate the imaginary numbers from i.

Why? In terms of physics, there's nothing physically meaningful which has value \(10^{-(10^{10^{10^{10^{10^{10}}}}})}\) but it's a real number.

It doesn't exist in the reals. But then there's no real solution to \(x^{2}+1=0\). Does this mean 'i' doesn't exist? There's no rational solution to \(x^{2}-2=0\). There's no polynomial which has pi as a solution.

You can always find a set of numbers which satisfies some broad definition but fails to include some concept you can construct. There's no number which fails a*b=b*a, but there's such things everywhere in physics.

The infinitesimal exists mathematically. But it's not in the set of Reals. Does this mean the infinitesimal isn't physical? Well since when did being a Real number mean it's literally real? Is there a planet of 5s and 6's raising little number 4's? :shrug:

 

But even just as a concept or theory it still seems to not exist. I mean, i doesnt exist but still, you can give it a description--the square root of -1. even though that doesnt really make sense on the surface and you could never have i in the physical world, at least you can imagine a solution. I can't even imagine or grant a description to the infinitessimal. how can something be so small that any smaller and it is zero. It is just nonsensical to me. Its like trying to zoom on a fractal until there is no new detail.

It just seems contradictory to me--it exists about as much as a square circle exists.

 

You've almost got it.
You're still connecting the idea of a limit (like the limit of a spread of throws of a set of dice, or a large number of poker hands, say), with the idea of a very small interval that can be divided even smaller.
A limit is only approached, an infinitisemal is always infinitesimal. A limit is different because you can say the interval is too small to "not be close" to a limit. It's to do with the possibility of being able to divide it further, or throw more dice, or whatever count you're doing to "approach" some value.

 

is there another thing which would be like the anti-infinitessimal which is a quantity that is so large that any larger and it would be infinite? It seems like the same reasoning

 

Well, infinity does exist - except we can't prove it does, other that by not being able to get to it.

 

Well this is a valid pont an dI wonder how the math people can address it....

or would they rather avoid the whole issue.
It seems like the same reasoning should apply but I bet it don't.

 

('infinitesimal sigh')

 

In the other thread on infinity alphanumeric brought up a point i forgot about.

It was related to the .9r=1 in that if two numbers are not the same, then there exists a number between them. since there is no number between .9r and 1, they must be the same number.

So if this infintessimal does exist and is nonzero, then there must be a number between it and zero, meaning it is not the smalles number. If you say there is no number between it and zero, then it must equal zero by the same logic .9r=1.

If we accept the initial premise, that any two distinct numbers must have a number between them (average), then the infinitessimal is impossible. I do not see how the argument can be defeated unless you say the initial premise is false