Do you think the human notion of 'infinite'/'infinity' is actually real infinity?

[Agent327]

The question

Hi, nice to see you reached this part of the post, the question which will greet you here is:

"Do you think that the polar opposite notions of finite and infinite are themselves tied to something inherent in infinities and non-infinities? Ie are there any real, not dependent on human thought, infinities/non infinities?

This whole question (both the ones in the title and the one just posed) revolves around language. Anything expressed in language will be 'a human notion'; there is no way around that. That's not to say language doesn't describe some reality, but it will always be 'a human notion'. Which then also applies to finity/infinity.

Infinity is a word trying to describe something that escapes human perception. There is no means to ascertain if there actually is something infinite. (But in the end, that's irrelevant. Infinity is a clear concept in mathematics, and there it is very 'real'.) The most we can achieve is that 'infinity' is an approximation of something in reality.

 

[Kyriakos]

Well... 'real' in this sense is juxtaposed to human (or other observer) particular notion or sense, so in that manner the question is just excellently worded, like all my sentences happen to be

 

[Agent327]

You can't juxtapose real and human. They are both human terms. As I just argued, anything we think or say will be a human notion. Which, again, is not to say that there is no reality to our words. But we have no means to determine exactly where our notions end and 'reality' begins. They are, in fact, quite inseparable. Schroedinger's cat comes to mind.

 

[Kyriakos]

Well, 'real' is not a notion that has volume. It is a singular point, the so-called 'thing-in-itself'. Ie one can always say "assuming object X has properties independent of any particular observer of it, those properties are there in object X as it is in itself". It is an ideal notion, and merely a point on the horizon. It still is the final point in a human thought's horizon, but alludes to stuff supposedly after that limit, without encompassing the 'after', cause no limit would either.

 

[Borachio]

Some of them indeed don't, due to bad aim.

Let's say you're firing a virtual arrow from the number 0 to the number 1. The only way for you to reach your target is to first mark the half-way point, write it down (0.5), and then write down the half-way point between where you end up and the target (0.75)

You repeat this exercise until you write down the number 1.

You will never write down the number 1. And so the virtual arrow will never reach its target, because you'll be stuck dividing things in half forever.

Real arrows of course reach their targets, if the aim isn't off and the wind and other factors are favourable. But real arrows aren't tasked with the division of distances into halves for all eternity, all they do is obey the laws of physics.

It's true that no arrow has been given the task of reaching a target by division of distances into halves for all eternity, by a human mathematician. Nonetheless they accomplish the feat with remarkable alacrity. As amateurs, they're remarkably gifted, imo.

 

[Kyriakos]

It's true that no arrow has been given the task of reaching a target by division of distances into halves for all eternity, by a human mathematician. Nonetheless they accomplish the feat with remarkable alacrity. As amateurs, they're remarkably gifted, imo.

It takes one purely oblivious of the difficulties, to complete the otherwise impossible task :sage:

 

[Borachio]

Those arrows have something Zen-like about them. How do they achieve their targets in the face of Zeno's paradox? Answer: just like this: thwonkkk!

 

[warpus]

It's true that no arrow has been given the task of reaching a target by division of distances into halves for all eternity, by a human mathematician. Nonetheless they accomplish the feat with remarkable alacrity.

Yes, because arrows reach their destinations via the laws of physics, not futile mathematical exercises.

But if you still don't understand my initial point, then I don't think there's any other way of me to explain it to you how it's impossible to add an infinite number of 9s to 0.99 .. You will start, but won't finish - you've given yourself an infinite number of steps to work on. An arrow does not have an infinite number of steps to work through, it flies through the air centimetre by centimetre - a finite amount of interactions with the physical world. Only via an irrelevant mathematical formulation, which has nothing to do with how the arrow travels, do you arrive at the "impossible" task of travelling arrows.

If you really want to, you could set up similar mathematical formulations for any task. You could break down pooping into an infinite number of steps, using math. But that doesn't mean anything as it relates to the real world of poopery.

 

[Borachio]

I agree it can't physically be done: adding one 9 at a time. I said as much earlier. (It would plainly take an infinite amount of time, if nothing else.)

But conceptually I don't see why it can't be done. There. I've just done it.

Here it is again in slo-mo, in case you missed it: 0.999... = 1.

 

[warpus]

The point was that you can't reach 1 by adding 9s to 0.99, while at the same time it is true that 0.99... (infinite number of 9s in a chain) is equal to 1.

 

[Borachio]

Tis a mind-bending paradox, then?

 

[Perfection]

Do you think the human notion of ________ is actually real _______?

 

[Kyriakos]

^Way to miss the REAL point 'Real' can be used as a notion that itself alters depending on people's views on it. So while if it is set defined as attached to a 'thing-in-itself' it can remain stable (cause it would be turned into a singular point and an ideal) it can be argued to also depend on the observer, or to be essentially true as real for notions that are argued to be particular-observer tied (part of the thread is already about arguing whether infinity falls into such a category or not).
For example, each individual human person's conscience has a reality of its own, regardless of how that may break up to other observers, or if examined in regards to ties to his/her unconscious mental world.

 

[Perfection]

It is if you actually accept that 1/9 = 0.111... .

I think it works better with 1/3 because almost everyone agrees that it is equal to 0.333... . Multiply by three and you have 1/1 on one side and 0.999... on the other.

Another popular proof goes like this: let x = 0.999..., then:

x = 0.999... | * 10
10x = 9.999... | -x
9x = 9 | / 9
x = 1

The only way x can equal both 1 and 0.999... is if they are equal themselves.

It's fascinating how people can still grasp at straws trying to deny something that can be proven in such a simple way.

Your proof does not work. The first line does not subtract evenly from the second line as shown in the third line. There is an infinitessimal remainer, which you simply dropped.

So how do you propose we handle said remainder?

In my mind .999... = 1 is a choice, an opinion based on axioms or whatevs. But it's a good choice. It allows for consistent math that makes intuitive sense if you adopt the proper mindset. You can do operations like Loreth's and get consistent results by presuming no such remainder exists.

If you make .999... ≠ 1 then you got problems with operations. Consider the following operations

.999...+.111...
1+.111...

How do you propose we handle them?

Then you got problems like what's .999...^2?

There are ways one can probably do it (it would be like the [wiki]Surreal number[/wiki] system), but it'd be really ugly and would negate the utility and simplicity of .999... = 1

^Way to miss the REAL point 'Real' can be used as a notion that itself alters depending on people's views on it. So while if it is set defined as attached to a 'thing-in-itself' it can remain stable (cause it would be turned into a singular point and an ideal) it can be argued to also depend on the observer, or to be essentially true as real for notions that are argued to be particular-observer tied (part of the thread is already about arguing whether infinity falls into such a category or not).
For example, each individual human person's conscience has a reality of its own, regardless of how that may break up to other observers, or if examined in regards to ties to his/her unconscious mental world.

Wow each individual human person's conscience? As opposed to inhuman persons or non-individual humans?

I have a confession. I very seldom understand exactly what the hell you're talking about. And I'm absolutely certain noone else does either.

Just thought someone should tell you I'm sick of the illusion of you being understandable continuing.

 

[Kyriakos]

^Some do, you just aren't able to understand, Amorpheus

 

[onejayhawk]

The way we have always dealt with the remainder is to ignore it. From the origins of calculus the approach has been "close enough". That is what the epsilon does. It says, "Figure out what tolerance you need. It will be closer than that."

The thing is this. In any interval, there are an infinite number of "usual" numbers (for lack of a good term). These are rational numbers, roots of polynomial equations and some others. There will be more (higher cardinality infinity) "unusual" numbers.

In effect, the rational numbers are like a screen. No matter how fine the mesh, it will never be a solid sheet. These "unusual" numbers fill in the gaps. The term for the unusual numbers is transcendental. Pi is one. The base of common logs, e, is another. They are extremely difficult to work with. The closest uses a device called a Dedekind Cut (after Richard Dedekind). Any transcendental number can be described (here' that "close enough" again) by the rational numbers around it, but it's complicated. Rather than confuse the students, we tell them that open endpoints don't exist.

J

 

[nc-1701]

The thing is this. In any interval, there are an infinite number of "usual" numbers (for lack of a good term). These are rational numbers, roots of polynomial equations and some others. There will be more (higher cardinality infinity) "unusual" numbers.

There's a word for these..."Algebraic Numbers", it means something very different that "Rational Numbers"

In effect, the rational numbers are like a screen. No matter how fine the mesh, it will never be a solid sheet. These "unusual" numbers fill in the gaps. The term for the unusual numbers is transcendental. Pi is one. The base of common logs, e, is another. They are extremely difficult to work with. The closest uses a device called a Dedekind Cut (after Richard Dedekind). Any transcendental number can be described (here' that "close enough" again) by the rational numbers around it, but it's complicated. Rather than confuse the students, we tell them that open endpoints don't exist.

Everything you said is true, up to that sentence at the end. It is certainly true that the suprema of a set exists, but it absolutely doesn't have to be inside the set. And in the case of an open set it cannot be inside the set or the set wouldn't be open topologically.

 

[onejayhawk]

There's a word for these..."Algebraic Numbers", it means something very different that "Rational Numbers"

I could not think of that expression. The Algebraic numbers are still countable.

Everything you said is true, up to that sentence at the end. It is certainly true that the suprema of a set exists, but it absolutely doesn't have to be inside the set. And in the case of an open set it cannot be inside the set or the set wouldn't be open topologically.

This is because the definition of an open set is that every point has a neighborhood in the set. Wow. Epsilon strikes again.

J

 

[nc-1701]

This is because the definition of an open set is that every point has a neighborhood in the set. Wow. Epsilon strikes again.

Indeed, but just because you don't like epsilon arguments doesn't mean you get to change the definition of an open set.
Of course there is a strictly topological definition of openness that doesn't require epsilon arguments, since it doesn't require the set to be a measure space. Of course it agrees with the epsilon definition on measure spaces

 

[onejayhawk]

Indeed, but just because you don't like epsilon arguments doesn't mean you get to change the definition of an open set.
Of course there is a strictly topological definition of openness that doesn't require epsilon arguments, since it doesn't require the set to be a measure space. Of course it agrees with the epsilon definition on measure spaces

Explain how a neighborhood is different than an epsilon argument. Something to the right, something to the left, arbitrarily small. I see no practical distinction. You may, of course, generalize to any number of dimensions. I leave that to you.

The definition of measure is similarly fuzzy. It has to be. After all 0.999... and 1 measure no distance from each other. By theorem 0.999... =/= 1. So, we have two distinct points zero distance apart.

J