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The Potential Infinity Contradiction

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This is the thing that I don't know how to explain to a 5 year old. What you said here is a clear contradiction on the surface.

Context 1 (spatial):

U = Infinite

Context 2 (entities):

U ≠ Infinite

That makes me think: U(s) ≠ U(e)

But U is U

Dog = Brown

Dog = Not Brown

(This can be true logically, but we know that we're talking about two different dogs)

Sure. Totally understandable that it would be tough to explain, and I'll be happy to elaborate.

In one sense, we talk about the universe as the sum of all things. It's as if you said, "I want to consider everything as one giant thing. It's everything. What shall I call that? How about 'the universe'. When I say 'the universe', that's my term for everything that is." (That also distinguishes it from everything that isn't, such as your 83-year-old granddaughter, George Washington's living body, ghosts, and the warp engine.)

In another sense, we talk about the physical size of this sum of all things. Physical things have size. How big is the thing that contains them all? Well, it's endless. It has no boundaries. If it did, it would be a thing that exists within the boundaries of something else, but since the universe is everything that exists, that can't be something that exists physically apart/outside the universe. It's all there is.

So you use the same word to mean two rather different, but massively useful, ideas. One sense focuses on the container of all existence, the other sense on the physical extent of this all-encompassing container.

So how to explain this to your son? Well, off the top of my head, I might say this. Let's look at your room. Your room has only so much stuff in it, right? You know all the things in your room. There's a certain amount of stuff there. So let's say your room is everything in your room all added up. Now let's imagine that your room had no walls, ceiling, or floor. Your stuff is just floating in the same spot, but in space. Does that change the amount of stuff there? Nope. It's still exactly the same amount of stuff in your room. But there's no container for your stuff anymore. Your stuff could go and be anywhere. It isn't limited spatially -- it has no walls. But it's still the same amount of stuff that you had all along. Two senses of the same word: amount of content and physical extent.

You could also explain how a word can have two different meanings. Such as, he's "five-years-old", which means the length of his life. But "old" can also mean a lot of length of life, such as someone who is 83. When you say that someone is "five-years-old", you're not saying that they've been around a long time, like they're old -- you're just using it to say how long they've been around. Two different senses. You can probably think of better examples.

When it comes to the universe, we're just talking about one universe, right?
Right, hence the "uni-", meaning "single". There's only one sum of everything. Mutliple sums of everything is a contradiction in terms.

Last month we touched on space and time a little bit (me and Nicholas). He wanted a synonym for circle, and the thesaurus said "sphere" in there. Explaining the differences got me to mentioning 2 dimensions vs 3 dimensions... After getting that, he wanted to know what the 4th dimension is.

My explanations included the thought experiment about the ant on the globe that could walk in one direction forever.

When he asked about the 5th dimension, I punted. I honestly can't explain it to myself.

It might be tough, but I think you can relate it in familiar terms. Show him a string. Explain what motion is possible along that string, and how that's called motion in one dimension. Then take a piece of paper. Now your range of movement expanded. Still no up and down, but you've added left and right to your original back and forth. Now take a box. You've got all three dimensions inside the box. (This will be the tough part.) Every time you added a new way to move, you did it at right-angles to your original way. That is, left and right on the sheet of paper is motion orthogonal (at right-angles) to the back and forth on the string. Likewise with the up and down of the box to the (effectively) two-dimensional paper.

Physicists talk about time being the fourth dimension, but I've never understood the justification for this. The first three refer to spatial extent. Time refers to extent of motion, not spatial extent of motion, but an entirely different concept: duration of motion. So time may warrant status as a dimension, and my understanding of general relativity confirms that, but not in the same sense as length, height, and width.

As for some purported 5th dimension, there is none. Easy. Hope this helps! :)

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There is a sub field of mathematics that can address that very question. It is called Point Set Topology. I should warn you, however, that if you wish to study it, you have (as you said) your work cut out for you. It is possible for metrically finite spaces to be unbounded in the sense they have no boundary points.

That was great actually... thank you for the reference. I may check out that book, but I am definitely a non-mathematician.

Edited by freestyle
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So how to explain this to your son? Well, off the top of my head, I might say this. Let's look at your room.

That's a good idea... I think I can make that work. It might even be possible to associate it to "his room" all throughout his life... meaning, even if we move to a new house, or when he goes to college, there still is a universe of his personal stuff in a finite space. I think that will help. Thanks.

You could also explain how a word can have two different meanings. Such as, he's "five-years-old", which means the length of his life.

This he definitely understands. And he has a handle on the concept of things being relative too (e.g. Which is tall, A Boy or a mouse? -- then Which is tall, A boy or a building? He ask so many questions and remembers with great recall that early on I swore to myself that I'd try to always give him correct and detailed answers.

That has led to looking up definitions of words a lot to help me explain definitions. He now REQUIRES me to look up the word of the day at dictionary.com every morning. It has been every day for two months now and there isn't one of the words he has forgotten.

Some of his favorites that he actually uses in conversation are: roborant, fatuous, cogitate, paroxysm & diaphanous... (he's even got his 3 year old little brother using diaphanous, clinquant and surreptitious). This morning I explained that taking things from other people is not really "selfish" because selfish means doing something that, in the end, is good for your SELF. And since taking things from others means they'll probably take things from you, then it isn't good for you, is it? He said, "That's ratiocination!" (I also explained to him that most people think only of "bad selfishness" which I think he currently understands as an oxymoron.)

In any event, I just try to feed them (both boys) the information they're interested in until they're ready to change subjects or do something else.

It might be tough, but I think you can relate it in familiar terms. Show him a string. Explain what motion is possible along that string, and how that's called motion in one dimension. Then take a piece of paper. Now your range of movement expanded. Still no up and down, but you've added left and right to your original back and forth. Now take a box. You've got all three dimensions inside the box. (This will be the tough part.) Every time you added a new way to move, you did it at right-angles to your original way. That is, left and right on the sheet of paper is motion orthogonal (at right-angles) to the back and forth on the string. Likewise with the up and down of the box to the (effectively) two-dimensional paper.

I've got to figure all that out... but I like the idea of showing motion on a string in one dimension. That will definitely work.

Physicists talk about time being the fourth dimension, but I've never understood the justification for this. The first three refer to spatial extent. Time refers to extent of motion, not spatial extent of motion, but an entirely different concept: duration of motion. So time may warrant status as a dimension, and my understanding of general relativity confirms that, but not in the same sense as length, height, and width.

I'm not sure if I did a good job on the 4th dimension or not... I actually said that it is a little complicated and he said "If you can explain it to me, I can understand it." I tried a variation on an Einstein thought experiment I think I remembered correctly (lighting and train tracks - Simultaneity is Relative Section here). I basically explained how people would see a single event at different times depending on what space they are standing in.

As for some purported 5th dimension, there is none. Easy. Hope this helps! :)

Ok... I just hope he doesn't ever hear someone mention string theory. :confused:

Thanks for the help, btw! Much appreciated.

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I'm not sure if I did a good job on the 4th dimension or not... I actually said that it is a little complicated and he said "If you can explain it to me, I can understand it." I tried a variation on an Einstein thought experiment I think I remembered correctly (lighting and train tracks - Simultaneity is Relative Section here). I basically explained how people would see a single event at different times depending on what space they are standing in.

Ok... I just hope he doesn't ever hear someone mention string theory. :confused:

Thanks for the help, btw! Much appreciated.

Hey, sweet, your link is to my alma mater, Syracuse University. If this doesn't click with him -- and that would be understandable, because the propagation of light is hard to relate to in everyday terms -- you might try an example using sound, to show how two people can see/hear the same thing at different times. Like how someone standing close to a loud boom hears it slightly earlier than someone standing far away, and how light travels through space like sound. That will give him the feel for how the things that we see come from light traveling through space, and that that traveling takes a certain amount of time, depending on how far away the thing is.

Yeah, I'm not really sure what you tell him about string theory, other than that it's something that physicists work on. I'm glad the ideas helped. You're most welcome!

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So we got into this again. My son brought up the ant on the earth analogy, from earlier, and said that it means that if you go one direction in space forever then you'll end up back in the same place some time (like the ant would). You wouldn't know how that happened.

I tried to tie the idea of unbounded in by changing the earth to a balloon. I told him that while the ant is walking, the balloon (i.e. universe) can also be blowing up (expanding). So, at any time you measured the balloon it would have an exact size, but as the ant keeps walking, it keeps getting bigger... so he could walk forever and never get back to where he started. He said "space gets bigger when you use it". Sounds good.

Then he said, "what is the opposite of expanding?"

I said, "contracting".

He asked if the universe could end up like an empty balloon.

Without understanding decimals, I'm not sure that relating infinite to smallness is going to be possible. I just told him that no matter how small something he can think of is, I can think about that thing cut in half... forever.

I think it may be time for me to redirect to sports or something, lol. I'll have to do some more studying.

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  • 1 year later...

So I'm major necroing this thread based on the conversation going on in the "god" thread.

I don't understand this idea that there are no physical infinities, there most definitely are.

Any continuos system is by definition an infinite set, even if it is bounded. The set of all lengths from zero to one foot is infinite. The only contrary position to this is to state that there are no continuous systems.

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Aristotle identified that infinity first arises in the case of continuous quantity. In the case of lengths from zero to one foot, you may continue to subdivide the unit until you reach the level of precision required. When you cease to subdivide the unit, you have a finite number of increments to quantify your length. Infinity in this regard, as a concept of method, says you may subdivide as many times as required to reach that level of precision.

While I've never seen one, a quark is considered to be the smallest particle known to man. If you established a quark as your unit of measurement to quantify your length of one foot, can the quark be subdivided? As a thought experiment, perhaps, and if we managed to subdivide a quark into other particles, can the process be repeated? If the unit of length is beyond the capability to measure it, can it be objectively validated?

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I don't see how the required precision of a measurement has any bearing on the reality of the space, so I'm going to skip that.

Let's talk about the quark (or whatever the smallest particle is). Can it only move in units of itself? Can it displace itself by 1/2 it's own width? What about a quarter?

Unless space is quantized any distance represents an infinite set of positions.

Bu hey, it's possible that space is quantized (although it's really weird, think about a diagonal motion for a second), so what about time? Not that time can go on ininitely, but that time is a continuous system. Is time quantized? Maybe, i honestly don't know what kinds o theories there are on that.

There's another one though that is sticking in the back of my head, which you may like more because it's philosophically based. Free will.

Can you have free will with a finite set of possible actions?

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I don't see how the required precision of a measurement has any bearing on the reality of the space, so I'm going to skip that.

Let's talk about the quark (or whatever the smallest particle is). Can it only move in units of itself? Can it displace itself by 1/2 it's own width? What about a quarter?

Why are we going from a set of lengths (which presupposes something that is one foot long) to motion and the complications that it adds to the measuring process?

The required precision of a measurement comes in, for if you cannot distinguish between two objects possessing length a and length b, can you determine if they are the same length or different lengths?

Or to try and organize the principle around the words of your bordering on thought experiment type like example: If you cannot objectively distinguish or objectively measure the distance the quark moved, to what data of sense are you appealing?

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So I'm major necroing this thread based on the conversation going on in the "god" thread.

I don't understand this idea that there are no physical infinities, there most definitely are.

Any continuos system is by definition an infinite set, even if it is bounded. The set of all lengths from zero to one foot is infinite. The only contrary position to this is to state that there are no continuous systems.

Where is the physical infinity in this example? What physically exists in infinite quantity?

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Wait, are you arguing that reality is defined by the precision of your measurements? So, before the invention of an optical microscope space and distance only existed within the realm that could be seen with the naked eye?

Actually, Grames captures it quite well here.

Where is the physical infinity in this example? What physically exists in infinite quantity?

What I am actually stating here is that our precision of measurement delimits our assessment of reality. Our identification of reality is delimited to the precision of the tools we have developed to quantify it with. The law of identity only establishes that if something exits, that it is something specific. What you appear to be specifying is that if we have not discovered the specificity of that which we have determined to exist is due to a mentally perceived limitation of a capacity for which our minds are yet unable for us to comprehend. The form/object distinction comes to mind here.

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What I am actually stating here is that our precision of measurement delimits our assessment of reality. Our identification of reality is delimited to the precision of the tools we have developed to quantify it with. The law of identity only establishes that if something exits, that it is something specific. What you appear to be specifying is that if we have not discovered the specificity of that which we have determined to exist is due to a mentally perceived limitation of a capacity for which our minds are yet unable for us to comprehend. The form/object distinction comes to mind here.

So your assesment is an approximation of reality? I'm not really sure what you are trying to say here.

Although you seem to touch on something that has always intrigued me, which is whether the finite mind can actually comprehend the infinite.

Where is the physical infinity in this example? What physically exists in infinite quantity?

I'm not exactly sure what you mean by physical. But my answer would be that in a non-quantized space there are an infinite quantum of space. That seems physical to me.

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So your assesment is an approximation of reality? I'm not really sure what you are trying to say here.

Although you seem to touch on something that has always intrigued me, which is whether the finite mind can actually comprehend the infinite.

I'm not exactly sure what you mean by physical. But my answer would be that in a non-quantized space there are an infinite quantum of space. That seems physical to me.

Perhaps that was because I was unclear on what you were trying to ask.

Are you looking to understand how Objectivism formulates and approaches infinity and space, or to put forth your view of how infinity and space ought to be understood?

You stated in your opening remarks that there are definately physical infinities. You should then be easily able to reduce or valdiate your claim showing the hierarchal chain(s) which link it back to the evidence or data of the senses.

After you do that, you will have arrived at just how the finite mind actually comprehends infinity.

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I'm not exactly sure what you mean by physical. But my answer would be that in a non-quantized space there are an infinite quantum of space. That seems physical to me.

By physical what I mean is not simply the material but everything that is causal. What exists in infinite quantity in a length of space that is capable of participating in a causal relation?

What you are positing is that the geometric abstraction known as a 'point' is real, and that there are infinitely many of them on a line segment. But geometry is not physics. As points have no properties or intrinsic attributes (such as mass, charge, spin, or whatever) they cannot be distinguished from something that does not exist. The so-called property of location is itself a relational property: it only has physical meaning by referencing at least two existents. No relational property can exist apart from something that exists. Therefore points do not exist.

This last bit is explicit in Rand's ethics. No relational property can exist apart from something that exists. From this, she refutes the intrinsic theory of value and the idea of duty. There can be no value apart from a valuer, a person or organism performing the action of valuing.

Taking the idea of points too seriously, meaning treating them as if they were existents, leads to Zeno's paradoxes.

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Is space a physical thing?

Yes. The space between the more obviously existing particles is suffused with a variety of fields and radiations emanating from those particles. On the other hand, this could be interpreted as meaning there is no such thing as truly empty space, just various extensions of the things that do exist. That is a version of the 'full plenum' metaphysics first advocated by Parmenides.

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First off, Zeno's Paradoxes aren't something I would consider evidence of a problem with a continuous system as there are a lot of arguments that there are no paradoxes to begin with and that the arguments to get to the paradoxes are themselves fundamentally flawed.

The arrow paradox, for instance, postulates as an assumption that at an infinitessimal division of time there is no motion. This is incorrect. At an infinitessimal point in time there is an infinitessimal motion. To further explore this you start getting into calculus, the math of continuous systems, wherein, there are most definitely infinites.

------------

As for the rest, I'm not talking geomtry, at least not in an abstract sense, I am talking about physical space. You are saying that position is a relational property and therefore is out of bounds, as points do not in fact exist.

Its an interesting point. There's something about the way that this argument is going that makes me think that both sides are in danger of running into tautologies.

But what about time? Is time a relational property?

As a side note I have caught myself multiple times thinking: A continuous space allows for an infinite number of potential positions. Which really comes back to the whole "potential of infinity."

Edit: Ok, something I just realized. There is a very important difference between an instance in time and an infinitessimal division of time. Same thing with space. A point is different from an infinitessimal division of space.

Edit2: To put it another way, there is a difference between 1/inf and Zero, and there is a difference between inf and 1/zero.

Edited by emorris1000
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Tautologies are not necessarily bad. In fact, every true statement must be reducible to a tautology. That is what it means for a proposition to be true. (The concept used also needs to be reducible to a percept.)

Time is relational. Even if you try to imagine just one thing existing and changing over time it is the parts of the whole changing in relation to each other. In a universe of a single fundamental particle that did not have parts there could be no memory of or consequences of a posited change, so time would be meaningless.

I think I agree with you that there is a difference between infinity and 1/zero, but I would like to attempt to spell out what the difference actually is.

In mathematics, the infinitesimals are infinitesimal non-zero spans. So long as they are non-zero there cannot be an infinite number of them in any finite interval. The gimmick in calculus is finding the derivative dy/dx by first learning how to define the limit of a function as dx approaches, but does not assume the value of, zero. The mathematics of continuous systems does not assert or require the actual existence of infinities to justify its method.

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Attributes exist. Ayn Rand explicitly states attributes are existents in ITOE. Attributes are not independent entities however, and the fact that they cannot be separated ("... for even a split-second" in her phrase) from some entity having the attribute is what makes them attributes and not parts.

Furthermore, although intrincism as a doctrine is false that does not mean there do not exist intrinsic attributes and relational attributes. For example mass is the intrinsic attribute inferred to be the cause of the relational attribute of weight. An entirely different chain of reasoning led to the inference the mass can also be the cause of energy (E=mc2). The redness of an apple is relational attribute, a combination of the intrinsic properties of the surface of the apple and the relation to a person viewing it. A colorblind man will not experience the redness of the apple as other men do but if he was a scientist he could measure the principal wavelength of light reflected off the apple with a spectrometer and come to the same objective, conceptual (albeit abstract for him) conclusion as other men concerning the color attributed to apples without having to reify redness as an intrinsic attribute.

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However, I do find that the number of restrictions placed on the definition make the definition somewhat useless.

I'm curious now. Useless? Towards what purpose? What is it that needs doing which 'me and my definitions' aren't helping with?

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Well, it can't be used in math, that's already understood, as infinities exist by definition.

It can be used in science, but not with regards to relational attributes as they have infinites, therefore in the treatement of a problem you have to divide all relational and non-relational attributes and appreciate infinities differently for each one. And since the definition only operates absent of time, another relational attribute, it can't be used in any application that uses time.

Now, it may have a distinct use in philosophy, which is pretty boss I guess. If it represents Knowledge with the big K, if you will, that's a pretty important thing.

So, for the last thing it is usefull, which is important.

also:

In mathematics, the infinitesimals are infinitesimal non-zero spans. So long as they are non-zero there cannot be an infinite number of them in any finite interval.

I don't think that's correct. The definition of an infintessimal is 1/inf, which implies they are all contained in a finite interval.

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