Total Posts:32|Showing Posts:1-30|Last Page
Jump to topic:

Is It This, or is It That?

s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 4:03:05 PM
Posted: 2 years ago
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end. In defining infinity, aren't our actions in opposition to its very meaning? The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.
Mhykiel
Posts: 5,987
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 6:41:45 PM
Posted: 2 years ago
At 5/24/2014 4:03:05 PM, s-anthony wrote:
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end. In defining infinity, aren't our actions in opposition to its very meaning? The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.

There are different kinds of infinity. Counting numbers start at 1 then go on to infinity. There is no end but there is a beginning and it is endless it a particular direction.

I think it was Aristotle who said there was greater infinity and small infinity. Greater being descriptive by addition and small being by division (which is now termed infinitesimal)

So I don't see a confusion. I know some people think infinity has to be endless in all directions and so immense that it can not be bound. There is a very special case and not usually the infinities we work with.
Sidewalker
Posts: 3,713
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 7:44:12 PM
Posted: 2 years ago
At 5/24/2014 4:03:05 PM, s-anthony wrote:
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end.

But as you said, it is defined as without end or unlimited, how is that setting a limit.?

In defining infinity, aren't our actions in opposition to its very meaning?

Not if it's defined properly.

The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.

No, because It isn't true that infinity is "not definable by definition".
"It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater
Cody_Franklin
Posts: 9,483
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 9:18:24 PM
Posted: 2 years ago
Well, that's kind of the thing about infinity. It's bounded by a definition, but it's not finite. In mathematics, there is a sharp distinction drawn between something's being bounded and its being finite. Consider: you could take some arbitrary subset of the real numbers--for convenience, we will play it safe and use the closed interval [0, 1]. Very clear boundaries. However, if you were asked to write, one by one, each member of the set of real numbers within this interval, you'd never make it further than zero: the interval is continuous, which means the members of the corresponding set are uncountable. In other words, you could write any specific member of the set, e.g., 0.0063784267846, but you can't write the next member of the set, because there are infinitely many down to infinitesimal size. So, you've got boundedness, but you're also stuck with infinity. This is why mathematical definitions for weird things often take the form "X is the thing such that..." It's s little kooky, but it checks out when you actually do the footwork.
Skyangel
Posts: 8,234
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 9:24:12 PM
Posted: 2 years ago
At 5/24/2014 9:18:24 PM, Cody_Franklin wrote:
Well, that's kind of the thing about infinity. It's bounded by a definition, but it's not finite. In mathematics, there is a sharp distinction drawn between something's being bounded and its being finite. Consider: you could take some arbitrary subset of the real numbers--for convenience, we will play it safe and use the closed interval [0, 1]. Very clear boundaries. However, if you were asked to write, one by one, each member of the set of real numbers within this interval, you'd never make it further than zero: the interval is continuous, which means the members of the corresponding set are uncountable. In other words, you could write any specific member of the set, e.g., 0.0063784267846, but you can't write the next member of the set, because there are infinitely many down to infinitesimal size. So, you've got boundedness, but you're also stuck with infinity. This is why mathematical definitions for weird things often take the form "X is the thing such that..." It's s little kooky, but it checks out when you actually do the footwork.

Defining something does not necessarily set boundaries on it. It simply describes it so everyone understands what concept the word conveys.
Lack of understanding creates the boundaries in peoples minds.
Poetaster
Posts: 587
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 9:27:24 PM
Posted: 2 years ago
At 5/24/2014 9:24:12 PM, Skyangel wrote:
Defining something does not necessarily set boundaries on it. It simply describes it so everyone understands what concept the word conveys.
Lack of understanding creates the boundaries in peoples minds.

Different kind of boundary.
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
Skyangel
Posts: 8,234
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 9:30:13 PM
Posted: 2 years ago
At 5/24/2014 9:27:24 PM, Poetaster wrote:
At 5/24/2014 9:24:12 PM, Skyangel wrote:
Defining something does not necessarily set boundaries on it. It simply describes it so everyone understands what concept the word conveys.
Lack of understanding creates the boundaries in peoples minds.

Different kind of boundary.

Yes, its the kind where people are stuck in a rut and cannot think outside the box.
Poetaster
Posts: 587
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 9:32:50 PM
Posted: 2 years ago
At 5/24/2014 9:30:13 PM, Skyangel wrote:
Yes, its the kind where people are stuck in a rut and cannot think outside the box.

And that's exactly the kind of "boundary" that isn't relevant to the immediate topic here.
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
Skyangel
Posts: 8,234
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 9:36:40 PM
Posted: 2 years ago
At 5/24/2014 9:32:50 PM, Poetaster wrote:
At 5/24/2014 9:30:13 PM, Skyangel wrote:
Yes, its the kind where people are stuck in a rut and cannot think outside the box.

And that's exactly the kind of "boundary" that isn't relevant to the immediate topic here.

Sure its relevant because if a person cannot understand the concept of infinity being something that never ends, the problem must be in their own understanding. Their own thinking is limited.
Poetaster
Posts: 587
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 9:42:59 PM
Posted: 2 years ago
At 5/24/2014 4:03:05 PM, s-anthony wrote:
However, in defining that which, by definition, is not definable, we are creating a paradox.

"By definition undefinable"? You're mistaking your own incoherent choice of words for an absurdity in the idea of infinity. Mathematical infinities exhibit plenty of weirdness, we can all agree, but you certainly haven't touched on it here.
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
Poetaster
Posts: 587
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 9:49:35 PM
Posted: 2 years ago
At 5/24/2014 9:36:40 PM, Skyangel wrote:
Sure its relevant because if a person cannot understand the concept of infinity being something that never ends, the problem must be in their own understanding. Their own thinking is limited.

Cody_Franklin was talking about the mathematical notion of a boundary, say B, of a set of points S, where B is the set of points generated by the closure of S intersected with the closure of its complement. Has nothing to do with psychology.
"The book you are looking for hasn't been written yet. What you are looking for you are going to have to find yourself, it's not going to be in a book..." -Sidewalker
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 10:48:55 PM
Posted: 2 years ago
At 5/24/2014 6:41:45 PM, Mhykiel wrote:
At 5/24/2014 4:03:05 PM, s-anthony wrote:
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end. In defining infinity, aren't our actions in opposition to its very meaning? The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.

There are different kinds of infinity. Counting numbers start at 1 then go on to infinity. There is no end but there is a beginning and it is endless it a particular direction.

The end is your inability to count indefinitely.


I think it was Aristotle who said there was greater infinity and small infinity. Greater being descriptive by addition and small being by division (which is now termed infinitesimal)

Either extreme is beyond experience and, therefore, meaningless.


So I don't see a confusion. I know some people think infinity has to be endless in all directions and so immense that it can not be bound. There is a very special case and not usually the infinities we work with.

No. Just endless.
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 11:06:44 PM
Posted: 2 years ago
At 5/24/2014 7:44:12 PM, Sidewalker wrote:
At 5/24/2014 4:03:05 PM, s-anthony wrote:
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end.

But as you said, it is defined as without end or unlimited, how is that setting a limit.?

Anytime you say something is this but not that, you are setting a limit.


In defining infinity, aren't our actions in opposition to its very meaning?

Not if it's defined properly.

The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.

No, because It isn't true that infinity is "not definable by definition".

In- means no; -finity means end.

De- means to set; -fine means an end.
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/24/2014 11:14:58 PM
Posted: 2 years ago
At 5/24/2014 9:18:24 PM, Cody_Franklin wrote:
Well, that's kind of the thing about infinity. It's bounded by a definition, but it's not finite. In mathematics, there is a sharp distinction drawn between something's being bounded and its being finite. Consider: you could take some arbitrary subset of the real numbers--for convenience, we will play it safe and use the closed interval [0, 1]. Very clear boundaries. However, if you were asked to write, one by one, each member of the set of real numbers within this interval, you'd never make it further than zero: the interval is continuous, which means the members of the corresponding set are uncountable. In other words, you could write any specific member of the set, e.g., 0.0063784267846, but you can't write the next member of the set, because there are infinitely many down to infinitesimal size. So, you've got boundedness, but you're also stuck with infinity. This is why mathematical definitions for weird things often take the form "X is the thing such that..." It's s little kooky, but it checks out when you actually do the footwork.

Infinity's boundaries are you, in other words, your inability to demonstrate this mathematical claim.
Cody_Franklin
Posts: 9,483
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 3:11:25 AM
Posted: 2 years ago
At 5/24/2014 11:14:58 PM, s-anthony wrote:
At 5/24/2014 9:18:24 PM, Cody_Franklin wrote:
Well, that's kind of the thing about infinity. It's bounded by a definition, but it's not finite. In mathematics, there is a sharp distinction drawn between something's being bounded and its being finite. Consider: you could take some arbitrary subset of the real numbers--for convenience, we will play it safe and use the closed interval [0, 1]. Very clear boundaries. However, if you were asked to write, one by one, each member of the set of real numbers within this interval, you'd never make it further than zero: the interval is continuous, which means the members of the corresponding set are uncountable. In other words, you could write any specific member of the set, e.g., 0.0063784267846, but you can't write the next member of the set, because there are infinitely many down to infinitesimal size. So, you've got boundedness, but you're also stuck with infinity. This is why mathematical definitions for weird things often take the form "X is the thing such that..." It's s little kooky, but it checks out when you actually do the footwork.

Infinity's boundaries are you, in other words, your inability to demonstrate this mathematical claim.

As has been mentioned, we are dealing here with two kinds of boundaries. I am speaking, not of conceptual boundaries, but of boundaries in the formal sense. You must understand, infinity is not monolithic: there are several different sizes of infinity, some with boundaries, some without. Some are countable, others are not. The integers are infinite, unbounded, and countable; the real numbers are infinite, unbounded, but uncountable. The subset of real numbers between zero and one is bounded by definition--zero on one side, one on the other--but also uncountable. It is for this same reason that the easy way of conceptualizing infinity--beginning at one and perpetually increasing--is as easy to understand as it is inaccurate. People like to imagine infinity temporally, but all of the members of the set are always already there. We simply cannot make each member explicit, requiring instead some kind of shorthand (e.g., aleph null, 0.999 repeating).

Anyway, I just wanted to make it clear that the boundaries of which I speak are of a technical nature, not a conceptual or psychological one.
Sidewalker
Posts: 3,713
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 6:08:09 AM
Posted: 2 years ago
At 5/24/2014 11:06:44 PM, s-anthony wrote:
At 5/24/2014 7:44:12 PM, Sidewalker wrote:
At 5/24/2014 4:03:05 PM, s-anthony wrote:
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end.

But as you said, it is defined as without end or unlimited, how is that setting a limit.?

Anytime you say something is this but not that, you are setting a limit.

Nonsense, if you say something is this (unlimited), but not that (limited), you are not saying it is limited. On the contrary, you are saying it is not limited.

In defining infinity, aren't our actions in opposition to its very meaning?

Not if it's defined properly.

The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.

No, because It isn't true that infinity is "not definable by definition".

In- means no; -finity means end.

De- means to set; -fine means an end.

You are contradicting yourself.

What is it about the concept of infinity that causes people to want to think it is so mystical and esoteric.

Mathematically, the idea of a repeating series is about as simple a mathematical concept as can be, yet people try to make 1/3 into some kind of inconceivable, transcendent, impossible thing, I just don't understand it.

A definition that says it doesn't end is an end, unlimited means limited, unbounded means bounded, someone said infinity and I hear the sound of one hand clapping....really?
"It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 8:36:26 AM
Posted: 2 years ago
At 5/25/2014 3:11:25 AM, Cody_Franklin wrote:
At 5/24/2014 11:14:58 PM, s-anthony wrote:
At 5/24/2014 9:18:24 PM, Cody_Franklin wrote:
Well, that's kind of the thing about infinity. It's bounded by a definition, but it's not finite. In mathematics, there is a sharp distinction drawn between something's being bounded and its being finite. Consider: you could take some arbitrary subset of the real numbers--for convenience, we will play it safe and use the closed interval [0, 1]. Very clear boundaries. However, if you were asked to write, one by one, each member of the set of real numbers within this interval, you'd never make it further than zero: the interval is continuous, which means the members of the corresponding set are uncountable. In other words, you could write any specific member of the set, e.g., 0.0063784267846, but you can't write the next member of the set, because there are infinitely many down to infinitesimal size. So, you've got boundedness, but you're also stuck with infinity. This is why mathematical definitions for weird things often take the form "X is the thing such that..." It's s little kooky, but it checks out when you actually do the footwork.

Infinity's boundaries are you, in other words, your inability to demonstrate this mathematical claim.

As has been mentioned, we are dealing here with two kinds of boundaries. I am speaking, not of conceptual boundaries, but of boundaries in the formal sense.

All boundaries are conceptual boundaries. You can't conceive something and, then, say it's not conceptual.

You must understand, infinity is not monolithic: there are several different sizes of infinity, some with boundaries, some without.

Infinity's only size is the size we give it. It is not any larger or any smaller than our imaginations.

Some are countable, others are not. The integers are infinite, unbounded, and countable; the real numbers are infinite, unbounded, but uncountable. The subset of real numbers between zero and one is bounded by definition--zero on one side, one on the other--but also uncountable.

You can't have the capacity, for doing something, and, then, say you're not able to do it.

It is for this same reason that the easy way of conceptualizing infinity--beginning at one and perpetually increasing--is as easy to understand as it is inaccurate. People like to imagine infinity temporally, but all of the members of the set are always already there. We simply cannot make each member explicit, requiring instead some kind of shorthand (e.g., aleph null, 0.999 repeating).

Since we are temporal beings, temporally is the only way we can imagine infinity, or anything else, for that matter.

The members that are there are the members that are explicit, to us.


Anyway, I just wanted to make it clear that the boundaries of which I speak are of a technical nature, not a conceptual or psychological one.

All boundaries are conceptual and psychological.
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 9:58:28 AM
Posted: 2 years ago
At 5/25/2014 6:08:09 AM, Sidewalker wrote:
At 5/24/2014 11:06:44 PM, s-anthony wrote:
At 5/24/2014 7:44:12 PM, Sidewalker wrote:
At 5/24/2014 4:03:05 PM, s-anthony wrote:
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end.

But as you said, it is defined as without end or unlimited, how is that setting a limit.?

Anytime you say something is this but not that, you are setting a limit.

Nonsense, if you say something is this (unlimited), but not that (limited), you are not saying it is limited. On the contrary, you are saying it is not limited.

Exactly. If I say something is unlimited, I am not saying it is limited.


In defining infinity, aren't our actions in opposition to its very meaning?

Not if it's defined properly.

The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.

No, because It isn't true that infinity is "not definable by definition".

In- means no; -finity means end.

De- means to set; -fine means an end.

You are contradicting yourself.

My point exactly, and so is anyone else who says infinity is indefinite and, then, sets out to define it.


What is it about the concept of infinity that causes people to want to think it is so mystical and esoteric.

I don't know.


Mathematically, the idea of a repeating series is about as simple a mathematical concept as can be, yet people try to make 1/3 into some kind of inconceivable, transcendent, impossible thing, I just don't understand it.

Neither can I.


A definition that says it doesn't end is an end, unlimited means limited, unbounded means bounded, someone said infinity and I hear the sound of one hand clapping....really?
dylancatlow
Posts: 12,242
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 11:02:06 AM
Posted: 2 years ago
But infinity does have bounds, for if it didn't, it would be nothingness. In defining infinity, we are not defining it to have bounds as a concept, we are placing bounds on what it is not. It is "bounded" in the sense that it can't have bounds. If infinity actually had no bounds in the sense you mean, it would be a meta-paradox. Why? Because if it didn't have bounds, it would have them, for it could not prevent itself from having them. But then it wouldn't be a paradox because nothingness cannot be anything, yet it would be a paradox becuase it can't "not" be anything either.
Cody_Franklin
Posts: 9,483
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 1:05:35 PM
Posted: 2 years ago
At 5/25/2014 8:36:26 AM, s-anthony wrote:
At 5/25/2014 3:11:25 AM, Cody_Franklin wrote:
As has been mentioned, we are dealing here with two kinds of boundaries. I am speaking, not of conceptual boundaries, but of boundaries in the formal sense.

All boundaries are conceptual boundaries. You can't conceive something and, then, say it's not conceptual.

Then we mean different things by conceptual.

You must understand, infinity is not monolithic: there are several different sizes of infinity, some with boundaries, some without.

Infinity's only size is the size we give it. It is not any larger or any smaller than our imaginations.

No, it isn't. Infinity has demonstrably different sizes, and some are larger than others. The real number line and the integers are both infinite; the integers are a subset of the real numbers; the cardinality of the set of integers is smaller than the cardinality of the set of real numbers, which is to say that the infinity of real numbers is larger. There is even a largest infinity, which is the set containing as members all the cardinalities of infinity.

Some are countable, others are not. The integers are infinite, unbounded, and countable; the real numbers are infinite, unbounded, but uncountable. The subset of real numbers between zero and one is bounded by definition--zero on one side, one on the other--but also uncountable.

You can't have the capacity, for doing something, and, then, say you're not able to do it.

What, counting? In the case of uncountable infinities, I said explicitly that you can only enumerate individual members of the set. I challenge you to give me the positive real number immediately adjacent to zero.

It is for this same reason that the easy way of conceptualizing infinity--beginning at one and perpetually increasing--is as easy to understand as it is inaccurate. People like to imagine infinity temporally, but all of the members of the set are always already there. We simply cannot make each member explicit, requiring instead some kind of shorthand (e.g., aleph null, 0.999 repeating).

Since we are temporal beings, temporally is the only way we can imagine infinity, or anything else, for that matter.

The act of thinking of infinity is temporal. It takes place on time. Infinity is not itself temporal--the mathematical sense of infinity is not "counting up forever". If you have even the most rudimentary understanding of set theory, you'd understand that members aren't being perpetually added in: they're always already there, even if we can't think or write them all. It isn't contingent on or cognitive access to it.

The members that are there are the members that are explicit, to us.


Anyway, I just wanted to make it clear that the boundaries of which I speak are of a technical nature, not a conceptual or psychological one.

All boundaries are conceptual and psychological.

Boundaries are a concept, but not always conceptual in the sense I mean it. This is why I draw the line between threes and mathematical boundaries. You're welcome to call them whatever you want, as long as you're clear about what sits behind the sign.
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 1:59:42 PM
Posted: 2 years ago
At 5/25/2014 1:05:35 PM, Cody_Franklin wrote:
At 5/25/2014 8:36:26 AM, s-anthony wrote:
At 5/25/2014 3:11:25 AM, Cody_Franklin wrote:
As has been mentioned, we are dealing here with two kinds of boundaries. I am speaking, not of conceptual boundaries, but of boundaries in the formal sense.

All boundaries are conceptual boundaries. You can't conceive something and, then, say it's not conceptual.

Then we mean different things by conceptual.

You must understand, infinity is not monolithic: there are several different sizes of infinity, some with boundaries, some without.

Infinity's only size is the size we give it. It is not any larger or any smaller than our imaginations.

No, it isn't. Infinity has demonstrably different sizes, and some are larger than others. The real number line and the integers are both infinite; the integers are a subset of the real numbers; the cardinality of the set of integers is smaller than the cardinality of the set of real numbers, which is to say that the infinity of real numbers is larger. There is even a largest infinity, which is the set containing as members all the cardinalities of infinity.

Demonstrate it.


Some are countable, others are not. The integers are infinite, unbounded, and countable; the real numbers are infinite, unbounded, but uncountable. The subset of real numbers between zero and one is bounded by definition--zero on one side, one on the other--but also uncountable.

You can't have the capacity, for doing something, and, then, say you're not able to do it.

What, counting? In the case of uncountable infinities, I said explicitly that you can only enumerate individual members of the set. I challenge you to give me the positive real number immediately adjacent to zero.

It is for this same reason that the easy way of conceptualizing infinity--beginning at one and perpetually increasing--is as easy to understand as it is inaccurate. People like to imagine infinity temporally, but all of the members of the set are always already there. We simply cannot make each member explicit, requiring instead some kind of shorthand (e.g., aleph null, 0.999 repeating).

Since we are temporal beings, temporally is the only way we can imagine infinity, or anything else, for that matter.

The act of thinking of infinity is temporal. It takes place on time. Infinity is not itself temporal--the mathematical sense of infinity is not "counting up forever". If you have even the most rudimentary understanding of set theory, you'd understand that members aren't being perpetually added in: they're always already there, even if we can't think or write them all. It isn't contingent on or cognitive access to it.

You know they're always there, even if you haven't the cognitive capacity to access all of them? In other words, you know, that which you don't know. Sorry, but that's called a contradiction.


The members that are there are the members that are explicit, to us.


Anyway, I just wanted to make it clear that the boundaries of which I speak are of a technical nature, not a conceptual or psychological one.

All boundaries are conceptual and psychological.

Boundaries are a concept, but not always conceptual in the sense I mean it. This is why I draw the line between threes and mathematical boundaries. You're welcome to call them whatever you want, as long as you're clear about what sits behind the sign.
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 2:09:57 PM
Posted: 2 years ago
At 5/25/2014 11:02:06 AM, dylancatlow wrote:
But infinity does have bounds, for if it didn't, it would be nothingness. In defining infinity, we are not defining it to have bounds as a concept, we are placing bounds on what it is not. It is "bounded" in the sense that it can't have bounds. If infinity actually had no bounds in the sense you mean, it would be a meta-paradox. Why? Because if it didn't have bounds, it would have them, for it could not prevent itself from having them. But then it wouldn't be a paradox because nothingness cannot be anything, yet it would be a paradox becuase it can't "not" be anything either.

If we weren't able to conceive it, it would have no meaning, to us.

Even nothingness has bounds, in that nothingness is not something.
Cody_Franklin
Posts: 9,483
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 2:38:05 PM
Posted: 2 years ago
At 5/25/2014 1:59:42 PM, s-anthony wrote:
At 5/25/2014 1:05:35 PM, Cody_Franklin wrote:
At 5/25/2014 8:36:26 AM, s-anthony wrote:
All boundaries are conceptual boundaries. You can't conceive something and, then, say it's not conceptual.

Then we mean different things by conceptual.

Infinity's only size is the size we give it. It is not any larger or any smaller than our imaginations.

No, it isn't. Infinity has demonstrably different sizes, and some are larger than others. The real number line and the integers are both infinite; the integers are a subset of the real numbers; the cardinality of the set of integers is smaller than the cardinality of the set of real numbers, which is to say that the infinity of real numbers is larger. There is even a largest infinity, which is the set containing as members all the cardinalities of infinity.

Demonstrate it.

I mean no disrespect, but I just did. You don't need to who out any fancy equations or technical vocabulary. If you know what the real numbers are, what the integers are, what the set theoretic relation between them is, and that they are both infinite, all you have to do is put it together. [5,8], as a sub-interval of [4,10], has to correspond to a set of lower cardinality. The set of integers is a smaller subset of the real number line, so it must be a smaller infinity. Cardinalities of sets are also themselves members of sets, which gives way to an infinity with a highest cardinality. You could search this stuff on Google; I don't just make this all up for my own amusement.


Some are countable, others are not. The integers are infinite, unbounded, and countable; the real numbers are infinite, unbounded, but uncountable. The subset of real numbers between zero and one is bounded by definition--zero on one side, one on the other--but also uncountable.

You can't have the capacity, for doing something, and, then, say you're not able to do it.

What, counting? In the case of uncountable infinities, I said explicitly that you can only enumerate individual members of the set. I challenge you to give me the positive real number immediately adjacent to zero.

It is for this same reason that the easy way of conceptualizing infinity--beginning at one and perpetually increasing--is as easy to understand as it is inaccurate. People like to imagine infinity temporally, but all of the members of the set are always already there. We simply cannot make each member explicit, requiring instead some kind of shorthand (e.g., aleph null, 0.999 repeating).

Since we are temporal beings, temporally is the only way we can imagine infinity, or anything else, for that matter.

The act of thinking of infinity is temporal. It takes place on time. Infinity is not itself temporal--the mathematical sense of infinity is not "counting up forever". If you have even the most rudimentary understanding of set theory, you'd understand that members aren't being perpetually added in: they're always already there, even if we can't think or write them all. It isn't contingent on or cognitive access to it.

You know they're always there, even if you haven't the cognitive capacity to access all of them? In other words, you know, that which you don't know. Sorry, but that's called a contradiction.

No, I'm afraid it isn't. You can know that they're all there without ever being able to write them all. A set always contains all its members from the beginning; we don't have to count out the natural numbers every time we want a subset of those. You could go on counting forever, if you want, but it wouldn't demonstrate anything we don't already know from doing real technical labor in which those and like entities are employed. There are, however, still infinities, like the real line, which you cannot count discretely. Again, I would challenge you to give me the first positive real number after zero. If I am bluffing, you will not find this request unreasonable.


The members that are there are the members that are explicit, to us.


Anyway, I just wanted to make it clear that the boundaries of which I speak are of a technical nature, not a conceptual or psychological one.

All boundaries are conceptual and psychological.

Boundaries are a concept, but not always conceptual in the sense I mean it. This is why I draw the line between threes and mathematical boundaries. You're welcome to call them whatever you want, as long as you're clear about what sits behind the sign.
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/25/2014 6:34:03 PM
Posted: 2 years ago
At 5/25/2014 2:38:05 PM, Cody_Franklin wrote:
At 5/25/2014 1:59:42 PM, s-anthony wrote:
At 5/25/2014 1:05:35 PM, Cody_Franklin wrote:
At 5/25/2014 8:36:26 AM, s-anthony wrote:
All boundaries are conceptual boundaries. You can't conceive something and, then, say it's not conceptual.

Then we mean different things by conceptual.

Infinity's only size is the size we give it. It is not any larger or any smaller than our imaginations.

No, it isn't. Infinity has demonstrably different sizes, and some are larger than others. The real number line and the integers are both infinite; the integers are a subset of the real numbers; the cardinality of the set of integers is smaller than the cardinality of the set of real numbers, which is to say that the infinity of real numbers is larger. There is even a largest infinity, which is the set containing as members all the cardinalities of infinity.

Demonstrate it.

I mean no disrespect, but I just did. You don't need to who out any fancy equations or technical vocabulary. If you know what the real numbers are, what the integers are, what the set theoretic relation between them is, and that they are both infinite, all you have to do is put it together. [5,8], as a sub-interval of [4,10], has to correspond to a set of lower cardinality. The set of integers is a smaller subset of the real number line, so it must be a smaller infinity. Cardinalities of sets are also themselves members of sets, which gives way to an infinity with a highest cardinality. You could search this stuff on Google; I don't just make this all up for my own amusement.

Just telling me something goes on indefinitely is called demonstrating infinity?



Some are countable, others are not. The integers are infinite, unbounded, and countable; the real numbers are infinite, unbounded, but uncountable. The subset of real numbers between zero and one is bounded by definition--zero on one side, one on the other--but also uncountable.

You can't have the capacity, for doing something, and, then, say you're not able to do it.

What, counting? In the case of uncountable infinities, I said explicitly that you can only enumerate individual members of the set. I challenge you to give me the positive real number immediately adjacent to zero.

It is for this same reason that the easy way of conceptualizing infinity--beginning at one and perpetually increasing--is as easy to understand as it is inaccurate. People like to imagine infinity temporally, but all of the members of the set are always already there. We simply cannot make each member explicit, requiring instead some kind of shorthand (e.g., aleph null, 0.999 repeating).

Since we are temporal beings, temporally is the only way we can imagine infinity, or anything else, for that matter.

The act of thinking of infinity is temporal. It takes place on time. Infinity is not itself temporal--the mathematical sense of infinity is not "counting up forever". If you have even the most rudimentary understanding of set theory, you'd understand that members aren't being perpetually added in: they're always already there, even if we can't think or write them all. It isn't contingent on or cognitive access to it.

You know they're always there, even if you haven't the cognitive capacity to access all of them? In other words, you know, that which you don't know. Sorry, but that's called a contradiction.

No, I'm afraid it isn't. You can know that they're all there without ever being able to write them all. A set always contains all its members from the beginning; we don't have to count out the natural numbers every time we want a subset of those. You could go on counting forever, if you want, but it wouldn't demonstrate anything we don't already know from doing real technical labor in which those and like entities are employed. There are, however, still infinities, like the real line, which you cannot count discretely. Again, I would challenge you to give me the first positive real number after zero. If I am bluffing, you will not find this request unreasonable.

Sorry, but saying you can know, that which you don't know, is a violation of the law of noncontradiction. If I could go on forever counting, I would be able to go on forever counting; however, since everyone else I know and I are temporal beings, I can assure you that's not going to happen. To say something can be done, that can't be done, again, is a contradiction.



The members that are there are the members that are explicit, to us.


Anyway, I just wanted to make it clear that the boundaries of which I speak are of a technical nature, not a conceptual or psychological one.

All boundaries are conceptual and psychological.

Boundaries are a concept, but not always conceptual in the sense I mean it. This is why I draw the line between threes and mathematical boundaries. You're welcome to call them whatever you want, as long as you're clear about what sits behind the sign.
Cody_Franklin
Posts: 9,483
Add as Friend
Challenge to a Debate
Send a Message
5/26/2014 1:24:49 AM
Posted: 2 years ago
Alright, so, your primary problem seems to be your excessive reliance on etymological arguments. Infinity means "without ends", definition is to set limits, so no dive. If mathematicians hadn't already discovered, and made substantial use of, the several permutations of infinity, I might even be inclined to agree. This kind of problem emerges in several places, but mathematicians usually find their solution, not on defining infinity in its own right, or "demonstrating" it, a strategy often purposely left by the wayside, but in merely marking the absence of an end. This is why I say there are two uses of the term "boundary" at play, and that we should not confuse them. The first is conceptual, and deals merely with black-boxing infinity, with specifying unlimitedness. The second kind of boundary, to which I refer, is technical or numerical bounding. In the case of the real numbers, there is an uncountable infinity between any two numbers you can think of. This is true for every number, and you are welcome to choose as many as you like. In this sense, you are right that infinity is never "demonstrated"; yet, it works perfectly well, produces usable results in the real world, and has yet to be discarded by those far more dedicated and intelligent than us, despite ceaseless scrutiny. And, in analysis of limits, we can just write a function that never terminates. Limit of y as x approaches infinity. We get top specify it, and its demonstration is that it *works*.

Even with the integers, infinity is not the act of counting--it is not a ceaselessly-increasing quantity--but the mere fact of the possibility of going on forever, and the implication that there must therefore always be something to move on to (i.e., that there exist sets of infinite cardinalities). The problem, with which I once struggled myself, is not limitlessness, but how we think about it. Once you have a suitable mental model, it is not difficult to see how it can be used to achieve results, even in the practical world. If you were correct, the engineers who have to employ even the simplest integrations would have been struck dumb and impotent a long time ago. A lot of mathematical definitions are goofy and abstract, I realize, but this is why a special kind of intuition is required to excel at it. It is an intelligence which, I sadly admit, I do not possess. Frankly, if I ultimately cannot make these things clear to you, I would advise you to do your own research and consult better thinkers than I.
wrichcirw
Posts: 11,196
Add as Friend
Challenge to a Debate
Send a Message
5/26/2014 2:50:09 PM
Posted: 2 years ago
At 5/24/2014 4:03:05 PM, s-anthony wrote:
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end. In defining infinity, aren't our actions in opposition to its very meaning? The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.

I think the key to this is to keep in mind that there are "many" infinities, some of which are mutually exclusive from each other.

For example, there are infinite numbers between 0 and 1, and all of them are mutually exclusive from the infinite numbers between 1 and 2.
At 8/9/2013 9:41:24 AM, wrichcirw wrote:
If you are civil with me, I will be civil to you. If you decide to bring unreasonable animosity to bear in a reasonable discussion, then what would you expect other than to get flustered?
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/26/2014 4:17:19 PM
Posted: 2 years ago
At 5/26/2014 1:24:49 AM, Cody_Franklin wrote:
Alright, so, your primary problem seems to be your excessive reliance on etymological arguments. Infinity means "without ends", definition is to set limits, so no dive. If mathematicians hadn't already discovered, and made substantial use of, the several permutations of infinity, I might even be inclined to agree. This kind of problem emerges in several places, but mathematicians usually find their solution, not on defining infinity in its own right, or "demonstrating" it, a strategy often purposely left by the wayside, but in merely marking the absence of an end. This is why I say there are two uses of the term "boundary" at play, and that we should not confuse them. The first is conceptual, and deals merely with black-boxing infinity, with specifying unlimitedness. The second kind of boundary, to which I refer, is technical or numerical bounding. In the case of the real numbers, there is an uncountable infinity between any two numbers you can think of. This is true for every number, and you are welcome to choose as many as you like. In this sense, you are right that infinity is never "demonstrated"; yet, it works perfectly well, produces usable results in the real world, and has yet to be discarded by those far more dedicated and intelligent than us, despite ceaseless scrutiny. And, in analysis of limits, we can just write a function that never terminates. Limit of y as x approaches infinity. We get top specify it, and its demonstration is that it *works*.

Even with the integers, infinity is not the act of counting--it is not a ceaselessly-increasing quantity--but the mere fact of the possibility of going on forever, and the implication that there must therefore always be something to move on to (i.e., that there exist sets of infinite cardinalities). The problem, with which I once struggled myself, is not limitlessness, but how we think about it. Once you have a suitable mental model, it is not difficult to see how it can be used to achieve results, even in the practical world. If you were correct, the engineers who have to employ even the simplest integrations would have been struck dumb and impotent a long time ago. A lot of mathematical definitions are goofy and abstract, I realize, but this is why a special kind of intuition is required to excel at it. It is an intelligence which, I sadly admit, I do not possess. Frankly, if I ultimately cannot make these things clear to you, I would advise you to do your own research and consult better thinkers than I.

Thank you, and not to be argumentative, but I'm not in anyway saying infinity doesn't exist; I believe it's just as real, as we are real. Yet, I see a contradiction in its meaning, a contradiction that may never be resolved.
sdavio
Posts: 1,798
Add as Friend
Challenge to a Debate
Send a Message
5/27/2014 12:32:30 AM
Posted: 2 years ago
At 5/24/2014 4:03:05 PM, s-anthony wrote:
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end. In defining infinity, aren't our actions in opposition to its very meaning? The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.

"Infinite" and "finite" in the metaphysical sense are indeed self-refuting concepts in that they invoke their opposite. I would define the infinite as something more like, that which entails an invalid concept in a given context, rather than 'without limit'. For instance, "The universe is infinitely large" translates to "the concept of size doesn't apply to the universe." "Numbers can go up to infinity" really means that 'number' is a game with a certain set of rules, and that 'end of game' doesn't apply in that context because the rules don't specify an end-point.
"Logic is the money of the mind." - Karl Marx
s-anthony
Posts: 2,582
Add as Friend
Challenge to a Debate
Send a Message
5/27/2014 4:01:05 PM
Posted: 2 years ago
At 5/27/2014 12:32:30 AM, sdavio wrote:
At 5/24/2014 4:03:05 PM, s-anthony wrote:
The word infinity basically means without end, or unlimited. To define something means to set a limit, bound, or end. In defining infinity, aren't our actions in opposition to its very meaning? The act of defining is to say something is this, as opposed to that: a dog is an animal; it is not a plant. However, in defining that which, by definition, is not definable, we are creating a paradox.

"Infinite" and "finite" in the metaphysical sense are indeed self-refuting concepts in that they invoke their opposite.

Being physical, I don't believe we can know anything in the metaphysical sense. Secondly, I don't believe that which is metaphysical has any knowledge of contradictions or oppositions.

I would define the infinite as something more like, that which entails an invalid concept in a given context, rather than 'without limit'. For instance, "The universe is infinitely large" translates to "the concept of size doesn't apply to the universe." "Numbers can go up to infinity" really means that 'number' is a game with a certain set of rules, and that 'end of game' doesn't apply in that context because the rules don't specify an end-point.

Size may not apply to the universe, but size applies to us. Saying we can know infinity is like saying a camera with a limited scope can take an infinitely large picture. We can know infinity, in as much as we can define it.