What is Real?

What is a human being? What is life? Can science give us reliable answers to such questions? The electricity of life. The meaning of human consciousness. Are we alone? Are the traditional contests between science and religion still relevant? Does the word "spirit" still hold meaning today?

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Antone
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Re: What is Real?

Unread post by Antone » Tue Dec 02, 2008 5:26 pm

There are many reasons why I believe that the DS theory is a better theory of reality than anything else out there. One of the reasons is that the same basic structure can be used to model virtually any thing or any field. This is true for very big areas and very small. For example: (1) quantum physics, (2)linguistics, (3)theory of numbers, (4)holographic structure of light, (5) definitions of words, (6) philosophy, (7) how computers can think like humans, (8) theory of light, (9) ethics and morality, (10) Logic, (11) epistemology, (12) how the muscles work, (13) cymatics: relationship between vibrations and sound, (14) how the brain works, (15) set theory, (16) theory of sound, (17) mathematics, etc.

Here's an example of how the reciprocal structure of the DS theory can be applied to simple words, such as [father] and [son].

The Father and Son Analogy
The words [father] and [son] are clearly opposites when seen [from within the framework of a particular relationship between two people]. For example, if [Charles1] is [my father] then it is clearly not also possible for [Charles1 to be my son] or for [me to be the father of Charles1]. The [relationship between father and son] necessitates that the later designations are incompatible with the first. However, it is just as obvious that [my father], [Charles1], must also be [someone else’s son], and that I, the [son of Charles1], can myself be a [father to someone other than Charles1].

Thus, there is clearly one sense in which is true.
And another sense in which is false.
With respect to [my father] I must be [son],
and with respect to [my daughter] I must be [father].
But I cannot be [both father and son] to the same person.

This simple fact is so universally understood at an intuitive level that we rarely if ever stop to consider its significance in the broader scheme of things. But imagine what would happen if we assumed (the way that philosophy routinely does for other things) that a person could be either a [father] or a [son], but never [both at the same time]. Given such an assumption, it would seem impossible for [fathers] or [sons] to exist at all. In the case of this analogy, the fallacious reasoning is ridiculously easy to identify and so we do not think of it as being a paradoxical results. Unfortunately, this is not always the case. In fact, I would suggest that the misunderstanding over the reciprocal nature of reality is the single thing most responsible for the inconsistencies and puzzling quandaries that are so common in areas such as set theory, mathematics and philosophical theories.

The DS theory avoids these problems by embracing the notion that every [thing] must express [reciprocal properties]. Otherwise, the thing simply could not exist—and we certainly could not accurately define it.

What keeps this principle of the DS theory from contradicting Aristotle’s (PNC) is Aristotle’s own stipulation that the things we are comparing must be compared
In the same respect and at the same time.

This is where the importance of being very precise comes into play, for clearly, when is true, it is because I am referring to the [multiple relationships with different people]. And when is false, it is because I am referring to a perspective that involves a [single, specific relationship between two specific people]. These perspectives are clearly reciprocal in nature. Thus, when we think precisely, it is obvious that these are two very different things. But when we define things in a vague way, we leave the door open for a vague interpretation, and this can appear to turn [two things that are different-in-an-absolute-sense] into [two things that are similar-in-a-relative-sense]—invalidating our conclusions, and making them seem paradoxical.

Aristotle’s principle of non-contradiction is not the problem itself, but it has become so firmly ingrained in our Western way of thinking that we commonly take tremendous pains not to violate the part that says
One cannot say of something [that it is] and [that it is not]...

While virtually ignoring the part that says
… in the same respect and at the same time.

In our mad rush to avoid the appearance of contradiction, we have refused to give a fair and unbiased hearing to the qualities that are absolutely necessary if we wish to accurately define and identify any given thing.

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Antone
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Re: What is Real?

Unread post by Antone » Tue Dec 02, 2008 6:17 pm

Antone wrote:There are many reasons why I believe that the DS theory is a better theory of reality than anything else out there. One of the reasons is that the same basic structure can be used to model virtually any thing or any field. This is true for very big areas and very small. For example: (1) quantum physics, (2)linguistics, (3)theory of numbers, (4)holographic structure of light, (5) definitions of words, (6) philosophy, (7) how computers can think like humans, (8) theory of light, (9) ethics and morality, (10) Logic, (11) epistemology, (12) how the muscles work, (13) cymatics: relationship between vibrations and sound, (14) how the brain works, (15) set theory, (16) theory of sound, (17) mathematics, etc.

Here's an example of how the reciprocal structure of the DS theory can be applied to mathematics
The Multiplication Analogy
Careful consideration will reveal that it is totally impossible to multiply 2 things that are defined by the same aspect. This is why, for example, if we wish to know the magnitude (i.e. number of band members) in a marching band we must multiply [columns] by [rows]. It is meaningless to multiply [rows] by [rows], because we would then have no way of knowing how many [band members] were in each [row].

Notice that [rows] and [columns] are reciprocal in nature. They are both part of the [marching band] but they are (generally) oriented perpendicular to one another. This allows the [magnitude of the columns] to be defined by the [number of rows it contains]; while the [magnitude of the rows] is defined by the [number of columns it contains]. In other words, each aspect contains reciprocal information about the other--and so defines it (at least in terms of magnitude).
Obviously, it is only the [magnitude property of the rows and columns] that is defined in this way. Other properties are defined using [other reciprocal aspects]. And sometimes several reciprocal properties are needed to accurately define a given [object or entity].

What is important to realize is that the reciprocal relationship that we saw with the [marching band] is absolutely necessary for any meaningful multiplication (or division) problem. For examlpe, consider MPH. We find MPH by [dividing distance by time]. But notice that [distance] and [time] are reciprocal aspects of reality--that's why Einstein's theory dealt with   and [time]... [Distance] is a spacial measure, and [time] is a temporal measure.

Notice also that we can easily change our perspective by focusing on various parts of the overall equation.
Distance = Time x MPH
MPH = Distance/Time
Time = Distance/MPH
In much the same way, we can choose a perspective for our [marching band]. If we orient in one direction, a [stationary marching band] may have [5 rows and 7 columns], but if we reorient at a 90 degree angle to that direction, the same band (which has not moved) may have [7 rows and 5 columns].

We can find the same fundamental principles in any multiplication problem. Any two things that we multiply or divide must have a reciprocal relationship. It is common to ignore this fact when we are multiplying simple numbers--but think back to the way your math teacher probably taught you to multiply. She probably said that Multiplying [3 x 4] is the same thing as [adding 3] [four times].

A slightly more advanced way to think of [3 x 4] is to imagine that we are multiplying the [number of sets] by the [number of elements in each of those sets]. Or vice versa. Thus we have
{0, 0, 0, 0}
{0, 0, 0, 0}
{0, 0, 0, 0}
or
{0, 0, 0}
{0, 0, 0}
{0, 0, 0}
{0, 0, 0}
Which number we choose to see as the [element] or as the [set] is entirely arbitrary.
And again, [3 sets multiplied by 3 sets] is meaningless--because we have no way of knowing how many elements are in each of those three sets.

Another way to understand multiplication is to think of it as the union (or intersection) of two RECIPROCAL things.

Addition, by contrast, is the union of two LIKE things. In other words, we can add elements, as in:
{0, 0, 0} + {0, 0, 0} = {0, 0, 0, 0, 0, 0}
or we can add sets, as in:
..{0, 0} + {0, 0} {0, 0} = {0, 0} {0, 0} {0, 0}
1 {0, 0} + ...2 {0, 0}...= 3 {0, 0}
But when we are multiplying, we must simultaneously consider both aspects at once.

Now... understanding this very simple and obvious (once it is pointed out) fact means that by observing which things we can multiply together tells us a great deal about the nature of those things--because it tells us that they are reciprocal in at least some way. By figuring out how they are reciprocal, we can figure out something new and useful about them--and about the structure of reality.

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Re: What is Real?

Unread post by rcglinsk » Tue Dec 02, 2008 11:10 pm

Hey Alton,

When you talk about a continuous object, I always get the sense that your considering the object to be solid and not liquid. Could a blob and not a bar be the atom?

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Re: What is Real?

Unread post by altonhare » Wed Dec 03, 2008 10:01 am

Antone wrote:There are many reasons why I believe that the DS theory is a better theory of reality than anything else out there.
Including your inability to define "object" in a non-circular, observer-free way?

Including the fact that, according to you, an object is defined as a concept which is defined in terms of non-objects which in turn is defined in terms of objects which are in turn defined in terms of concepts... ad infinitum.

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DS Theory: A concept that is not defined in terms of objects

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Common sense: A concept defined in terms of objects, the concept motion.

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DS Theory:An "object" without shape.

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DS Theory: This object never touches anything, so it doesn't exist?

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DS Theory: Now it exists?

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Common sense: An object has shape whether anything ever touches/interacts with it or not.

I appreciate the examples you've given but we can't get into that until we resolve these very basic, fundamental issues of object, concept, and exist.
rcglinsk wrote:Hey Alton,

When you talk about a continuous object, I always get the sense that your considering the object to be solid and not liquid. Could a blob and not a bar be the atom?
Solid and liquid are poor terms not because they're wrong (it is the referent not the symbol that is important) but because people associate them with everyday experience and try to make "continuous" synonymous with "dense", which is incorrect in the common usage/conceptualization of dense. A lump of metal is considered dense relative to a balloon from the carnival. But they are not fundamentally different, it's just a matter of degree. The constituents of the lump of metal are at smaller distances from each other than the constituents of the balloon.

The difference between continuous and discontinuous is *not* a matter of degree. A continuous object is fundamentally different from a discontinuous object. A continuous object IS a constituent. It is a single piece. The "density" of a continuous object is just one, one object. A discontinuous object is one comprised of other parts. So a link of chain may be continuous and the chain itself is discontinuous. Make sense?

The word's "blob" and "bar" are only important insofar as they specify the structure of what you're referring to. If they are both continuous then they cannot deform or break by definition. If you think the atom is a blob or bar you will need to describe and/or draw these structures for me and justify how these structures justify our observations.
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Re: What is Real?

Unread post by rcglinsk » Wed Dec 03, 2008 5:19 pm

Solid and liquid are poor terms not because they're wrong (it is the referent not the symbol that is important) but because people associate them with everyday experience and try to make "continuous" synonymous with "dense", which is incorrect in the common usage/conceptualization of dense. A lump of metal is considered dense relative to a balloon from the carnival. But they are not fundamentally different, it's just a matter of degree. The constituents of the lump of metal are at smaller distances from each other than the constituents of the balloon.

The difference between continuous and discontinuous is *not* a matter of degree. A continuous object is fundamentally different from a discontinuous object. A continuous object IS a constituent. It is a single piece. The "density" of a continuous object is just one, one object. A discontinuous object is one comprised of other parts. So a link of chain may be continuous and the chain itself is discontinuous. Make sense?

The word's "blob" and "bar" are only important insofar as they specify the structure of what you're referring to. If they are both continuous then they cannot deform or break by definition. If you think the atom is a blob or bar you will need to describe and/or draw these structures for me and justify how these structures justify our observations.
What do you mean by deform when you say "they cannot deform or break by definition?" Does that mean change shape?

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Re: What is Real?

Unread post by Antone » Wed Dec 03, 2008 6:34 pm

altonhare wrote:... according to you, an object is defined as a concept which is defined in terms of non-objects which in turn is defined in terms of objects which are in turn defined in terms of concepts... ad infinitum.
Often times the definitions of the DS theory aren't so much circular as they are shaped like a spiral. Yes, an [object] is defined in terms of a [concept], but that concept is then defined in terms of a [collection of objects], which are defined by a [new and different concept], which is defined by [another new and different collection of objects].

This, however, is not significantly different from the fact that [words in a dictionary] are defined by [other words that are in the dictionary]--but we do not consider such words to be too circular.

In order to help explain how this works, I will include excerpts from my DS theory.

But first, I need to define what a limit point is. According to the DS theory, [0] is a limit point on the number line because it separates two reciprical aspects--i.e. the positive numbers and the negative numbers. Like all limit points, [0] has the characteristics of neither the [positive] nor the [negative] numbers. It is, in essence the toggle switch between [being positive] and [being negative]. In much the same way, all reciprocal aspects are separated by a [limit point]. I won't go into explaining this relationship in great detail here... perhaps later.

The DS Theory of Tropes
1) There are two reciprocal aspects of reality:
a. Concrete reality: is physical in nature, meaning:
i. It is spatially and temporally dependant
ii. It is physically divisible but not conceptually divisible and
iii. It is publicly accessible.
We typically call things that exist in concrete reality objects or conditionals, and understand them to be physically real.
b. Abstract reality: is not physical in nature, meaning:
i. It is not spatially or temporally dependent
ii. It is conceptually divisible but not physically divisible and
iii. It is not publicly accessible.
We typically call things that exist in abstract reality entities or processionals, and generally understand them to be conceptually real.
2) Objects and entities each have two reciprocal aspects:
a. A property is an aspect of a quality or a relation that is not singular in nature, in part because it has two aspects that define it.
b. A trope is a property as seen from the inverse perspective, so it is singular in nature, with respect to the trope itself. However, it still has a reciprocal nature because:
i. With concrete objects, the trope is more singularly unique; and
ii. With abstract entities, the trope is more singularly holistic.
3) The same object or entity can be thought of as a property or a trope.
This does not change the essential nature of what the object or entity is, only whether we are referring to its singular or its plural aspect; and whether we chose to perceive it as the parent or the child in the web structure of hierarchical qualities that define it.
4) The following asymmetrical relationship exists between tropes and properties.
a. It is necessary that all concrete tropes have a functionally equivalent abstract property.
b. It is possible that any abstract trope will have a functionally equivalent concrete property.
5) The following functional equivalencies exist:
a. A concrete property is functionally equivalent to an abstract trope. For instance, the concrete property, [my cat at time t1], is functionally equivalent to the abstract trope, [my cat at time t1].
b. A concrete trope is functionally equivalent to an abstract property. For instance, the concrete trope, [my cat], is equivalent to the abstract property, [my cat].
c. An abstract property is functionally equivalent to the set of all concrete tropes that exemplify (or share) the abstract property in question. For example, if we think of [cat] as an abstract property, it will be functionally equivalent to the set that individually lists all real cats. [My cat], [your cat], and all other concrete tropes which exemplify [cat-ness] will belong to this set.
d. An abstract trope is functionally equivalent to the set of all the time-dependant instances (or concrete properties) of the abstract trope. For example, [my cat at exactly 1 second old], [my cat at exactly1 year old], and [my cat at the exact moment of death], as well as all the other possible instances in time when my cat is uniquely different, are all concrete properties (or instances of) the abstract trope, [my cat].

The Nature of ExistenceAccording to John D. Barrow, the question of whether or not [existence] is a property has proven problematic to a number of philosophers, for example:
…Kant… pointed out…that ‘existence’ is [not] a property of things… it is really a precondition for something to have properties. For example, while we can say that ‘some white tigers exist’, it is conceptually meaningless to say that ‘some white tigers exist and some do not’. This suggests that while whiteness can be a property of tigers, existence cannot. …it is not logically correct to assert that because something is a logical possibility, it must necessarily exist in actuality.
By the same logical structure, we could argue that [Whiteness] may be a property of [tigers], but it is not a property of [white tigers], because it is not logically correct to assert:
Some [white tigers] are [white] and [some are not].
The problem, however, is that we clearly do think of [whiteness] as being a property of [white tigers]—it’s simply a property that all white tigers necessarily have. Lack of exceptions does not exclude it from being a property. In fact, if you could choose one and only one answer to the following question, what would it be?
Is it more appropriate to say that whiteness is a property of:
a) tigers, or
b) white tigers?
Since not [all tigers] are [not white], it intuitively seems to make sense to think of [whiteness] as being a more dominant property of [white tigers] than of [all tigers]. I would suggest that Kant's criticism only seems problematic to those philosophers who refuse to acknowledge the role that concepts play with respect to defining reality.

The term [white tigers] has the more encompassing term [tigers] which includes cases that are [white] and cases that are [not white]. But if we refuse to acknowledge the existence of concepts, then physical objects would not appear to have a similar term that includes cases that are [physical] and cases that are [not physical].

The DS theory of tropes resolves this dilemma with (axiom 4), which basically states that [all physical things] have a corresponding [concept aspect]—but [not all concepts] have a corresponding [physical aspect]. This means that we can now say (in a sense) that [some concepts are physical] and [some concepts are not].

Dead Ends in the Chain of Limit Points?Conceptual and physical aspects form chains that progress by passing through a succession of limit points--so it might seem that, eventually, there would have to be a point beyond which the chain can progress no further. Fortunately, the reciprocal structure of the DS theory provides us with a way to get around this conundrum.
image mentioned in text
image mentioned in text
The table above shows how the mind forms an unending and circular chain of tropes and properties. Keep in mind that with [physical objects], a property is less specific than a trope, (axiom 2). Thus, if we start with the concrete property [white tigers], we can work our way up the chain, transitioning from [one level in the hierarchy] to the [next] by passing through one limit point after another. This works quite well until we reach the top and encounter [all physical things]. At this point, it might appear that we have come to a dead end. After all, what could be “less specific” than [all physical things]?

As we’ve just seen, however, [all abstract things] is a larger set than [all physical things]. Thus, by the rules of the DS theory of tropes, [all physical things] is the concrete trope with respect to [all conceptual things], because it is more “singularly unique”.

Now, because this is the limit point between the [abstract and concrete] aspects, there is a necessary inversion of characteristics. What was up now becomes down. On the concrete side of the chain, the trope was more [singularly unique] (or in other words more “specific” than the property). With the inversion of the limit point, we find that the tropes on the abstract side of the chain are more [holistically unique] (or in other words, they are more “all encompassing” than the property. Notice how this means (from the abstract perspective) that [all abstract things] is the trope with respect to [all physical things]—which of course makes [being physical] a [property of abstract things].

Once again, we can work our way down the abstract side of the circular chain until we reach what appears to be a dead end. If we simplify by ignoring the [temporal aspect], then we can represent this as [white tiger1 at the Cincinnati Zoo]. What could be less “all encompassing” than that?

Well, since a concept is something that each person holds privately, each person will hold a concept of the [white tiger1 at the Cincinnati Zoo] that is (at least) slightly different. Thus, there are many such concepts—but there is only one [physical object] that exemplifies all of these concepts. Thus, the physical aspect of [white tiger1 at the Cincinnati Zoo] is less “all encompassing” than the conceptual aspect. Notice that, once again, [being physical] is a property that the abstract [white tiger1 at the Cincinnati Zoo] possesses. And, because we have passed through another limit point, when we consider the relationship from the [concrete side of the chain], the physical tiger is a trope with respect to the abstract aspect of [white tiger1 at the Cincinnati Zoo].

Thus, we have a circular progression that never ends—and so can never lead us into a logical dead end. Like the words defining words in a dictionary, the reason this circular structure isn't a problem is because the circle in question is a very, very large circle.

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Re: What is Real?

Unread post by altonhare » Wed Dec 03, 2008 7:03 pm

rcglinsk wrote:What do you mean by deform when you say "they cannot deform or break by definition?" Does that mean change shape?
That's exactly what it means. A continuous object's shape is immutable by definition.
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Re: What is Real?

Unread post by Antone » Wed Dec 03, 2008 7:28 pm

The circular structure that I mentioned in my last post has practical applications in math and set theory as well.

The Principle of Elemental RedundancyEverything can be understood as either a [whole] or as a [collection of parts]. This is the basic distinction we are making when we differentiate between the [element and set] aspects of a class. And this simple fact can help us to understand Cantor's paradox (which I believe) states that the [cardinal value of a set] is always smaller than the [cardinal value of that set's powerset].
Definition The Principle of Elemental Redundancy states that a set can only contain one instance of an absolutely unique element.
For example, if the following enumeration set contains only [counting numbers], then:
{1, 1, 1, 2, 2, 3} is identical to {1, 2, 3}
In other words, these are simply two names for the same set.
To see how this works, imagine a class that contains the three physical apples sitting on my desk. Let’s call this class (A) and express it as:
(A) {A1, A2, A3}={x: x is an apple on my desk}
The power set of A would be:
(PSA) {[A1], [A2], [A3], [A1, A2], [A1, A3], [A2, A3], [A1, A2, A3]}={x: x is a grouping of elements in A}
[PSA] unit of measure is [groupings] not [apples], and there is nothing physical about the [idea of a grouping]. The objects that are grouped are physical, but the [notion of grouping] is not. Thus while the A1 in [A1, A2] refers to a physical apple, the elementized set [A1, A2] does not. As a set {A1, A2} focuses on the parts that make up the whole, but when we elementize this set aspect, we are changing our focus to the holistic aspect; thus, in this sense, the element [A1, A2] has no parts. It is the concept of a particular grouping of parts, but if you add or subtract parts from that concept, you change the concept. Thus, [A1, A2] literally has no parts.

We can, of course, turn the elements of the power set back into physical objects by removing the square brackets, which are used to identify elements. This produces the set:
{A1, A2, A3, A1, A2, A1, A3, A2, A3, A1, A2, A3}
Now, we can apply the Axiom of Elemental Redundancy, to remove elements, as follows:
{A1, A2, A3, A1, A2, A1, A3, A2, A3, A1, A2, A3}
{A1, A2, A3}
In a sense, the set {A1, A2, A3, A1, A2, A1, A3, A2, A3, A1, A2, A3} and the set {A1, A2, A3} are just two different names for identifying the [same class], just as {A1, A2, A3} and {x: x is an apple on my desk} are two different ways of naming the same class. And the principle of sign differentiation tells us that the [name of the set] is not the same thing as the [set] itself—thus, while we are dealing with two different names which are not themselves identical, the [underlying concept to which they refer] is identical.

By way of analogy, if a man’s full name was [John Paul Smith] and he said, “My name is John. John Smith. John Paul Smith.” It would probably not be his intention to indicate that he had six different names. Similarly, the extra [1 and 2]s in {1, 1, 1, 2, 2, 3} are a bit like the redundant sounds that someone who stutters makes. And like those redundant sounds, the redundant elements in a set can be ignored.

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Re: What is Real?

Unread post by Antone » Wed Dec 03, 2008 7:59 pm

altonhare wrote:Including your inability to define "object" in a non-circular, observer-free way?
Since concepts are observer-dependent, by definition, and my claim is that object are defined in terms of concepts--the inescapable result is that objects are defined by observers. This is common sense, as I'll demonstrate shortly. But the irony is that you do exactly the same thing with your definition of an object as a shape--because there isn't any shape that you can define without observing what it is. Define a circle, without using anything that relies on an observation. You can't do it. Try...
A circle is a closed plane arc that is equidistant at all points from a center point.
But what is an [arc]? How do you figure that out without any observations? You define it with something else, which again you have to define. All the definitions in the world tell you nothing unless somewhere along the line you've experienced some observations that make the terms used in the definition mean something.

Sure, an object has a unique shape. Fine... but what is it? How do you compare it to other shapes? How do you say anything at all about it... with entirely observer-free language. It is impossible. So all you've said is that [x] is [x].

First of all, that doesn't actually tell me anything... and secondly it is the exact same thing that I've said repeatedly from my DS perspective. So once again, your criticism is lame and I don't think going over this again and again is getting us anywhere.

How We Learn
Composite Photography is a technique whereby an ordinary piece of film is exposed with several overlapping images. Each image is of a similar [type of object]. For instance, composite photography is sometimes used to create a composite image of all the members in a given family. Each portrait is taken from the same distance and angle so that when the film is developed the same basic features of each person will appear (as much as possible) in the same place on the film.

The result is that those features that the family members hold in common will be reinforced, while those features that are different will be deemphasize. If the family has very similar features, the resulting image will be relatively sharp and well defined; if their features vary widely, then the image will be a fuzzier blend of all the shades and colors that were exposed to that part of the film. Pictures made with such negatives may look as if they were taken slightly out of focus, but for the most part, if you place the [composite picture] beside [ordinary pictures] of all the family members, you might have to look rather closely to pick out which picture was the composite photograph—particularly if you aren’t familiar with the family.

I believe that a very similar mental technique is used (primarily by children) to learn the meaning of concepts and the names of objects. Suppose, for instance, that a young child is being raised in a family that has a Persian [cat]. If this is the only [cat] that the child has ever seen, then the collection of images that the child uses to define [what it means to be a cat] is quite sharp—since the images are entirely unmuddied by examples of any other [types of cats]. But the child’s actual concept of [what it means to be a cat] is still quite vague, because the [one sampling of cat] is insufficient to clarify what it is about this particular animal that makes it a cat.

Suppose, however, that the child sees a small [dog] for the very first time. He can certainly recognize that the [dog] is different from the other animals he’s learned to call cats, but several of the cats were quite different from one another as well. For instance, to his young mind, the [Siamese cat] that he saw recently, may have looked considerably less like the [Persian cat] than this [dog] does.

But when he points at the [dog] and says, “Cat!” his mother gently corrects him by saying, “No honey, that’s a dog,” he is forced to make new evaluations about [what a cat is] and [what a cat is not]. The [image of the dog] goes into the [collective whole that is the child’s concept of NOT cat], forcing him to reexamine his collective memories and decide what exactly it is about this new animal that prevents it from being a [cat].

As the child sees more examples of [cats] the composite image of what it means to be a cat becomes stronger, as the essential qualities are reinforced and the non-essential qualities become more and more blurred. And because this process of composite holography is primarily irrational in nature, it is also largely automatic. It is not something we have to think about in order to do. Instead, it is a simple matter of the overlapping images producing a sharper and sharper mental composite. In this way, the child learns to associate certain [groupings of irrational mental images] with a [particular rational term]. And while the term itself, may be rational (meaning that it is under concious control), the composite image created by all the mental images is not.

Perhaps what is most important to recognize is that the individual [mental images] and the [composite of those mental images] are reciprocal aspects of thinking. We can visualize these reciprocal aspects using the illustration below. (sorry about the poor quality.
Class: with set and element aspects
Class: with set and element aspects
pix-1.JPG (3.65 KiB) Viewed 12697 times
The larger circle, labeled (I), represents the composite image that is associated with a single term, such as the word [cat], while collectely, all of the smaller circles, labeled (R), represent the [individual mental images of a cat] that the child has seen.

The larger circle, (I), is a singular, abstract thing, which we can represent using the abstraction set {x: x is y}, where [y] is a unique property or set of properties that only the smaller circles possess. Keep in mind that the larger circle necessarily includes all of the smaller circles, but it does not include them as individuals—just as a [composite photo ]includes [images of all the photos that were used to create it], but is not itself an image of any of the [individual photos]. The result is that, just as we can add new photos to a [composite photo], so too we can mentally add (or subtract) the smaller mental circles from the larger mental circle without changing the basic nature of the larger mental circle.

For instance, if we let (I) stand for the concept, [parts of my body], then the smaller circles that comprise (R) would be such things as [my head], [my arms], [my legs], and so forth.

Now then, we can represent the larger circle using the abstraction set:
(I)B {x: x is a part of my body}
And the smaller circles can be represented by the enumeration set:
(R)B {[my head], [my right arm], [my left arm], [my torso],…}
As a set, (I)B, contains but a single element, [parts of my body]; while (R)B contains several different elements. If we add another element to (R)B—say by distinguishing for the first time between the [upper arm] and the [fore arm]—we do not change the element that (I)B contains. Its single element is still [parts of my body]. Thus, in a very real sense, (I) and (R) are independent of one another, since one can change while the other does not; and yet in another sense, they are intimately interconnected--because one defines the other.

When we are thinking properly, each [kind of mind] serves as feedback to keep the other [kind of mind] working at top efficiency. This is true at virtually every possible level: for the two, [reciprocal kinds of mind] share an intimate and co-dependent relationship, with both aspects being equally necessary. Notice also that each of the smaller circles in [R], is its own larger circles in a different diagram—while the large circle is a small circle in another diagram.

In addition, each of these diagrams are interconnected in a vast and complex network of ideas, which we might call a mind set. The illustration below gives a vastly over-simplified visual representation.
mindset: a network of overlapping conceputal classes
mindset: a network of overlapping conceputal classes
pix2.JPG (5.17 KiB) Viewed 12699 times
The larger boxes (which are labeled with letters) represent [terms the child has learned and associated with a composite mental image]; while the smaller circles (labeled with numbers) represent the [individual mental images that go into creating those composite images].

Clearly, these [composite images] overlap and interact in rather complex ways. For example, while a dog and cat may share such characteristics as four legs and fur, a dog may be the same color as the family’s pet turtle, while Snowball, their pet cat, may be the same color as a child’s ball. A child first begins to develop these mental correlations because each time his mother corrects him she is teaching him which mental images belong together in the same composite images.

Common Sense
At the same time that the child is learning to match names and objects together, they are learning other, subtler relational correlations. This learning occurs in much the same way: by repeatedly encountering similar things (which are none-the-less different) and different things (which are somehow the same), and building from these individual instances a collective, irrational concept that defines the whole picture. We sometimes call this collective body of irrational concepts common sense.

For the most part, these relational rules are so obvious (and in some cases, so subtle) that we rarely stop to think about what they really are. If asked, we would generally be hard pressed to put them into words—not surprising, given their holistic and irrational nature. Common sense is constructed from the composite “images” of our mental concepts, so just as we can recognize a person’s face even though we can’t adequately describe what they look like in words, we can generally recognize common sense when we see it, even though we are quite often unable to put it adequately into words. When we try, our words frequently tend to fall far short of the mark. Moreover, when philosophers try to examine their attempts closely enough, it always seems that they eventually lead to paradox.

According to the DS theory, most paradoxes are produced because we are trying to take the [holistic nature of common sense] and break it into [rational bits] that we can examine as individual elements of the whole. When we do this we commonly produce two apparently incompatible [bits of common sense] which are both equally satisfied by the same scenario. Since neither [bit of common sense] gives rise to a chain of logic that is less appropriate than the other, we are unable to simply discount one bit in favor of the other. But trying to define the scenario strictly in terms of one of these bits (but not the other), invariably leads to one paradox or another.

This idea, that common sense is less than obvious, is not an entirely new thought. For example, Sorensen expresses a very similar idea when he says:
… common sense is reactive. We do not bother to defend (or even think about) the proposition that the future resembles the past until David Hume formulates the problem of induction. Paradoxes illuminate common sense by provoking bits of it into consciousness. As more paradoxes are discovered, more of common sense becomes visible. Without a provocateur, common sense is faceless.
Apparently, common sense is faceless precisely because it is the type of thought which is most hidden from our rational minds, and it is our rational mind that we are typically most aware of. What is not so apparent, however, is why common sense should primarily be a [product of our irrational mind] in the first place.

Modern men are creatures who are dominated by their rational minds. We utilize our reason to puzzle out the consequences of believing in and acting on our common sense notions. In the real world, the reason this is such a crucial skill is because humans are weaker and slower than many other animals. Our main advantage is our faculty for applying rational thought to hypothetical situations that can allow us make decisions not on [what is happening], but rather on [what might happen]. By making such predictions, we are able to avoid potentially dangerous situation and invent new and better ways to do things.

In this sense, we might think of reason as the [rules by which we apply common sense to create new mental structures], which can in turned be modified by reason to produce newer and yet more subtle concepts. Often times, however, the complex interrelated, and cyclic nature of this process makes it very difficult to distinguish between what is common sense and what is reason. For example, according to Roy Sorensen, “G. E. Moore, admitted that common sense underestimates the distance from the earth to other heavenly bodies.”

I would suggest, however, that Sorensen and Moore seem to be confusing common sense with simplistic (or perhaps even flawed) reasoning. I don’t think there is anything at all about common sense that would lend itself to predicting the distance to the planets or the stars.

What common sense can tell us is,
1. Because a [small object that is close to us] can appear larger than a [large object that is far away], we can’t possibly determine how far heavenly bodies are from us without knowing the relative size of the object in question, and
2. Because it becomes increasingly difficult to judge the size of an object the further away from us it is, if astronomical objects are very far from us, it will not be possible to accurately judge their relative size or distance.
To me, these are bits of common sense.

Trying to estimating the distance to heavenly bodies (despite these bits of common sense) is an exercise of flawed reason.

This may seem a bit backwards. We commonly suppose that rational thinking is the source that gives accuracy and clarity to our mental ruminations, but just as we saw with the Broken Clock analogy, where it is only possible to deal with something in an absolute sense when we have defined [what it is to be absolute] in a relative way. It is only possible to [think rationally] because we have used our irrational minds to define [our rational] ideas.

In Western philosophy, the goal seems to be to express everything in increasingly rational terms. Thus, we rely on our reason to help us figure out what our common sense is telling us—but because these faculties stem from very different ways of thinking, this process is often a complete failure, or it meets with very limited success.

For example, our common sense tells us that every proposition is either [true] or [false]. But, as we’ve seen, this only names one aspect of that particular bit of common sense—the very same application of common sense also tells us that virtually every [ostensibly true] proposition is [true in one way and false in another]. So when I say, “It’s exactly twelve o’clock,” the statement is [true] but it is also [false]—each in its own way. It is true in relative terms--but false in absolute terms.

The apparent paradox results because we are not looking at the whole picture of common sense, but only at partial snippets of the whole. As we’ve seen, treating these incomplete snippets as if they were the whole frequently leads to paradoxical conclusions—it can also sometimes lead to deductions that are the opposite of what they should be. This is not unlike the situation that can occur when you’re looking out the front window of a moving car on an exceptionally humid day. You see fat beads of water splattering on the windshield, so you assume that it is raining. But if you change your perspective, by stopping the car (or by looking out the side window), you suddenly realize that it isn’t raining at all.

What keeps these observations from being truly inconsistent is that they involve a change in our perspective. As Aristotle pointed out, no [single thing] is ever both [true and false] in exactly same way at exactly same time. By changing our perspective from looking out the front window to looking out the side window, we have changed something about the specific scenario and thus changed whether we must believe the [claim that it is raining] to be true or false.

Collective Common Sense
What we normally tend to refer to as [common sense] (in everyday speech) is actually the [set of more-or-less shared beliefs that are common to the majority of people]. This kind of [collective common sense] is the opposite of an individuals [common sense] in many ways. In fact, about the only thing they have in common is that they both involve a [holistic aspect]. Individual [common sense] is holistic because of the [way a person’s irrational mind processes it]. The [collective common sense], on the other hand, is holistic in terms of being a statistical averaging (so to speak) of the rational ideas of all the individuals in the whole community.

Another important thing to keep in mind is that these two bodies of belief are not identical. In fact, I would suggest that there are probably more [ways in which they are different] than there are [ways in which they are the same]. Certainly, there are some ways in which they capture the same information: the longer you stick your hand into a flame, the worse you get burned. Experience has taught us this general principle, and it is easy enough to put it into rational thought.

Other experiences, however, are not so easy to convert from irrational to rational thought—as we saw with the Broken Clock Analogy, for instance. For example, there was a time when the [collective common sense] held that the [earth was flat]. This was such a commonly held belief that it was rarely if ever questioned. To the rational mind of these ancient people it seemed to make sense, after all whenever the surface of the earth is free from valleys or mountains (which are just surface features, after all) the earth appears to be flat for as far as we can see.

What individual [common sense] actually tells us is more like the following, however:
1. What we can see of the world appears to be more or less flat.
2. Something that has a particular characteristic at one place will sometimes continue to have that characteristic at another place.
3. The world extends beyond what we can actually see.
4. We cannot know for certain what is beyond what we’ve seen.
5. Sometimes our senses can appear to tell us things that are not true.
Using these elements of [individual common sense], we can make a flawed rational deduction that [the world is flat], but only because we choose to focus on certain parts of the whole. The rational deductions we should make from these [common sense] principals are as follows:
1. the world may be flat (1, 2)
2. it may not be flat (3, 4, 5)
The reason we tend not to reach these conclusions is because the foremost nature of the rational mind is primarily to see things in absolute certainties—not relative possibilities.

This goes back to the two needs of decision making that I mentioned a few posts back: speed and accuracy. The rational and the irrational mind both have their strengths. But the way they work together is that the irrational mind sees things in terms of the big picture--or in shades of gray. While the rational mind focuses on the individual parts--or in terms of black and white. Since the rational mind is not suited to seeing the subtlties of the relative perspective, it is not adept at accepting that both possibilities might be valid. It wants to pick one possibility or the other so that it can make a decision about the world. That, after all, is its job.

rcglinsk
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Re: What is Real?

Unread post by rcglinsk » Wed Dec 03, 2008 10:13 pm

altonhare wrote:
rcglinsk wrote:What do you mean by deform when you say "they cannot deform or break by definition?" Does that mean change shape?
That's exactly what it means. A continuous object's shape is immutable by definition.


Thanks for clearing that up. Sorry if you might have said that already. There are 14 pages of posts...

I have a few questions about your rope theory of light. I hope you'll indulge my curiosity.

Are there two kinds of atoms? Like, one that magnetic threads are made of and another that electric threads are made of?

What causes the connection between atoms in a thread?

If two threads cross paths, do they get tangled? Does one or more of them break?

altonhare
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Re: What is Real?

Unread post by altonhare » Thu Dec 04, 2008 11:07 am

rcglinsk wrote:
altonhare wrote:
rcglinsk wrote:What do you mean by deform when you say "they cannot deform or break by definition?" Does that mean change shape?
That's exactly what it means. A continuous object's shape is immutable by definition.


Thanks for clearing that up. Sorry if you might have said that already. There are 14 pages of posts...

I have a few questions about your rope theory of light. I hope you'll indulge my curiosity.

Are there two kinds of atoms? Like, one that magnetic threads are made of and another that electric threads are made of?

What causes the connection between atoms in a thread?

If two threads cross paths, do they get tangled? Does one or more of them break?
Go to the "problems with thread theory" or "details of thread theory" threads.
Physicist: This is a pen

Mathematician: It's pi*r2*h

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Antone
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Re: What is Real?

Unread post by Antone » Sat Dec 06, 2008 10:49 am

Here is the traditional understanding of Berry's Paradox:
Berry's Paradox
Consider the set containing all natural numbers that are definable in English in fewer than twenty syllables. It will contain [twenty-six] (only three syllables), the [two-millionth prime number] (only seven syllables) and [ten raised to the ten raised to the ten raised to the ten raised to the ten plus fourteen] (nineteen syllables). Since there are only a finite number of syllables, this set must be finite in size. Since this set has only a finite number of members, it must have a largest member.

Consider the number defined by: the [least number not definable in English in twenty syllables]. This number is the smallest number not in the set; yet it is definable in fewer than twenty syllables, for it has just been defined in eighteen! This is Berry's paradox, proposed by Berry in 1906 as a simplified version of Richard's paradox, which is as below.
Addording to the DS theory, the supposed paradox occurs for two main reasons: (1) it fails to distinguish between a [conceptual entity] and a [physical object] and, (2) it fails to distinguish between a [name] and the [thing named].

The first problem can be addressed by realizing that the [least number not definable in English in twenty syllables] is a concept that is distinctly different from the [actual number it supposedly describes]. By way of analogy, consider the [number of fingers a human has on his hands]. The obvious supposition would be that this so-called number is [10]--but some humans have [fewer than 10 fingers] and some have [more than 10 fingers]. Obviously then the [number of fingers a human has on his hands] does not refer to a specific number. It is a concept of something--and concepts may or may not have the property of physical existence. By a similar argument, the so-called number the [least number not definable in English in twenty syllables] is actually a concept--not a number--and like the [square circle] it is a concept that cannot be actualized in physical reality.

One way for critics of DS theory to attack my comments might be to suggest that the [number of fingers a human has on his hands] can simultaneously refer to more than one number. But if this is so, then we no longer have a paradox, because we can say things like the [counting numbers] or an [even number] both of which refer to infinitely many numbers. Thus, the numbers that we can name are also infinite. Thus, the [least number not definable in English in twenty syllables] is still a concept that does not actually name a number, because there are infinitely many numers--and the nature of infinity demands that there can be no largest number. So there can be no [least number not referrable to] in the set.

Similarly, we can attack the problem from the other perspective. A [name] is the [thing that is named]--it is not the [conceptual entity named] nor the [physical object named]. It refers to both--but since the concept and the object are logical opposites, the name cannot be either one because something cannot be both [A] and [not A].

Names are bridges that allow us to tie [concepts and objects] together in a meaningful way. The unfortunate consequence is that having only [1 name] for two things that are [logical opposites] makes it much easier to confuse the [logical opposites] when we are working with philosophy and logic. And it is just this confusing that is so often responsible for paradox.

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Antone
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Re: What is Real?

Unread post by Antone » Sat Dec 06, 2008 12:16 pm

The Infinite Hotel
There’s an analogy (sometimes called Hilbert’s Hotel) that is commonly used by philosophers and mathematicians to help explain the nature of infinity, as it is understood by mainstream theories. As the story goes, there is this [imaginary hotel] that has an [infinite number of rooms]. According to one version:
…A travelers arrives at the Infinity Hotel, only to have the hotel manager tell him there is no vacancy. All of the rooms are full. “But I thought you had an infinite number of rooms,” the traveler says. “We do,” the manager replies, “but we currently have an infinite number of guests, and currently, all of our infinite rooms are filled by our infinite number of guests. There isn’t even a single vacant room.” The traveler scratches his head for a few moments and then suggests: “Why don’t you try this? Move the person in room one to room two; then move the person in room two to room three. Since you have infinitely many rooms, you can continue doing this until all your guests are in a new room. And room one will then be available for me.”
According to the DS Theory, the erroneous assumption here is not that you can move the guests in this way—but that an actual, physical hotel (which by the nature of physical reality means that it must necessarily have a definite number of rooms) can satisfy both of the following conditions at the same time:
1. Having an infinite number of rooms, and
2. Being physically completed.
According to the DS theory, there are two ways to get around the [incompatibility problem between physical existence and infinity]: We can conceive of an [infinity that changes over space], which we might call spatial infinity, or we can conceive of [an infinity that changes over time], which we might call temporal infinity.
1. Spatial Vagueness: When we allow the aspect of time to be indefinite, we are essentially saying that the hotel is actualized in a perpetual now. The only way to accomplish this is to suppose that the infinite hotel has an unspecified number of physical rooms. Since we cannot state precisely how many rooms the hotel contains, we cannot state how much space those rooms occupy—even if we know the exact dimensions of every room.
2. Temporal Vagueness: By contrast, when we allow the aspect of space to be definite, we are ensuring that there are a specific number of rooms—but now, the infinite hotel must be spread out over time. In other words, the specific number of rooms changes as time passes. It can change very quickly, or it can change very slowly—but regardless of its current size, we must assume that eventually the hotel will contain more rooms than it currently does.
From the [perspective of spatial vagueness], the [DS version of the Infinity Hotel] is almost identical to the story recounted above, except, of course, that it would make no sense for the manager to claim that all the rooms were full. After all, if you can’t specify how many rooms the hotel has, how can you specify how many of them are full?

The nature of infinity, however, is much easier to understand when we approach it from the [perspective of temporal vagueness]. Too make the story of the Infinite Hotel temporally vague, all we have to do is imagine that it is necessarily a [work in perpetual progress]. Since each room is necessarily a discrete unit, if the hotel were [completely finished being built] then the hotel would necessarily have a [specific and unchanging number of rooms]. No matter how large that number was, by definition it would not be infinite—nor could it change as more guests arrived.

On the other hand, if new rooms are continually being built, then we have a situation where the hotel is constantly spreading out (and getting larger) as time passes. This means that there is no [single point in time] that defines the infinity hotel’s temporal aspect. It is literally meaningless to speak of the infinity hotel as existing at a given point in time. The hotel is defined by the [incomplete-able process of building the hotel].

Now we can retell the story of the Infinity Hotel in such a way that the hotel could conceivably be a physical object that can theoretically exist in the real world:
An incredibly wealthy man decides (for whatever reason) that he is going to build the largest hotel in the whole universe, but since he realized that this will take a very long time, he decides not to wait until the hotel is completed to start renting out the rooms. To accomplish his goal, he decides to build his hotel out in space so that he will have an endless amount of room to grow. Then, each new room of the hotel is built on earth, in a modular fashion, so that it can be shipped up to the hotel when it is done and attached to the space hotel as each new room is finished. Each time a new room is added to the hotel, work is begun on building two new rooms.

The owner of the infinity hotel is so happy with the results of this project that before he dies he has his lawyers create a perpetual will. All of the owner’s money and all the profits of the hotel are to be used to continue building new rooms for this hotel, forever and ever.

…Now, because of the incredible (and ever growing) popularity of the infinity hotel, it is still possible that when guests arrive all the rooms might be full, but because new rooms are being built all the time (and at a rate that is regularly doubling) by the time the bellhop takes the guests to their room, several new rooms will have been completed and the quests will have their choice of accommodations.
Notice that in this version of the story (if you ignore certain impossibilities, such as the notion that such a large hotel could be financially feasible) nothing the least bit unusual or impossible is going on. The story still captures the notion that the rooms of the infinite hotel might be full and yet the waiting guests (no matter how many of them there are) will all eventually get a room, but it does not require the absurd notion of moving guests into rooms that simultaneously [do and yet do not exist]. Instead, it is understood that the bellhop will simply move guests into rooms that [don’t exist—yet].

[Temporally vague infinity] is one method of “conceptually actualizing” the notion of infinity. It should be realized, however, that this kind of infinity is no more “actual” than [spatially vague infinity]—and the reason is because no such [building process] could ever be kept going indefinitely; and the moment the building process is stopped, the hotel ceases to be infinite in nature.

The Infinity Hotel and the Negative Integers: Another problem with the analogy of the Infinite Hotel is that it ignores the concept of the [negative numbers]. However, we can expand the analogy slightly so that it does account (semi-adequately) for the negative numbers. After all, one of the reasons that our [first revision of the analogy] couldn’t actually work is that all things will eventually decay. This is also true of the rooms in any hotel.

Now, if the building process expands the Infinite hotel in all directions, there will eventually come a point in time when the first rooms that were built (at the very center) will begin to decay and fall apart. As more time passes, rooms further from the center will decay. Thus, we can think of the Infinity Hotel as two expanding rings. The outer ring is defined by the [size of the Infinity Hotel’s expansion], as new rooms are built. The inner ring is defined by the [size of the Infinity Hotel’s decay], as old rooms fall apart.

Notice that (since construction is begun on 2 rooms for every 1 that is finished) both the [rate of expansion] and the rate of decay] will continue to increase in size for as long as the Infinity Hotel continues to be the Infinity Hotel.

Rates of Progression
Suppose that each time a room of the Infinite Hotel is built, construction is started on two new rooms. Further, suppose that when the first room of the Infinity Hotel is built a guest immediately moved in and called four of his friends, who showed up promptly.
Rates of Progression
Rates of Progression
Infinity Hotel.JPG (14.28 KiB) Viewed 12603 times
When the two rooms that are being built are finished four new rooms are started and two of the waiting guests are allowed to take up residency in the newly completed rooms; and they in turn each call four more friends. Now we have four rooms that are in the process of being built and ten friends waiting for rooms (two that didn’t get rooms the last time and four each for the two that just moved in: 2+4+4=10).If this process continues the number of people waiting for a room will continue to increase, but since new rooms will never stop being built, every guest who waits long enough will eventually get a room.

What is important to realize is that the reason everyone who shows up will eventually receive a room is because the rooms will continue to be built forever. The waiting period, of course, will also increase forever, but since there are no temporal limitations, everyone who waits long enough will have a room.

We can make the Infinity Hotel an analogy for a sequence by imagining that a record is being kept of the hotel’s progress. When the first room is completed a [2] is recorded in the ledger under the [number of rooms being built] column—and when the first guest moves in and calls four friends who immediately show up, a [4] is placed in the [occupants waiting] column. When the two rooms are completed a [4] is entered under the [number of rooms being built] column, and a [10] is entered under the [occupants waiting] column. The next line in the ledger would be the numbers [8] and [22], and so forth.

Keep in mind that when it comes to a meaningful comparison between the [number of rooms being built] and the [number of waiting guests], what we are comparing is not the size of infinity. Infinity is the concept of [endlessness] and so has no cardinal value and thus can not be measured that way. We can compare the Rate of Progression towards infinity, however, which I like to write using the abbreviation: ROP. In this case, the ROPs are not linear. Thus, the only meaningful thing we can say about [how quickly the rooms are being built] as compared to [how quickly the guests are arriving], is that the guests are arriving faster.

Baseline Infinity: I believe the [ordinal number sequence] is arguably the best sequence to choose as a baseline for comparing other infinite sequences. Other than the [ names that we give to numerical order], the [ordinal number sequence] contains no information other than the basic concept of [endlessness].

The [even number sequence], for example, skips some of the elements in the [ordinal number sequence]. In fact, it skips [1 number] for every [2 numbers] in the [ordinal number sequence]. Thus, we can say that the [even numbers have] an ROP of [2:1]—typically written as [ROP-2:1]. Similarly, the [improper fractions] would have [ROP1:oo], because there are infinitely many fractions between each whole number.

Notice that the sequence that makes the fastest progress towards the limit point (which in this case is the concept endlessness) is the one that has the larger Rate of Progress. This may seem a little counter intuitive, since in [actualized set terms] the set that appears to be smaller is the one that has the larger ROP, but keep in mind that we are measuring a [sequence’s Rate of Progress]—not the [size of the set]. The one that gets their faster is traveling at the higher speed, so too speak, but this is only because it is skipping over elements that the other set contains.

Thus, the sequence with the higher ROP can be said to identify a set with fewer unique elements.

Comparing Without a Baseline: Sometimes, it is useful to compare two infinite sequences when neither one of them has [ROP1]. For example, consider the following sequences:
(A) {2, 4, 6, 8, 10, 12, 16, 18… }
(B) {3, 6, 9, 12, 15, 18…}
Neither of these sequences has an [ROP1], and they do not share the same elements. For example, (A) contains [2] but not [3], and (B) contains [3] but not [2]. However, they do share some of the same elements—and we can use this fact to create a comparison as follows:
(A)....2........4...6....8.......10...12...14.........16...18
.............................................
(B).........3........6........9........12.........15.........18
The red numbers indicate the [elements which both sequences share]. Now, If we end the sequence at any one of these red numbers and count the number of elements in both (A and B)—the result will be a ratio.
At [6], (A) has [3 elements] and (B) has [2 elements].
At [12], (A) has [6 elements] and (B) has [4 elements].
At [18], (A) has [9 elements] and (B) has [6 elements].
All of these ratios, are equivalent (or can be reduced) to [3:2]. Thus, if we want to compare (set A) to (set B), we can say that the [ROP of A:B] is [3:2].

Containment and Domination: A set with a smaller ROP can be said to contain or dominate a set with a larger ROP. This is true for any two comparable sets, even if they are finite. For instance, if we define two finite sets as follows:
(T) {10, 20, 30, 40… 1000}
(O) {1, 2, 3, 4… 1000}
We can say that the [ROP of T:O] is [10:1], and the [ROP of O:T] is [1:10].

Of course, not all [infinite sequences] are so as easy to compare as those we've looked at so far. Suppose, for example, that we have three infinite progressions as follows:
(A) {1, 2, 3, 4, 5, …}
(B) {2, 4, 6, 8, 10, …}
(C) {1, 4, 9, 16, 25, …}
Clearly the [ROP A:B is 1:2], and the [ROPB:A is 2:1]. The sequence (C), however, does not lend itself to such an easy comparison with either [A] or . The reason, of course, is that its ROP is not a linear progression.

It might be possible to compare such non-linear ROPs (using non-linear algebraic notations, for instance) but that is not something I can attempt here.

However, many non-linear infinite sequences will still lend themselves to some sort of simple comparison relationship. For instance, while we cannot create an exact comparison ratio for [ROP A:C] we can still indicate the following relationships:
[ROP C > A]

Understanding How ROPs Fit into the Other Numbers: All infinite progressions will continue forever, because the [very nature of infinity] is to be [endless]. Thus, it is totally meaningless to say that the [finale entry in one sequence] will be larger than the [finale entry in another]. However, if we invert our thinking from that of a sequence—or a potential set—to that of an actualized set, then we can say that we have two potential sets which contain a specific number of elements. Because these sets are infinitely large, it is not possible to specify their [exact cardinal number]—but we can compare their cardinal numbers by comparing their ROPs.

This, of course, is the exact opposite of the conclusion that George Cantor reached. But according to the DS theory, virtually every conclusion that Cantor reached about infinite sets was the exact opposite of what he should have concluded.

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Antone
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Re: What is Real?

Unread post by Antone » Sat Dec 06, 2008 1:13 pm

The Gas Pressure Analogy
According to Charles Seife, the definition of the infinite is as follows:
It is something that can stay the same size even when you subtract from it.
But what exactly does this imply? What kind of real thing can possibly stay the same size when you subtract from it? Clearly, there is no solid that has this property—but what about something that is the logical opposite of a solid?

What about a Gas?

If we have a gas tank with a volume of [1 cubic meter] and we fill the tank as full as we can with gas, we might say that we have [1 cubic meter of gas]. However, this is a rather strange (and largely meaningless) way of speaking about the gas in the tank, for if we remove half of the gas from the tank, the gas in the tank still occupies [1 cubic meter]. Here then, we seem to have found a physical candidate which (at least in an analogous way ) satisfies Seife’s definition of being infinite. The reason, of course, is that gas doesn’t have a specific size in the same sense that a solid does—just as a [conceptual entity] doesn’t have a specific size in the same sense that a [physical object] does. Thus, [gas] is to an [infinite set] as a [solid] is to a [finite set].

Outside of the confines of a tank, the typical gas would quickly expand and dissipate, spreading itself over a larger and larger area until parts of it might well be scattered throughout the whole atmosphere of the earth. Conversely, if we could find a way to force more gas in the tank, we could keep doing so until the tank literally exploded. (And this doesn't even include the influence that heat has on the gas). Thus we see that what has size (in the sense of volume) is the [tank], not the [gas in the tank].

There is a characteristic of gas, however, that lets us determine when we have more or less gas in a given space. It’s called air pressure. The basic formula for determining the amount of gas we have in a given space is pretty simple: the higher the pressure on the inner sides of the tank, the more gas we have inside the tank. And if the pressure is higher on the outer sides of the tank, then we have a parallel to the negative numbers.

We might think of [infinity] (i.e. endlessness) as being analogous to the [tank] while the “size of infinity” (i.e. ROP) is analogous to the [air pressure]. In other words, the ROP (or air pressure) is determined by [how many numbers we stuff into that same infinite enclosure]. For example, the [counting numbers] might be said to have an infinite size that stuffs twice as many elements into the same sized infinite enclosure as the [even counting numbers].

Perhaps the most important point of the analogy, however, is that (just as we cannot measure gas in the same way that we measure a solid) we cannot measure the infinite numbers in the same way that we measure the finite numbers. As Calvin C. Clawson says, “infinity… flows out, always leaving you behind.” Or in other words, left in its natural, (conceptual) environment infinity will always expand to fill any numerical space no matter how large (just as gas eventually expands to fill the whole atmosphere of the earth).

This [expanding aspect of infinity] is possible only because [infinity] does not refer to magnitude at all--rather it is the [concept of expanding] itself. After all, [endlessness], when combined with the [idea of adding physical objects] is functionally equivalent to expanding forever. And like the concept [heap], the concept [endless] doesn’t change when you [add to it] or [subtract from it]. Thus, if you add twice as many physical objects per unit of time, it's still endless adding that is occurring.

Magnitude, of course has two reciprocal aspect: (1) largeness, and (2) smallness. So far, we've been thinking of infinity only in terms of the [largeness] aspect of magnitude. When we think of infinity in terms of the smallness aspect, then instead of adding endlessly many objects we are dividing a single object into infinitely smaller and smaller parts. And since the DS theory defines a natural number as a line segment--an infinitely small part of the line segment is a specific location on the line segment.

Thus, in terms of the number line analogy at least, [finite numbers] refer to [magnitude] and [infinite numbers] refer to [location].

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Re: What is Real?

Unread post by Antone » Sat Dec 06, 2008 2:42 pm

The easiest way to demonstrate why Cantor's assumptions are wrong is to attack his principle of one-to-one-correspondence. Demonstrating the fallacy of this principle when dealing with infinite sets is the purpose of this post.
Countable infinities: The first commonly held belief I’d like to look at is the notion that certain kinds of infinity are enumerable—and more importantly, that this is essentially the same thing as being countable.

According to Amir D. Aczel,
The natural numbers can be counted, even though they are infinite. It is the process of counting that matters, not the actual counting, since such counting will never end. The natural numbers can be counted, since it is possible to call them one after the other, 1, 2, 3, 4,…and so on. Thus—while infinite—the natural numbers are countable.
According to the DS theory, the problem with this assumption is that it ignores the distinction between [sequences] and [sets]. Clearly, it is possible to complete the [process of counting a relatively small subset of the natural numbers]. But no matter [how many people] were engaged in the [process of counting the natural numbers] or [how long they were allowed to count]—it would never be possible to complete the process, because by definition the process is [endless] and so must go on forever.

Since there can never be an [actual end to the process], there can never be an actual point at which the [process is completed]. This clearly indicates that the [process of counting the elements of an infinite sequence] is something that must be occurring—at least in part—at some point in the future. If we invert our thinking, we might say that the time of completion is perpetually in the future, but however we look at it, it is obvious that such an “infinite set” can never actually be [counted], past tense. And since it cannot be [counted], it is obviously [not countable], just as a [distance that cannot be walked] is [not walk-able], a [name that cannot be said] is [not say-able] and so forth. If you can't complete the counting of an infinite set, then it is not countable.

This is a very important distinction when we want to considering the difference between using one-to-one correspondence to [count a finite set] and to [count an infinite set]. Obviously, the [process of counting a finite set] is something that can be completed. Even if the set is astronomically large, it contains a finite number of elements, and so if we had enough people doing the counting for a long enough period of time, it is conceivable that [any finite set could be counted].

Trying to [count an infinite set] is analogous to [trying to count a finite set where an Evil Trickster keeps randomly taking random sized handfuls out of the set and putting random sized handfuls back in]. Trying to count elements under such conditions would be pointless, and thus it would truly be impossible to finish the counting process. Even if we did reach a last element, having done so would tell us nothing at all about the actual number of elements in the finite set.

Infinite sets are uncountable for a different reason. There is no Evil Trickster, but we are still unable to reach the last element, and so we are unable to [assign a cardinal number to the set].

This is why the DS Theory makes the following distinction:
A set is enumerable if we can engage in a sort of [meaningless counting process that tells us nothing]. This is possible any time the numbers can be ordered in such a way that for any given number, the next number in the set is always predictable.

A set is countable if-and-only-if the counting process has the potential of giving us meaningful and specific information about the number of element in the set.
One-to-One-Correspondence: The second commonly held mathematical belief that I would like to examine is the notion that one-to-one correspondence is all that is necessary to ensure that two infinite sets have the same [cardinal value]. (Called Cardinal Number in traditional theories).

When we are dealing with finite sets, there are two important points that make one-to-one correspondence an effective method of counting:
1. [Each element in one set] must match to [one element in a second set],
2. Neither set can have any extra elements left over.
The second point (not having elements left over) is at least as important as the first point about matching aspect, since this is the part of the process that actually determines whether or not the two sets have a one-to-one correspondence. If nothing is left over, the sets have the same cardinality. If something is left over, they do not. According to the DS theory, this clearly implies that until the matching process has been successfully completed it is not possible to determine whether or not the two sets have a one-to-one correspondence.

Suppose, for example, that we have two barrels of marbles with slots that allow one marble to come out at a time; only bucket [A] has a known quantity, which happens to be [100 marbles]. The quantity of bucket is unknown. Now, we can use the process of one-to-one correspondence to determine if has the [same], [more] or [less] than 100 marbles. If runs out before we’ve taken out [100 marbles] it has [fewer than 100 marbles], and if marbles are still coming out of after [A] has run dry, has [more than 100 marbles].

However, if we stop the one-to-one correspondence process at [50 marbles] it doesn’t actually tell us anything about the [number of marbles in B]. Similarly, suppose we’re counting the contents of two very large grain silos—each of which holds an unknown, but astronomically large number of grains. We can match grains from these two silos for as long as we want, and the process of one-to-one correspondence won’t tell us anything at all about whether or not the silos contain the same number of grains or not—unless we can completely empty at least one of the silos.

Matching Infinite sets
By now it should be obvious why it is absurd to presume that placing the elements of two infinite sets into arbitrary one-to-one correspondence can tell us anything at all about the size of the infinite sets. Quite simply, since [counting an infinite set] is a [never ending process], it is a [process that cannot be completed].

In terms of our analogies, an [infinite set] is functionally equivalent to a [finite set that is changing over time]. In effect, an Evil Trickster is arbitrarily adding elements to the sets—making it impossible to count them using one-to-one correspondence.

Furthermore, there is no reason to believe that matching the elements from two uncountable sets suddenly makes the process of one-to-one correspondence meaningful. Unless we can devise some (as yet unspecified) method of ensuring that we have completely matched one-and-only-one [element from the first set] with one-and-only-one [element from the second set]—and when we’re done we have no elements left over—the process of one-to-one correspondence tells us nothing about the size of an infinite set.

Given this, it seems to me that accepting one-to-one correspondence as the [fundamental principle behind a proof that compares the sizes of two infinite sets] should be viewed with intense skepticism, at best. Especially when the results of such proofs lead directly to some of the most counter intuitive results in all of mathematics.

Cantor used just such a counter-intuitive construction to supposedly prove that an [infinite set that is a subset of another infinite set] could have the same number of elements as the [infinite set that was not a subset]. The logic that Cantor used to justify his strategy was essentially this: Since both sets are infinite (and will thus never run out of numbers), there will always be more numbers to match with the next number from the other set. To Cantor (and the majority of mathematicians since Cantor) this meant that the two sets were in a one-to-one correspondence; and thus that they were the same size.

The problem, however, is that Cantor’s logic is circular reasoning. Assuming that two infinite sets can be placed into one-to-one correspondence in such a way is to assume (a priori) that they are the same size. Thus, Cantor is clearly making an assumption, and then using that assumption to prove that what he has assumed is true.

As the one-to-one-correspondence of two astronomically large sets demonstrates, just because we will never run out of elements does not mean that [what remains uncounted] is the same size.

Counter-proof for Arbitrary One-to-One Correspondence. Traditionally, a variation of Cantor’s diagonalization method has been used to show that the [power set of a set] has a larger cardinal value than the [set] itself. However, if we deem [Cantor’s typical use of arbitrarily matching elements in an infinite set] sufficient to determine that [two sets have the same cardinal value], then we must also allow that [any infinite set that can be well ordered can also be placed into a one-to-one correspondence with any other well ordered infinite set]. Thus, to prove that the [power set of C], [psC] (Where C stands for the counting numbers] can be placed into an arbitrary one-to-one correspondence with [C], all we have to do is create a set of rules that define a precise order for the elements in [psC].

This is not very difficult to do.

Consider the following [rules of formation] for accomplishing the task, the process uses three sets. The first set is [C]; the second set is [psC’], (the well ordered power set of C), and the third set is called a construction set. The [construction set] is a staging area which allows us to work with a changing collection of elements from [C]. We use the elements in the [construction set] to help us determine the next elements to be added to [psC’].

Now the rules of actually creating [psC’].
Step One: From the set of counting numbers, select a new number, (this will the first number, starting with the smallest number possible, which has not been previously selected). Add this number to the current construction set.
Step Two: Turn this new number into an element and add it to the power set.
Step Three: If possible, form pairs with this new number with all possible combinations of the old numbers currently in the construction set, starting with the largest numbers first and working towards smaller numbers. If this is not possible, return to (step one). If it is possible then turn each of these new groupings into elements and add them to the power set; then move on to the next step.
Step Four: If possible, form triplets with this new number with all possible combinations of the old numbers currently in the construction set, starting with the largest numbers first and working towards smaller numbers. If this is not possible, return to (step one). If it is possible then turn each of these new groupings into elements and add them to the power set; then move on to the next step.
.
.
.
Step n: If possible, form n-lets with this new number with all the old numbers currently in the construction set, starting with the largest numbers first and working towards smaller numbers. If this is not possible, return to (step one). If it is possible then turn each of these new groupings into elements and add them to the power set; then move on to the next step.

Once we’ve created [psC’] we can match each element in [C] with an element in [psC’], using Cantor’s own rules of engagement.

Lets work through the [rules of formation] for a few cycles:
Cycle one: The first counting number is [1], so according to (step one) we start by creating the construction set {1}, according to step two, we elementized it as [1], then we place it in [psC’] as the first element—and match it with the first element in [C]. Since there are no other numbers in our construction set, we return to step one.

Cycle two: The next counting number is [2], so according to the first two rules of formation, we add it to the construction set to create the set {1, 2…}; then we elementized [2] and add it to [psC’]. According to (rule three) we pair (2) with any other numbers currently in the construction set, creating the set {1, 2} which we elementize, as [1, 2] and add to [psC’]. We now match these two numbers with the next two elements in [psC]. Since there are no other numbers in our construction set, we return to step one.

By the way, the elements currently in [psC’], in the order added, are {[1], [2], [1, 2]}.

Cycle three: The next counting number is [3], so according to the first two rules of formation, we add it to the construction set so that we’re now working with {1, 2, 3…}; then we elementize [3] and add it to our [psC']. According to (rule three) we pair (3) with any other numbers currently in the construction set, creating sets that we elementize as [3, 2] and [3, 1] and add to [psC’]. Following the next step, we form the triplet [3, 2, 1]. The next step asks us to form quadruplets with (3), but, since this is not yet possible, we can pair the four new numbers in [psC’] with the next four numbers in [C] and return to step one.

Once again, the elements currently in [psC’], in the order added, are:
{[1], [2], [1, 2], [3], [3, 2], [3, 1], [3, 2, 1]}.

We can repeat this process indefinitely creating a precise ordering for the elements of [psC’]. And, given this fact, it is an unavoidable conclusion (using Cantor’s own rules) that the [elements in [psC’] can be placed in a one-to-one correspondence with [C]. And since [psC’] necessarily contains the same elements as [psC]—it is also unavoidable that [psC] can be placed in a one-to-one correspondence with [C].

Thus, (if we accept Cantor’s rules) it would appear that [C] and its powerset must have the same cardinal value.

The problem is that [Cantor’s diagonalization proof], mentioned at the beginning of this proof, reaches a different conclusion than the proof just given. Obviously, any set with more than one element will have a power set with a larger cardinal value—but (in the DS theory) the cardinal value of a set has a very limited range of practical applications. Of much more importance is a set’s cardinal number.
In the DS theory, a sets cardinal number is the union of the [number of elements] and [the unit of measure] so a set that contains 3 apples would have a cardinal number of [3 apples]. A set that contains [2 apples and 1 orange] could be said to have a cardinal value of [2 apples] or [1 orange] or [3 fruit].


Specific One-to-One Correspondence
Whenever we want to use one-to-one correspondence to demonstrate that two enumeration sets have the same cardinal number, we must employ specific one-to-one correspondence, rather than [arbitrary one-to-one correspondence].

Lets consider our astronomically large sets again. Using the following qualifiers to [limit the nature of the sets in question], we can easily ensure that both astronomically large sets will have the same cardinal number. This will be possible even though there is no way to actually pair every element from both sets. All we have to do is make certain that both sets follow a pattern that satisfies the following conditions:
1. Both sets must be well ordered, and absolutely predictable. This prevents arbitrary sets like
{1, 3, 4, 55, 101, 107, 179,…} or {10, 2, 5, 3, 99, -1, 4,…}, but does not prevent sets like
{2, 4, 6, 8, 10,…}, or {2, 4, 8, 16, 32,…}.
The next element in the first two sets cannot reliably be predicted, while the next element in the last two sets can be reliably determined. One increases by [2] for each element. And the last one always doubles the next element.
2. All elements must be given exactly the same weight, or in other words, have proportional spacing. This prevents errors caused by the earlier elements in a set being [closer together] or [further apart] than those later in the same set. This disallows sets like
{2, 4, 8, 16, 32,…} or {2, 4, 9, 16, 25,…}, but does not disallow sets like
{2, 4, 6, 8, 10,…} or {100, 200, 300, 400,…}
3. If an element occurs in both sets, it will have the same relative order in relation to other shared elements. For example, the [counting numbers] and the [even counting numbers] both contain the elements [2], [4] and [6]. The fact that one of these sets also contains [1], [3] and [5] does not change the fact that in both sets [2] will occur before [4] and [4] will come before [6], etc.
4. If one set has a greater number of elements than the other set, the extra elements will be evenly distributed among the shared elements. This prevents comparisons between a set like
{1, 2, 3, 4, 5} and {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, but not between sets like
{2, 4, 6, 8, 10} and {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
because in both cases [2] represents [1/5] of the enumeration of the whole set.

Now, we can compare any two sets, regardless of their size, and if these four conditions are meet, we can be absolutely certain (generally after comparing only a very small sampling) whether or not they have the same cardinal number or not.

For example, (set 1) has 500 elements, and (set 2) has 1,000 elements, although the person counting the sets does not know the number of elements in each set. If both sets simply enumerated the [counting numbers] we would have {1, 2, 3,…500} and {1, 2, 3,…1,000}. But this would violate (rule 4) because, for example, the numbers [1-10] in (set 1) would only represent 1/50 of the whole set whereas in (set 2) [1-10] would only represent 1/100 of the whole set.

It is very simple to make sure we do not violate (rule 4). For example, we can make the elements of (set 1) [even number] only—then when we compare like numbers to like numbers we get:
(A)..........2,....4,....6,....8,....10,… 1,000
(B).......1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …1,000

This construction satisfies all four of our rules, for now, the numbers [1-10] refer to 1/100 of the whole for both sets. However, because (B) has extra numbers left over, we can be certain that it is a larger set without enumerating all the elements in either set.

Notice that it doesn’t matter how many elements are in the set, as long as the last number is the same and the numbers are evenly distributed. It wouldn’t matter if the last number was the [number of atoms in the universe] or [infinity]. As I’ve said, infinity basically means [endless]. So if we change the example to:
..(N)..........2,....4,....6,....8,....10, … ∞
(2N)..........1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … ∞
We know that the two sets will continue on in this pattern forever. And since both sets will have the same largest number, (i.e. none, since they continue forever) there will be one identically matching element in each set--but (2N) will have a great many elements left over.

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