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How are concepts of measurement formed?

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RSalar
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I think I have a basic understanding of how measurement omission generally works in concept formation but there are some concepts, namely concepts of measurement, that measurement omission is impossible. How are concepts of measurement formed?

By concepts of measurement I mean concepts such as: inch, yard, pound, ton, quart, gallon, etc. Take an inch for example. What measurement is omitted in the concept “inch?”

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What measurement is omitted in the concept "inch?"

What is omitted is not necessarily quantitative.

In the case of "inch", you omit: what is being measured; where and when it is being measured; what kind of instrument is measuring it; how many other inches are being measured; etc..

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What is omitted is not necessarily quantitative.

In the case of "inch", you omit: what is being measured; where and when it is being measured; what kind of instrument is measuring it; how many other inches are being measured; etc..

So what you are saying is that the word "measurement" does not always mean measurement? If, to form some concepts, I omit something other than measurement, then I have not omitted a measurement in that specific instance. I think of measurement as the dimension, quantity, or capacity determined by measuring. I do not think of "measurement" as "a type of instrument used to measure" or "an object being measured." As far as where an object is when it is being measured (like the longitute, latitude measurement or country, city, etc.). I agree that location measurement (or variable) is being omitted (everything that exists must exist in some specific place at a specific time) ... but is that what she means by "measurement omission?" It seems like a rather arbitrary measurement to omit ... it's like omitting the fact the moon goes around the earth as part of the process of forming the concept "sand."

I think the measurement that is being omitted must be a "variable attribute" (my words) and the attribute that is variable must be one that other concepts within the same class all share and you are acknowledging the fact that the difference between the various existents within that class is that variable. (Lets call it an essential variable attribute). Like "blue" represents all shades of blue but does represent any of the shades of red. Blue things share the essential variable attribute of blueness. All tables have a flat horizontal surface but the size of the surface varies--the flat horizontal surface is the essential variable attribute--it doesn't matter what country the table is in. The location of the table has been omitted but location is not an essential variable attribute of tables.

The concept "inch" has only one essential attribute, that of length, but the length is not a variable. The length of all inches are exactly the same. Since "inch" is a concept and all concepts are formed through measurement omission there must be some other variable attribute that all inches have ... but I can not think what that attribute is. Can you? Could it be that there is only one inch? And if so how do we form concepts when they refer to a single thing?

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All tables have a flat horizontal surface but the size of the surface varies--the flat horizontal surface is the essential variable attribute--it doesn't matter what country the table is in. The location of the table has been omitted but location is not an essential variable attribute of tables.

The concept "inch" has only one essential attribute, that of length, but the length is not a variable. The length of all inches are exactly the same. Since "inch" is a concept and all concepts are formed through measurement omission there must be some other variable attribute that all inches have ... but I can not think what that attribute is. Can you? Could it be that there is only one inch? And if so how do we form concepts when they refer to a single thing?

The answer is simple: you are omitting all the measurements of the thing that is an inch long, and focusing solely on the attribute length. Just as in your table example, the location of the thing that is an inch long is omitted, as is its color, weight, orientation, material, etc. Consider an interim concept you have to form before forming "inch": that of length. "Length" is a concept formed by focusing solely on one attribute (namely, length--linear extent in space) and omitting the "measurements" of every other attribute of things. Well, inch has a similar omission, although it has the further specification that the length is a fixed measurement.

So there is not "one inch" any more than "one table." That would be Platonism.

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"Length" is a concept formed by focusing solely on one attribute (namely, length--linear extent in space) and omitting the "measurements" of every other attribute of things.

Isn't it true that the concept "lenght" includes all lengths -- we know that length has some quantity but that quantity is not specified (length is the variable -- the specific measurement of length has been omitted). We do not form the concept "length" by omitting measurements of color, weight, temperature and other arbitrary and irrelevant attributes, we omit the essential measurement--that of lenght, i.e., the length is x number of inches, feet, miles, etc. We know that length exists in some quantity but the specific quantity is omitted. You make it sound as though we form concepts by eliminating all non-essential attributes. I think we only omit the specific measurements of the essential attributes.

I would say, "the 'table' must have a horizontal surface (that is one of the essential attributes) but that surface can be of any size, of any shape, and of any color (the essential attribute has a variety of measurements)." You woulod say, "the 'table' is a table because it does not have the attribute of reason, it does not have the attribute of velocity, and it doeas not have the attribute of temperature--these are the measurements that have been omitted." I do not think that we form concepts by omitting all of the things that the concept is not. Instead we elliminate the specific measurements of the attributes that make it what it is.

What is man? He is a rational animal--an animal that has the attribute of reason--reasoning ability in some quantity. but not a specific amount of reasoning ability--the measurement of reasoning ability has been omitted. It's the measurement of the essential attribute that has been elliminated. This method of measurement omission works in all cases that I can think of except when the concept's essential attribute is a specific measuremt--like the concept "inch." The concept inch has only one essential attribute and it is a specific measurement--that essential measurement can not be elliminated. Yes we can omit all kinds of irrelevant measurements, like rotational velocity, molecular weight, electrical current, etc. but they are meaningless omissions.

Is man a man because he is not a cat, not a mouse, not an elephant, not a truck, not a mountain, not a spaceship, and not a brick? Or is he man because he IS an animal with some degree of rationality? I think the later. So it is not about what an inch in not, it is about what an inch IS. An inch is a specific lenght--a length of one inch (1/12 of a foot). I do not think that the concept inch is formed by omitting everything that it is not.

I don't know if it's Platoism or not ... and even if it is that wouldn't bother me, so long as the answer I derive is correct. The more I think about it there is only a single inch and that same inch can be used over and over again anywhere in the universe. It is a man made unit of measure--an idea--a non-physical construct of the mind. How we form the concept "inch" is still not known to me but I do not think it is by omitting measurement.

If someone can prove me wrong I will gladly change my mind, for no man is harmed by the truth, but he is harmed by his own ignorance and deception. So if someone can offer an explaination it will be highly appreciated.

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First it is important to point out that we are discussing inch as a concept and not as a concrete. I think the goal in this discussion is to find the "conceptual common denominator" of "inch." I am taking a short cruise through pertinent parts of The Ayn Rand Lexicon. I will report back after my trip.

Edited by FeatherFall
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I don't know whether to be grateful or annoyed that your post caused me to spend almost two hours reading ITOE and thinking about this problem. :)

We do not form the concept "length" by omitting measurements of color, weight, temperature and other arbitrary and irrelevant attributes, we omit the essential measurement--that of lenght, i.e., the length is x number of inches, feet, miles, etc. We know that length exists in some quantity but the specific quantity is omitted. You make it sound as though we form concepts by eliminating all non-essential attributes. I think we only omit the specific measurements of the essential attributes.

You are right both in that only measurements of the CCD are omitted, and you are right that I made it seem like every irrelevant measure is omitted. (I have been corrected on that before, I am dismayed to admit.) However, after thinking about it for quite a bit, I may have been more right than I realized, with a poor explanation.

To know what measurements are omitted, one has to know what the CCD of inch is (I see FeatherFall has beaten me to this while I've been editing). I am not entirely sure I can answer that question, so let me first try a related concept: entity. For the purposes of this discussion, the CCD of "entity" is the characteristic of existing as an independent physical thing. Therefore, all measurements related to existing (not all measurements whatsoever, which I previously suggested) are omitted. Thus one retains no measurements of the entities that make up the concept besides the fact of their independent existence ("independent" being epistemological here, not metaphysical).

(Of course, I summarized this poorly and all but said that one even omits the measurement of the name of the person who owns the table. Since that is unrelated to the CCD, it is not omitted.)

In the related case of numbers:

Prof. J: What measurements are omitted in forming concepts of particular numbers, for example, the concept of "seven"?

AR: ... What you omit are the measurements of any existents which you count. The concept "number" pertains to a relationship of existents viewed as units. ... Therefore, what is it that you retain? The relationship. What do you omit? All the measurements of whichever units you are denoting or counting by means of the concept of any given number.

That said, I am still not sure exactly what the CCD (or the genus, for that matter) of "inch" is. I will have to think more on this.

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The following is a quote from Introduction To Objectivist Epistemology, Page 53:

"The rules of correct definition are derived from the process of concept-formation. The units of a concept were differentiated -- by means of a distinguishing characteristic(s) -- from other existents possessing a commensurable characteristic, a Conceptual Common Denominator. A definition follows the same principle: it specifies the distinguishing characteristic(s) of the units, and indicates the category of existents from which they were differentiated.

The distinguishing characteristic(s) of the units becomes the differentia of the concept's definition; the existents possessing a Conceptual Common Denominator become the genus."

In my previous post, I made a mistake by not simply asking for the genus and differentia.

The genus of an inch is a line. A line's length is the differentia that forms an inch; specifically, 1/12 a foot. So what has been omitted? Several things.

First and foremost, it's form. The form of an inch can be an arc, a curve, a spiral or straight. Also, orientation has been omitted. In the case of some lines, such as straight lines and arcs the orientation is a specific part of a specific plane.

Edit: Tag, Douglas, you're it!

Edited by FeatherFall
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This all directly relates to the first chapter of ITOE2, Cognition and Measurement, viz:

Measurement is the identification of a relationship—a quantitative relationship established by means of a standard that serves as a unit. Entities (and their actions) are measured by their attributes (length, weight, velocity, etc.) and the standard of measurement is a concretely specified unit representing the appropriate attribute. Thus, one measures length in inches, feet and miles—weight in pounds—velocity by means of a given distance traversed in a given time, etc.

It is important to note that while the choice of a given standard is optional, the mathematical rules of using it are not. It makes no difference whether one measures length in terms of feet or meters; the standard provides only the form of notation, not the substance nor the result of the process of measuring. The facts established by measurement will be the same, regardless of the particular standard used; the standard can neither alter nor affect them. The requirements of a standard of measurement are: that it represent the appropriate attribute, that it be easily perceivable by man and that, once chosen, it remain immutable and absolute whenever used. (Please remember this; we will have reason to recall it.)

So, a particular length serves either as a unit standard for that length, or has a particular fixed linear relationship to the unit standard that one is using. So "inch" is both a subset of the broader concept of length, and a tie between the abstraction of length and actual specific real world instances of length. Without a unit standard you couldn't relate an abstraction back to real world measurements.

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This all directly relates to the first chapter of ITOE2, Cognition and Measurement, viz:

So, a particular length serves either as a unit standard for that length, or has a particular fixed linear relationship to the unit standard that one is using. So "inch" is both a subset of the broader concept of length, and a tie between the abstraction of length and actual specific real world instances of length. Without a unit standard you couldn't relate an abstraction back to real world measurements.

This all makes sense ... and I thank all those who have spent time on this problem ... but (you knew there had to be a but) I am still unsure about which specific attribute of "inch" has had its measurement omitted. Yes "inch" is a unit of meaure and it shares that characteristic with all units in the same catagory--namely those units used to measure length. So we know the CCD is "units used to measure length." What distinguishes an inch from a foot? I think the differentia is that an "inch" is a specific length that is different than a "foot," or any other unique unit used to measure length. I suppose you could say we are omitting all units of length that are not exactly an "inch" in length ... but isn't that the same as only including units that measure exactly an "inch?"

All units that measure an "inch" in length are exactly the same -- they are all mental constructs (there is no concrete "inch"--there are concretes that are an inch long but no "inch" exists in reality) that, because they are exactly an inch are one and the same. That's why I think that the concept "inch" is a singular thing. There is only one of them, duplicated an infinate number of times in the minds of people.

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This all makes sense ... and I thank all those who have spent time on this problem ... but (you knew there had to be a but) I am still unsure about which specific attribute of "inch" has had its measurement omitted.

Try looking at it this way: an inch is a specific distance between two points. The exact location of these two points is omitted.

All units that measure an "inch" in length are exactly the same -- they are all mental constructs (there is no concrete "inch"--there are concretes that are an inch long but no "inch" exists in reality) that, because they are exactly an inch are one and the same. That's why I think that the concept "inch" is a singular thing. There is only one of them, duplicated an infinate number of times in the minds of people.

You can apply that manner of thinking to all abstract concepts, though I don't see how it's particularly useful.

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All units that measure an "inch" in length are exactly the same -- they are all mental constructs (there is no concrete "inch"--there are concretes that are an inch long but no "inch" exists in reality) that, because they are exactly an inch are one and the same. That's why I think that the concept "inch" is a singular thing. There is only one of them, duplicated an infinate number of times in the minds of people.

I'm not really sure what you're trying to say here, but it might help to keep in mind that units of length all have a linear relationship to each other. i.e. 12 inches = 1 foot, 5280 feet = mile, etc. etc. Length is an abstraction but it is not "made up" by the human mind, it measures something very real in the external world.

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Length is an abstraction but it is not "made up" by the human mind, it measures something very real in the external world.

I think the concept inch exists only in the mind because if there were real metaphysical inches out there we would be able to see, feel, and/or taste them. Yes there are real things that have length--this we all know--but this thing we call "inch" can not be found in the real world.

You can apply that manner of thinking to all abstract concepts, though I don't see how it's particularly useful.

What is the definition of an abstract concept? Is there such a thing as a non-abstract concept? What I was trying to say is that some concepts refer to things that exist -- ex.: The concept "apple" refers to the real fruit that gows on real trees and these real apples can be seen, touched, and tasted. Have you ever seen, touched or tasted an "inch?" Obviously there is no such thing as an inch in reality. It refers to a measurement of length but it's only purpose is to measure actual things -- unless there is something to measure (something that has length as one of its attributes) the concept "inch" is absurd.

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The genus of an inch is a line. A line's length is the differentia that forms an inch; specifically, 1/12 a foot. So what has been omitted? Several things.

First and foremost, it's form. The form of an inch can be an arc, a curve, a spiral or straight. Also, orientation has been omitted. In the case of some lines, such as straight lines and arcs the orientation is a specific part of a specific plane.

I've never seen, nor can I imagine, a curved inch. I can picture a curved line, but an inch, because it is (exclusively) a specific distance between two points, has no form. In order to measure the distance between two points we have to measure the imaginary straight line that extends between the two points. To meausre the distance between two points in an arc would give us an erroneous answer.

Since an inch has no form, and never had a form, how can we omit it? Wouldn't that be like omitting the wave length from the concept table? Since tables do not have wave lengths there is no need to omit that measurement. I really think we are stretching this measurement omission thing when we simple say we are omitting all the measurements of attibutes that are nonexistent. We should be focusing on an attributes that do exist but that have no specific measurement. A table is a table no matter what size or color is. All tables must have some size and some color but the specific size and color does not change the fact that it is a table. (The attribute exists but its measurement has been omitted.)

If "form" was the attribute that we are not measuring, there would have to be multiple and various forms that an inch could be, and no matter which form it took it wouldn't change the fact that it still is what it is--in this case an "inch." You are right, an inch has no form, because inches do not exist--if they did they would be lines.

We know the concept "inch" exists but no actual hard object "inch" exists in reality. That's ok, that doesn't change the fact that we had to form that concept in our mind. When we form the concept, "inch" we retain the only attribute that it has--the fact that an inch is specific distance between to points--that distance being an inch. We can not omit the measurements of nonexistant attributes (like omitting the wave length of tables), because before you can omit something, that something has to exist!

Since the concept "inch" has only one attribute and that attribute has a very specific measurement I find it impossible to omit it. When I omit the measurement from "inch" I am left with nothing. So rather than helping me form the concept, measurement omission destroys the very concept I am try to form.

Edited by RSalar
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We know the concept "inch" exists but no actual hard object "inch" exists in reality.

That's true of all attributes. "Hardness" doesn't exist either (by itself). Only objects which are hard exist.

As for what is omitted when we form the concept "inch", you were very close to answering the question yourself when you said, "When we form the concept, "inch" we retain the only attribute that it has--the fact that an inch is specific distance between to points--that distance being an inch."

Now just add the following "It can be ANY two points" and you have your answer. You may want to add as you noted that it has to be a straight line.

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I've never seen, nor can I imagine, a curved inch. I can picture a curved line, but an inch, because it is (exclusively) a specific distance between two points, has no form. In order to measure the distance between two points we have to measure the imaginary straight line that extends between the two points. To meausre the distance between two points in an arc would give us an erroneous answer.

....

There are curved inches, in that the length of a curved line can be measured in inches.

For example, one speaks of the the length of the circumference of a circle as being a certain number of inches. If one were to mark off those inches on the circle, they would be curved.

Another example of a curved unit of length measurement: when I drive my car a certain distance which registers a total of 1 mile on my odometer, it's likely that the mile I just drove was not straight.

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While "inches" do exist in reality, they are not entities. Other things also exist in reality as non-entities, like emotions. They are aspects of things, but we omit the specific things of which they are aspects.

Inch only applies to things that have length. An inch is a distance (1/12 foot) between two unspecified points, along an unspecified path. This means that you omit the two points being measured, and the path.

The example where you omit "wavelength" from "table" is inappropriate. It would be a better comparison if we were discussing the omission of emotions from inches. But we don't form either concept based on the absence of wavelength or the emotional state of the concept.

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There are curved inches, in that the length of a curved line can be measured in inches.

Maybe the "line" is curved but I don't think the "inch" is curved. A line, if drawn on paper, exists in reality -- it has form, but "inch" is nothing but a conceptual construct refering to a specific length. "Inches" exist only in the mind. My question is: What measurement is omitted when I form the concept "inch." Others have said that I omit all kinds of things from where the inch is, to all the objects that could be measured. I don't think "measurement omission" means omitting "attributes" that are not possessed by that which we are referring. These attributes are all the attributes that "inch" does not posses. I don't think that's quite it. I believe that "measurement omission" refers to omitting the "measurement" of an actual possessed attribute. But I have been wrong before!

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That's true of all attributes. "Hardness" doesn't exist either (by itself). Only objects which are hard exist.

As for what is omitted when we form the concept "inch", you were very close to answering the question yourself when you said, "When we form the concept, "inch" we retain the only attribute that it has--the fact that an inch is specific distance between to points--that distance being an inch."

Now just add the following "It can be ANY two points" and you have your answer. You may want to add as you noted that it has to be a straight line.

Are you saying that the measurement that is bing omitted is the actual two points? Besides the fact that "points" do not actual exist any more than "inches" exist, these two points must be exactly an inch apart but they can be anywhere, so you are saying that the location of these two points is the measurement that is being omitted? Couldn't you likewise and by the same logic say that we omit the location measurement of "table" when we form that concept? Why is location a distinguishing attribute in this type of concept? I think "location measurement" is omitted from the concepts like "inside, outside, beside, beneath," etc., because location is an intigral part of the concept but the specific measurement has been omitted. When something is "outside" we know that is is somehere out of doors but we do not know exactly where it is outdoors--that measurement has been omitted. Maybe I am way mixed up here--but I think we must only omit measurements that are existing variables of the concept (in other words the attribute is required to form the concept but it can exist in any quantity). We don't just omit random attributes that have no bearing on the concpet being formed.

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While "inches" do exist in reality, they are not entities. Other things also exist in reality as non-entities, like emotions. They are aspects of things, but we omit the specific things of which they are aspects.

Inch only applies to things that have length. An inch is a distance (1/12 foot) between two unspecified points, along an unspecified path. This means that you omit the two points being measured, and the path.

The example where you omit "wavelength" from "table" is inappropriate. It would be a better comparison if we were discussing the omission of emotions from inches. But we don't form either concept based on the absence of wavelength or the emotional state of the concept.

I was using the example of omitting a table's wavelength as an illistration of why omitting an irrelevant attribute is not what Ayn Rand meant by measurement omission in concept formation. I believe (and I am fully aware of how wrong I may be) that she meant that we omit the "specific" measurement of an attribute possessed by the existent. The attribute MUST be one that is required in order to form the concept. Wavelength in NOT an attribute of table so omitting it's measurement is meaningless.

The location of the two points that define the limits of "inch" is NOT an attribute of the existent. These points, like the concept "inch," exist only in the mind. "inch" IS a measurement, and it is the only attribute possesed by the concept, so it can not be omitted. There are no other attributes whose measurement can be omitted. The location of this "inch" is not an attribute of inch so omitting its measurement is as useless as omitting the wavelenght in forming the concept table. On the other hand, we know that all tables have tops and all tops have measurements but we omit the measurement of the tops when we form the concept "table." The location of the table is omitted but the fact that the table has a location is not an attribute whose measurement is being omitted.

Location is no more a relevant attribute of "inch" than it is of "table" so omitting its measurement is equally useless. Let's remember that we are not omitting attributes, we are omitting the specific measurement of an attribute. The attribute must exist in some quantity before we can omit its measurement.

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Because color is an attribute of tables, and because color is an aspect of light which conforms in many ways to wave motion, tables could be considered to have wavelengths. But I understand your point and don't think that this is material to the discussion. I just had to get that off my chest.

I agree that non-applicable attributes are not involved in concept formation, even during measurement omission. What I disagree with is the assertion that a lack of substance means that something isn't real.

Distances are real; they do not only exist in your mind. To declare otherwise is an error of logic, and it may be due to the "primacy of consciousness" position. Location and path are necessary attributes for specific inches.

Points exist. The distance one traces between them is real if the points are real. This distance can be infinite, depending on the path that is traced. A straight line is only the shortest path between two points.

When we say that two points are one foot apart, it is typically assumed that the distance is being measured in a straight line.

However, man is not always capable of traveling the space between two points in a straight line, so distance is often measured by circular paths, parabolic lines, or even routes that are so erratic that they don't have commonly used definitions (like if I were to give you directions to the library). Distance applies to all these different forms of lines.

When we discuss two points that are 60,000 miles apart, we may even think of them as being in a straight line. But if these are two points on earth, and there is a "straight" road connecting the two points, these points are actually a little closer than 60,000 miles. However, because we are slaves to physical constraints and must travel along the road by avoiding the intervening mass of solid earth, we do not trace a straight line when we follow that road.

Point one: The North Pole. Point two: The South Pole. The points exist, the distance exists, and it is important how you draw your travel path. Will you fly? Will you drive and swim? Will you bore a hole through the earth?

RSalar, I maintain that it is the specific and concrete points and paths that are omitted when one forms the concept "inch." To continue this discussion, you will have to offer a coherent arguement that points and distances do not exist in reality. Please do not simply state that they are not entities, as I have already accepted this.

The claim that they don't exist is implicit in the posts where you assert that points do not exist outside of theoretical exercise, or "in your own mind". Offer an argument to back up your assertion, please.

-edited for clarity

Edited by FeatherFall
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I don't think "measurement omission" means omitting "attributes" that are not possessed by that which we are referring. These attributes are all the attributes that "inch" does not possess. I don't think that's quite it. I believe that "measurement omission" refers to omitting the "measurement" of an actual possessed attribute.

A concept is a single abstraction which represents and unites many concretes. When we form the concept we must mentally discard (or "omit") all the variation between those concretes. So the things which are omitted are attributes of the concretes, NOT attributes of the concept. What is retained are those attributes which are shared by all the concretes belonging to the concept. This includes those attributes which are necessary to the purpose for which the concept was formed.

... we must only omit measurements that are existing variables of the concept (in other words the attribute is required to form the concept but it can exist in any quantity). We don't just omit random attributes that have no bearing on the concept being formed.

On the contrary, we especially want to omit any attribute which has no bearing on the concept.

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

What is an inch? I have a three-foot ruler before me as I write this. Now I go and get a book. I lay the ruler on top of the book to measure its length. I get 9.5 inches, by placing the beginning of the ruler at one end of the book and observing that the other end of the book coincides with the 9.5 mark on the ruler.

I could have measured the book with a compass whose opening was set to one inch and walking it over the book. This would show that its length is between 9 and 10 inches.

Generalizing, an inch is a rigid object with two marks on it separated by a distance of one inch. This object is used to measure the length of other objects by counting how many times its length (between the marks) goes into the length of the other object. How such an object is verified to be an inch is beyond the scope of this thread -- see:

http://nist.gov

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Generalizing, an inch is a rigid object with two marks on it separated by a distance of one inch. This object is used to measure the length of other objects by counting how many times its length (between the marks) goes into the length of the other object. How such an object is verified to be an inch is beyond the scope of this thread -- see:

http://nist.gov

An inch isn't a rigid object, neither is a foot or a yard. A ruler is a rigid object that is also a foot in length. A yardstick is rigid and is three feet long.

Tailors use measuring tools that are not rigid, to measure things like waistlines - in inches. An inch has no mass, and the tool one uses to reference the length of an inch is meaningless. There are concepts for things that don't exist as entities. In this thread, the entities that we use to reference length are being confused for the actual lengths.

By the way, I apologize for the hasty example in my previous post. If you traveled 60,000 miles to get to another point on Earth, then you definitely took the scenic route. If I had cut the distance down to 6,000 miles, it would have made more sense.

Edited by FeatherFall
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I thought it might help clarify certain aspects of this discussion if I quoted AR on the subject. This is from the Appendix of ITOE. The example given here is the concept "seven" but I think the principle would be the same for "inch":

"Prof. J: What measurements are omitted in forming concepts of particular numbers, for example, the concept of "seven"?

AR: In a certain sense the measurements omitted from the concept of numbers are the easiest to perceive. What you omit are the measurements of any existents which you count. The concept "number" pertains to a relationship of existents viewed as units—that is, existents which have certain similarities and which you classify as members of one group. So when you form the concept of a number, you form an abstraction which you implicitly declare to be applicable to any existents which you care to consider as units. It can be actual existents, or it can be parts of an existent, as an inch is a part of a certain length. You can measure things by regarding certain attributes as broken up into units—of length, for instance, or of weight. Or you can count entities. You can count ten oranges, ten bananas, ten automobiles, or ten men; the abstraction "ten" remains the same, denoting a certain number of entities viewed as members of a certain group according to certain similarities.

Therefore, what is it that you retain? The relationship. What do you omit? All the measurements of whichever units you are denoting or counting by means of the concept of any given number.

Here the omission of measurements is perceived almost at its clearest. And I even give the example in the book—it's an expression I have heard, I did not originate it—that an animal can perceive two oranges and two potatoes but cannot conceive of the concept "two." And right there you can see what the mechanics are: the abstraction retains the numerical relationship, but omits the measurements of the particulars, of the kind of entities which you are counting."

The example I thought of was viewing 10 very diverse objects lying on a table, e.g. a piece of candy, a paper clip, etc., and all having only one thing in common: that they are an inch long, which btw would be very noticeable, especially if they were lined up next to each other. That would be similar to AR's example of 10 oranges, 10 men, etc. where the abstraction is "ten".

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