Consider a square. Look at a side. Now look at the diagonal. The diagonal is clearly longer than the side.

What common unit is omitted in comparing the diagonal of a square to its side?

Answer: None. The diagonal and the side have no common unit of measure.

Ba'al Chatzaf

Bob,

Then how do you know one is longer?

(I don't even know why I bother. I'm done... )

Michael

From Euclid's axioms. One can prove that the longest side of a triangle lies opposite the largest angle of the triangle. In the case of the square you have a 45-45-90 degress right triangle and the diagonal is opposite the 90 degree angle (the corner angle). In general the way you prove line segment A is shorter than line segment B (whether or not they are co-measurable) is to construct a line segment A' lying entirely B where A' is derived from A by constructing equilateral triangles. I think that is Prop 3 in Book I. See Heath's translation of Euclid's Elements.

The existence of a common unit is not necessary (and in general, not possible) to compare lengths of line segments within Euclid's geometry.

If you are upset with the non co-measurement of the diagonal and the side of a square, think of how Pythagoras must have felt, when one of his students discovered this disconcerting fact. The story goes (I think it is just a story) that this student was murdered to keep news of his finding from being spread far and wide. The bottom line (sic!) is that ratios of integers (rational numbers) are not sufficient to deal with measurement. Which is why we have real numbers. And so it goes.

One line is bigger than the other. There are exactly two common units of measurement right there. Actually there are three if you want to use the stub left over.

How you quantify all this is another matter, but you can certainly measure both according to the same standard. If you can use it on both, it is common to both.

You can't say, there is no common unit of measurement because the comparison doesn't add up.

How I would prove one line is longer than another and in what quantity is take a string... then... er... "measure" one of the lines by making it the exact length (1 = 1) and then line that up with the other line. It will be less than that one or more. You have a fraction right there. This is really basic. Why try to talk your way around a basic with sophistication?

Are you bothered because the math of one standard does not work exactly the same as for the other and there is a piece left over?

(Why am I doing this? Apparently people are getting murdered over it... )

u is a unit of measure for a line segment x if and only if the length u divides the length x an integral number of times.

Take out a common ruler. Do you see all those intervals a 1/16 or 1/32 of an inch long? Sure you do. 1/16 of an inch is the unit of measure ment for that rules.

Now consider the diagonal of a square d and its side s. If there were a common unit u then u would go into d an integral number of times, say m. u would go into s an integral number of times, say n.

The ratio of the length of d to the length of s is m:n. But also notice d is the diagonal of the square (by hypothesis) so d^2 = s^2 + s^2 (Pythagoras theorem). So that d^2/s^2 = (m/n)^2, But d^2/s^2 = 2, hence (m/n)^2 = 2. We may assume without loss of generality that m and n have only 1 as a common divisor. Do a little algebra and get m^2 = 2*n^2. Since m^2 is even (a multple of 2) it follows that m is also even, hence m = 2^k for some integer k. Going back to the equation we get 4*k^2 = 2*n^2. Divide both sides by 2 and get 2^k^2 = n^2. But this implies that n is also even hence 2 divides both m and n. But we assumed that m and n have no other common divisor than 1. Contradiction. Therefore there exists no such integers m, n for which the presumed unit u divides d m times and s n times. Hence no common unit.

Q.E.D.

I can tell that you are upset. Think of how Pythagoras must have felt when he got this bad news. He believed all linear lengths were co-measurable. But he was wrong. Pythagoras believed in unit omission, but he was wrong.

u is a unit of measure for a line segment x if and only if the length u divides the length x an integral number of times.

Bob,

Really? I was right. That's a weird definition of unit of measurement.

The length used as a standard of measurement, whether the small one or large one, can and must be set by humans, not by an accident. You are leaving out the really fundamental part. "u" is a measurement of space in general, not just the line. Space is much bigger than your line and contains your line within it. There is no "u" measurement of a line that does not measure the rest of space.

But even on a metaphysical level, a violin string, for instance, will issue overtones according to nodes starting at exactly one half the length of the string (defined by the pressure points at each end).

So even if metaphysics boiled down to a line with a beginning, middle and end, I still don't see how I could agree with you. But space is not a line. It is infinite and eternal, at least from our human perspective.

That is, unless you do some tricks with math. But those who claim things like the beginning of the universe based on math projections are also quick to claim that math is not based on reality. I find that an odd form of reasoning.

No, I'm not upset, at least not with you. I'm exasperated at me for letting myself get into a discussion that promises to go nowhere. I am behind in my work.

I'm with you, Ba'al, not even all lengths are commensurable, as every high-school student knows (or should know.)

Universal commensurability is not possible. (That doesn't create a problem for Objectivist epistemology, though.) I'd go on, but this is science, etc., and going on is epistemology...

I'm with you, Ba'al, not even all lengths are commensurable, as every high-school student knows (or should know.)

Universal commensurability is not possible. (That doesn't create a problem for Objectivist epistemology, though.) I'd go on, but this is science, etc., and going on is epistemology...

--Mindy

I disagree. The -fact- of non comeasurability of length blows the notion of measurement omission to smithereens. One will have to look elsewhere to find out how we conceptualize.

Just a question from a person who left high-school long ago.

Other than infinity, which lengths cannot be measured? I have always thought length was a standard of measurement. That's what any online dictionary will tell you.

To me, saying length cannot be measured is a stolen concept, since length is a form of measurement.

That fact kinda blows some affirmations to smithereens.

Just a question from a person who left high-school long ago.

Other than infinity, which lengths cannot be measured? I have always thought length was a standard of measurement. That's what any online dictionary will tell you.

To me, saying length cannot be measured is a stolen concept, since length is a form of measurement.

That fact kinda blows some affirmations to smithereens.

Michael

The normal measures; length, area, volume, hypervolume are measured by real numbers. However real numbers require a limiting process to define.

Start with the integers. Now consider ratios of integers, the so-called rational numbers. It turns out that lengths of Euclidean lines cannot be measured (in general) with rational numbers. So what is required? Infinite sequences of rational numbers which converge to a real (but non-rational) limit.

Length are are rational numbers or non-rational reals.

Measurement means (literally) to apply a unit length an integral and finite number of times to a given length. It has been known for over 2000 years that the there is no such unit length that will divide the side of a square and its diagonal an integeral number of times. The square root of two was the first number to be proved non-rational.

Measurable lengths are a proper subset of general lengths (those given by real numbers).

So I give you the side of a square and its diagonal and I ask you what is an omitted measurement applicable to both? Do you have an answer?

You just made a salad of concepts. Do you understand what measurement omission means in concept formation? Your comments show clearly that you are talking about something else.

You certainly did not answer my doubt, which is how a standard of measure cannot be measured. You even called this a fact: "The -fact- of non comeasurability of length...".

How can red not be a color? How can a goose not be a bird?

How can length not be a measurement when it is defined as a type of measurement? You even stated: "The normal measures; length..."

I don't understand how A can be A and not A. I use a different standard of logic.

How can length not be a measurement when it is defined as a type of measurement? You even stated: "The normal measures; length..."

In general length requires an infinite process to define, to with the convergence of a sequence of rational numbers to an irrational and real limit. That is not what Rand hand in mind.

By the way, what you call word salad is mathematical precision. Apparently to you are complete stranger to mathematical precision otherwise you would not have confused it with a jumble of words. There is a realm of knowledge to which you are a stranger. Either ignore it or enter it with the work necessary to do so, but do not denigrate it.

I am tone deaf and musically illiterate and I cannot carry a tune in a bushel basket, but I do not mock or denigrate music.

You simply talked about one thing and said it disproved something else entirely. Blew it to smithereens you said. Heh. That's quite a lot of bluster, so a minimum requirement to take that seriously is for you to be talking about what you are disproving. But you were not.

All this leads me to believe you do not understand what measurement omission means in Objectivist concept formation and are going on the term as you understand it outside of context.

All measurement omission means in Objectivist concept formation is that a measurement can exist in any quantity in the concept, but no quantity is specified. In other words, to establish a concept, a measurement standard is stipulated but not used. Stipulating the standard helps define the concept, not any specific use of the standard.

After the standard is stipulated, math can be applied but is not at that moment.

Many things are measured in a concept, including importance of a characteristic, and many different standards chosen. Measurements can be also be cardinal or ordinal. In this system, even things like love can be measured, as Rand stated explicitly in ITOE.

None of that is what you have been talking about or blowing to smithereens.

And you still not have explained how a standard of measurement cannot be measured, if you really believe that like you said.

Of course I agree that a word can have more than one meaning, but they were talking specifically about measurement.

Yes, I know your dispute is with Bob. I was again pointing out how Rand confounded the two meanings in ITOE and assumed knowing the first meaning implies knowing the second meaning.

And you still not have explained how a standard of measurement cannot be measured, if you really believe that like you said.

I hold that consistency is important.

Michael

Quite so. Ok, consider the straight line connecting the point (x0, y0) to (x1,y1). The length of that line is sqrt ((x1 -x0)^2 + (y1 -y0)^2), a number that is in general not rational and cannot be specified (in general) by a finite decimal representation of a number. Now I ask you again what measurement is omitted from the lengths (x1 - x0) and (y1 - y0). I keep asking, but you keep on not answering. If it turns out that (y1-y0)/(x1-x0) is the ratio of two integers m and n, the there exists a unit length that divides (y1-y0) m times and (x1-x0) n times. but if the quotient (y1-y0)/(x1-x0) is not such a ratio then what can we say?

It is unfortunate that Rand did not no much in the way of mathematics. If she had, she would not have come up with measurement omission.

The mathematical theory of numbers is very different from everyday intuitive understanding. The foundation of the real numbers is necessary for building integral and differential calculus and has very little to do with measuring the length of something. If you have a right triangle with sides=1 and you measure the diagonal with your ruler you will actually be estimating dimensions and you might say the diagonal is 1.414 in length, depending on what you read on your measuring device. Measuring physical objects is not the same as calculating theoretical dimensions in pure mathematics. Of course we know the real (mathematical) dimensions are 1,1, and sqrt(2) but the sqrt(2) actually represents an infinite process that we are free to terminate whenever we want to.

The mathematical theory of numbers is very different from everyday intuitive understanding. The foundation of the real numbers is necessary for building integral and differential calculus and has very little to do with measuring the length of something. If you have a right triangle with sides=1 and you measure the diagonal with your ruler you will actually be estimating dimensions and you might say the diagonal is 1.414 in length, depending on what you read on your measuring device. Measuring physical objects is not the same as calculating theoretical dimensions in pure mathematics. Of course we know the real (mathematical) dimensions are 1,1, and sqrt(2) but the sqrt(2) actually represents an infinite process that we are free to terminate whenever we want to.

Aux Contraire. To define length precisely one needs integral calculus.

The length of that line is sqrt ((x1 -x0)^2 + (y1 -y0)^2), a number that is in general not rational and cannot be specified (in general) by a finite decimal representation of a number.

Many rational numbers cannot be specified by a finite decimal representation either. For example, 1/3 is 0.333333333....

It is unfortunate that Rand did not no much in the way of mathematics. If she had, she would not have come up with measurement omission.

Aux Contraire. To define length precisely one needs integral calculus.

I don't follow, could you be more specific? Seems to me you only need a metric space to define length.

That works for straight line segments. To do the same for curves you need an integral. Pythagoras' theorem as at the base for both the length of straight line segment and length of curves.

I'm with you, Ba'al, not even all lengths are commensurable, as every high-school student knows (or should know.)

Universal commensurability is not possible. (That doesn't create a problem for Objectivist epistemology, though.) I'd go on, but this is science, etc., and going on is epistemology...

--Mindy

I disagree. The -fact- of non comeasurability of length blows the notion of measurement omission to smithereens. One will have to look elsewhere to find out how we conceptualize.

Ba'al Chatzaf

I agree that it "blows" measurement omission! I'm just saying we don't need "measurement omission" to "validate" concept-formation.

You keep on asking and challenging about what type of measurement, not that measurement does not exist. The type of measurement is not the issue if you want to blow Rand's process of measurement omission to smithereens. And so far you haven't come closer than an unfounded opinion.

Look at it this way (to oversimplify). The existence of measurement is a metaphysical issue. Using the different types of measurement is a scientific one. The standards of measurement (length, intensity, importance, etc.) pertain to both.

Just because one kind of measurement does not work, that does not mean another kind of measurement cannot work, i.e., that measurement per se does not exist. You can't measure a curve by a straight line? OK. Then measure it with curve units.

Note that for Rand to omit measurement, it had to exist in the first place.

What kind of measurements are omitted when we form conjunctions, or form concepts such as "justice"? What concepts are omitted when we learn how to ride a bike?

I'm not surprised that Objectivist discussions of this are generally limited to chairs and tables.

[EDIT: I haven't been following this thread, so I hope my post isn't off-topic.]

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## Michael Stuart Kelly

Bob,

Then how do you know one is longer?

(I don't even know why I bother. I'm done... )

Michael

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## BaalChatzaf

AuthorFrom Euclid's axioms. One can prove that the longest side of a triangle lies opposite the largest angle of the triangle. In the case of the square you have a 45-45-90 degress right triangle and the diagonal is opposite the 90 degree angle (the corner angle). In general the way you prove line segment A is shorter than line segment B (whether or not they are co-measurable) is to construct a line segment A' lying entirely B where A' is derived from A by constructing equilateral triangles. I think that is Prop 3 in Book I. See Heath's translation of Euclid's Elements.

The existence of a common unit is not necessary (and in general, not possible) to compare lengths of line segments within Euclid's geometry.

If you are upset with the non co-measurement of the diagonal and the side of a square, think of how Pythagoras must have felt, when one of his students discovered this disconcerting fact. The story goes (I think it is just a story) that this student was murdered to keep news of his finding from being spread far and wide. The bottom line (sic!) is that ratios of integers (rational numbers) are not sufficient to deal with measurement. Which is why we have real numbers. And so it goes.

Ba'al Chatzaf

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## Michael Stuart Kelly

Bob,

You have a weird definition of unit of measure.

One line is bigger than the other. There are exactly two common units of measurement right there. Actually there are three if you want to use the stub left over.

How you quantify all this is another matter, but you can certainly measure both according to the same standard. If you can use it on both, it is common to both.

You can't say, there is no common unit of measurement because the comparison doesn't add up.

How I would prove one line is longer than another and in what quantity is take a string... then... er... "measure" one of the lines by making it the exact length (1 = 1) and then line that up with the other line. It will be less than that one or more. You have a fraction right there. This is really basic. Why try to talk your way around a basic with sophistication?

Are you bothered because the math of one standard does not work exactly the same as for the other and there is a piece left over?

(Why am I doing this? Apparently people are getting murdered over it... )

Michael

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## BaalChatzaf

Authoru is a unit of measure for a line segment x if and only if the length u divides the length x an integral number of times.

Take out a common ruler. Do you see all those intervals a 1/16 or 1/32 of an inch long? Sure you do. 1/16 of an inch is the unit of measure ment for that rules.

Now consider the diagonal of a square d and its side s. If there were a common unit u then u would go into d an integral number of times, say m. u would go into s an integral number of times, say n.

The ratio of the length of d to the length of s is m:n. But also notice d is the diagonal of the square (by hypothesis) so d^2 = s^2 + s^2 (Pythagoras theorem). So that d^2/s^2 = (m/n)^2, But d^2/s^2 = 2, hence (m/n)^2 = 2. We may assume without loss of generality that m and n have only 1 as a common divisor. Do a little algebra and get m^2 = 2*n^2. Since m^2 is even (a multple of 2) it follows that m is also even, hence m = 2^k for some integer k. Going back to the equation we get 4*k^2 = 2*n^2. Divide both sides by 2 and get 2^k^2 = n^2. But this implies that n is also even hence 2 divides both m and n. But we assumed that m and n have no other common divisor than 1. Contradiction. Therefore there exists no such integers m, n for which the presumed unit u divides d m times and s n times. Hence no common unit.

Q.E.D.

I can tell that you are upset. Think of how Pythagoras must have felt when he got this bad news. He believed all linear lengths were co-measurable. But he was wrong. Pythagoras believed in unit omission, but he was wrong.

Ba'al Chatzaf

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## Michael Stuart Kelly

Bob,

Really? I was right. That's a weird definition of unit of measurement.

The length used as a standard of measurement, whether the small one or large one, can and must be set by humans, not by an accident. You are leaving out the really fundamental part. "u" is a measurement of space in general, not just the line. Space is much bigger than your line and contains your line within it. There is no "u" measurement of a line that does not measure the rest of space.

But even on a metaphysical level, a violin string, for instance, will issue overtones according to nodes starting at exactly one half the length of the string (defined by the pressure points at each end).

So even if metaphysics boiled down to a line with a beginning, middle and end, I still don't see how I could agree with you. But space is not a line. It is infinite and eternal, at least from our human perspective.

That is, unless you do some tricks with math. But those who claim things like the beginning of the universe based on math projections are also quick to claim that math is not based on reality. I find that an odd form of reasoning.

No, I'm not upset, at least not with you. I'm exasperated at me for letting myself get into a discussion that promises to go nowhere. I am behind in my work.

Michael

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## Mindy Newton

I'm with you, Ba'al, not even all lengths are commensurable, as every high-school student knows (or should know.)

Universal commensurability is not possible. (That doesn't create a problem for Objectivist epistemology, though.) I'd go on, but this is science, etc., and going on is epistemology...

--Mindy

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## BaalChatzaf

AuthorI disagree. The -fact- of non comeasurability of length blows the notion of measurement omission to smithereens. One will have to look elsewhere to find out how we conceptualize.

Ba'al Chatzaf

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## Michael Stuart Kelly

Just a question from a person who left high-school long ago.

Other than infinity, which lengths cannot be measured? I have always thought length was a standard of measurement. That's what any online dictionary will tell you.

To me, saying length cannot be measured is a stolen concept, since length is a form of measurement.

That fact kinda blows some affirmations to smithereens.

Michael

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## BaalChatzaf

AuthorThe normal measures; length, area, volume, hypervolume are measured by real numbers. However real numbers require a limiting process to define.

Start with the integers. Now consider ratios of integers, the so-called rational numbers. It turns out that lengths of Euclidean lines cannot be measured (in general) with rational numbers. So what is required? Infinite sequences of rational numbers which converge to a real (but non-rational) limit.

Length are are rational numbers or non-rational reals.

Measurement means (literally) to apply a unit length an integral and finite number of times to a given length. It has been known for over 2000 years that the there is no such unit length that will divide the side of a square and its diagonal an integeral number of times. The square root of two was the first number to be proved non-rational.

Measurable lengths are a proper subset of general lengths (those given by real numbers).

So I give you the side of a square and its diagonal and I ask you what is an omitted measurement applicable to both? Do you have an answer?

Ba'al Chatzaf

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## Michael Stuart Kelly

Bob,

You just made a salad of concepts. Do you understand what measurement omission means in concept formation? Your comments show clearly that you are talking about something else.

You certainly did not answer my doubt, which is how a standard of measure cannot be measured. You even called this a fact: "The -fact- of non comeasurability of length...".

How can red not be a color? How can a goose not be a bird?

How can length not be a measurement

when it is defined as a type of measurement? You even stated: "The normal measures; length..."I don't understand how A can be A and not A. I use a different standard of logic.

Michael

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## BaalChatzaf

AuthorIn general length requires an infinite process to define, to with the convergence of a sequence of rational numbers to an irrational and real limit. That is not what Rand hand in mind.

By the way, what you call word salad is mathematical precision. Apparently to you are complete stranger to mathematical precision otherwise you would not have confused it with a jumble of words. There is a realm of knowledge to which you are a stranger. Either ignore it or enter it with the work necessary to do so, but do not denigrate it.

I am tone deaf and musically illiterate and I cannot carry a tune in a bushel basket, but I do not mock or denigrate music.

Ba'al Chatzaf

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## merjet

It's a matter of "A and A'." From the

Merriam-Webster Online Dictionary:length - 1 a: the longer or longest dimension of an object, b: a measured distance or dimension

The first refers to an attribute of an object per se. The second refers to that attribute in relation to a measuring tool such as a ruler.

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## Michael Stuart Kelly

Bob,

I wasn't mocking or denigrating.

You simply talked about one thing and said it disproved something else entirely. Blew it to smithereens you said. Heh. That's quite a lot of bluster, so a minimum requirement to take that seriously is for you to be talking about what you are disproving. But you were not.

All this leads me to believe you do not understand what measurement omission means in Objectivist concept formation and are going on the term as you understand it outside of context.

All measurement omission means in Objectivist concept formation is that a measurement can exist in any quantity in the concept, but no quantity is specified. In other words, to establish a concept, a measurement standard is stipulated but not used. Stipulating the standard helps define the concept, not any specific use of the standard.

After the standard is stipulated, math can be applied but is not at that moment.

Many things are measured in a concept, including importance of a characteristic, and many different standards chosen. Measurements can be also be cardinal or ordinal. In this system, even things like love can be measured, as Rand stated explicitly in ITOE.

None of that is what you have been talking about or blowing to smithereens.

And you still not have explained how a standard of measurement cannot be measured, if you really believe that like you said.

I hold that consistency is important.

Michael

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## Michael Stuart Kelly

Merlin,

Of course I agree that a word can have more than one meaning, but they were talking specifically about measurement.

They were talking about length that cannot be measured. Something any high-school student should know.

At the best, this is an attribute for which a standard of measurement cannot be established.

I finished high-school, but I have no idea what that is.

Michael

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## merjet

Yes, I know your dispute is with Bob. I was again pointing out how Rand confounded the two meanings in ITOE and assumed knowing the first meaning implies knowing the second meaning.

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## BaalChatzaf

AuthorQuite so. Ok, consider the straight line connecting the point (x0, y0) to (x1,y1). The length of that line is sqrt ((x1 -x0)^2 + (y1 -y0)^2), a number that is in general not rational and cannot be specified (in general) by a finite decimal representation of a number. Now I ask you again what measurement is omitted from the lengths (x1 - x0) and (y1 - y0). I keep asking, but you keep on not answering. If it turns out that (y1-y0)/(x1-x0) is the ratio of two integers m and n, the there exists a unit length that divides (y1-y0) m times and (x1-x0) n times. but if the quotient (y1-y0)/(x1-x0) is not such a ratio then what can we say?

It is unfortunate that Rand did not no much in the way of mathematics. If she had, she would not have come up with measurement omission.

Ba'al Chatzaf

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## tjohnson

The mathematical theory of numbers is very different from everyday intuitive understanding. The foundation of the real numbers is necessary for building integral and differential calculus and has very little to do with measuring the length of something. If you have a right triangle with sides=1 and you measure the diagonal with your ruler you will actually be

estimatingdimensions and you might say the diagonal is 1.414 in length, depending on what you read on your measuring device. Measuring physical objects is not the same as calculating theoretical dimensions in pure mathematics. Of course we know the real (mathematical) dimensions are 1,1, and sqrt(2) but the sqrt(2) actually represents aninfinite processthat we are free to terminate whenever we want to.## Link to comment

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## BaalChatzaf

AuthorAux Contraire. To define length precisely one needs integral calculus.

Ba'al Chatzaf

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## merjet

Many rational numbers cannot be specified by a finite decimal representation either. For example, 1/3 is 0.333333333....

If you gnu how to spell, I'd agree even more.

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## tjohnson

I don't follow, could you be more specific? Seems to me you only need a metric space to define length.

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## BaalChatzaf

AuthorThat works for straight line segments. To do the same for curves you need an integral. Pythagoras' theorem as at the base for both the length of straight line segment and length of curves.

See:

http://en.wikipedia.org/wiki/Curve_length

Ba'al Chatzaf

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## Mindy Newton

I agree that it "blows" measurement omission! I'm just saying we don't need "measurement omission" to "validate" concept-formation.

Mindy

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## Michael Stuart Kelly

Bob,

You keep on asking and challenging about

what type of measurement, not that measurement does not exist. The type of measurement is not the issue if you want to blow Rand's process of measurement omission to smithereens. And so far you haven't come closer than an unfounded opinion.Look at it this way (to oversimplify). The existence of measurement is a metaphysical issue. Using the different types of measurement is a scientific one. The standards of measurement (length, intensity, importance, etc.) pertain to both.

Just because one kind of measurement does not work, that does not mean another kind of measurement cannot work, i.e., that measurement

per sedoes not exist. You can't measure a curve by a straight line? OK. Then measure it with curve units.Note that for Rand to omit measurement, it had to exist in the first place.

Michael

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## Neil Parille

What kind of measurements are omitted when we form conjunctions, or form concepts such as "justice"? What concepts are omitted when we learn how to ride a bike?

I'm not surprised that Objectivist discussions of this are generally limited to chairs and tables.

[EDIT: I haven't been following this thread, so I hope my post isn't off-topic.]

-NEIL

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