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Do Algorithmically Non-Trivial Definitions Refute Measurement-Omission Theory?

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I define an "almost equilateral triangle" to be a triangle whose side-lengths are each within 10% of the average length of all the sides. I claim that this concept, although a perfectly natural and simple one, cannot possibly be accounted for by Rand's theory of measurement-omission and measurement-range-restriction. There are at least four reasons for this.

1) Any definition must be finite, and to give a Randian definition of the above concept would require us to make an infinitely long list of ranges for the measurements of the sides to be appropriately restricted.

2) The required ranges must vary jointly with the sides in order to work correctly, but Rand's theory requires that measurements be restricted to ranges independently of the measurements of other characteristics.

3) But even if Rand's theory allows for the joint variation of measurements and ranges, serious inconsistency issues in the definitions can arise. For example, if a measurement of one characteristic determines the allowed measurement range of another, and this other characteristic also determines the allowed measurements of the first, unless you are extremely careful, it's likely that certain measurements could end up excluding themselves from the ranges that they themselves determine.

4) Even if something like the procedure in point 3 could be done, it would be so exceedingly complex that it can in no way be considered a realistic representation of human psychology.

The essence of the problem seems to be that some concepts have what I call, "algorithmically non-trivial" definitions. These definitions require us to compute some non-trivial property of the entity or entities in question in order to be able to classify them.

I therefore propose the following improvement to Rand's theory of concepts. The change that I propose is that a concept is really just an algorithm which, given some measurement-data about an entity, outputs a "yes/no" answer as to whether the entity specified by the data belongs to the concept or not. In other words, that the process of concept-formation is really just informal computer programming.

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When we form the concept "table" some of the measurements we omit are the substance(s) of which the table is made and the exact size and shape of the level surface and support(s).  The substance can vary but can not be butter or anything caustic.  The size and shape can vary but some sizes and shapes are unsuitable.  

The point about measurement omission is that the measurements can exist in any combination consistent with the definition.  There does not have to be a simple description of the exact ways in which the individual measurements can vary.  

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24 minutes ago, Doug Morris said:

When we form the concept "table" some of the measurements we omit are the substance(s) of which the table is made and the exact size and shape of the level surface and support(s).  The substance can vary but can not be butter or anything caustic.  The size and shape can vary but some sizes and shapes are unsuitable.  

The point about measurement omission is that the measurements can exist in any combination consistent with the definition.  There does not have to be a simple description of the exact ways in which the individual measurements can vary.  

The trouble with this point is that in order to be able to have a definition in the first place, you have to know which ranges of measurements are allowed. As far as I know, Ayn Rand gave no details about the joint variation of measurements beyond the idea of an essential characteristic, whereby some special characteristic is functionally related to some other characteristics in the sense that its measurements are sufficient to determine the measurements of those non-essential characteristics.

But in those cases, the restriction of the range of measurements of the essential characteristic is then also sufficient to determine the allowed ranges of the characteristics so determined. This, however, is still not sufficient to explain the existence of the concept of almost-equilateral triangles and other similar concepts, because there is still no variation of ranges with respect to measurements.

I also have another pathological example whereby even the essential characteristic of the concept varies with the measurements of its units. I'll post it here later, I have to go to work.

Edited by SpookyKitty
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 the definition of table (“An item of furniture, consisting of a flat, level surface and supports, intended to support other, smaller objects”), 

(From the Ayn Rand lexicon).

This is a good definition of "table".  We can formulate it without knowing any exact ranges of measurements.  The range of a measurement can be affected by the values of other measurements.  For example, how thin the supports can be depends on the number of supports, the substance of which they are made, the weight of the top that provides the level surface being supported, and the weight of the other objects we expect to be able to support.  All these are among the measurements we are omitting.  We don't have to know anything specific about these ranges or relationships to form the concept "table".

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20 minutes ago, Doug Morris said:

 the definition of table (“An item of furniture, consisting of a flat, level surface and supports, intended to support other, smaller objects”), 

(From the Ayn Rand lexicon).

This is a good definition of "table".  We can formulate it without knowing any exact ranges of measurements.  The range of a measurement can be affected by the values of other measurements.  For example, how thin the supports can be depends on the number of supports, the substance of which they are made, the weight of the top that provides the level surface being supported, and the weight of the other objects we expect to be able to support.  All these are among the measurements we are omitting.  We don't have to know anything specific about these ranges or relationships to form the concept "table".

The problem with this definition of "table" is that it is an adult definition of table. Notice that it makes use of terms far more abstract than "table" itself. This kind of adult definition of a concept, according to Rand, is only possible after a "child's" definition of the concept has been given in terms of restricted measurement values. Once other, more abstract concepts are integrated, it is only in this new context of knowledge that the child's definition can be abandoned in favor of the adult one.

The trouble with algorithmically non-trivial concepts is that no "child's definition" of them is ever possible.

Edited by SpookyKitty
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4 minutes ago, SpookyKitty said:

a "child's" definition of the concept has been given in terms of restricted measurement values.

The child's definition would probably be at least partly perceptual and ostensive.  It would not have to involve any precise ranges.  The child could use perceptual, ostensive methods to get some feel for how the value of one measurement can affect the range of another.  A child would not need an adult understanding of exactly how to draw the line between a table and a non-table.

13 minutes ago, SpookyKitty said:

The trouble with algorithmically non-trivial concepts is that no "child's definition" of them is ever possible.

A child would not need to form the concept of an "almost equilateral triangle" as you have defined it.  However, a child could form a similar concept using perceptual and ostensive methods, especially if it was in an environment where it was often confronted with triangles and the effects of their shapes.

 

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31 minutes ago, Doug Morris said:

The child's definition would probably be at least partly perceptual and ostensive.  It would not have to involve any precise ranges.  The child could use perceptual, ostensive methods to get some feel for how the value of one measurement can affect the range of another.  A child would not need an adult understanding of exactly how to draw the line between a table and a non-table.

A child would not need to form the concept of an "almost equilateral triangle" as you have defined it.  However, a child could form a similar concept using perceptual and ostensive methods, especially if it was in an environment where it was often confronted with triangles and the effects of their shapes.

 

Except for axiomatic concepts, ostensive or perceptual definitions are impossible for the simple reason that non-axiomatic concepts are never perceived, only concretes are. (Strictly speaking, axiomatic concepts are never perceived either, but rahter, they are implicit in every act of cognition)

Secondly, regarding "child's definitions", we are not talking about a literal child and the way that a literal child's mind works. This is merely a metaphor. Here, as in Rand's works, it is referring to any context of knowledge in which all Man has to go on is raw concept formation. In these contexts, we might call them the frontiers of knowledge, the process of concept formation proceeds wordlessly and measurements are performed through comparisons where precise quantities cannot be given. This is the context in which brand new concepts are formed. An adult definition can only arise from integration and never from concept formation. And at no point in Rand's works are measurement-dependent ranges in these contexts ever mentioned or implied.

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Consider all "almost equilateral triangles" with an average "side-length" of 1. Then all possible "almost equilateral triangles" are just off by a scaling factor from the previously described triangles. However, the scaling factor is not important for determining whether some triangle fits our definition or not, so the scaling factor is just one measurement that's omitted.

 

On 12/8/2021 at 12:11 PM, SpookyKitty said:

2) The required ranges must vary jointly with the sides in order to work correctly, but Rand's theory requires that measurements be restricted to ranges independently of the measurements of other characteristics.

There are no "required ranges" for a triangle that already exists. A triangle either fits the definition or it doesn't. Its other sides are probably not going to vary just because you measured one side. In the case of triangles with an average "side-length" of 1, all 3 sides are within 1±0.1. The three sides are not "correlated" and we don't need a long list. Other triangles are just off by a scaling factor (which is not important and can be omitted).

 

On 12/8/2021 at 12:11 PM, SpookyKitty said:

3) But even if Rand's theory allows for the joint variation of measurements and ranges, serious inconsistency issues in the definitions can arise. For example, if a measurement of one characteristic determines the allowed measurement range of another, and this other characteristic also determines the allowed measurements of the first, unless you are extremely careful, it's likely that certain measurements could end up excluding themselves from the ranges that they themselves determine.

Mostly irrelevant. Measurement of one side of a triangle doesn't "determine" the allowed measurement range of another. A triangle is what it is and doesn't change during measurement. Such concerns are only relevant if you're constructing a triangle to fit the definition, but it's irrelevant to the problem of defining a triangle. For a triangle that actually exists, its average "side-length" is well defined and the side-lengths are not correlated.

 

On 12/8/2021 at 12:11 PM, SpookyKitty said:

4) Even if something like the procedure in point 3 could be done, it would be so exceedingly complex that it can in no way be considered a realistic representation of human psychology.

What you described previously is the process of constructing a triangle. This is obviously complex and has nothing to do with concepts or measurement omission. The first measurement that is omitted is the scale of the triangle, since the only thing we need to know is whether the side-lengths are within a certain percentage point of the average. You can scale any triangle (that already exists) so that the average "side-length" is 1. Then all sides must be within ±0.1 of 1.

The next measurement that's omitted is the exact percentage point by which the sides are off from the average (since it doesn't matter if it's 4.3% or 8.7%).

The sides aren't "correlated". The only reason they appeared correlated was because you were describing a non-existent triangle with an undefined (and changing) average "side-length". Hence, you arrived at contradictions like "it's likely that certain measurements could end up excluding themselves from the ranges that they themselves determine". You were not talking about things that exist (or could exist). You were talking about the process of constructing a triangle (which does not exist yet). This is completely irrelevant to Rand's idea about concept formation.

Edited by human_murda
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@human_murda

That's a valiant effort there, but your omission of scaling factor idea is an instance of circular reasoning. How do you go about forming the concept of "triangles whose side lengths are such that their average is 1 while the lengths are all within 10% of that average"? Forming even this concept is not simply a matter of keeping each side length between 0.9 and 1.1. As a counterexample to a Randian definition of such a concept, consider the equilateral triangle whose side lengths are all 1.09. The given measurements fall within the allowed ranges, but the average side length is not equal to 1. The point is that any attempted definition of almost equilateral triangle concepts which considers the side lengths independently is always subtly vulnerable to extreme-case counterexamples.

Your claim that I was considering triangles that cannot possibly exist is a strawman. I did not mention the triangle inequality explicitly, but that is something that could reasonably be inferred from my consistent use of the word "triangle" and not "a triple of positive real numbers".

EDIT: In addition, the scaling factor is not a characteristic of a triangle. In order to count as a measurement, a comparison with a standard must be possible. So which of the many possible average-side-length-1 triangles is the standard? Note that they are not all similar to each other. This means that whatever standard is chosen, it will fail to account for almost equilateral triangles that are not similar to the standard one.

Edited by SpookyKitty
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Are you leaning toward the isosceles solutions of ±10% shown to the left, or the scalene versions offered to the right? Why ±10% on the length (upper) and not ±10% on the angle (lower) as illustrated in the provided graphic?

image.png

Personally, the borderline case offered up in Introduction to Objectivist Epistemology should extend to any of the "±10% almost equilateral" triangles illustrated above.

The horizontal line shown between the upper linear ±10% and the lower angular ±10% is only has a 1° change of direction at its center. Less than 1° is difficult for some folk to detect.

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35 minutes ago, dream_weaver said:

Are you leaning toward the isosceles solutions of ±10% shown to the left, or the scalene versions offered to the right?

It doesn't matter. Any triangle that fits the definition, whether it is isosceles or scalene is fine.

Quote

Why ±10% on the length (upper) and not ±10% on the angle (lower) as illustrated in the provided graphic?

 

Because three angles do not determine a unique triangle whereas three side lengths do.

Quote

Personally, the borderline case offered up in Introduction to Objectivist Epistemology should extend to any of the "±10% almost equilateral" triangles illustrated above.

This has nothing to do with borderline cases. There are no borderline cases here. Every triangle either is almost equilateral or it isn't.

Quote

The horizontal line shown between the upper linear ±10% and the lower angular ±10% is only has a 1° change of direction at its center. Less than 1° is difficult for some folk to detect.

For larger triangles, even small differences in degrees will be detectable with the naked eye.

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6 hours ago, SpookyKitty said:

How do you go about forming the concept of "triangles whose side lengths are such that their average is 1 while the lengths are all within 10% of that average"?

The only way to do that would be to compute the average first (the average can be taken as the scaling factor in the previous example). Only once you have gotten rid of all the triangles whose average is not 1 would you even need to check whether the side lengths are between 0.9 and 1.1.

 

6 hours ago, SpookyKitty said:

As a counterexample to a Randian definition of such a concept, consider the equilateral triangle whose side lengths are all 1.09

Here, the average (or scaling factor) is 1.09, not 1.

If the side lengths were 9.87, 10.199, 10.88: the scaling factor would be 10.3163. After omitting the scaling factor, the side lengths would be 0.9567,0.9886,1.0546. After omitting the exact side-length, it would be (yes,yes,yes). What's "circular" about this?

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@human_murda

I will try to make myself clearer.

If, in the course of concept formation, one is allowed to perform arbitrary computations (I mean arbitrarily chosen but specific well-defined computations such as finding averages, as opposed to computations that don't make sense) on the measurements of the characteristics prior to any sort of measurement omission/restriction (as in your proposal), then this concedes the whole point in my favor. Because at that point, most concepts would be defined primarily by the algorithms applied to the measurements, and the measurement omission/restriction steps are just secondary steps that may or may not appear in the overall algorithm that defines the concept.

Stated slightly differently, Rand held that the only constructs necessary to form any non-axiomatic concept are measurement omission and differentiation. If now further constructs (such as computing averages) are required, as in your example above, then Rand's theory is false.

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1 hour ago, SpookyKitty said:

If, in the course of concept formation, one is allowed to perform arbitrary computations (I mean arbitrarily chosen but specific well-defined computations such as finding averages, as opposed to computations that don't make sense) on the measurements of the characteristics prior to any sort of measurement omission/restriction (as in your proposal), then this concedes the whole point in my favor.

I don't think it matters to Rand's theory what quantities are directly measurable or not. Side-length is a measurement, average of side-lengths is a measurement, angles are measurements, sines and tangents of angles are measurements. These are all characteristics of a triangle, even if we might need to perform some computations to find them out.

It's possible to restate your definition in terms of quantities that are "directly measurable": we just need the ratio of a triangle's side-length to its perimeter (which is "directly measurable") to be between 0.9/3 and 1.1/3. However, even here we need to "compute" the ratio (which isn't directly measurable). The measurement omission here is the fact that only the ratios matter, not the actual lengths.

 

1 hour ago, SpookyKitty said:

...one is allowed to perform arbitrary computations on the measurements of the characteristics prior to any sort of measurement omission/restriction...

This isn't actually necessary. It was just the easiest way. Since we know for a fact that only the ratios matter, we can discard all length measurements as a first step (and instead just look at angles). Thus, even without computing averages, we can omit all length measurements (since they're just indicators of scale).

Then, based on the law of sines, we can apply the following conditions:

0.9/3 < sin(A)/(sin(A)+sin(B)+sin(B)) < 1.1/3

0.9/3 < sin(B)/(sin(A)+sin(B)+sin(B)) < 1.1/3

0.9/3 < sin(C)/(sin(A)+sin(B)+sin(B)) < 1.1/3

Even after this, there are additional measurement omissions (only ratios of sines matter, not the actual values of the sines. The exact value of the ratio also doesn't matter and only a certain range matters).

The idea that we need to compute averages before any measurement omission is incorrect. It's possible to get rid of length measurements first and then do other computations. However, calculating averages first is easier (and it honestly doesn't matter. The average is as much a property of a triangle as a side-length).

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Yes, specifying two angles would still require one length to control the size.

Ok. So the length of the sides are X±10% to be to print. Presumably the intent is to exclude triangles that have three sides that measure X. What about a triangle that all sides are X+10%. That is still an equilateral triangle, albeit not one with three sides of X length specifically? Is it to be considered an "almost equilateral triangle with sides being X±10%", as long as the sides are not exactly X? How much more complexity need be added to fully obfuscate the matter?

 

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On 12/8/2021 at 12:50 PM, SpookyKitty said:

Except for axiomatic concepts, ostensive or perceptual definitions are impossible for the simple reason that non-axiomatic concepts are never perceived, only concretes are.

From the Ayn Rand Lexicon:

With certain significant exceptions, every concept can be defined and communicated in terms of other concepts. The exceptions are concepts referring to sensations, and metaphysical axioms.

Sensations are the primary material of consciousness and, therefore, cannot be communicated by means of the material which is derived from them. The existential causes of sensations can be described and defined in conceptual terms (e.g., the wavelengths of light and the structure of the human eye, which produce the sensations of color), but one cannot communicate what color is like, to a person who is born blind. To define the meaning of the concept “blue,” for instance, one must point to some blue objects to signify, in effect: “I mean this.” Such an identification of a concept is known as an “ostensive definition.”

On 12/8/2021 at 12:50 PM, SpookyKitty said:

any context of knowledge in which all Man has to go on is raw concept formation. In these contexts, we might call them the frontiers of knowledge, the process of concept formation proceeds wordlessly and measurements are performed through comparisons where precise quantities cannot be given.

Regardless of who is doing the concept formation, there might be an initial stage in which the working definition is at least partly ostensive.

 

 

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14 hours ago, SpookyKitty said:

Stated slightly differently, Rand held that the only constructs necessary to form any non-axiomatic concept are measurement omission and differentiation. If now further constructs (such as computing averages) are required, as in your example above, then Rand's theory is false.

Measurement omission is usually not part of definition or differentiation.  Definition codifies and clarifies differentiation.  Differentiation may include complicated computations.  It may also include observing differences that at first are poorly understood.

Measurement omission is implicit in any subsuming of concretes with different measurements under a single concept.  But it is not necessarily an explicit part or logical step of the differentiation.

Again, 

On 12/8/2021 at 8:33 AM, Doug Morris said:

The point about measurement omission is that the measurements can exist in any combination consistent with the definition. 

For each concrete that is included under the concept,  the measurements must exist in some combination consistent with the definition. 

 

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16 hours ago, human_murda said:

This isn't actually necessary. It was just the easiest way. Since we know for a fact that only the ratios matter, we can discard all length measurements as a first step (and instead just look at angles). Thus, even without computing averages, we can omit all length measurements (since they're just indicators of scale).

Then, based on the law of sines, we can apply the following conditions:

0.9/3 < sin(A)/(sin(A)+sin(B)+sin(B)) < 1.1/3

0.9/3 < sin(B)/(sin(A)+sin(B)+sin(B)) < 1.1/3

0.9/3 < sin(C)/(sin(A)+sin(B)+sin(B)) < 1.1/3

 

This is actually a very sexy calculation. I am impressed.

Edited by SpookyKitty
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