This is a criticism of the Objectivist theory of concept-formation. Tell me what you think.
objCrit.pdf (146 KB)
Not sure why you have to use examples from math. Could you usea âconceptâ that is more familiar, say: people with height 5" 11" living in Nebraska and having moderated OO.com in the past". It would give you as single referent⌠QED
18 hours ago, SpookyKitty said:
This is a criticism of the Objectivist theory of concept-formation. Tell me what you think.
objCrit.pdf (146 KB)
What I see missing from your paper is an adequate discussion of Perception, which Rand spends a good deal of time doing in ITOE.
That is, the ability to differentiate and integrate objects in oneâs field of perception is the metaphysically given and forms the necessary precondition from which concept formation then proceeds. Infants, long before they form words can distinguish between similar and dissimilar objects.
From ITOE, p. 5:
âA percept is a group of sensations automatically retained and integrated by the brain of a living organism.â - Note, this is not unique to humans, but is applicable to animals in general.
p. 16:
(As far as can be ascertained, the perceptual level of a childâs mind is similar to the awareness of the higher animal: the higher animals are able to perceive entities, motions, attributes, and certain numbers of entities. But what an animal cannot perform is the process of abstraction - of mentally separation attributes, motions or numbers from entities. It was been said that an animal can perceive two oranges or two potatoes, but cannot grasp the concept âtwoâ.)
p. 164:
Suppose a child is forming a concept âtable.â First he has to isolate a table from the rest of his perceptual concretes, then integrate it with other tables. Now, in this process words are not present yet, because he is merely observing, and performing a certain mental process. It is after he has fully grasped that these particular objects (tables) are special and different in some way from all other objects he perceives - it is then that he has to firm up, in effect, his mental activity in his own mind by designating that special status of these particular objects in some sensory form [i.e., by means of a word].
Edit: The capacity to engage in basic Arithmetic is a given, and follows from such observed perceptual mechanisms as subitizing and chunking. Without these basic perceptual mechanisms we could not perform arithmetic, geometry, algebra, analytic geometry, trigonometry, calculus, mechanics, etc.
@New Buddha, Iâm not sure what your point is.
1 hour ago, SpookyKitty said:
@New Buddha, Iâm not sure what your point is.
Concepts such as Primes are abstractions from abstractions - not abstractions from concretes. Thus, in your paper where you state the followingâŚ
_Consider the following question. Given the Objectivist theory of concept formation, how could we form the concept of prime number? Well, first off, by condition 1, we need at least two numbers that are referents of the concept, say 3 and 5 (but pretend we donât know that 3 and 5 are prime yet). Still the question remains just how many examples of a prime number we would need in order to form the concept. _
âŚyou are reducing the conceptualization of Primes to an ostensive, first-level percept/concept relationship â without taking into account the many antecedent steps necessary to arrive at the concept. This is true for most mathematical ideas such as irrational numbers, limits in calculus, etc. Itâs also true for such abstractions as âpropertyâ or âmarriageâ.
Just now, New Buddha said:
âŚyou are reducing the conceptualization of Primes to an ostensive, percept/concept relationship â
No. Higher order concepts such as primes are formed by treating lower-order concepts as though they were percepts.
This is abundantly clear from the Objectivist concept of conceptual hierarchy.
4 minutes ago, SpookyKitty said:
No. Higher order concepts such as primes are formed by treating lower-order concepts as though they were percepts.
All concepts are give perceptible form. This is the purpose of numbers and words.
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Consider the concepts of ânatural number greater than 0â and ânatural number which dividides at least one natural numberâ. These two concepts have the same referents since 0 is the only number which divides no number whatsoever, and every other number is greater than 0. But they are different, hence theorem 6 is false. Therefore, by modust tollens, there exist some concepts which cannot be formed
by the Objectivist process of concept formation.
You have not invented two concepts, but two descriptions for various particular things. âNatural number greater than 0â merely describes individual numbers like 5 or 50,000. You have essentially created a complex proper name for certain, unidentified numbers. Are we playing Jeopardy? âAlex, what is 523?â
I could say, âHuman being who is not dead.â
âAlex, who is SpookyKitty?â
This wouldnât be a concept. It would be a proper name for every living human entity. Instead of saying, âHi, Bob!â I could say, âHi, human being who is not dead!â
To create your concept, youâd have to define it as âthe set of natural numbers greater than zero,â which is how real mathematical concepts work. This idea would include all the various numbers, or referents, in the set.
In fact, if you had only used the plural numbers instead of the singular number in your own description, you may have solved the problem yourself. Because the set part would have been implied.
Just now, MisterSwig said:
You have not invented two concepts, but two descriptions for various particular things. âNatural number greater than 0â merely describes individual numbers like 5 or 50,000. You have essentially created a complex proper name for certain, unidentified numbers. Are we playing Jeopardy? âAlex, what is 523?â
The difference between a âmereâ proper name and a descrption of a concept is that the description of the concept contains conceptual content.
âNatural number greater than 0â is only superficially the same as a complex proper name such as âmigwit swamy blubberâ (which we can define as âa natural number greater than 0â).
The key difference is that ânatural number greater than 0â allows you to determine all by yourself which numbers it does and does not refer to, for an unlimited number of numbers, whereas âmigwit swamy blubberâ does not. One can understand the concept of ânatural number greater than 0â provided that one understands the concepts of ânatural numberâ, âgreater thanâ, and â0â.
None of this holds true for a mere complex proper name like âmigwit swamy blubberâ. In fact, one can define âmigwity swamy blubberâ as âa natural number greater than 0â but you canât define âa natural number greater than 0â as âmigwity swamy blubberâ.
Now, you might claim that ânatural number greater than 0â and ânatural number which divides a natural numberâ are still the same concept. But they are not. The first depends on the concept of âgreater thanâ while the second depends on the concept of divisibility. Someone who understands one might not understand the other. The difference in understanding can only be caused by a difference in the concepts that each has available to him.
Thus, if one person utters ânatural number greater than 0â but does not understand divisibility, and another person, who does not understand âgreater thanâ utters ânatural number which divides a natural numberâ, although they would be referring to precisely the same numbers, they would not be mutually intelligible.
29 minutes ago, SpookyKitty said:
The key difference is that ânatural number greater than 0â allows you to determine all by yourself which numbers it does and does not refer to, for an unlimited number of numbers, whereas âmigwit swamy blubberâ does not.
No, it doesnât. It doesnât refer to ânumbersâ (plural). It refers to one ânumberâ (singular). And that number is unidentified, so no matter how many times you insist otherwise, your phrase will never allow anybody to determine to which specific number it refers.
Play word games all you like. A is still A.
Just now, MisterSwig said:
No, it doesnât. It doesnât refer to ânumbersâ (plural). It refers to one ânumberâ (singular). And that number is unidentified, so no matter how many times you insist otherwise, your phrase will never allow anybody to determine to which specific number it refers.
Play word games all you like. A is still A.
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The basic principle of concept-formation (which states that the omitted measurements must exist in some quantity, but may exist in any quantity) is the equivalent of the basic principle of algebra, which states that algebraic symbols must be given some numerical value, but may be given any value. In this sense and respect, perceptual awareness is the arithmetic, but conceptual awareness is the algebra of cognition.
âAyn Rand
ânatural number greater than 0â cannot possibly refer to a singular number precisely because it denotes the concept and not some specific number.
EDIT: Furthermore, if you insist that ânatural number greater than 0â must refer to a single specific number, then you run into the following contradiction:
P1: 4 is a natural number greater than 0
P2: 5 is a natural number greater than 0
P3: Two things the same as a third are the same as each other
Conclusion: 4 is 5.
8 minutes ago, SpookyKitty said:
P1: 4 is a natural number greater than 0
P2: 5 is a natural number greater than 0
P3: Two things the same as a third are the same as each other
Conclusion: 4 is 5.
Itâs not my contradiction. Itâs yours. I just helped you see it. Have fun with your new âconcept.â I have nothing more to say here.
Just now, MisterSwig said:
Itâs not my contradiction. Itâs yours. I just helped you see it. Have fun with your new âconcept.â I have nothing more to say here.
Thereâs no shame in admitting you were wrong about something, MisterSwig.
23 hours ago, SpookyKitty said:
This is a criticism of the Objectivist theory of concept-formation. Tell me what you think.
objCrit.pdf (146 KB)
I think itâs incoherent babble. Same as always.
1 Like
13 minutes ago, SpookyKitty said:
Thereâs no shame in admitting you were wrong about something, MisterSwig.
You crack me up, Kitty. I fear what would happen if we ever met in person.
Just now, Nicky said:
I think itâs incoherent babble. Same as always.
What specifically doesnât make sense to you, Nicky?
Theorem 5: If you are distinguishing redness, it would only need to be at least two colors, e.g. red and non-red. Youâd want a similar entity, too, which could have the same exact hue. Rather than what you wrote, there must be a conceptual common denominator such that multiple values are possible.
Theorem 6: Do you mean no two distinct concepts share all of the same referents in a 1:1 relation? I donât see why. If it were something directly perceivable, couldnâ t factors besides those of the referent establish the need for multiple concepts?
Theorem 7: Even if youâre right about 6, your example here shows reasons to reject, alter, or reorganize concepts you hold. Or, simply, they -arenât- different concepts. You didnât establish that only one definition per concept is valid. If you insist on them being different, it is a reason to say that concepts improve over time, or that some definitions are found to be wrong.
Theorem 11: What makes that even a concept? Or why not say lacking a referent makes a concept impossible to form, thus contradicting that it is a (valid) concept?
Corollary 12: Proper nouns arenât considered to be concepts for this very reason.
You seemed to overlook aspects of fallibility here, reasoning as if you were an omniscient viewer.
@Eiuol said,
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Theorem 5: If you are distinguishing redness, it would only need to be at least two colors, e.g. red and non-red. Youâd want a similar entity, too, which could have the same exact hue. Rather than what you wrote, there must be a conceptual common denominator such that multiple values are possible.
That would make sense, but not within Randâs theory. Itâs clear that what Rand is going for is something like this âthis match has some lengthâ, âthis other match has another lengthâ, then we form the concept âlengthâ by observing that things have length regardless of what that length actually is. And this is true for all of the examples of concept formation that Rand provides.
An artificial, but totally clear example is, take the âperceptsâ â1001101â and â1011101â. We can think of each position in the sequence as a characteristic. The sequence of characteristic values determines the percepts. One observers that the two percepts differ in the third characteristic, and forms the unique concept corresponding to that characteristic â10x1101â. The referents of this concept are then obtained by substituting one of the possible values of a characteristic for âxâ in the âdefinitionâ of the concept, i.e., â10x1101â.
Rand is right in requiring that her theory of concept formation requires the isolation of âcommensurable characteristicsâ. If the given percepts were â1001101â and â01101â, then we wouldnât be able to form a concept from them because it isnât possible to find a natural matching âcommensurable characteristicsâ. This is because one could match the first characteristic in the second string with with the either the first characteristic in the first string, and get a matching like:
1 0 0 1 1 0 1
0 1 1 0 1 - - (the hyphens stand for blanks)
And then you might be able to (assuming that integration over multiple characteristics is possible) get the concept
1xxx101
Or the second characteristic in the second string with the first characteristic in the first string, and get
1 0 0 1 1 0 1
- 0 1 1 0 1 -
10x1xx1
Or some other strange matching such as
1 0 0 1 1 0 1
0 - 1 - 1 0 1
Regardless, the uniqueness of concept formation fails because in each of the above cases we are attempting integration by using incommensurable characteristics.
Another way to look at this is, by forming the concept of redness by abstracting from things that are different shades of red, one can then say âthis entity is red regardless of the shade of redâ. One cannot do concept formation by comparing red things to things of different colors. That is, you canât say that âthis entity is red regardless of its colorâ. Because the first sentence makes sense and the second one doesnât, that means that shades of red are a commensurable characteristic pertaining to the concept of redness, while different colors are not. Objects of different colors are commensurable, but then one is led to the concept of color, and not redness.
I realize this is kind of confusing, but essentially, the concept of red is supposed to refer to all red things regardless of the specific measurement of the shade of red. It cannot refer to blue things because no shade of blue is a shade of red.
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Theorem 6: Do you mean no two distinct concepts share all of the same referents in a 1:1 relation? I donât see why. If it were something directly perceivable, couldnâ t factors besides those of the referent establish the need for multiple concepts?
Yes.
Can you expand on the second sentence? I donât understand what youâre trying to say there.
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Theorem 7: Even if youâre right about 6, your example here shows reasons to reject, alter, or reorganize concepts you hold. Or, simply, they -arenât- different concepts. You didnât establish that only one definition per concept is valid. If you insist on them being different, it is a reason to say that concepts improve over time, or that some definitions are found to be wrong.
Given the two concepts âa natural number greater than 0â and âa natural number which divides at least one other natural numberâ, which one should I reject or alter? Neither is better than the other, they are simply different concepts (although with the same referents).
I did not establish that only one definition per concept is valid because the same concept can have multiple definitions. For example âa natural number which is a multiple of 7 or divisible by 5â is the same concept as âa natural number which is divisible by 5 whenever it is not a multiple of 7â.
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Theorem 11: What makes that even a concept? Or why not say lacking a referent makes a concept impossible to form, thus contradicting that it is a (valid) concept?
According to Randâs theory, the concept âa number greater than 1 but which divides both 17 and 6â is invalid because it has no referents. But the concept is perfectly intelligible and can be meaningfully used in a true sentence as in,
ââ8 is a number greater than 1 which divides both 17 and 6â is falseâ.
The above statement is true and can be understood by anyone precisely because the statement â8 is a number greater than 1 which divides both 17 and 6â is intelligible but false. Every such statement is intelligible regardless of what number is placed at the beginning. Therefore, the definition âa number greater than 1 which divides both 17 and 6â denotes a concept (with no referents).
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You seemed to overlook aspects of fallibility here, reasoning as if you were an omniscient viewer.
Can you expand on this?
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Corollary 12: Proper nouns arenât considered to be concepts for this very reason.
But âoneâ is a proper noun. Is âoneâ, then, not a concept?
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You seemed to overlook aspects of fallibility here, reasoning as if you were an omniscient viewer.
Can you explain what you mean by this?
16 hours ago, SpookyKitty said:
That would make sense, but not within Randâs theory. Itâs clear that what Rand is going for is something like this âthis match has some lengthâ, âthis other match has another lengthâ, then we form the concept âlengthâ by observing that things have length regardless of what that length actually is. And this is true for all of the examples of concept formation that Rand provides.
I donât see how within Randâs theory that all measurements must be different, only that at least one must be different. Even more, there needs to be at least 3 entities: the entity used for comparison, an entity with a different measurement, and an entity with a similar measurement. There can be red, i.e. what looks red within a range on the color spectrum, and non-red, i.e. anything outside that part of the spectrum. So when Rand says âanother lengthâ, I take her to mean simply âsome lengthâ, not that sheâs excluding identical lengths.
I think youâre totally missing Randâs idea of a conceptual common denominator. You seem to suggest that by Randâs theory, different colors are incommunsurable. No, this isnât true. Blue and red are commensurable, because they are the same type of characteristic; they are measured in the same way on the same scale. If you were then to form a concept out of this commonality, you get the concept âcolorâ - the CCD. The concept âredâ just points out a difference from entities on the same scale of measurement.
Rand doesnât go into detail about commensurability really past things like shape being incommensurable with color. Iâd say as long as the same mode of perception or cognition is used, then the characteristic is commensurable. Clearly, âwhat shape is red?â makes no sense.
16 hours ago, SpookyKitty said:
Can you expand on the second sentence? I donât understand what youâre trying to say there.
Consider that color is more than just a hue. What you see also depends on the lighting. That lighting isnât produced by the referent, the entity that is that color. This isnât a problem for Randâs theory, since the referent is still measurable, and the measurement is still of something objectively valid.
16 hours ago, SpookyKitty said:
Given the two concepts âa natural number greater than 0â and âa natural number which divides at least one other natural numberâ, which one should I reject or alter? Neither is better than the other, they are simply different concepts (although with the same referents).
You could keep both of the definitions in that case. They arenât -concepts- though. See the next bit.
16 hours ago, SpookyKitty said:
The above statement is true and can be understood by anyone precisely because the statement â8 is a number greater than 1 which divides both 17 and 6â is intelligible but false. Every such statement is intelligible regardless of what number is placed at the beginning. Therefore, the definition âa number greater than 1 which divides both 17 and 6â denotes a concept (with no referents).
According to Randâs theory, itâs not even a concept anyway. âA number greater than 1 but which divides both 9 and 30â isnât a concept either, even though it has some referent. They are statements. Intelligibility is not sufficient to makes something a concept. Somewhere in ITOE Rand talks about how merely seeing an assortment of characteristics (e.g. blonde woman with a blue dress and curly hair) is not a good enough reason to form a concept.
16 hours ago, SpookyKitty said:
But âoneâ is a proper noun. Is âoneâ, then, not a concept?
How is âoneâ a proper noun? Thereâs only one Eiffel Tower; there are many "one"s.
16 hours ago, SpookyKitty said:
Can you explain what you mean by this?
I meant it seemed like a valid epistemology to you is one based on what an omniscient viewer would imagine.
Just now, Eiuol said:
I donât see how within Randâs theory that all measurements must be different, only that at least one must be different. Even more, there needs to be at least 3 entities: the entity used for comparison, an entity with a different measurement, and an entity with a similar measurement. There can be red, i.e. what looks red within a range on the color spectrum, and non-red, i.e. anything outside that part of the spectrum. So when Rand says âanother lengthâ, I take her to mean simply âsome lengthâ, not that sheâs excluding identical lengths.
Rand says in ITOE:
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A concept is a mental integration of two or more units which are isolated according to a specificcharacteristic(s) and united by a specific definition.
Therefore, there need not be at least three entities, but at least two.
Later on, she also says:
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Let us now examine the process of forming the simplest concept, the concept of a single attribute
(chronologically, this is not the first concept that a child would grasp; but it is the simplest one
epistemologically)âfor instance, the concept âlength.â If a child considers a match, a pencil and a stick, he observes that length is the attribute they have in common, but their specific lengths differ. The difference is one of measurement. In order to form the concept âlength,â the childâs mind retains the attribute and omits its particular measurements. Or, more precisely, if the process were identified in words, it would consist of the following: âLength must exist in some quantity, but may exist in any quantity. I shall identify as âlengthâ that attribute of any existent possessing it which can be quantitatively related to a unit of length, without specifying the quantity.â
Itâs not as if one of the things has no length at all, nor is it that one of the objectâs areas or volumes is being compared to the lengths of some others.
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The same principle directs the process of forming concepts of entitiesâfor instance, the concept âtable.â
The childâs mind isolates two or more tables from other objects, by focusing on their distinctive
characteristic: their shape. He observes that their shapes vary, but have one characteristic in common: a
flat, level surface and support(s). He forms the concept âtableâ by retaining that characteristic and
omitting all particular measurements, not only the measurements of the shape, but of all the other
characteristics of tables (many of which he is not aware of at the time).
Which again means that at least two tables are necessary, and not at least three.
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I think youâre totally missing Randâs idea of a conceptual common denominator. You seem to suggest that by Randâs theory, different colors are incommunsurable. No, this isnât true. Blue and red are commensurable, because they are the same type of characteristic; they are measured in the same way on the same scale. If you were then to form a concept out of this commonality, you get the concept âcolorâ - the CCD. The concept âredâ just points out a difference from entities on the same scale of measurement.
No, Iâm not saying that at all. What I think Rand is saying is that one cannot form the concept of color before one has formed a concept of each individual color. As in here:
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The first stages of integrating concepts into wider concepts are fairly simple, because they still refer to
perceptual concretes. For instance, man observes that the objects which he has identified by the concepts
âtable,â âchair,â âbed,â âcabinet,â etc. have certain similarities, but are different from the objects he has
identified as âdoor,â âwindow,â âpicture,â âdrapesââand he integrates the former into the wider concept
âfurniture.â In this process, concepts serve as units and are treated epistemologically as if each were a
single (mental) concreteâalways remembering that metaphysically (i.e., in reality) each unit stands for an
unlimited number of actual concretes of a certain kind.
EDIT: Point of all this is, although it may be the case that for some concepts one needs three or more entities, at least one of them has to have a different measurement from the others. But prime numbers all have the same amount of primeness.
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Rand doesnât go into detail about commensurability really past things like shape being incommensurable with color. Iâd say as long as the same mode of perception or cognition is used, then the characteristic is commensurable. Clearly, âwhat shape is red?â makes no sense.
To reiterate, âwhat shade of red is blue?â also makes no sense.
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Consider that color is more than just a hue. What you see also depends on the lighting. That lighting isnât produced by the referent, the entity that is that color. This isnât a problem for Randâs theory, since the referent is still measurable, and the measurement is still of something objectively valid.
How does that lead to multiple concepts? I mean you are claiming that Randâs theory allows multiple concepts with all of the same referents, right?
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According to Randâs theory, itâs not even a concept anyway. âA number greater than 1 but which divides both 9 and 30â isnât a concept either, even though it has some referent. 1. They are statements. 2. Intelligibility is not sufficient to makes something a concept. Somewhere in ITOE Rand talks about how merely seeing an assortment of characteristics (e.g. blonde woman with a blue dress and curly hair) is not a good enough reason to form a concept.
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âa number greater than 1 but which divides both 9 and 30â is not a statement. It cannot be true or false. Itâs not even a sentence in English.
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But it is necessary. Are you claiming that there are intelligible abstractions which are not concepts? What are the other conditions needed to make an intelligible abstraction a concept?
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How is âoneâ a proper noun? Thereâs only one Eiffel Tower; there are many "one"s.
I have one apple. And I have one orange. Is the number of oranges the same as the number of apples? If so, then there is only one number of both, i.e., âoneâ.
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I meant it seemed like a valid epistemology to you is one based on what an omniscient viewer would imagine.
Well I imagined it, but Iâm not omniscient.