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I had a question about the way in which we form concepts.  The definition of a concept is “a mental integration of TWO OR MORE units possessing the same distinguishing characteristics, with their particular measurements omitted.”  However, when I listen to Peikoff’s lectures about how to form concepts, his examples always involve something more than just the two units. It involves something else to differentiate the units from.  For example, he explains how to form the concept blue by observing two particular instances of blue, one shade of blue and a slightly different shade of blue and omitting their particular shades while retaining the characteristic “BLUE” and distinguishing them from red.  This third unit serves as what you use to differentiate the former two units from and there are rules that have to be followed regarding this unit. He mentions seeing “two blue objects AS OPPOSED TO RED.” So there’s always something more included in the conceptualization process than just the two or more units.  The rule that stands out the most to me is that the characteristic that you are using to differentiate the former two units from the third has to be “commensurable” with the former two units.

I am trying to see how the concepts I have measure up against Rand’s theory of concept formation but I am having difficulties doing this with many of the concepts I have.  My biggest problem so far is I don’t understand what commensurable existents I am differentiating the units of my concepts from.

How did I form the concept “color?”  At first I thought I saw a red object and I saw a blue object and I omitted the particular colors in question and integrated them into the concept “color.”  But I’m missing a key piece in the process of concept formation (which by the way is not mentioned in the definition of the concept, as a I stated above). I am missing the COMMENSURABLE existent that I need to differentiate the two units FROM.

I am trying to figure out what that existent is.  I was thinking maybe it could be a *ring of a phone or a particular instance of “sound.”  So I observed blue and red and I heard a *ring and I realized that blue and red were far more similar to each other than they both are to a *ring, so I integrate them into the concept “color” and I differentiate them from a *ring of a phone.  But am I wrong about this because a *ring is not COMMENSURABLE with red or blue? And this is just one example. I don’t understand how I formed the concept “sound” for the same reasons.  I was thinking I heard two different sounds, like a *ring and a *thud, but I dont know what existent I differentiated them from?

The other day I read that Rand formed the concept “entity” by observing two separate entities and DISTINGUISHING THEM FROM THEIR ATTRIBUTES.  How does she know that their attributes are the commensurable existents that she is supposed to use to differentiate them? I mean now that I think about it, I can maybe understand that an entity is commensurable with its attributes because it could be thought of as a collection or more precisely an integration of its attributes, but I’m still struggling with this.

For pretty much every concept I have, I can’t identify what existent I distinguished the units from and this is making me wonder “If I can’t do this, does this mean that I haven’t actually formed the concepts that I thought I formed?”  And if I haven’t formed the concepts that I thought I formed, how am I able to even think about the referents that I am thinking about? This seems to me to be a dumb question but I’m seriously wondering about this.

I think I’ve succeeded with a couple concepts, the concept of “pushes” and “pulls.”  You observe pushes and pulls, you observe Push1 and Push2 and omit the particular magnitudes of the pushes while distinguishing them from a pull to form the concept “push.”  You do the same thing vice versa for the concept “push.” But that’s as far I can go. I can’t even form the concept “force” in a physical context because I can observe a “push” and then a “pull” but again I don’t know what commensurable existent to differentiate them FROM to form the concept force?

I was hoping somebody could help me solve these problems?  And also, could someone tell me, why isn’t that third existent that you are using to differentiate the “two or more units” from included in the definition of a concept?  It seems to be an essential piece of the conceptualization process and therefore shouldn't it be included in the concept's definition?

Edited by [email protected]
Forgot to mention the last question

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To form the concept X, you need instances of the category -- call them x's -- and contrasts -- call them non-x's. To limit the number and variety of non-x's, both the x's and non-x's should satisfy a more abstract, wider, less specific, category. For example, to form the concept red instances of red are differentiated from instances of blue, green, etc. However, all the instances -- red, blue, green, etc. -- are instances of the wider concept color.

I suggest "The Sim-Dif Model and Comparison" in The Journal of Ayn Rand Studies, Vol. 11, No. 2 (December 2011), pp. 215-232. The entire essay can be read on jstor.org with a free membership.

Edited by merjet

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@merjet I'll check out that document.  But I wanted to respond first to what you mentioned:

Quote

To limit the number and variety of non-x's, both the x's and non-x's should satisfy a more abstract, wider, less specific, category.

How do you know that the x's and non-x's satisfy "a more abstract, wider, less specific, category" if that category corresponds to a higher-level concept which you have not formed yet? In your example, you're trying to form the concept of X, but you're using the higher-level concept that subsumes the X's and Non-X's to guide you in forming the concept X.  And you're using that higher-level concept when you impose the requirements that the x's and non-x's "should satisfy a more abstract, wider, less specific category."  It seems like you're stealing the higher-level concept to form the lower level one...

And by the way, since you mentioned color, can you tell me what existent did you differentiate red, blue, and green FROM to form the concept "color?"

Edited by [email protected]

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2 hours ago, [email protected] said:

It seems like you're stealing the higher-level concept to form the lower level one...

And by the way, since you mentioned color, can you tell me what existent did you differentiate red, blue, and green FROM to form the concept "color?"

That was decades ago, so I don't remember very clearly. When my parents or somebody else helped me learn red, blue, green, and so on, somehow or another I reckoned those words didn't refer to a thing, shape, motion, sound, etc. They probably also often used the word "color" along with "red", "blue", "green" and so on. Eventually I learned that "color" named the category to which red, blue, green, and so forth belonged. If that was stealing from my parents or somebody else, then I must plead guilty.  🙂

The first paragraph of my earlier post was an attempt to describe the essence of the process, not the whole story and context.

Edited by merjet

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8 hours ago, [email protected] said:

And also, could someone tell me, why isn’t that third existent that you are using to differentiate the “two or more units” from included in the definition of a concept?  It seems to be an essential piece of the conceptualization process and therefore shouldn't it be included in the concept's definition?

There is no third unit, at least not regarded as a unit of the concept you are trying to make.

The definition of a concept requires a genus and a differentia.  You are having trouble identifying the genus of force.  But it is a very elementary or fundamental thing which can only be identified as an existent, one of the things that exists. Force, mass, space, electric charge and a few other concepts refer to what existence is.  Existence (as concept or as being) itself does not have a genus, a larger category or background from which it is abstracted.  When dealing with such primary existents and the subjective experience of sensations resort to the ostensive definition.

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In that quote in your response I called it an existent.  But I do see earlier in my post I called it a "unit" and this was a mistake on my part.  I thought calling it an existent was a better choice of words because I didn't know what other word to refer to it by.  What would you call the "redness" when you are using redness to differentiate two shades of blue from in the process of forming a concept of blue? I think of it as "what you are using to differentiate blue from."  And it doesn't have to be red it could be green right?  Rand said you need something COMMENSURABLE that you can look at and use to differentiate the units of your concept from.  So I'm wondering what did we use to differentiate the units of various concepts from?  What did we differentiate red and blue from when we formed the concept "color?"  What did we differentiate a thud and a ring from when we formed the concept sound?  I'm wondering if I'm being unclear in my questions or if I misunderstood Rand or Peikoff about concept formation.  When you're dealing with ostensive units it seems to be easy to identify "what you are using to differentiate the units of your concept from."  Like when you form the concept "cat" by observing two cats and a dog and omitting parricular measurements and differentiating the cats from the dog.  But "color" and "sound" are more abstract.  So I'm wondering what did we use to differentiate various colors such as "red" and "blue" from that is commensurable with red and blue in order to be able to form the concept "color" in the first place?  I can't think of it.

Also isn't a force primarily a push or a pull?  You can't form the concept force unless you form the concept of push and pull first right?  It would be like forming the concept furniture before forming the concept chair and table, right?  I agree it's very primary.  Force is directly perceptible and that's why I was saying you experience two different particular pushes and a pull and after omitting measurements you differentiate the pushes from the pull and integrate them into the concept "push" and vice versa for the concept "pull."  Then after that you're ready to form the concept force, which is on a level that is higher from "push" and "pull."  That's my understanding.

Edited by [email protected]

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1 hour ago, [email protected] said:

So I'm wondering what did we use to differentiate various colors such as "red" and "blue" from that is commensurable with red and blue in order to be able to form the concept "color" in the first place?

Perception? Similarity but not being identical?

Edited by merjet

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3 hours ago, [email protected] said:

Then after that you're ready to form the concept force, which is on a level that is higher from "push" and "pull."  That's my understanding.

You didn't say explicitly that learning higher level concepts must come after learning lower level ones. Regardless, learning higher level concepts may precede learning lower level ones. For example, we probably all learned the concept car before we had the concepts Ford, Chevy, Toyota, sedan, convertible, etc.

Edited by merjet

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15 hours ago, [email protected] said:

The other day I read that Rand formed the concept “entity” by observing two separate entities and DISTINGUISHING THEM FROM THEIR ATTRIBUTES.

I like that. Do you recall where you read this?

Another element to consider in reference to the order of learning concepts is chronological versus logical order. A great many concepts we automatized prior to discovering concepts have logical inter-connections. I like how Harry Binswanger's Abstractions from Abstractions presentation described higher level concepts derived from yet higher as well as lower level concepts.

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Page 132 of ITOE.  She doesn't say this herself but she does respond "That's right" after the person she is having a discussion with states "For instance, in the case of concrete versus entity, the units are the same, but the concept entity distinguishes entities from attributes, while the concept concrete distinguishes entities from abstractions."  To this statement, Rand responded "That's right.

Yes it is possible to form higher-level concepts before you form lower-level ones and Rand does discuss the possibility of a child doing this at some point in ITOE.  I think she also mentions that this is possible only up to a certain level in the conceptual hierarchy.  I think she says the determining factor regarding the level that you start at when you form concepts is going to be "what is available to your observation."  But again, I think she said, there is a limit at how high of a level you can start at.

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17 hours ago, [email protected] said:

How did I form the concept “color?”  At first I thought I saw a red object and I saw a blue object and I omitted the particular colors in question and integrated them into the concept “color.”  But I’m missing a key piece in the process of concept formation (which by the way is not mentioned in the definition of the concept, as a I stated above). I am missing the COMMENSURABLE existent that I need to differentiate the two units FROM.

Color is an attribute of an entity. So you differentiate it from other attributes like shape, size, width, length.

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7 hours ago, [email protected] said:

What would you call the "redness" when you are using redness to differentiate two shades of blue from in the process of forming a concept of blue? I think of it as "what you are using to differentiate blue from."  And it doesn't have to be red it could be green right?

If all that existed was an identical shade of red except for that one patch of blue (how weird) then the genus of the definition of blue would equally valid as "red" or "existence".  If it were imagined as green instead of red it would make no difference as long as there was the phenomena of not-blue and blue.

I agree with MisterSwig about attributes.  We never actually encounter colors on their own but only things that have colors.  So the colors as a category are distinguished from the other aspects of those things, their size and weight and roughness and softness and smell etc.

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20 hours ago, [email protected] said:

The other day I read that Rand formed the concept “entity” by observing two separate entities and DISTINGUISHING THEM FROM THEIR ATTRIBUTES.

4 hours ago, dream_weaver said:

I like that. Do you recall where you read this?

See ITOE Expanded Second Edition p. 274.

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This is an important question that I've been thinking about as well. The more I introspect, the more I realize that my brain just doesn't work the way that Rand describes. For me, if I merely know how to classify a thing, I don't feel like I understand it. But if I can see its structure and the structures it forms in relation to other things and how that structure might be changed and so on, only then do I feel like I truly understand it, and only then can one come up with truly non-arbitrary classification schemes.

As an example from algebra, if you were to just tell me that a group is just a monoid where every element has an inverse, I wouldn't understand the concept of "group". But if you were to then show me a rotating triangle and how those rotations relate to the group operation, then I would easily understand the concept of a group.

Even the concept of "concept" (the one that Rand describes) I currently understand primarily in terms of space. For instance, when thinking about the relationship of dogs to all other animals, I see a big circle in my mind labeled "animals", and within that circle, a smaller circle labeled "mammals", and within that circle, a smaller circle labeled "dogs".

So I think that all of the concepts in my mind are spatial ones. Therefore, I strongly suspect that the process of concept formation is some kind of operation on spaces.

At the very least, this is true in my case. other people's brains might work differently. Some people can't see anything in their mind's eye. Others don't hear an internal monologue. And some think only in terms of pictures of things they've actually seen. It's too simplistic to think that everyone's brain works just like yours does.

EDIT: I should probably mention some of the research on this topic that I've been doing. I've been studying category theory, and I think the idea of adjunctions may hold the key to concept formaiton. By using adjunctions, one can "mechanically" derive significant mathematical concepts from totally trivial ones. I just need to find an interpretation of adjunctions that makes sense.

Edited by SpookyKitty

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39 minutes ago, SpookyKitty said:

For me, if I merely know how to classify a thing, I don't feel like I understand it. But if I can see its structure and the structures it forms in relation to other things and how that structure might be changed and so on, only then do I feel like I truly understand it, and only then can one come up with truly non-arbitrary classification schemes.

This suggests a metaphor. Classifying pertains to the skeleton. A fuller understanding adds flesh.

39 minutes ago, SpookyKitty said:

Even the concept of "concept" (the one that Rand describes) I currently understand primarily in terms of space. For instance, when thinking about the relationship of dogs to all other animals, I see a big circle in my mind labeled "animals", and within that circle, a smaller circle labeled "mammals", and within that circle, a smaller circle labeled "dogs".

So I think that all of the concepts in my mind are spatial ones. Therefore, I strongly suspect that the process of concept formation is some kind of operation on spaces.

Your use of circles suggests Venn diagrams, which are depicted spatially. My essay '"The Sim-Dif Model and Comparison" cited above uses Venn diagrams extensively.

39 minutes ago, SpookyKitty said:

I've been studying category theory, and I think the idea of adjunctions may hold the key to concept formaiton. By using adjunctions, one can "mechanically" derive significant mathematical concepts from totally trivial ones.

Interesting.

Edited by merjet

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On 2/20/2020 at 5:49 PM, [email protected] said:

I had a question about the way in which we form concepts.  The definition of a concept is “a mental integration of TWO OR MORE units possessing the same distinguishing characteristics, with their particular measurements omitted.”  However, when I listen to Peikoff’s lectures about how to form concepts, his examples always involve something more than just the two units. It involves something else to differentiate the units from.  For example, he explains how to form the concept blue by observing two particular instances of blue, one shade of blue and a slightly different shade of blue and omitting their particular shades while retaining the characteristic “BLUE” and distinguishing them from red.  This third unit serves as what you use to differentiate the former two units from and there are rules that have to be followed regarding this unit. He mentions seeing “two blue objects AS OPPOSED TO RED.” So there’s always something more included in the conceptualization process than just the two or more units.  The rule that stands out the most to me is that the characteristic that you are using to differentiate the former two units from the third has to be “commensurable” with the former two units.

I am trying to see how the concepts I have measure up against Rand’s theory of concept formation but I am having difficulties doing this with many of the concepts I have.  My biggest problem so far is I don’t understand what commensurable existents I am differentiating the units of my concepts from.

How did I form the concept “color?”  At first I thought I saw a red object and I saw a blue object and I omitted the particular colors in question and integrated them into the concept “color.”  But I’m missing a key piece in the process of concept formation (which by the way is not mentioned in the definition of the concept, as a I stated above). I am missing the COMMENSURABLE existent that I need to differentiate the two units FROM.

I am trying to figure out what that existent is.  I was thinking maybe it could be a *ring of a phone or a particular instance of “sound.”  So I observed blue and red and I heard a *ring and I realized that blue and red were far more similar to each other than they both are to a *ring, so I integrate them into the concept “color” and I differentiate them from a *ring of a phone.  But am I wrong about this because a *ring is not COMMENSURABLE with red or blue? And this is just one example. I don’t understand how I formed the concept “sound” for the same reasons.  I was thinking I heard two different sounds, like a *ring and a *thud, but I dont know what existent I differentiated them from?

The other day I read that Rand formed the concept “entity” by observing two separate entities and DISTINGUISHING THEM FROM THEIR ATTRIBUTES.  How does she know that their attributes are the commensurable existents that she is supposed to use to differentiate them? I mean now that I think about it, I can maybe understand that an entity is commensurable with its attributes because it could be thought of as a collection or more precisely an integration of its attributes, but I’m still struggling with this.

For pretty much every concept I have, I can’t identify what existent I distinguished the units from and this is making me wonder “If I can’t do this, does this mean that I haven’t actually formed the concepts that I thought I formed?”  And if I haven’t formed the concepts that I thought I formed, how am I able to even think about the referents that I am thinking about? This seems to me to be a dumb question but I’m seriously wondering about this.

I think I’ve succeeded with a couple concepts, the concept of “pushes” and “pulls.”  You observe pushes and pulls, you observe Push1 and Push2 and omit the particular magnitudes of the pushes while distinguishing them from a pull to form the concept “push.”  You do the same thing vice versa for the concept “push.” But that’s as far I can go. I can’t even form the concept “force” in a physical context because I can observe a “push” and then a “pull” but again I don’t know what commensurable existent to differentiate them FROM to form the concept force?

I was hoping somebody could help me solve these problems?  And also, could someone tell me, why isn’t that third existent that you are using to differentiate the “two or more units” from included in the definition of a concept?  It seems to be an essential piece of the conceptualization process and therefore shouldn't it be included in the concept's definition?

When the grandson was a toddler, we were at a restaurant, and I walked him around at one interval to a small Christmas tree with those old-fashioned colored lights. Pointing to a particular light, I asked him “What color is that?” He always answered correctly as I continued to point and cover the gamut on the tree. I’m pretty sure that the ability to identify individual colors in a grouped array is not originally the seeing those colors as individuals subsumed under the concept color. One has learned the proper relation of words blue and color in learning some language and sharing the world through it. (On cognitive developmental psychology of learning the various sorts of attributes of things, see 28-33 here.)

I’d be pretty surprised if learning the distinction between entities and their attributes did not require learning subject-predicate relation in one’s native language. To be sure, with further cognitive development, on learns how to put any category in the subject spot, not just entity, but at first I’d bet a coke it’s only entity there. Learning concepts and language seems to be a hand-over-hand sort of deal.

To reach our modern basic concept of force, basically Newtonian, took a lot adult thinking and controversy. Though push and pull are examples of it, earlier in intellectual history, for example, not only certain changes in state of motion required some sort of force, but keeping something moving (even in vacuum) was thought to require force. I imagine beginning to get the elementary concept of force (which happens to fall under the precised Newtonian one approximately) is already underway in learning first words. When a young child says ba (ball), turning it over in her hands, the object being isolated is known to have various ways it can be made to act by the child.

Talk of “concept formation” (and “conceptualization process”) in epistemology did not begin with Rand. However, in epistemology, I think it’s better to stop using that phrase if one is not literally talking about cognitive development in time, and if one is talking about that, it must be informed by the relevant contemporary psychological research to have any serious traction. However, the epistemological business (a philosophical business) of offering ways of seeing how various sorts of concepts are organized in their relations to each other and to the things they are about is just fine, mighty fine. That is analysis, not mainly theory of the genesis of concepts, and we should stop calling it a theory of concept formation. The enduring fertile innovation of Ayn Rand in theory of concepts, I say, is an analysis: the proposal of a mathematical structure to concepts in their taxonomic relations.

Edited by Boydstun

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Thanks. Pg. 240 on the research CD. My copy of ITOE doesn't include this.

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5 hours ago, SpookyKitty said:

For me, if I merely know how to classify a thing, I don't feel like I understand it.

I don't think Rand ever claimed that knowing how to classify something is sufficient to understand it. Classification is necessary to begin with, and only then can understanding start. That isn't so different than Aristotle, the difference is how Rand explains the way classification works. What you're talking about is a theory of induction, which she did not develop, a problem that she was aware of. I think she successfully defended why classification schemes may begin as arbitrary (in the sense there is no rule to determine which measurements or attributes must be used for a classification). I agree though that Rand did not provide a theory or process that "fills" a concept with structure or relationships. It's one thing to form a concept, it's another to fill it with content.

Rand was wrong to put so much focus on measurement omission. She only talks about quantitative measurements, nothing qualitative. Spatial measurements in your mind are more qualitative than anything, not to mention that spatial visualization is an important skill for cognition. That's why I really like what you mention about thinking of concept classifications in a spatial sense. This is what I do as well. And when I compare two things, I usually do it in strictly qualitative terms. I think the only time I make quantitative comparisons is when the concepts are literally concepts of measurement or quantitative concepts. But it isn't too hard to reformulate Rand's theory of classification or create a new one to allow for spatial relationships.

Personally, I don't know how to think of concepts in any way besides spatially. For hundreds of years, people have known that spatial thinking vastly helps memory, and takes an important role when contemplating anything abstract. Roman philosophers wrote about it, Aristotle wrote about it. It's unfortunate that Rand was pretty ignorant about spatial thinking and memory, her theory of classification would be even better if she wasn't.

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10 hours ago, Eiuol said:

Rand was wrong to put so much focus on measurement omission. She only talks about quantitative measurements, nothing qualitative.

When you omit the measurements a quantitative measurement becomes qualitative.  That is what a quality is: a certain range of measurements.  The quality of red means (refers to) any of the various shades and intensities of color within the range of red, and it does so open-endedly (all reds near, far, past, future, known, unknown).   Quality is itself a concept, not a concrete.  The philosophical problem is relating concepts to concretes.  Once a method of handling concretes conceptually is found, handling qualitative thinking is just more of the same.

And I don't understand how any of this other grumbling by others about spatial thinking is at all well founded either.  Space has measurements.  Measurements of distance can be omitted to form concepts of directions, directions can be omitted to form the concepts of near and far, both types of measurements can be omitted to specify relationships such as "on top of" or "to the left of".

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17 hours ago, SpookyKitty said:

For me, if I merely know how to classify a thing, I don't feel like I understand it. But if I can see its structure and the structures it forms in relation to other things and how that structure might be changed and so on, only then do I feel like I truly understand it, and only then can one come up with truly non-arbitrary classification schemes.

Your post made me struggle with the differentiation of "knowledge" and "understanding". Isn't to know include to understand? You may say that the unfamiliar is the not-understood. But even then, you understand that it exists, very low level, but to know of its existence is to understand something about it.

So it was a little confusing to see you say that you know but do not understand. Furthermore "to see it's structure" to me implies to examine how it fits in with everything, which inevitably will cause searches for types/category of things it could be. So I can't completely separate them.

Classification as in memorizing the class/category that something belongs to is obviously a hollow understanding of something. Where does objectivist epistemology imply that understanding is primarily classification/categorization? (although it may indicate "some" understanding". In other words, what did Rand say that does not correspond to how you "know"?

Understanding of something has continuum, from no knowledge to "full understanding" (however that is determined). Full understanding of something is not determinable unless one has an objective test to use. And of course that is based on different contexts. The main experience of "not understanding" is "it does not make sense" or hard to imagine.

I am also curious to know how you would define, or know that you have fully understood something.

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5 hours ago, Grames said:

When you omit the measurements a quantitative measurement becomes qualitative.  That is what a quality is: a certain range of measurements.  The quality of red means (refers to) any of the various shades and intensities of color within the range of red, and it does so open-endedly (all reds near, far, past, future, known, unknown).

1. Consider different contents of books. Some books are fiction and others are non-fiction. Types of fiction are mystery, romance, children’s stories, etc. Types of nonfiction are history, science, mathematics, music, food recipes, etc. These various contents are congruous but not commensurable. The differences between them that need to be omitted to form the concept book are qualitative, not measurable.

2. Consider different kinds of boats. There are rowboats, some with an outboard motor, some are steamboats, some have an inboard diesel engine, some have paddle wheels, some airboats/fanboats, and so forth. The different kinds of locomotive power are qualitative differences.  That some aspect of the different kinds, e.g. horsepower, is measurable does not overturn the fact the some differences between the kinds are only qualitative.

3. A child capable of identifying different colors is not measuring anything. People who lived before it was known that colors reflect different light wave lengths/frequencies were not measuring anything. Also, a wave length is a length, not a color.

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17 hours ago, Eiuol said:

Rand was wrong to put so much focus on measurement omission. She only talks about quantitative measurements, nothing qualitative.

You might like my essay "Omissions And Measurement" in The Journal of Ayn Rand Studies, Volume 7, No. 1 – Fall 2005, pp. 383-405. The entire essay can be read on jstor.org with a free membership.

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I agree with a lot of what you said. I also think qualitative distinctions are more fundamental than quantitative ones.

I will write out some rough thoughts I've been having and some research I've been doing on this subject. Hopefully it will make things clearer.

Imagine that you have two entities where the measurements of the first entity with respect to some characteristics (that we care about) are (1.0,1.0,1.0) and the measurements of the second entity are (2.0,1.0,1.0). Now, at the end of the day, Objectivism allows you to form exactly one "big" concept from this data set:

A = (x,1.0,1.0)

where the use of the variable "x" means that the entities belonging to this concept must have some measurement value for the first characteristic, but may have any measurement value. But also, we can use differentia to get many more "small" concepts by specifying ranges that the variable is allowed to take. For example:

B = ([1.0, 12.0], 1.0, 1.0)

means that the value of the first characteristic can be anything between 1.0 and 12.0, but the values of the other two characteristics must be exactly 1.0 and 1.0, respectively. So B is a sub-concept of A. You can take concepts like these and make further restrictions to get sub-concepts of sub-concepts, for instance

C = ([2.3, 4.6], 1.0, 1.0).

And, I don't think it would be too much of a stretch to say that you can do disjunctive sums of intervals to get even more complex concepts such as:

D = ([1.0, 12.0] + [26.5, 123.4], 1.0, 1.0)

where the notation "+" means that the entity is allowed to have values either in the interval [1.0, 12.0] or in the interval [26.5, 123,4]. Furthermore, by using these two entities in combination with others, again, I don't think it would be an unreasonable interpretation of Objectivism to say that you can have concepts that look like this:

E = ([1.0, 12.0] + [26.5, 123.4], y, [0.1, 2.6])

or even ones that allow for infinite collections of allowed intervals like this:

F = ( [1,0, 2.0] + [4.0, 5.0] + [7.0, 8.0] + ..., y, z).

If you spend some time graphing these examples, you will notice that all of the concepts formed by these methods look like rectangular prisms arranged in rectangular prisms arranged in rectangular prisms, etc., all of which have edges that are parallel to at least one of the axes. Furthermore, these are the only kinds of shapes that concepts in Objectivism are allowed to have. I hope that this will make clear what I meant when I said that Rand's theory of concepts is "merely classifying things".

The problem with her theory is that there will always be real-world collections of entities that completely confound this sort of scheme. That is, when you plot the measurements of all the entities in the collection, they might form a shape which is very simple but which cannot be a combination of rectangular prisms.

For instance, consider the collection of entities:

{(0,2),(1,1),(2,0),(3,1.1)(4,2.1),(3.1,3),(2.1,4),(1.1,3)}

Any combination of measurement omissions and restrictions to intervals will result in just two kinds of interpretations of the data. On the one hand, you will have very simple interpretations which underfit the data. And on the other hand, you will have very complicated and counter-intuitive interpretations which overfit the data. And in no case whatsoever will you obtain a system of concepts which notices the super-simple underlying pattern you would have gotten had you simply plotted some points and used your spatial intuition to play connect-the-dots.

Even though none of the entities in the above example have any measurements in common, by using your spatial intuition, you can form a simple network of similarities:

(0,2) ~ (1,1) ~ (2,0) ~ (3,1.1) ~ ... ~ (1.1,3) ~ (0,2)

which your brain would immediately recognize as a one-dimensional loop. A one-dimensional loop is a notion that:

1) Is highly abstract, and can be applied to just about any data whatsoever to yield tons of non-trivial information about that data.

2) Is super easy to understand. It's almost concrete in how easy it is to understand.

3) Captures the essence of how the entities in the example are related in a very simple and accurate way, even though they are all different from each other.

I say "accurate" in the above because saying that the data is described by a one-dimensional loop also implies a network of dissimilarities among the given entities. For instance, we can say that (0,2) is dissimilar to (2,0), because, on the one-dimensional loop, there is no shortcut from (0,2) to (2,0) which allows you to skip the entity (1,1).

The process of fitting a manifold to a set of data-points is studied in the field of Persistent Homology:

In my opinion, I think that Rand was trying to do something like this when she formulated her theory of concept formation.

Rand's theory also suffers from another problem which I've been trying to address. Basically, it smuggles huge portions of mathematics (at the very least the non-zero rational numbers), which themselves have highly non-trivial spatial structure, into the notion of measurement. This, in addition to the above, is why I don't find Grames' account of how concepts of space can be derived from measurement omission at all convincing.

I believe that this problem can be remedied by claiming that the human brain comes equipped with a very small number of simple spatial ideas and operations which can be used to form any mathematical concept including the concepts of logic.

All this has lead me to investigating the theory of simplicial sets. These are very simple and very interesting mathematical gizmos that can encode combinatorial and topological information simultaneously. Furthermore, they constitute what is called a "topos" in category theory, which means that they are capable of serving as a foundation for all of mathematics. Additionally, every topos has its own internal logic, (and these logics are, in general, higher-order intuitionistic type theories). So there is a conception of logic out there somewhere which can be derived entirely from spatial concepts.

The main problem is that the standard theory of simplicial sets allows for simplices of arbitrarily high finite dimension, and the human brain can handle only 3. However, as it turns out, it's very easy to prove that simplicial sets restricted to at most 3 dimensions also constitute a topos.

I am currently trying to figure out the ins-and-outs of all of this stuff, but I think that Rand's dream of a mathematical epistemology is on the horizon.

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2 hours ago, merjet said:

1. Consider different contents of books. Some books are fiction and others are non-fiction. Types of fiction are mystery, romance, children’s stories, etc. Types of nonfiction are history, science, mathematics, music, food recipes, etc. These various contents are congruous but not commensurable. The differences between them that need to be omitted to form the concept book are qualitative, not measurable.

2. Consider different kinds of boats. There are rowboats, some with an outboard motor, some are steamboats, some have an inboard diesel engine, some have paddle wheels, some airboats/fanboats, and so forth. The different kinds of locomotive power are qualitative differences.  That some aspect of the different kinds, e.g. horsepower, is measurable does not overturn the fact the some differences between the kinds are only qualitative.

3. A child capable of identifying different colors is not measuring anything. People who lived before it was known that colors reflect different light wave lengths/frequencies were not measuring anything. Also, a wave length is a length, not a color.

1.  'Book' is a first level concept that even an illiterate child or adult can have and use correctly.  One can also understand that the words (content) are the essential characteristic of books that cause the other characteristics and that the book only exists at all for the sake of the words within, all while being unable to read them or read at all.  Certainly one understands books better if one were literate but not a different set of objects.

2. This boat example is merely taking for granted the quality of the locomotive power, treating it as a primary when it is not.  Oared boats differ in the number and placement along a spectrum from a one person dinghy to a competitive crew boat for racing to an ancient trireme. Paddle wheel boats differ in the size, placement midships or stern, and whether single or split port and starboard.  Outboard motor boats differ in horsepower, size, weight, and fuel type and whether one or more motors are used.  Each of these different kinds of locomotive power do form a qualitative difference between boats but each is also a conceptual category itself.

3. Rand's point with measurements is that they were always there in the form of the attributes of entities.  Recall that Rand also teaches that entities are their attributes, not that entities have their attributes.  Thus there is no underlying substratum, no place for an Aristotelian essence to be (nor a quality) and no role for a metaphysical essence (or a quality) in knowledge.  The attributes are all that exists, they exist in definite specific form amenable to measurement, and even when not mathematically measured merely perceiving a thing creates a particular percept  (perceptual form in Kelley) in the subject.  The percept is a measure, caused by and corresponding to the thing perceived and the means of perceiving it (and by Kelley the environment in which it was perceived).

"Implicit measurement" is a phrase Rand uses on page 13 of ITOE 2nd when writing about color.  Perceiving similarity among percepts is also given directly by the perceptual faculty.  One does not need to measure explicitly or know how to measure explicitly to perceive similarity between percepts or an entire range of percepts.  The measurements involved are implicit and automated by the sensing and perceiving structures of the body.  For example we know scientifically how color perception works, how the eye has structures named rods and cones and there is a set just for red light.  Redness and similarity of redness means the neurons connected to the red-responding rods and cones being excited and then the same ones being excited again.

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2 hours ago, SpookyKitty said:

I hope that this will make clear what I meant when I said that Rand's theory of concepts is "merely classifying things".

I hope you may someday come realize that "merely classifying things" is precisely what concepts do and are for.  Knowing in depth about what the concepts refer to is a different problem.

That a string of numbers in one format seems meaningless but in another is easily recognizable has little to do with Rand's theory of concepts but is rather about the human body's means of perceiving.  The implicit measurements of perception are actually easier to work with conceptually than the explicit measurements of an abstract set of numbers.  That said, there is no reason why an artificial structure cannot be made to work as Rand describes directly with those numbers in the explicit form.

You need to become more familiar with the capabilities of networks of neurons.  They can recognize complex patterns not merely rectangular ranges.  There is thread here that brought his theory of Hierarchical Temporal Memory to my attention.

edit: searching youtube for "Jeff Hawkins" or "hierarchical temporal memory " brings up lots of hits to the present, he is continuing his work.  Example

Edited by Grames

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