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Concept-Formation & Mathematical Process

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With the grasp of the (implicit) concept "unit" man reaches the conceptual level of cognition, which consists of two interrelated fields: Conceptual and the Mathematical. The process of concept-formation is, in large part, a mathematical process.

Without the implicit concept of "unit," man could not reach the conceptual method of knowledge. Without the same implicit concept, there is something else he could not do: he could not count, measure, identify quantitative relationships; he could not enter the field of mathematics. Thus the same (implicit) concept is the base and start of two fields: the conceptual and the mathematical. This points to an essential connection between the two fields. It suggests that concept-formation is in some way a mathematical process.

Rand moves into measurement at this point, where Peikoff begins to describe the conscious processes.

Rand is so emphatic in the pronunciation here, whereas Peikoff appears to apply much less stress to this aspect.

Peikoff specifically mentions ITOE as a reference for those who are interested in delving into concept-formation more deeply as he begins his section on concept-formation as an introduction to Miss. Rands introduction.

I just happened to have both texts open and was reading them when this noticed this difference. Asserting that something is, has so much more emphasis that just suggesting that it might be. I do like the identification of the shared concept as the base of the two fields. I guess after such a powerful lead-in, I am left wondering why is there so much less degree of emphasis.

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Weaver it's funny because your quotes emphasize something burning a hole in my back burner of things to get back to when my time gets freed up.

While once again researching induction and the personalities on both sides of a couple mostly implict differences, I came to wonder about something. If you look at both quotes you'll notice in ITOE the 2nd edition, That "implict" is in parenthesis while in OPAR it is not. I don't want to go into why I found this importannt ,but I wonder if anyone knows if those parenthesis, or the word itself within,where added by later editors and not Mrs. Rand ??

Edit: I realize that it in parenthesis once in OPAR.

Edited by Plasmatic
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Weaver it's funny because your quotes emphasize something burning a hole in my back burner of things to get back to when my time gets freed up.

While once again researching induction and the personalities on both sides of a couple mostly implict differences, I came to wonder about something. If you look at both quotes you'll notice in ITOE the 2nd edition, That "implict" is in parenthesis while in OPAR it is not. I don't want to go into why I found this importannt ,but I wonder if anyone knows if those parenthesis, or the word itself within,where added by later editors and not Mrs. Rand ??

The parentheses exist in the original articles printed in The Objectivist - July 1966, the final editorial authority then being Ayn Rand.

p.s. Nitpicking, but that is a pronouncement by Rand, not a pronunciation.

Edited by Grames
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Now that you mention it, Peikoff left them off two of three places cited. A little earlier in ITOE, Rand stepped through the (implicit) concept "entity" --> the (implicit) concept "identity" and followed with:

The third stage consists of grasping relationships among these entities by grasping the similarities and differences of their identities. This requires the transformation of the (implicit) concept "entity" into the (implicit) concept "unit."

The implicit concept of "unit" is "the (implicit) concept 'unit'" arrived at in the third stage being used to reach both the conceptual level, and to count, measure and identify quantitative relationships.

Notice, after making these two distinctions, he returns to stating "the same (implicit) concept" is at the base and start of the two fields.

(An aside: Never as a kid, did I ever think that I would ever find a use for the limited exposure, I have pretty much long since forgotten, of sentence diagramming.)

edit to add: I agree with the nitpick. Thanks for pointing that out.

Edited by dream_weaver
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  • 1 month later...

The semantics of parenthetical objects aside, when I read The Logical Leap and checked ITOE, I found assumptions and assertions not validated by empirical research. I refer specifically to the investigations of Denise Schmandt-Besserat into the origins of counting and the origins of writing. They are the same. Writing began as inventory lists. As noted, these items were conceptual units: a sheep, a bottle of beer. But also, "a metal." That is an abstraction, not a concrete.

In most languages, counting is 1-2-3-many. "Higher" numbers 5, 6, 7, ..., were invented only about 8,000 years ago. The earliest accounting (tax) records use 2-2-1 for 5, and so on. Realize that this meant the existence of cities, political structure, temples (gods), taxes, and all that, and still no large numbers. And likely there was still no written poetry. The Gilgamesh appears about 1000 years after the first inventories in cuneiform.

So, numbering is, indeed, critical to abstraction. Realize though, that "metal" is a second-level generalization. Before the invention of cuneiform, there was a token symbol for that.

Ayn Rand (and Leonard Peikoff) worked as have many other frontline researchers - von Mises was another - by imagining hypothetical constructs to explain the development of our present state. The historical record is somewhat different. The distinctions are not fatal. I look at geometry as perhaps the perfect example of a study with deep historical roots, which nonetheless comes to us as an integrated body of knowledge, whose presentation now is decidedly not historical, but which remains valid and true.

(You can google Denise Schmandt-Besserat easily enough. On my website, two articles on the origins and history of money cite her work.)

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  • 2 months later...

<snip> Writing began as inventory lists. As noted, these items were conceptual units: a sheep, a bottle of beer. But also, "a metal." That is an abstraction, not a concrete.

<snip>

So, numbering is, indeed, critical to abstraction. Realize though, that "metal" is a second-level generalization. Before the invention of cuneiform, there was a token symbol for that.

<snip> I look at geometry as perhaps the perfect example of a study with deep historical roots, which nonetheless comes to us as an integrated body of knowledge, whose presentation now is decidedly not historical, but which remains valid and true.

(You can google Denise Schmandt-Besserat easily enough. On my website, two articles on the origins and history of money cite her work.)

That writing began as inventory lists is very plausible. I believe that Rand's observation 'unit' is at the base of both abstraction and mathematics is still in its infancy.

Understanding that 'bird', 'dog', 'cat' are considered first-level concepts, and observing my grandson playing with some magnetic toys recently would lead me to realize that "metal" is a first-level concept as well. The steel to which the magnet stuck had already been conceptualized already as "metal". The gold ring I wear and aluminum frames on some furniture were also conceptualized already as "metal" as well.

Pat Corvini in The Crisis of Principles in Greek Mathematics questions if Euclid's Elements are fully validated in this lecture (note: not challenging that the Elements principles outlined work.)

How Writing Came About looks like an interesting history.

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Understanding that 'bird', 'dog', 'cat' are considered first-level concepts, and observing my grandson playing with some magnetic toys recently would lead me to realize that "metal" is a first-level concept as well. The steel to which the magnet stuck had already been conceptualized already as "metal". The gold ring I wear and aluminum frames on some furniture were also conceptualized already as "metal" as well.

I do not understand what you mean by calling "metal" first-level.

There are metallic objects, but there is no such thing as "metal" as such independent of the objects.

A child would first learn that certain objects are called "toys". Later the child would compare and contrast his toys with other kinds of objects like "table" and "car" and "fork" to isolate their respective materials.

First you distinguish objects from the background.

Then you isolate their attributes.

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I do not understand what you mean by calling "metal" first-level.

There are metallic objects, but there is no such thing as "metal" as such independent of the objects.

A child would first learn that certain objects are called "toys". Later the child would compare and contrast his toys with other kinds of objects like "table" and "car" and "fork" to isolate their respective materials.

First you distinguish objects from the background.

Then you isolate their attributes.

Yes. Thanks, Vic. My assessment is missing that step. Hermes is correct. I was focused on differentiations within the family of metals.

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  • 2 years later...

Back along the lines of the opening comments:

from Salmieri's "Conceptulization and Justification" in "Concepts And Their Role In Knowledge"

"Measurement," writes Rand, "is the identification of a relationship - a quantitative relationship established by means of a stand that serves as a unit" (ITOE7) To grasp that existents are similar is already to measure them in a crude form, establishing their nearness by a series of pairwise comparisons (in which each of the compared items serves as a standard against which the other is measured).

 

So in seeing two or more men, we see the similarites by taking one of them to serve as a unit and "measure" the other(s) in a wordless they both have heads (though they may vary slightly is shape, size, etc.) arms, torsos, legs, etc., and integrate the basic aspects they have in common into the concept "man".

 

Consider a triangle. Seeing a triangular object off in the distance, we can approach it. As we do, the object in our visual field looms larger as we approach it. The triangular shape is the same, while the size appears to vary. We can see that other triangles are similar along the variation of size, and encountering other triangles recognize the proportions of the lengths of the sides can vary along a range as well.

 

It also ties in nicely with Miss Rand's observation: "In this sense and respect, perceptual awareness is the arithmetic, but conceptual awareness is the algebra of cognition." (ITOE)

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This sentence from the same work is most important to Salmieri's thesis.(which influenced Dr. Peikoff on his induction thesis...)

"One can raise questions about whether various people who are on the cusp of grasping some of the above points have knowledge or not, or about what degree of methodological sophistication is needed to know a given conclusion"

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Back along the lines of the opening comments:

from Salmieri's "Conceptulization and Justification" in "Concepts And Their Role In Knowledge"

"Measurement," writes Rand, "is the identification of a relationship - a quantitative relationship established by means of a stand that serves as a unit" (ITOE7) To grasp that existents are similar is already to measure them in a crude form, establishing their nearness by a series of pairwise comparisons (in which each of the compared items serves as a standard against which the other is measured).

 

So in seeing two or more men, we see the similarites by taking one of them to serve as a unit and "measure" the other(s) in a wordless they both have heads (though they may vary slightly is shape, size, etc.) arms, torsos, legs, etc., and integrate the basic aspects they have in common into the concept "man".

 

Consider a triangle. Seeing a triangular object off in the distance, we can approach it. As we do, the object in our visual field looms larger as we approach it. The triangular shape is the same, while the size appears to vary. We can see that other triangles are similar along the variation of size, and encountering other triangles recognize the proportions of the lengths of the sides can vary along a range as well.

 

 

i think you're extrapolating too far for that quote. Keep in mind the quote is specific to measurement, not concept formation. Measurement is necessary, but probably is not sufficient for concept formation. I wouldn't use differentiation into parts as good example, at least because that's an entity of an entity more or less. At the very least, it's too complex if you want to illustrate measurement. The triangle is better because it presumes no kind of concept to perform the quantitative comparison, it only presumes noting a change in visuals. Indeed, you can reason about it, but even without cognition, a mathematical relationship is still there and indicating some kind of relationship. From there, you could explain where concept formation arrives.

 

Many animals implicitly use mathematical relationships for navigation. Not conceptually of course, but their perceptual systems are quite complex, enough to do measurement as you are describing, which animals are aware of to some extent. See studies with bees, ants, and locusts to understand what I mean.

 

(Or this [link] book for the science on all that)

Edited by Eiuol
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I'm not beyond being put in check, Eiuol.

 

Moving toward or away from a triangular object provides a key visual observation supporting measurement omission, or being disregarded as an essential element of the concept.

 

Pairing in math the concept 3 is precisely the same every time, while pairing in concept formation does so along similarity allowing the particulars to be different.

 

Pairing in concept-formation is the identification, though not always explicitly, that the range of varience of a particular measurement is not relevant to the object being what it is.

 

The basic shapes of a coffee cups can vary. Using cup 'A' as a unit and comparing cup 'B' and 'C' gives the similarity and allows us to disregard the differences observed in specific diameters or heights, or variations observed in the handles. These can be used to differentiate between cups 'A', 'B', and 'C', but are not relevent to them being coffee cups. We see the differences, yet disregard them as being relevant, or integrate them as the algebra, a wordless formulation of formulas such as diameter=range 2"-3", height=range4"-5", handle=range of shapes and sizes, etc.

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  • 9 months later...

Plas,

 

Yes, Bergson thought we have an intellectual power he called intuition, and it is in contrast and in some tension with (but also in some support of) what we call the intellect, our power of concepts, judgments, and inference.

 

In Creative Evolution, he writes:

 

That is a bit odd.

 

In addition to Bergson’s resonance with Rand on differentiation and integration, at least in a certain consciousness we have in metaphysics, I see also the following rough resonance:

 

Integration of this perspective with the circumstance of our broader geometries such as ordered geometry or affine geometry and with Rand’s cast of similarity classes of qualities in terms of measure-value suspensions could pay dividends.

Ordered geometry means 'unmeasured'--or using algebra--, of which 'affine' (parallels) are a part.Can Rand's concepts be reduced to math?

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Ordered geometry means 'unmeasured'--or using algebra--, of which 'affine' (parallels) are a part.Can Rand's concepts be reduced to math?

 

I think that one can overplay one's hand here. One can and does quickly arrive at floating abstractions within mathematics. I believe that this discussion should be limited to a discussion of natural numbers. Primitive societies, such as existed in Europe during the dark ages, can lose sight of what "two means". As a result, you end up with differing words for this concept, such as a yoke of oxen, brace of geese, pair of threes in poker, a couple, duo, dyad, etc. Through a process of unit reduction, we can identify each of these concepts as representing and instance of "two". Such an identification might have seemed remarkable to a primitive consciousness.

 

This does not mean that there is an isomorphism between Randian concepts and those within mathematics.

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Instead of trying to turn concept formation into a form of mathematics, doesn't it make more sense to observe, concept formation and mathematics utilize abstraction in the same way?

 

It's like observing the streets are formed in the same way as they are found on a map and saying the streets are an expression of the map or an eight year old seeing that after putting five marbles in a bag and putting another three in the bag he counts eight in the bag after which he exclaims marbles obey the addition tables he learned in school.  Streets do not express maps and marbles do not obey math, maps and math are things we have developed in order to help us describe/understand streets and marbles.

 

 

Mathematics is an abstract language which has as its basis rational thought and specifically concept formation. Concept formation is not built on math (although it resembles it) math is built on concepts.  

Edited by StrictlyLogical
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I think that one can overplay one's hand here. One can and does quickly arrive at floating abstractions within mathematics. I believe that this discussion should be limited to a discussion of natural numbers. Primitive societies, such as existed in Europe during the dark ages, can lose sight of what "two means". As a result, you end up with differing words for this concept, such as a yoke of oxen, brace of geese, pair of threes in poker, a couple, duo, dyad, etc. Through a process of unit reduction, we can identify each of these concepts as representing and instance of "two". Such an identification might have seemed remarkable to a primitive consciousness.

 

This does not mean that there is an isomorphism between Randian concepts and those within mathematics.

All concepts in math are justified by 'formal proofs' wich do not involve arithmetic (numbers).

 

Kindly, moreover, clarify your 'primitive society comment for Europe's 'dark Ages'...

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