# Universals and Measurement

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The category I have targeted as reflecting the metaphysical magnitude structure implied by Rand's measurement-omission form of concepts is the hybrid category resulting from combination of the algebraic category of lattices and the topological category of uniform spaces. In this category, the objects are lattices, and the morphisms are uniformly continuous lattice homomorphisms. The category of lattices (and uniform lattices) includes the binary operations of meet and join as well as morphisms, called lattice homomorphisms. These map one lattice to another, are order-preserving, and satisfy category requirements.

Exactly what Ayn Rand had in mind. Not!

Clever construction Stephen. If Ayn Rand knew more mathematics than she learned in the Petrograd School for Tots. I am sure she might have come up with something like this.

Ba'al Chatzaf

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This essay was first published in V5N2 of The Journal of Ayn Rand Studies.

Universals and Measurement

Stephen Boydstun

I. Orientation

All concretes can be placed within some concept class(es). All concretes can be placed under concepts. Supposing those concepts are of the Randian form, then all concretes must stand in some magnitude relation(s) such that conceptual rendition of them is possible. What is the minimal magnitude structure (minimal ordered relational structure) that all concretes must have for them to be susceptible to being comprehended conceptually under Rand's measurement-omission formula?

That is to say, What magnitude structure is implied for metaphysics, for all existence, by the measurement-omission theory of concepts in Rand's epistemology? My core task in the present study is to find and articulate that minimal mathematical structure. With that structure in hand, we shall have as well the fuller articulation of the class character of universals implied by Rand's theory of concepts.

Such mathematical structure obtaining in all concrete reality is metaphysical structure. It is structure beyond logical structure; constraint on possibility beyond logical constraint. Yet it is structure ranging as widely as logical structure through all the sciences and common experience.

The minimum measurement and suspension powers required of the conceptual faculty by Rand's theory of concepts calls for neuronal computational implementation. Is such implementation possible, plausible, actual? This is a topic for the future, bounty beyond the present study.

We must keep perfectly distinct our theoretical analysis of concepts and universals on the one hand and our theory of the developmental genesis of concepts on the other. Analytical questions will be treated in the next section, and it is there that I shall discharge the core task for this study.

The logicomathematical analysis of concepts characterizes concepts per se. It characterizes concepts and universals at any stage of our conceptual development, somewhat as time-like geodesics of space-time characterize planetary orbits about the sun throughout their history. The analysis of concepts and universals offered in the next section constrains the theory of conceptual development, as exhibited in §III.

I am reminded of Rand's statement in ITOE that there is no "Conceptual Common Denominator shared by all existents as such, being that there are no non-existents from which to differentiate the similar unit of "existence" shared by all existents. Still, the question of, "What magnitude structure is implied for metaphysics, for all existence, by the measurement-omission theory of concepts in Rand's epistemology?" is intriguing. May I offer my brainstormed response?

Perhaps another concept that is in the same CCD as existence is the void of space. Objectivism, as you know, does not hold that matter can be created. But one may counter this by saying that just as life was created from non-life, so the first existent in the universe, assuming cosmic existents are born from previous existents, was created from a non-existence. Maybe there was a time where all that existed was a vast void, total "black". The void was so large that it could not stand to remain in entropy, and basic atomic structure just had to keep moving or "growing" in some way. Is it possible that imperceptible, fleeting energy particles composes your "minimal mathematical structure", and that it is the common denominator between existents and potential existents?

John

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Stephen,

It was my pleasure to read the root post for this string today. My first response was to wonder what specific examples you might be considering so-as to address those examples individually, but I can see this is not optimal for a theoretical answer.

Let us suggest that the minimum magnitude of measurement is the presence of any magnitude currently measureable. In that case, we need no strict minimums per se. Now if we address concrete categories of measurement (and I apologize for not having ITOE in front of me to be definitively accurate), it seems to me that we must first establish the category of measurement methodology prior to defining what it means for a measurement to exist as a finite and detectible magnitude.

Example: it must have the property "red." There is the perceptual minimum, which is detectible through biological mechanisms and could be measured in a laboratory across subjects. Example 2: It must have the property of being with XXX-YYY wavelengths. In this case, the instrument used to measure the wavelengths must be validated as sufficient to capture accurate readings under different conditions. Example within example: to establish that the instrument detects light within the wavelengths that make up the color blue, humans might first establish instrument and wavelength magnitude validity by comparing instrument readings to perceptual observations. Sure, 99% of objects probably exhibit some magnitude of the given wavelength(s). Starting with a concrete point (perception), humans might set the rule that any measurements from the instrument under a magnitude Z are considered background noise within the catergorization process that man focuses on.

Thinking about this example, it appears that we must establish validity of the instruments in order to recognize what readings should be considered for categorization. Likewise, when determining what readings should be considered for conceptual categorization, the process seems to be founded on a pragmatic value of being useful. Humans set conceptual boundaries in order to effectively understand reality; therefore, minimum measurements which no longer fit with usefulness of identification should be discarded as "noise." ... Hardly a complete answer, but purpose certainly has a place.

Christopher

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John and Christopher,

It should be just a couple days. I am nearing completion of the next installment in the Nietzshce v. Rand thread of Corner, and it will be efficient wrap to that up first.

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Stephen,

It was my pleasure to read the root post for this string today. My first response was to wonder what specific examples you might be considering so-as to address those examples individually, but I can see this is not optimal for a theoretical answer.

Let us suggest that the minimum magnitude of measurement is the presence of any magnitude currently measureable. In that case, we need no strict minimums per se. Now if we address concrete categories of measurement (and I apologize for not having ITOE in front of me to be definitively accurate), it seems to me that we must first establish the category of measurement methodology prior to defining what it means for a measurement to exist as a finite and detectible magnitude.

Example 2: It must have the property of being with XXX-YYY wavelengths. In this case, the instrument used to measure the wavelengths must be validated as sufficient to capture accurate readings under different conditions. Example within example: to establish that the instrument detects light within the wavelengths that make up the color blue, humans might first establish instrument and wavelength magnitude validity by comparing instrument readings to perceptual observations. Sure, 99% of objects probably exhibit some magnitude of the given wavelength(s). Starting with a concrete point (perception), humans might set the rule that any measurements from the instrument under a magnitude Z are considered background noise within the catergorization process that man focuses on.

Christopher

First of all, Christopher, when you write, "Example within example: to establish that the instrument detects light within the wavelengths that make up the color blue. . . .", I am assuming that you meant to say the instrument "detects light within the wavelengths that compose the color blue." I wasn't sure if you meant that the instrument in your sentence was detecting "light within the wavelengths" within the category of blue, thereby categorizing one shade of blue apart from other shades of blue, or if you were categorizing the wavelength of "blue" from all other wavelengths of colors.

Keeping that assumption in mind, the category of other "wavelengths" are indeed useful despite their imperceptibility at a given time, given that they are the fundamental broader category from which "blue" is differentiated.

This seems to me a sketchy answer--I am in a hurry to get to an appointment.

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Stephen,

It was my pleasure to read the root post for this string today. My first response was to wonder what specific examples you might be considering so-as to address those examples individually, but I can see this is not optimal for a theoretical answer.

Let us suggest that the minimum magnitude of measurement is the presence of any magnitude currently measureable. In that case, we need no strict minimums per se. Now if we address concrete categories of measurement (and I apologize for not having ITOE in front of me to be definitively accurate), it seems to me that we must first establish the category of measurement methodology prior to defining what it means for a measurement to exist as a finite and detectible magnitude.

Example 2: It must have the property of being with XXX-YYY wavelengths. In this case, the instrument used to measure the wavelengths must be validated as sufficient to capture accurate readings under different conditions. Example within example: to establish that the instrument detects light within the wavelengths that make up the color blue, humans might first establish instrument and wavelength magnitude validity by comparing instrument readings to perceptual observations. Sure, 99% of objects probably exhibit some magnitude of the given wavelength(s). Starting with a concrete point (perception), humans might set the rule that any measurements from the instrument under a magnitude Z are considered background noise within the catergorization process that man focuses on.

Christopher

First of all, Christopher, when you write, "Example within example: to establish that the instrument detects light within the wavelengths that make up the color blue. . . .", I am assuming that you meant to say the instrument "detects light within the wavelengths that compose the color blue." I wasn't sure if you meant that the instrument in your sentence was detecting "light within the wavelengths" within the category of blue, thereby categorizing one shade of blue apart from other shades of blue, or if you were categorizing the wavelength of "blue" from all other wavelengths of colors.

Keeping that assumption in mind, the category of other "wavelengths" are indeed useful despite their imperceptibility at a given time, given that they are the fundamental broader category from which "blue" is differentiated.

This seems to me a sketchy answer--I am in a hurry to get to an appointment.

Color is wavelength and here is one way a measuring it for visible light:

http://www.practicalphysics.org/go/Experiment_124.html

Ba'al Chatzaf

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John and Christopher,

The hierarchy of types of matter and fields, the hierarchy from elementary to complex, is one hierarchy. The hierarchy of physical magnitude structures, from sparse structure to more structure (to which we match the hierarchy of measurement scales), is another hierarchy and does not echo the first hierarchy.

Whatever are the most elementary forms of matter and fields, they will be mass-energy. The magnitude structure of mass-energy is one for which one-dimensional ratio measurement scales are appropriate, which is not the most elementary (least-structure) scale. This is all physics and mathematics, parts that are definite and settled, which I always take as given in my work in philosophy. In addition to matter and fields, we have the physical reality of space, time, spacetime, and energy-momentum. These each have a magnitude structure to which a certain measurement form is appropriate; none of these physical realities have the magnitude structure for which the most elementary measurement scale is appropriate. Other physical realities do have that most elementary form of measurement: such are the number of birds at the feeder, and those existents are far from the most elementary forms of matter and fields.

The one-dimensional hierarchy of measurement scales is: absolute, ordinal, hyperordinal, interval, and ratio. Those are all the scale-types there are; that result was proven, mathematically, in the late 80’s. (Our ordinary and scientific practice was illuminated by new formal theory.) There are sub-types within those general types. The definitive reference, treating both one-dimensional and multidimensional scales, is Foundations of Measurement I, II, III, by Suppes, Krantz, Luce, and Tversky. Other measurement-theory and mathematical work I relied on in “Universals and Measurement” will be found in Notes and References.

On the application of measurement scales to magnitudes in psychophysics, see references in Note 16. Also of note: Trout 1998 and Heidelberger 2004, Chapter 6.

Thanks for the link, Bob! See also Note 23 (but beware the unfortunate dual meaning of the term absolute as applied to one-dimensional measurement-scale types and to synthetic geometries).

I should mention that although the absolute measurement scale, which includes counts (e.g., number of birds at the feeder, number photons registered in a photodetector), is the most elementary one-dimensional scale, it is not always thought of as a form of measurement outside the measurement-theory discipline. Rand’s thesis is about forms of measurement above that one, and it is only because she was making a claim about measurement forms above that level in her analysis of concepts that her analysis (and, separably, her theory of concept genesis) is distinctive and substantial.

Rand gave three definitions of concept. I shall tie them all together in the next section, but for the present section, we need this one alone: Concepts are mental integrations of "two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted" (Rand 1966, 13)[6].

The units spoken of in this definition are items appropriately construed as units by the conceiving mind. They are items construed as units in two senses, as substitution units and as measure values (Rand 1969, 184, 186–88). As substitution units, the items in the concept class are regarded as indifferently interchangeable, all of them standing as members of the class and as instances of the concept. Applied to concept units in their substitution sense, measurement omission means release of the particular identities of the class members so they may be treated indifferently for further conceptual cognitive purposes[7]. This is the same indifference at work in the order-indifference principle of counting. The number of items in a collection may be ascertained by counting them in any order. Comprehension of counting and count number requires comprehension of that indifference.

The release of particular identity for making items into concept-class substitution units is a constant and necessary part of Rand's measurement-omission recipe. But this part is not peculiar to Rand's scheme. What is novel in Rand's theory is the idea that in the release of particular identity, the release of which-particular-one, there is also a suspension of particular measure values along a common dimension.

The substitution-unit standing of concepts under their superordinate concepts is a constant and necessary part of Rand's measurement-omission recipe as applied to the superordinate-subordinate relationship. But this part is not peculiar to Rand's scheme for that relationship. Here is what is novel in Rand's measurement-omission theory for superordinate constitution, as I have dissected it: Whichever concept is considered as an instance of the superordinate concept, not only will that subordinate concept and its instances stand as a substitution instance of the superordinate, each instance of the subordinate will have some particular measure value along a specific dimension. And that particular value is suspended for the concept, thence for the superordinate concept.

~~~~~~~~~~~~~~~~

PS

An issue raised by Daniel concerning Section III of “Universals and Measurement” is addressed here and here. A question from Neil is addressed here.

Edited by Stephen Boydstun
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So let me get your understanding of "two omissions" straight, Stephen. Rand's theory of concept-formation, at the stage when we are "releasing" (which I am understanding as synonomous with "omitting") the particular identities that are not part of the similar units (in two or more entities) under mental isolation, not only are those irrelevant (to the concept) identity-attributes omitted, but also "some particular measure value along a specific dimension"? For example, "fluffy coat" and "threadbare coat" are isolated according to the "distinguishing characteristic" (Rand's term) of "cloth that fits the torso in order to retain body heat", in order to form the concept of "coat"; the identity of the coats as respectively "fluffy" and "threadbare" are released, AS WELL AS the respective manner of classifying the coats, i.e. the standard of thickness, i.e. the "measure value according to a specific dimension." Given that the fit of the cloth and its thermostatic purpose was used to form the concept of coat, the conceptualizer of the coat has no recourse from then on to form the concept of coat by releasing another irrelevant identy-attribute, such as "thickness" of the coat? Does this approximate the "second omission" you are referring to?

Sincerely,

John

Edited by ValueChaser
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Example: it must have the property "red." There is the perceptual minimum, which is detectible through biological mechanisms and could be measured in a laboratory across subjects. Example 2: It must have the property of being with XXX-YYY wavelengths. In this case, the instrument used to measure the wavelengths must be validated as sufficient to capture accurate readings under different conditions. Example within example: to establish that the instrument detects light within the wavelengths that make up the color blue, humans might first establish instrument and wavelength magnitude validity by comparing instrument readings to perceptual observations. Sure, 99% of objects probably exhibit some magnitude of the given wavelength(s). Starting with a concrete point (perception), humans might set the rule that any measurements from the instrument under a magnitude Z are considered background noise within the catergorization process that man focuses on.

Thinking about this example, it appears that we must establish validity of the instruments in order to recognize what readings should be considered for categorization. Likewise, when determining what readings should be considered for conceptual categorization, the process seems to be founded on a pragmatic value of being useful. Humans set conceptual boundaries in order to effectively understand reality; therefore, minimum measurements which no longer fit with usefulness of identification should be discarded as "noise." ... Hardly a complete answer, but purpose certainly has a place.

Christopher

Christopher, I believe I misunderstood you in my comment. I thought you were saying that certain wavelengthsof color should be disregarded when measuring color, because they are not useful in identifying percepts of color; to which I had been thinking that this way is effective if identifying colors in the spectrum of visible light, but not if identifying invisible light, such as "ultraviolet."

I was thinking, after my reply, that only imperceptible light such as "ultraviolet"--light that need not be known when identifying the category of light per se--is indeed useless "noise" when differentiating "blue" from other colors within the visible spectrum, for obvious reasons. But, I continued in my thoughts, we need to invoke the spectrum of invisible wavelengths of light in order to differentiate, say, "blue" from "ultraviolet," leading me to conclude that in that case, imperceptible light is NOT useless "noise".

Now I realize I misunderstood what you meant by useless "noise." You are saying, I gather, that the various shades of a color such as blue, are useless when identifying "blueness", since the multitude of shades does not need to be invoked in order to arrive at "blueness" per se.

Yours in thinking "noisily,"

John

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. But, I continued in my thoughts, we need to invoke the spectrum of invisible wavelengths of light in order to differentiate, say, "blue" from "ultraviolet," leading me to conclude that in that case, imperceptible light is NOT useless "noise".

Some of that useless "noise" will give you a wicked sunburn.

The word color has several meanings:

1. The frequence or c/wavelength of the light whether it is perceived or not

2. The energy of the photon whether it interacts with our senses or not

3. The name we give to the experience of sensing light of a given wavelength, frequency, energy etc.

The last is subjective the first two are objective.

Ba'al Chatzaf

Edited by BaalChatzaf
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• 4 months later...

John,

Thanks for the reflective post #158. Your paragraph there seems not quite faithful yet, so far as I understand you.

The two releases (or suspensions, or omissions) for the concept coat are like this. I have two particular coats. One is of fabric, only lightly insulating, and is army green. The other is of leather, more highly insulating, and is bluish-black. They are two particular coats. Taken simply as coats, they are both, as you would say, “material that fits a torso in order to retain body heat.” Each of the various characteristics that are different between the two coats, following Rand’s theory, is a characteristic having measurable dimension(s). The suspension of the distinctness of my one particular coat and my other particular coat is the substitution-unit suspension that accompanies each of the particular value-unit suspensions along each of the measurable dimensions of the various characteristics: material, thermal conductivity, and optical reflectance properties. Let my two coats be named H and R. H has the particular values hm, ht, hc along the three dimensions we are considering (where hm is some particular value in a complex measure of kinds of material workable for coats, ht is some particular value of thermal conductivity, and hc is some particular value in a complex measure of reflectance properties within the optical range). Similarly, for R. The dual suspensions are these:

H-R, hm-rm

H-R, ht-rt

H-R, hc-rc

I should add my usual caveat about analysis v. theory-of-genesis, since you used the term concept-formation. Just to be sure. In Rand’s essay, formation is the process of formation. We use the word formation in other ways too, as when we speak of the results of processes, such as structures of nature (rainbow; canyon) or artifact (the Bean in Chicago; football formation for the hike). Rand’s proposal was focused on the genesis of concepts, and that is what is usually meant by people who speak of concept-formation in connection with her theory of concepts. My study was on something presupposed by such a conception of concept-formation, assuming every concrete can be brought under some concept or other: every concrete stands in measurable relations with some other concretes such that they can be placed (by me the analyst, with all the power of science, mathematics, and measurement theory) under some concept or other having the Randian structure. So my talk about dual suspensions (of units in the two senses) is not about or dependent on any alleged suspensions in one’s historical formations of concepts; it is about programmatic explicit measurement renditions of concepts by the scientifically advantaged analyst. And the thesis that all concretes can be placed under such concepts using measurement dimensions having ordinal-scale level or higher (ordered-space level or higher for multi-dimensioned characteristics) is a substantial claim.

~~~~~~~~~~~~~~~~~

In the Concluding Historical Postscript section of the oral exchanges (c. 1970) that Binswanger transcribed for the Appendix to Rand’s Introduction to Objectivist Epistemology, Rand says a bit about how she came to the germ of what would later become her measurement-omission theory. She had been discussing essences and universals with a Jebbie who was a moderate realist. After the discussion, she thought to herself that she was dissatisfied with her dissenting position, as she also did not agree with the standard alternative, nominalism. Over the course of a half-hour personal reflection, she got onto her fresh approach. She said this occurred “somewhere in the 1940’s, so it’s over twenty years ago now.”

That sounds offhand like the episode occurred in the late 40’s. Well, maybe so. But if so, she had perhaps a precursor of that germinal idea by 1943. Dominique had just attended a revolting play No Skin Off Your Nose with her husband Gail. To herself she is thinking, comparing the play and what it epitomizes in the world with Roark’s Temple of the Human Spirit. “This was not a comparison between two mutually unmeasurable entities, a building and a play . . . .”

To be sure, there is ancestry in Plato and Aristotle. Another day.

Edited by Stephen Boydstun
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• 11 months later...

This is only a sidebar of the essay, but I thought it might be of some interest.

Physical temperature, certain aspects of sensory qualities, and certain aspects of utility rankings are examples of concretes whose magnitude structures afford what are now called interval measures, but evidently do not afford ratio measures[16].

Physical temperature certainly affords ratio measurements when proper units are used, like the Kelvin scale.

Notes

16. It might be thought that temperature was found to afford ratio scaling once absolute zero was conceived and the “absolute thermodynamic temperature scale” was constructed. That is incorrect. The interval units of the absolute thermodynamic temperature scale (˚K) are the same as the interval units of the Celsius scale (˚C). Like the Celsius and Fahrenheit scales, construction of the absolute thermodynamic temperature scale requires not only that an interval unit be chosen, but that a fundamental fixed point be chosen and assigned a value. The fixed point selected for the absolute thermodynamic temperature scale is the triple point of water (unique temperature and pressure at which water, ice, and vapor coexist). Absolute zero is then defined to be 273.16 ˚K below the triple point exactly.

What if, contrary to my supposition, temperature were found to be a physical quantity that affords ratio measures? That would not change the outcome of my core task in this study. I am to delineate and put aside the richer types of magnitude structures affording measurement until we arrive at the minimal structure required for Rand’s measurement-omission recipe. The physical examples presented need be, for our purpose, only hypothetical illustrations of types of magnitude structures.

Bradford Skow argues in the latest issue of Philosophy of Science (V78N3, July 2011) that Kelvin’s absolute temperature scale, which was independent of choice of medium for the thermometer, did not entail that the magnitude of the attribute being measured afforded ratio scales. The laws of thermodynamics (and equations of state) do not require a ratio-scale magnitude structure of the thermodynamic temperature. Only when thermodynamics was explained by statistical mechanics, wherein temperature is defined as the derivative of internal energy with respect to entropy, with entropy now defined as the logarithm of the possible microstates of the system, did we actually come to know that temperature is a magnitude rightly affording ratio scales.

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• 1 month later...

. . .

Concepts can be analyzed, according to Rand's theory, as a suspension of particular measure values of possible concretes falling under the concept. Items falling under a concept share some same characteristic(s) in variable particular measure or degree. The items in that concept class possess that classing characteristic in some measurable degree, but may possess that characteristic in any degree within a range of measure delimiting the class (Rand 1966, 11–12, 25, 31–32)[5]. This is Rand's "measurements-omitted" theory of concepts and concept classes.

. . .

. . .

5. Pale anticipations of this idea of Rand’s may be found in James (1890, 270), Johnson (1921, 173–92), and Heath (1925, 132–33). For relations to Aquinas and Hume, see Boydstun (1990, 24–27).*

. . .

. . .

Boydstun, S. 1990. Capturing Concepts. Objectivity 1(1):13–41.

Heath, T. [1925] 1956. Euclid’s Elements (Vol. 1). New York: Dover.

James, W. [1890] 1950. The Principles of Psychology (Vol. 1). New York: Dover.

Johnson, W.E. [1921] 1964. Logic (Part 1). New York: Dover.

Rand, A. 1966. Introduction to Objectivist Epistemology. 2nd ed. 1990. New York: Meridian.

. . .

I have learned of a “pale anticipation” of Rand’s measurement-omission perspective on concepts way back in the fifth or sixth century. My studies of Roger Bacon (* **), a contemporary of Aquinas, led me to study Bacon’s mentor and model Robert Grosseteste (c. 1168–1253). The latter mentioned that Pseudo-Dionysus (an influential Neoplatonic Christian of the fifth or sixth century*) had held a certain idea about the signification of names. From James McEvoy’s The Philosophy of Robert Grosseteste (1982): “[Grosseteste] reminds us that Pseudo-Dionysius himself at one point introduced the hypothesis that the names signify properties held in common, but subject to gradation in the order of intensity. Thus the seraphim, for instance, are named from their burning love; but it goes without saying that love is a universal activity of spirit” (141–42). Angels were thought to exist and to have ranks, I should say. Some kinds have burning love; others do not have that kind of love. The thought of Pseudo-Dionysius and of Grosseteste was that angels in the different ranks, angels of different kinds, all shared some properties (e.g. their participation in being, their knowledge, or their love) that the various types possessed in various degrees.

I have located the pertinent text of Pseudo-Dionysius. It is in chapter 5 of his work The Celestial Hierarchy. The heading of that chapter is “Why the Heavenly Beings Are All Called ‘Angel’ in Common.” Dionysius writes: “If scripture gives a shared name to all the angels, the reason is that all the heavenly powers hold as a common possession an inferior or superior capacity to conform to the divine and to enter into communion with the light coming from God” (translation of Colm Luibheid [1987]).

Edited by Stephen Boydstun
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• 2 weeks later...

Do our senses allow us to observe the physical universe, or do we observe our senses?

What is observation without sense or memory of sensation?

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What is observation without sense or memory of sensation?

Marxist political and economic theory and practice?

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Sorry, I don't know anything about that...

I was just wondering. I just find the word "consciousness" is thrown around a lot, but really, I have some doubt to whether that is what best describes "us".

Observation is a choice, isn't it? Not being force-fed information.

We are then more of a force than an observer... I think?

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Calvin,

Rand took consciousness most fundamentally to be awareness of existence. She took it to be the most important biological power of animals possessing it, paramount in their manner of survival. In Rand’s conception of human cognition, there is no divide between perceptual observation and active exploration with all the power of mind and body.*

Some notes on Rand’s conception of perceptual consciousness and the place of memory in it are here: A, B

Concerning Rand’s direct realism in perception: A, B, C

Welcome to OL.

–Stephen

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Do our senses allow us to observe the physical universe, or do we observe our senses?

Do you taste your tongue or taste things with your tongue? Do you see your eyes or see other things with your eyes. (You don't see your eyes in a mirror, but a reflected image of them.)

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Do our senses allow us to observe the physical universe, or do we observe our senses?

What is observation without sense or memory of sensation?

If we sensed our senses we would enter in infinite loop. We would sense our senses sensing our senses and so on.

Keep it simple. We sense what is Out There. Sometimes our senses leave out some of what is happening. For example our vision. We can only sense light in a very narrow frequency band.

Ba'al Chatzaf

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Thanks for the info, Stephen.

To be aware of your own existence, you must identify with something... But you cannot identify with something of which you are not aware.

To identify with anything is to mislabel that thing "I" or "me".

This was not what I was getting at, but it's something.

And to Ba'al: There's no way of knowing whether or not the information our brain sends us is an accurate representation of anything, is there? If we are observing anything, wouldn't it be the mind?

Edited by Dglgmut
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Universals and Measurement

I. Orientation

II. Analysis

—Affordance of Ratio or Interval Measures

II. Analysis

—Affordance of Ordinal Measures

—Superordinates and Similarity Classes

—Amended Measure-Definitions of Similarity and Concepts

III. Genesis

—Elaboration of Identity

—First Words, First Universals

—Analytic Constraint

Notes

References

~~~~~~~~~~~~~~~~

. . .

Concepts can be analyzed, according to Rand's theory, as a suspension of particular measure values of possible concretes falling under the concept. Items falling under a concept share some same characteristic(s) in variable particular measure or degree. The items in that concept class possess that classing characteristic in some measurable degree, but may possess that characteristic in any degree within a range of measure delimiting the class (Rand 1966, 11–12, 25, 31–32)[5]. This is Rand's "measurements-omitted" theory of concepts and concept classes.

. . .

. . .

5. Pale anticipations of this idea of Rand’s may be found in James (1890, 270), Johnson (1921, 173–92), and Heath (1925, 132–33). For relations to Aquinas and Hume, see Boydstun (1990, 24–27).*

. . .

. . .

Boydstun, S. 1990. Capturing Concepts. Objectivity 1(1):13–41.

Heath, T. [1925] 1956. Euclid’s Elements (Vol. 1). New York: Dover.

James, W. [1890] 1950. The Principles of Psychology (Vol. 1). New York: Dover.

Johnson, W.E. [1921] 1964. Logic (Part 1). New York: Dover.

Rand, A. 1966. Introduction to Objectivist Epistemology. 2nd ed. 1990. New York: Meridian.

. . .

I have learned of a “pale anticipation” of Rand’s measurement-omission perspective on concepts way back in the fifth or sixth century. My studies of Roger Bacon (* **), a contemporary of Aquinas, led me to study Bacon’s mentor and model Robert Grosseteste (c. 1168–1253). The latter mentioned that Pseudo-Dionysus (an influential Neoplatonic Christian of the fifth or sixth century*) had held a certain idea about the signification of names. From James McEvoy’s The Philosophy of Robert Grosseteste (1982): “[Grosseteste] reminds us that Pseudo-Dionysius himself at one point introduced the hypothesis that the names signify properties held in common, but subject to gradation in the order of intensity. Thus the seraphim, for instance, are named from their burning love; but it goes without saying that love is a universal activity of spirit” (141–42). Angels were thought to exist and to have ranks, I should say. Some kinds have burning love; others do not have that kind of love. The thought of Pseudo-Dionysius and of Grosseteste was that angels in the different ranks, angels of different kinds, all shared some properties (e.g. their participation in being, their knowledge, or their love) that the various types possessed in various degrees.

I have located the pertinent text of Pseudo-Dionysius. It is in chapter 5 of his work The Celestial Hierarchy. The heading of that chapter is “Why the Heavenly Beings Are All Called ‘Angel’ in Common.” Dionysius writes: “If scripture gives a shared name to all the angels, the reason is that all the heavenly powers hold as a common possession an inferior or superior capacity to conform to the divine and to enter into communion with the light coming from God” (translation of Colm Luibheid [1987]).

. . .

Whatever are the most elementary forms of matter and fields, they will be mass-energy. The magnitude structure of mass-energy is one for which one-dimensional ratio measurement scales are appropriate, which is not the most elementary (least-structure) scale. This is all physics and mathematics, parts that are definite and settled, which I always take as given in my work in philosophy. In addition to matter and fields, we have the physical reality of space, time, spacetime, and energy-momentum. These each have a magnitude structure to which a certain measurement form is appropriate; none of these physical realities have the magnitude structure for which the most elementary measurement scale is appropriate. Other physical realities do have that most elementary form of measurement: such are the number of birds at the feeder, and those existents are far from the most elementary forms of matter and fields.

The one-dimensional hierarchy of measurement scales is: absolute, ordinal, hyperordinal, interval, and ratio. Those are all the scale-types there are; that result was proven, mathematically, in the late 80’s. (Our ordinary and scientific practice was illuminated by new formal theory.) There are sub-types within those general types. The definitive reference, treating both one-dimensional and multidimensional scales, is Foundations of Measurement I, II, III, by Suppes, Krantz, Luce, and Tversky. Other measurement-theory and mathematical work I relied on in “Universals and Measurement” will be found in Notes and References.

On the application of measurement scales to magnitudes in psychophysics, see references in Note 16. Also of note: Trout 1998 and Heidelberger 2004, Chapter 6.

. . .

I should mention that although the absolute measurement scale, which includes counts (e.g., number of birds at the feeder, number photons registered in a photodetector), is the most elementary one-dimensional scale, it is not always thought of as a form of measurement outside the measurement-theory discipline. Rand’s thesis is about forms of measurement above that one, and it is only because she was making a claim about measurement forms above that level in her analysis of concepts that her analysis (and, separably, her theory of concept genesis) is distinctive and substantial.

. . .

~~~~~~~~~~~~~~~~

PS

An issue raised by Daniel concerning Section III of “Universals and Measurement” is addressed here and here. A question from Neil is addressed here.

In “Universals and Measurement,” one of the ways in which I characterized the distinctive magnitude structure for metaphysics implied by Rand’s measurement-omission analysis of concepts was according to mathematical category. I reached this result:

The magnitude structure entailed by Rand’s theory has the algebraic character of a lattice, which has more structure than a partially ordered set (or a directed set) and less than a group (or a semi-group). In terms of the mathematical categories, Rand’s magnitude structure for metaphysics is a hybrid of two: the algebraic category of a lattice and the topological category of a uniformity. Rand’s structure belongs to the hybrid we should designate a uniform topological lattice. (JARS 5[2], 280; AOM)

What qualifies as the mathematical structure that is called a category? A category consists of three things: (i) a class of elements, which are called objects (ii) a set of morphisms from any one of those objects to another of them, (iii) a rule for composing a new morphism from any two morphisms, where this composition is associative. Included in the set of morphisms under (ii), there must be an identity morphism. There are certain further notions that are available in every category, such as the notions of monomorphism, epimorphism, isomorphism, and subobjects.

To specify a category, we must specify the objects, their morphisms, and the compositions of morphisms; and we must show that all the requirements of a category are met. Here are some categories. In the category of sets, the objects are sets, the morphisms are mappings from one set to another, the monomorphisms are one-to-one mappings, the epimorphisms are onto mappings, and so forth. In the category of vector spaces, the objects are vector spaces, the morphisms are linear mappings from one vector space to another, and so forth.

Examples of algebraic categories: sets, partially ordered sets, lattices, Boolean algebras, semigroups, groups, abelian groups, rings, fields, vector spaces, associative algebras, Lie algebras. Examples of topological categories: topological spaces, Hausdorff topological spaces, and uniform spaces. (Topological spaces have a notion of “points sufficiently close; neighborhood of point.” Uniform spaces have in addition a notion of “comparatively close; comparative size of neighborhoods.”) Examples of hybrid categories: topological groups and topological vector spaces.

The category I have targeted as reflecting the metaphysical magnitude structure implied by Rand’s measurement-omission form of concepts is the hybrid category resulting from combination of the algebraic category of lattices and the topological category of uniform spaces. In this category, the objects are lattices, and the morphisms are uniformly continuous lattice homomorphisms. The category of lattices (and uniform lattices) includes the binary operations of meet and join as well as morphisms, called lattice homomorphisms. These map one lattice to another, are order-preserving, and satisfy category requirements.

My program for the development of Ayn Rand’s theoretical philosophy could be called the with-measurement program. To every topic in metaphysics or epistemology, my program requires I add the phrase with measurement. Here are some topics: The Theory of Predication; The Nature of Entity; Existence is Identity; Causality and Natural Law; The Synthetic-Analytic Distinction; and The Computational Mind and Perception. To each such area of philosophical investigation, my task is to add with measurement. Each topic is to be recast in the forge of modern logic, set theory, and mathematics as well.

Why? Because Rand’s measurement-omission analysis of concepts implies a distinctive magnitude structure for metaphysics. The argument supporting that claim was given in the sixth paragraph of my essay “Universals and Measurement.” What is that argument? It is simple: All concretes can be placed under concepts. All concretes can be placed within some concept-class or another. For all concretes, some of those concept-classes will be of the Randian measurement-omission form. Then all concretes must stand in some magnitude relations such that the Randian form of conceptual rendition is applicable to them.

The preceding argument does not rely on a supposition that all one’s concepts are formed by a process of measurement omission. The argument says only that any concept subsuming concretes can be reformed into a measurement-omission form, or if not, at least this much is true: all the concretes falling under the given concept fall under some concept(s) or other for which we can discover its measurement-omission form. That last, modest premise is all I need to conclude that Rand’s measurement-omission form of concepts implies that all concretes must stand in certain minimal magnitude relations.

Rand’s concept theory—not her formation theory, but her analysis theory—implies a specific, meager (but nontrivial) magnitude structure for all concretes, which is to say, a distinctive magnitude structure for metaphysics. In my “Universals and Measurement,” I uncovered that minimal structure and characterized it in three ways: by its automorphisms, by its mathematical category, and by the types of measurement it affords.

What is meant by a magnitude structure? That means an ordered relational structure. These structures are not only abstractions. They can be concretely realized relations. The most accessible example is geometry. Not analytic geometry (with coordinate systems, calculus, and all that), but the plain old synthetic geometry such as one learns to weave in a high school geometry course. The various geometries are all ordered relational structures, but some have all the structure of others plus more. Here are some geometries in their cumulative hierarchy of structure:

{[(Ordered) Affine] Euclidean}

{[(Ordered) Absolute] Euclidean}

{[(Ordered) Absolute] Hyperbolic}

{[(Projective) Affine] Euclidean}

[(Projective) Elliptic]

These layers of ordered relational structure are an objective matter. They have been discovered, not arbitrarily constructed.

(The epigrammatic strings I improvised above concerning geometry are nothing but abbreviations of English statements. One true rendering of the string {[(Ordered)Affine]Euclidean} would be “Axioms that imply ordered geometry can be joined with certain further axioms to imply affine geometry, and those combined axioms can be joined with certain further axioms to imply Euclidean geometry.” Another true rendering would be “A Euclidean plane (or space) is composed of affine structure and specific additional structure, and the affine structure is composed of order structure and specific additional structure.”)

The minimal magnitude structure implied (by measurement-omission concept analysis) for all concretes is metaphysical structure. It is structure beyond logical structure; it is constraint on possibility beyond logical constraint. Yet it is structure ranging as widely as logical structure through all the sciences and common experience.

A word on what the with-measurement program is not: In “Universals and Measurement,” which was the first accomplishment in the program, I sought and found the specific minimal magnitude structure the world must have such that Randian conceptual rendition of the world is possible.

To ask for such structure conditions for the possibility of conceptual rendition sounds suspiciously similar to Kant’s quest for the conditions of possible experience or his quest for the conditions of possible cognition (KrV B138). There is a great difference between what I was seeking in “Universals and Measurement” and what Kant was seeking in his famous questions. The magnitude structure I captured is not something that our cognitive system (specifically, our conceptual faculty) prescribes for the knowable world. Rather, however pervasively that structure is in the world, it is there independently of our cognitions.

Ours is not a Kantian program. We do not say that because our conceptual faculty works necessarily in such-and-such way we must find the world everywhere conforming to that way. We do not say that our concepts must all necessarily be susceptible to being cast in measurement-omission form, and that therefore we must find the world affording that form. Rather, we leave open to trial whether we shall find the world everywhere congenial to the measurement-omission form of concepts.

We take as our thesis that the world is that way, and we run with it. We run it to all the traditional and current topics of epistemology, philosophy of mind, metaphysics, and philosophy of science, and we run it fully concordant with modern science and mathematics.

Edited by Stephen Boydstun
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• 4 weeks later...

In the decade after the publication of Atlas Shrugged, Rand made notes for a projected non-fiction book to be titled Objectivism – A Philosophy for Living on Earth. In a note of June 20, 1958, she wrote: “The philosophy which I now will have to present is, in essence, the ‘rules of thinking’ . . . . This will be the main part of my job: my theory of universals—the hierarchical nature of concepts—the ‘stolen concept’ fallacy—the ‘context-dropping’ fallacy— . . . —the rules of induction (and definitions)— . . . —the proof that ‘that which is empirically impossible is also logically impossible (or false)’—etc” (Harriman 1997, 699–700).

By the following April, she had gotten onto the importance to her theory of concepts the idea of unit, not only in the sense of a substitution unit (which she had months earlier), but in the sense of a measure value on the magnitude of a dimension(s) possessed by different members falling under a concept (ibid. 701–2).*

. . .

In the Concluding Historical Postscript section of the oral exchanges (c. 1970) that Binswanger transcribed for the Appendix to Rand’s Introduction to Objectivist Epistemology, Rand says a bit about how she came to the germ of what would later become her measurement-omission theory. She had been discussing essences and universals with a Jebbie who was a moderate realist. After the discussion, she thought to herself that she was dissatisfied with her dissenting position, as she also did not agree with the standard alternative, nominalism. Over the course of a half-hour personal reflection, she got onto her fresh approach. She said this occurred “somewhere in the 1940’s, so it’s over twenty years ago now.”

That sounds offhand like the episode occurred in the late 40’s. Well, maybe so. But if so, she had perhaps a precursor of that germinal idea by 1943. Dominique had just attended a revolting play No Skin Off Your Nose with her husband Gail. To herself she is thinking, comparing the play and what it epitomizes in the world with Roark’s Temple of the Human Spirit. “This was not a comparison between two mutually unmeasurable entities, a building and a play . . . .”

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• 5 months later...

I have moved this sequence of the last few days (between Calvin and me) to this thread for a better fit of topic. Sorry, but the links for all the block quotes got nixed somehow.

Calvin wrote

I hope this is not too far removed from your topic, but I have a question about the application of logic in real life scenarios: as in, “What should I do right now?”

A situation in the present is not completely conceptualized. Another example would be a personality; it is never a complete concept. We can never refer to our concept of a person to draw a logical answer to any question involving that concept. We can fill in the blanks, but it is not based on reality, rather it is our best guess at a reality we have not observed.

So my question is: How do we apply logic to unfinished concepts?

Our moral code helps us live a rewarding life, but for that code to be followed we have to know what we’re applying it to. We encounter so many situations that its impossible to have the answers predetermined, so it seems the closest thing to a solution is to sharpen our ability to form these incomplete concepts in critical areas.

I think of logic as an automatic force intrinsic to thought. Two and two, when identified simultaneously in our minds, do not go through a “process” when becoming four, they are four the instant they are acknowledged in the same mind space. The size of the “spotlight” of focus we can shine on multiple concepts is our degree of intelligence; it is what allows for non-contradiction. The wider this blanket of non-contradiction can be spread, the more logical the conclusion to any question that corresponds to the content.

I think what I’ve said contradicts Rand’s “sum total of knowledge” idea in that I disagree that all of our knowledge is present or represented in any moment of consciousness.

Well it looks like I actually brought up two things. I’m looking for an answer to the question, any simple corrections, or comments if you feel like it.

Stephen wrote

Calvin,

In her Introduction to Objectivist Epistemology, Rand put forth a theory in which concepts are open-ended and ready to be revised with the further growth of knowledge. There is no need to have “completed” knowledge or concepts in order to be correctly conceiving the world in the meantime. Also, at any given occasion of thinking, we may need to make a decision so quickly that we may forget something pertinent we know or make some other kind of error. Rand’s epistemology (and ethics) recognizes always that we conceptual animals are neither omniscient nor infallible. I think the more expert we are in an area of knowledge, the faster we will be able to bring our knowledge correctly to bear.

Logic as the art of non-contradictory identification is broad enough to include deduction, induction, and abduction we do at leisure. Sure. The same art enables us to drive a car or make some cookies.

“Two plus two is four” is a sentence and takes time. To borrow Rand’s term, we automatize. When I was younger, I could do arithmetic very fast. The smaller steps seemed to progress in an instant, though the full calculation took obvious time. But when I was learning that two and two is four, I’m sure it was a slow thinking process. That would have been in first or second grade, I think. Before that, one would have needed to learn to count—not just recite the right sequence of names of numbers (like saying the alphabet), but apply the basic rules of counting. That would not have begun until about age three and a half. [note] One would have been speaking sentences for about a year and a half by then. What a lot of work it takes to get our knowledge. And what a joy in ourselves climbing up and then helping others up.

There can be a simultaneity in the elements we are thinking about, even though it takes time to think them and some relationship in which they stand. The five fingers on my left hand and my two eyes make seven body parts. Though it takes time to think that through, they are all still here all at once.

Note: Calvin, I did not have access to my own computer and to my books when I made replies in this sequence. I would like to add (from the “Universals and Measurement” Notes):

The child has gone far beyond learning first words (roughly months 12 to 18) by the time she is learning to count. By 30 months, the basic linguistic system has become established and is fairly stable (Nelson 1996, 106). Not until around 36 months or beyond does the child have an implicit grasp of the elementary principles of counting: assign one-label-for-one-item, keep stable the order of number labels recited, assign final recited number as the number of items in the counted collection, realize that any sort of items can be counted, and realize that the order in which the items are counted is irrelevant (Gelman and Meck 1983; Butterworth 1999, 109–16).

At 22 months, a child in my family could “say his numbers.” This competence is not essentially different than being able to “say his ABC’s (Bloom 2000, 215). Rand may have mistaken the onset of recitation of count-word sequences with onset of ability to count.

~~~~~~~~~~~~~~~~

However one might go about imagining 19 items, we will not know if one has succeeded unless we count the items sooner or later. Similarly, in calibrating our counting machines, we have to make some counts to check them. The process you describe, and counting too, would be cases of what Rand called the principle of unit-reduction in cognition, as you may know.

Concerning your suggestion that thought includes recreation of a concept from memories linked to a concept, I do not know. I’m pretty sure there is research in cognitive science related to your idea. I’ll have to leave it to you to research. Our memory for words (which some of us are losing as we get older), and the extent to which we use a particular, typical instance of a concept when using the concept on the fly are matters on which there has been much research. You will find those programs of research and their results so far, at least those, I’m pretty sure.

If I ask you to imagine a dot or imagine 2 dots, there&amp;#39;s no issue. But if I ask you to imagine 19 dots, how do you do that?

Describing the process seems very tricky... It seems we create, sort of, compound concepts that allow us to build even larger ones.

Would you agree that thought is the recreation of a concept, in the moment, from memories linked to that concept?

Calvin wrote

Cheers for that. Didn’t realize you had replied to my previous post.

I don’t know if my definitions are wrong, but it seems to me that aside from concepts, we also store a lot of information comprising the relationships between concepts.

So, the abstract concept of “five” is relative, but so is “red” and “plastic” and any other abstraction. We don’t just remember what red is, but what it is relative everything else. I believe this is the automatizing that Rand was talking about. We ingrain the relationship between 2 and 4, and we can quickly jump from “2 + 2” to “4” without further effort. I have not ingrained the relationship between 1523 and 348, so I cannot look at the two numbers and automatically realize their sum, but I can break each number down into parts that I can relate to one another.

These relationships are not concepts themselves, but a way of organizing them. This is how we “organize” our brains, and it’s the essence of intelligence.

Have I gone horribly wrong here?

Calvin, concerning your last post in that sequence:

Even before one’s first word—for example, saying “ba” for ball—the object is already connected in mind to floor and to rolling and to making roll, all by perceptual and action schemata. From “Universals and Measurement” (2004):

For an older child or an adult, of course, “a concept identifying perceptual concretes stands for some implicit propositions” (Rand 1966, 48, 21). For a single-words toddler, no propositions can be adduced. Actions can be adduced. A ball is something that can be handled and thrown down. It bounces and rolls. These things are clearly known of balls even by the one-year-old whose first and only word is ba. The concept ball is likely held in mind in the form of image and action schemata as well as by the term ball (Rand 1966, 13, 20, 43; 1969, 167–70)[38].

38. See further Boydstun (1990, 16–18); Minsky (1974, 111–17); Johnson (1987, 23–30, 102¬–4); Iverson and Thelen (1999); Nelson (1996, 16–17).

Boydstun, Stephen. 1990. Capturing concepts. Objectivity 1(1):13–41.

Iverson, Jana, and Esther Thelen. 1999. Hand, Mouth, and Brain: The Dynamic Emergence

of Speech and Gesture. Journal of Consciousness Studies 6(11–12):19¬–40.

Johnson, Mark. 1987. The Body in the Mind. Chicago: University of Chicago Press.

Minsky, Marvin [1974] 1997. A framework for Representing Knowledge. In Mind Design II, edited by J. Haugeland, 111–42. Cambridge: MIT Press.

Nelson, Katherine. 1996. Language in Cognitive Development. Cambridge: Cambridge University Press.

We continue to operate with such schemata alongside our network of conceptual representation at a later stage of development. Hence I am able to execute correct motions of the pen as my mind is thinking about right words and thoughts for this note.

Then too we have in mind 3D models of objects experienced since childhood that enable our ongoing recognition of them. From “Capturing Concepts” (1990):

Quite plausibly, 3D models not only enable recognition of objects but also augment language-dependent cognition. We employ 3D models in concert with phonological and syntactic structures in at least some of our reasoning. Knowing the meaning of a word that denotes a physical object or action generally entails knowing what those objects or actions look like. Concepts of functional objects, too, are sustained in part through 3D models. One important characteristic of a chair is its affordance of a place to sit. Our definition of a chair would include the fact that it was something to sit in. Our concept chair relies in part on our 3D model of sitting (Jackendoff 1987, 200–202; see further Shaw and Hazelett 1986).

Jackendoff, R. 1987. Consciousness and the Computational Mind. Cambrige, MA: MIT Press.

McCabe, V., and G.J. Balzano 1986, editors. Event Cognition. Hillsdale, NJ: Lawrence Erlbaum.

Shaw, R.E., and W.M. Hazelett, 1986. Schemas in Cognition. In McCabe and Balzano 1986.

Coming to our adult conceptual knowledge, our concepts are connected by their definitions and more generally by all propositions in which different concepts are used correctly for definite and communicable thought. That is not all.

From "Capturing Concepts" again:

Children as young as fourteen months form thematic as well as taxonomic categories (Mandler, Fivush, and Reznick 1987). Adults, too, organize their knowledge of the world thematically as well as taxonomically (Mandler and Bauer 1989, 158; Jenkins, Wald, and Pittenger 1986). These perspectives are mutually supporting.

The relation of being a part of is a type of thematic relation and is especially important. Part-whole relations are given in perception along with similarity relations. We learn to form powerful semantic networks from is a part of and is a kind of. “A pear is a kind of fruit which is a part of a tree which is a kind of plant which (with others) is a part of the biosphere.” Parts are natural units of form and natural units of function. Grouping objects according to perceptually salient common parts may engender the child’s transition from classifications (of artifacts, procedures, and biological kinds) according to appearances to classifications according to functions (Tversky 1989, 983–95; see also Keil 1989, 104–9, 111–15, 166–72). Moreover, conceiving of things as systems would surely be impossible without a prior grasp of part-whole relations; similarity relations are not sufficient.

Jenkins, J., Wald, J., and J.B. Pittenger 1986. Apprehending Pictorial Events. In McCabe and Balzano 1986.

Keil, F. 1989. Concepts, Kinds, and Cognitive Development. Cambridge, MA: MIT Press.

Mandler, J., and P.J. Bauer 1988. The Cradle of Categorization: Is the Basic Level Basic? Cognitive Development 3:247–64.

Mandler, J., Fivush, R., and J.S. Reznick 1987. The Development of Contextual Categories. Cognitive Development 2:339–54.

Tversky, B. 1989. Parts, Partonomies, and Taxonomies. Developmental Psychology 25(6):983–95.

Lastly, I should mention that associational relationships and mechanisms in mind are surely essential for human memory, imagination,* and thought. See also Touchstone 1993 and §§ X “Mental Representations” and XI “Neural Networks” in Touchstone 1996.

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• 4 months later...

I have previously compiled historical anticipations of Rand’s analysis of concepts in terms of measurement omission, which compilation I quote below. To those I would like to add the case of John Duns Scotus. These notes, old and new, do not go to the truth or importance of Rand’s theory (and its presuppositions), only to its originality or uniqueness and its relations to other theories in the history of philosophy.*

Continuing from Aristotle and Porphyry, medieval thinkers reflecting on universals and individuation held specific differentia added to a genus make a species what it is and essentially different from other species under the genus. Similarly, individual differentia added to a species make an individual what it is and different from other individuals in the species. Scotus held individuals in a species to have a common nature. That nature makes the individuals the kind they are. It is formally distinct from the individual differentia, a principle that accounts for the individual being the very thing it is. The individual differentia, in Scotus’ conception, will not be found among Aristotle’s categories. Individual differentia are the ultimate different ways in which a common nature can be. Individual differentia are modes of, particular contractions of that uncontracted common nature (cf.).

“The contracted nature is just as much a mode of an uncontracted nature as a given intensity of whiteness is a mode of whiteness, or a given amount of heat is a mode of heat. It is no accident that Scotus regularly speaks of an ‘individual degree’ (gradus individualis)” (King 2000)

Reference

King, Peter 2000. The Problem of Individuation in the Middle Ages. Theoria 66:159–84.

. . .

Concepts can be analyzed, according to Rand's theory, as a suspension of particular measure values of possible concretes falling under the concept. Items falling under a concept share some same characteristic(s) in variable particular measure or degree. The items in that concept class possess that classing characteristic in some measurable degree, but may possess that characteristic in any degree within a range of measure delimiting the class (Rand 1966, 11–12, 25, 31–32)[5]. This is Rand's "measurements-omitted" theory of concepts and concept classes.

. . .

. . .

5. Pale anticipations of this idea of Rand’s may be found in James (1890, 270), Johnson (1921, 173–92), and Heath (1925, 132–33). For relations to Aquinas and Hume, see Boydstun (1990, 24–27).*

. . .

. . .

Boydstun, S. 1990. Capturing Concepts. Objectivity 1(1):13–41.

Heath, T. [1925] 1956. Euclid’s Elements (Vol. 1). New York: Dover.

James, W. [1890] 1950. The Principles of Psychology (Vol. 1). New York: Dover.

Johnson, W.E. [1921] 1964. Logic (Part 1). New York: Dover.

Rand, A. 1966. Introduction to Objectivist Epistemology. 2nd ed. 1990. New York: Meridian.

. . .

I have learned of a “pale anticipation” of Rand’s measurement-omission perspective on concepts way back in the fifth or sixth century. My studies of Roger Bacon, a contemporary of Aquinas, led me to study Bacon’s mentor and model Robert Grosseteste (c. 1168–1253). The latter mentioned that Pseudo-Dionysus (an influential Neoplatonic Christian of the fifth or sixth century*) had held a certain idea about the signification of names. From James McEvoy’s The Philosophy of Robert Grosseteste (1982): “[Grosseteste] reminds us that Pseudo-Dionysius himself at one point introduced the hypothesis that the names signify properties held in common, but subject to gradation in the order of intensity. Thus the seraphim, for instance, are named from their burning love; but it goes without saying that love is a universal activity of spirit” (141–42). Angels were thought to exist and to have ranks, I should say. Some kinds have burning love; others do not have that kind of love. The thought of Pseudo-Dionysius and of Grosseteste was that angels in the different ranks, angels of different kinds, all shared some properties (e.g. their participation in being, their knowledge, or their love) that the various types possessed in various degrees.

I have located the pertinent text of Pseudo-Dionysius. It is in chapter 5 of his work The Celestial Hierarchy. The heading of that chapter is “Why the Heavenly Beings Are All Called ‘Angel’ in Common.” Dionysius writes: “If scripture gives a shared name to all the angels, the reason is that all the heavenly powers hold as a common possession an inferior or superior capacity to conform to the divine and to enter into communion with the light coming from God” (translation of Colm Luibheid [1987]).

~~~~~~~~~~~~~~~~

Unrelated to the preceding part of this post, I also wanted to carry over from another thread a post very pertinent to this thread.

How does one do measurement omission on the class of non-metric topological spaces?

. . .

Bob, the conjecture I endorse, a conjecture implicitly presupposed in Rand’s stronger conjecture, is that one can classify all concretes into one-dimensional or multidimensional concept classes having at least the structure of a uniform topological lattice. Though every dimension will afford at least ordinal scaling, it need not afford metric structure.

Merlin would not want to count ordinal scales as measurement scales. Similarly, on his view, ordered geometry and affine geometry should not pass muster as multidimensional measurement systems. I do count ordinal and other scales on up to ratio scale as measurement scales, I count ordered geometry and affine geometry as measurement systems, and in all of that I’m in league with the principal measurement theorists of the last few decades.* Even if one did not think of ordinal ranking as measurement, it would remain that it takes the set structures the theorists have found for it, going beyond the structure for counting (absolute scaling). This makes Rand’s conjecture (her analysis conjecture presupposed by her formation conjecture) and mine (weaker than hers) an addition to the simple substitution-unit standing of instances under a concept that is common to pretty much all theories of concepts or universals.

. . .

I remarked previously, taking issue with Rand, as follows:

. . .

There are indeed some indispensable concepts we should not expect to be susceptible to being cast under a measurement-omission form of concepts. Among these would be the logical constants such as negation, conjunction, or disjunction. The different occasions of these concepts are substitution units under them, but the occasions under these concepts are not with any measure values along dimensions, not with any measure values on any measure scale having the structure of ordinal scale or above. Similarly, it would seem that logical concepts on which the fundamental concepts of set theory and mathematical category theory rely have substitution units, but not measure-value units at ordinal or above. The membership concept, back of substitution units and sets, hence back of concepts, is also a concept whose units are only substitution units. Indeed, all of the logical concepts required as presupposition of arithmetic and measurement have only substitution units. Still, to claim that all concretes can be subsumed under some concept(s) other than those, said concept(s) having not only substitution units, but measure values at ordinal or above, is a very substantial claim about all concrete particulars.  . . .

I said earlier in this addendum that in Rand’s philosophy objective meaningfulness requires the setting of identity by definition. I say further: Some logical and set-theoretic concepts—not, or, and, all, some, set—are defined by implicit definitions, a specification of their roles supporting meaning and truth of propositions, displayed most essentially in the propositions of logic and mathematics themselves. To be sure, these concepts are rooted in structures of action and situation learned in child development. (Notice also that some functions of these concepts can be implemented in machines.) Later they are rarified for use in language and abstract thought. (On action origins, see a, b. On Piaget’s perspective, see the contributions of Smith, Boom, and Campbell here. On acquisition of logical notions in language acquisition, see a, b, c, .)

The concept collection is not the very same as the concept set as the latter is used in logic or mathematics. Then too, logical class membership, which is used in Rand’s explication of conceptual class, is not the very same as natural-species membership such as Silver’s belonging to the horse species. Validity of the concept natural-species membership is not the full warrant for the concept class membership.

Perfecting the meaning and warrant of the concept class membership will not rely on a measure-value omission. Substitution units are not to be analyzed in terms of measure-value units. The concepts from logic and mathematics that are required for an analysis of measurement, thence measurement omission, are not to be analyzed in terms of the latter. That is why I have held, contrary to Rand, that certain logical and set-theoretic concepts are not to be analyzed in terms of measure-value omission, which is the explanatory structure distinctive of Rand’s analysis of concepts. It remains that the proposal to analyze all other concepts in terms of measurement omission (at least to the level of ordinal measurement) is a substantial, definite, meaningful, and meaning-giving proposal.

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• 1 year later...

Stephen,

Wonderful and thought provoking essay! I have a couple questions, that may or may not have been covered by other people, as I did not read all of the comments in this thread.

1. Did you say the "one-dimensional" is the most fundamental mathematical integration between all concepts? Does "dimensional" here imply a beginning and end?

2. If I understand you correctly you do not posit "the possible" as a/the conceptual staring point spawning all other concepts (I think you address this in post #171)? Could you explain this further, because for me, the possible is conceptually parallel the concept of existence--whatever was, is, or can be.