Historical addendum to “Universals and Measurement”

Grandson Julian, who shared his development with me from age one to two while I was producing U&M


Firstly, my usual caveat on the theory.

There are, I say, 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.

The notes below 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.

10 thoughts on “Historical addendum to “Universals and Measurement”

  1. In the years after composing this paper (2002–03), I 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).
    To the preceding compilation of historical anticipations of Rand’s analysis of concepts in terms of measurement omission, I should add the case of John Duns Scotus. 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. “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)” (Peter King 2000—The Problem of Individuation in the Middle Ages. Theoria 66:159–84).


  2. My paper “Universals and Measurement” (U&M) was published in the spring of 2004 in JARS. In December of that year, there was a paper read and discussed at the session of the Ayn Rand Society, and that was the paper “Rand and Aquinas on the Problem of Universals” by Douglas Rasmussen.* The commentator on that paper was Robert Pasnau.* Prof. Pasnau stated that he had not studied Rand’s theory of concepts directly, so he was only working from what Prof. Rasmussen had related concerning Rand’s theory. Rasmussen’s paper was hefty, and you could get a good deal of Rand’s thought in the area from that paper.
    Rasmussen argued some problems for Rand’s theory and proposed they could be resolved with a little help from the moderate realism of Aquinas, or by a moderate realism in that vicinity. Rand had opposed her theory to realism in universals (moderate or immoderate), that is, she had rejected any sense in which universals are in the world apart from processing by mind of the world. That processing begins with perception and ends with concepts. Many contemporary philosophers, including Pasnau, question traditional accounts of that processing, accounts in which the processing is abstracting. Nevertheless, as Pasnau pointed out, the abstraction as process which Rand elaborated can be viewed also as the proposed structure of abstract, universal concepts in their standing to the mind-independent particulars as taken up in perception, regardless of whether the process by which the structure was arrived at resulted from the supposed abstraction process. This corresponds to my distinction in U&M between genesis and analysis.
    In her treatise, and as shown in U&M, Rand gives a measurement-omission analysis of concepts and additionally a measurement analysis of the similarity relations (and comparative similarity relations) between objects and relations given in perception. There is turbidity in Rand’s account, some of which I turn clear in U&M by keeping memberships sharply distinct from measure-values along a dimension, terming the former, the members, ‘substitutional units’ and the latter ‘measure-values’ (where some dimensions of themselves afford ‘measure units’ [in specifying their measure values] and ratio-measure scaling, and whether they do so afford that much structure of measurement scale or afford properly only scales with less structure is in the nature of the dimensions, not simply a matter of our scaling facility). Having gotten hold of in the mind dimensions in the world and some magnitude relations in the world, Rand has the mind in position to apply measurement-omission to particular measure values along dimensions and, in the same stroke, release of particular instances into interchangeable substitutional units, while not disconnecting abstract general concepts of things and relations given in perception from those things and relations as given in perception.


    • (oops -this continues the immediately preceding comment)
      Repetitions of same or similar things in the mind-independent world does not a universal already make; Rand and Aquinas evidently agree there are no readymade universals just lying there in the world. How close is Rand’s Objectivist account to Aquinas’ of universal concepts such as sunrise or man, under my U&M lights, I dare not take on at this time in my studies. I just wanted to record here, partly for my own future return to it, that Pasnau has written a short study issued just this year under the title “Qualitative Change” as an entry in the collection The Routledge Companion to Medieval Philosophy, and it is highly pertinent to Rand’s measurement-omission theory of universal concepts (assimilation for us to do, not likely Prof. Pasnau).
      “Medieval natural science was largely conducted in imprecise, non-quantitative terms, in part because no one had a reason to suppose that quantitative precision could be fruitful, and in part because it was unclear how to measure, and so give meaningful numerical values to, the sorts of qualities that were fundamental to the theory. But once [in time for the fourteenth century] these qualities are conceived of as themselves complex, and built up out of an aggregation of modes, then in principle those modes can be measured. Once that happens, qualitative theories can be formulated in quantitative terms. . . .
      “Going beyond this historical context, the problem of qualitative change should have enduring relevance to philosophers today, given that what goes for qualities would seem to go just as much for properties or for any modern analogue of modes or forms. . . . Aristotelians . . . wanted such qualities to play an ineliminable causal role in natural philosophy. But the problem [of qualitative change] is very real for anyone who believes in such familiar properties such as whiteness or heat . . . .” (Pasnau 2021, 199)


  3. I don’t understand the caveat you issue at the beginning of the original post:

    There are, I say, 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.

    Your comment turns on the assumption that measurement has to be ordinal or above, but Rand explicitly rejects that assumption in chapter 4 of IOE, and then discusses grammatical/logical particles in that very chapter:

    These concepts are formed by retaining the distinguishing characteristics of the relationship [between thoughts] and omitting the particular thoughts involved (IOE, p. 37).

    I can’t quite tell whether you’re agreeing or disagreeing with Rand. Rand would agree that there is no ordinal measurement involved here, but insist that there is still measurement involved. Where there are “substitution units,” there is commensurability, i.e., measurement, though not necessarily ordinal measurement. There is no ordinal measurement in teleological measurement, either, despite there being comparative judgments of better and worse involving commensurables (IOE. pp. 32-33). But if color can be omitted measurement of perceived objects (whether or not we have a scientific theory that reduces color to a quantitative measure), particular thoughts can be omitted measurements of logical or grammatical concepts like conjunction and disjunction.


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    • The passage you quote from Rand, is only a substitution-unit case. Rand did not understand that that suspension of characteristics making things into interchangeable members of a class is not the same as and does not accomplish—nothing more needing to be said—the maneuver that is suspension of particular measure-value along a dimension. That the lift in the first suspension always or in some broad range of concepts comes also with the lift(s) in the second suspension is a thesis, one not clearly seen by Rand, and one which is the novelty of her theory under my light.

      I don’t know if Rand would think there is no ordinal measurement involved in that characterization of grammatical/logical particles. But there is none. There is indeed an even lower sort of measurement at hand in mere substitution-unit characterization, and that is known as absolute measurement, and that means counting. Rand may very well have thought of counting as a type of measurement (and measurement theorists have some reasons to agree with that, though some such a Patrick Suppes decline the opportunity for that bit of cuteness). But because it is only measurement with the structure of ordinality and above for concepts (and because that much measurability is staked by Rand for a wide, very wide, range of concepts) that is novel in Rand’s theory, I declined to speak of absolute measurement as among the sorts of measurement (and their appropriate underlying magnitude structures) worthy of pursuing in further developing Rand’s measurement-omission model for analysis of concepts (not for her use of it in speculation of processes of forming those concepts).

      I do not understand, Irfan, your thought that there is no ordinal measurement in teleological measurement, where there are judgments of better and worse. I gather from page 33, Rand thought that sort of judgment is fully sufficient to have landed in the park of ordinal measurement: diamond harder than quartz harder than gypsum (scratch-hardness, which scratches which). And she seems right to take rankings of better and worse and still worse as occasions of ordinal measurement.

      Rand of course did not have the idea that there are different mind-independent magnitude structures to which different measurement-scale types are appropriate. That was my re-conception articulated in U&M (2004). Contra Boydstun, and later, Binswanger (2014): Rand would suppose any scale we are using for measuring any physical property can start at ordinal because we don’t yet know enough about the property, and ultimately, with more knowledge, we could devise suitable ratio scales (or as she would say “extensive measurement”). David Kelley supposed her same picture on this, when we talked about my paper at a summer advanced seminar of his. I also had that common assumption when I began researching for this paper in 2002. I learned a lot more about measurement theory (which was continuing to make significant advances to the late 1980’s at least), modern geometry, and mathematical physics than I’d known before, and the profit went into the 2004 paper.


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