Guyau

Universals and Measurement

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Dean,

How do you know she's not just busting your balls over there?

:)

Some people do that when that when hostility spirals the way it has gone in the threads I have read.

I can't say one way or another as I only have her words to go on, but I'm more inclined to believe that than anything else.

I've seen some people call her a viper.

In my experience so far, she's a pussycat.

Will it stay that way? I don't know. I suspect it will. (But if it doesn't, it doesn't.)

After interacting with her, I think she's a good kid.

But then I'm a ball-buster myself. :)

Michael

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Dean,

How do you know she's not just busting your balls over there?

:smile:

Some people do that when that when hostility spirals the way it has gone in the threads I have read.

I can't say one way or another as I only have her words to go on, but I'm more inclined to believe that than anything else.

I've seen some people call her a viper.

In my experience so far, she's a pussycat.

Will it stay that way? I don't know. I suspect it will. (But if it doesn't, it doesn't.)

After interacting with her, I think she's a good kid.

But then I'm a ball-buster myself. :smile:

Michael

Michael,

Actually, vipers are rather peaceful, only striking back when stepped upon. So yes, that's me.

Otherwise, several ex-RoR writers have, in private correspondence, referred to an inner group of name-callers as 'fissiles', thereby resigning. I tend to ignore them, and respond only to several intelligent people who should be posting with you.

EM

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An excellent case in point is the ruler-- a culturally-defined artifact is there ever was one. Because we assume a given standard of inch and meter, we likewise assume that the markings accord from ruler to ruler. If use of common objects were not of assumed qualities, we'd all go crazy.

The marks on the measuring stick on convention. But we expect the property of length (however denominated) to be unchanged when we carry the measuring stick from where it is to where it is applied to thing measured. We also expect the -length- of the measuring stick not to be noticeably changed when we apply the stick to the object to be measured. NOTE: That is why we do not use metal rulers to apply to very hot objects. In this case we have to resort to other means of length measurement.

So the units of length are conventional but the property of length (a property of both the measuring stick and the thing measured by the measuring stick) is hedged and ringed by all sorts of theoretical assumptions. That is the load the theory imposes on the property.

Ba'al Chatzaf

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An excellent case in point is the ruler-- a culturally-defined artifact is there ever was one. Because we assume a given standard of inch and meter, we likewise assume that the markings accord from ruler to ruler. If use of common objects were not of assumed qualities, we'd all go crazy.

The marks on the measuring stick on convention. But we expect the property of length (however denominated) to be unchanged when we carry the measuring stick from where it is to where it is applied to thing measured. We also expect the -length- of the measuring stick not to be noticeably changed when we apply the stick to the object to be measured. NOTE: That is why we do not use metal rulers to apply to very hot objects. In this case we have to resort to other means of length measurement.

So the units of length are conventional but the property of length (a property of both the measuring stick and the thing measured by the measuring stick) is hedged and ringed by all sorts of theoretical assumptions. That is the load the theory imposes on the property.

Ba'al Chatzaf

Yes, but only if you're concerning yourself with 'theoretical load'. My ponit is that,in our daily lives, we don't.

EM

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Yes, but only if you're concerning yourself with 'theoretical load'. My ponit is that,in our daily lives, we don't.

EM

Consider the difference between a carpenter and a physicist.

Ba'al Chatzaf

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Yes, but only if you're concerning yourself with 'theoretical load'. My ponit is that,in our daily lives, we don't.

EM

Consider the difference between a carpenter and a physicist.

Ba'al Chatzaf

Actually, Kuhn's whole point re 'paradigms' is that by far the greater portion of any science is procedural within a given standard of measure.

EM

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Actually, Kuhn's whole point re 'paradigms' is that by far the greater portion of any science is procedural within a given standard of measure.

No, it was not his whole point -- link.

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Actually, Kuhn's whole point re 'paradigms' is that by far the greater portion of any science is procedural within a given standard of measure.

No, it was not his whole point -- link.

When a new coefficient is discovered, the old and new standards of measurement are considered to be 'incommensurate'. For example, Pauli's Principle and QED are 'incommensurate'.. It's therefore said that Lamb's discovery engendered in a new paradigm.

EM

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. . .

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 as a uniform topological lattice.

. . .

Category Theory for Scientists (and for mathematically inclined epistemologists)

Click on olog.

We admit from the beginning that category theory is not intended to provide formulas that take in initial data and make predictions about the future. This is the domain of differential equations, linear algebra, and other well-known subjects in applied mathematics; it is not something we are attempting to improve upon with category theory. Instead we look into the possibility that some of the very structure of our thinking can be adequately represented and articulated in the language of category theory. To the degree that it can, the infusion of mathematics into our thinking will afford additional rigor, which should lead us to new insights. –David Spivak

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Opening Chapter 3 of the textbook Category Theory for Scientists:

In this chapter we begin to use our understanding of sets to build more interesting mathematical devices, each of which organizes our understanding of a certain kind of domain. For example, monoids organize our thoughts about agents acting on objects; groups are monoids except restricted to only allow agents to act reversibly. We will then study graphs, which are systems of nodes and arrows that can capture ideas like information flow through a network or model connections between building blocks in a material. We will discuss orders, which can be used to study taxonomies or hierarchies. [Emphasis added.]

"Ah, but a man's reach should exceed his grasp,

Or what's a heaven for?" –Robert Browning

(Thank you, David.)

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Category Theory for Scientists (and for mathematically inclined epistemologists)

Click on olog.

Thanks. The graphic where you linked to and the one on Wikipedia (olog) reminded me of 'object models' in object-oriented programming (OOP) for computers. OOP images. In a similar vein the Wikipedia article likens an olog to a database schema.

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Similarity – Sameness and Free Range

~just musing~

In Euclidean geometry, one is taught that similar triangles are those having same corresponding angles. These will also have same ratios of corresponding pairs of sides. Less specifically, we say similar triangles have the same shape but have a free range of sizes as well as locations and orientations. It is commonly said that in hyperbolic plane geometry or in spherical plane geometry there are no similar triangles. That is to say, triangles on those surfaces having same corresponding angles and same ratios of pairs of corresponding sides are not free to range in size; rather, they are all of the same size. Leaving aside external specifications such as location and orientation in the surface, we say such triangles are not properly similar, rather, they are congruent or same-triangle.

It occurs to me, however, that circles on a sphere remain properly similar. They retain freedom in range of radius while being same by virtue of being a locus of points equidistant on the surface from some point or other on the surface. In the Euclidian, flat plane, all circles have the same ratio in their diameter to their circumference. On the surface of the sphere, that is not so. Let the diameter of a circle on a sphere be the segment of a great circle of the sphere that divides an arbitrary circle (choose a circle less than a great circle, please) on the sphere into two equal semicircles. Then let the radius of the arbitrary circle expand until it itself becomes a great circle. The ratio of the diameter to the circumference has then become 1:2. We know that prior to expansion into a great circle, the second number in that ratio for our arbitrary circle had been greater than 2 because shortest distance between two points on a sphere are segments of great circles.

Open question: In a mensural model of concepts and their coordinate model of simililarity (models such as Rand’s), is there any analogy between some specific sort of shift in context of a concept to shift of a triangle or circle from one sort of surface to another? Any such analogy, to be sure, will not in its essentials engage ratios in the realm of concepts and their coordinate similarities, only measure values along dimensions.

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

~keeping track~

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. 
. . .

(See also here.)



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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Circles on a sphere have radii which are the lesser portion of a great circle. However areas of the "caps" (a spherical circle) have a factor that does not show up in the plane counterparts. Also the areas of the caps depend on the radius of the underlying sphere.

Please see: http://mathworld.wolfram.com/SphericalCap.html

On different size spheres caps with equal area can differ in how bent or convex they are.

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BTW, when you measure light directly, how do you ~know~ that it is a constant? Not simply by enumerative "induction," but by genuine, conceptual induction, the kind that DB insists on calling something else -- anything but induction, of course.

REB

1. All measurements of the speed of light are indirect They require a clock or some kind of harmonic oscillator.

2. We don't know for absolutely sure that the speed of light is constant in the vacuum of space. So far every measurement and experiment every done so far has not falsified that proposition. Recently the OPERA measurement of the speed of neutrinos indicated some massless neutrinos going a bit faster than the speed of light. The entire physics community turned about and paid attention to this, because the entire structure of physics was in peril. FORTUNATELY it was later found out that this reading was do to a loose wire! Would you believe it????

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To Duns Scotus in #174 I’d like to note the following from Jan Aertsen’s Medieval Philosophy as Transcendental Thought (2012).

[scotus’] theory of “the intension (intensio) and remission (remissio) of forms” maintains that an accidental form or quality has a certain “latitude” (latitudo) within which it can be increased or decreased without a change in the essence or species of the form itself. Scotus’ example is “white”: the color can be found with various degrees (gradus) of intensity, which are real differences but not specific differences constituting different species of “white”. (428)

Also of interest, the foray of the Scotist Antonius Andreae, as reported by Marek Gensler.*

The thesis Antonius presents after the distinctions is interesting too; in what seems to be a partial retreat from the earlier full commitment to the doctrine of remission and intension of substantial forms states that at least sonic substantial forms accept grades. The arguments and examples which follow concern solely the remission and intension in elementary forms. He argues that elements can contain opposite forms, e.g. calidity and frigidity, one of the pair being in esse intenso, the other - in esse emisso. The form the intension of which is greater is dominant and determines what the substance is like, i.e. whether it is water or fire, etc. Thus, in a process of transmutation of elements, one can observe successive remission of one attribute with simultaneous intention of its opposite. The moment the two attributes are equally intensive is the moment of substantial change.

. . .

We can see here the final position of Antonius on the problem of remission and intension of substantial forms. He seems to have come back one more step to the point of view presented by Scotus. Contrary to what he said before, Antonius denies remission and intension in substantial forms; he concedes it in elementary forms but finds them to be another type of forms altogether.

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