SpookyKitty

I never understood the desire to predicate location of abstractions. If you wanted to be poetical about it you could say they are "ideas in the mind of God".
But to put it another way, they are inherent in the nature of reality itself, all the way down at the lowest level.
For example a materialist might say, at the lowest possible level, what the nature of reality defines is the simple, universal, mathematical laws of the fundamental particles in physics, and everything else we see around us is composed of these particles carrying out their basic behaviors (a la Conway's game of life). To avoid this reductionist materialism (and the logically incoherent absurdities it implies), and without introducing a magical element of "emergence" which accounts for the metaphysical existence and causal efficacy of the objects we see around us, then we must posit these natures as inherent in reality at the lowest possible level, that what is there a priori are not mathematically simple, universal laws of fundamental particles, but the nature of every universal that has metaphysical existence and causal efficacy.
If you take gravity for instance, "where" is it located? Well it's just inherent in the nature of reality, it's all throughout. If you want to identify where gravity is active/instantiated, the answer is "in bodies of mass". Likewise with hammer, it's nature is inherent in the nature of reality itself, it is throughout, and where it is active/instantiated is in actual hammers.

https://activeobjectivism.com/2020/11/24/the-argument-for-metaphysical-universals/
- Introduction to Objectivist Epistemology (bold emphasis mine)
 
“Epistemic universals”
Rand denies metaphysical universals quite explicitly, as quoted above. She believes everything in reality is concrete and particular, that there is in reality no “manness” in man which applies universally for all men at all times, but rather the concept “man” is merely man’s way of organizing the concretes seen around him into a mental grouping.
“The metaphysical referent of man’s concepts is … the facts of reality he has observed.”
If one holds that concepts are only “universal” over the total set of of one’s prior, concrete observations, then this is not the universal set! This isn’t guaranteed by any metaphysical principle to hold at all times and for all instances in reality. Concepts in this view aren’t describing something that holds abstractly in reality, they are just describing something that holds abstractly over the particular, delimited set of observations which one has accumulated thus far.
If “universals” are merely referring to sets of observed particulars, then one cannot interpret anything observed, predict the future, or classify anything new, when nothing in general about reality can be referred to. The “man” classified today might have nothing to do with the next “man” observed. The ball observed in one moment tells one nothing about what might be observed in the next moment. Any particular, any moment yet to be observed, nothing can be said about it, because the classifications are all retrospective, they only refer to the particulars already observed.
The “epistemic universal” of “length” one invents today can say nothing about the “length” observed tomorrow, because no necessary connection is being induced, nothing general about reality itself, it is just the cataloging of regularities in experience. They are just retrospective statistical observations – the moment one starts talking about length – every property of length in all places and all times – then one is talking about a universal property out in reality, a metaphysical universal, which is exactly what has been rejected.
No inference can be extended to particulars outside of the cherry-picked set of concretes previously observed. If a concept “stands for” an unlimited range of things abstractly, but concretely it only refers to some particular set of items already identified, then there is no way to know if the abstraction actually does apply to the full range of things that it stands for.
One can define a category of “winged things” which is open-ended, and therefore includes all winged things yet to be observed. Obviously any new instance added to the set will have wings, but nothing else can be said of it besides that. Without such a thing as a natural class, then what is formed is merely a nominal category, in other words the category is merely analytical, and the only thing that can be inferred from classifying something as a “winged thing” is that it has wings. Which is of course useless.
If there is no natural kind backing the concept, then there’s no justification for inferring anything beyond what has already been defined. If on the other hand concepts are identifying a natural kind, then there’s a necessary connection between all particulars in the set, from which one can justifiably infer things like “any new particulars added to the set will behave as the rest of the set”.
If one holds that “any new particulars added to the set will behave as the rest of the set”, then one is apparently identifying a universal in reality. It functions as a universal, and abstractly identifies something in reality that is timeless and essential, something where instances at all times and in all cases will behave in that same way. If an abstraction is be extended across all instances at all times, and out into reality (in the sense that it will predict the future behavior of things in reality), then the abstraction is something that is metaphysical and universal. A nominalist is someone who rejects that any such thing is metaphysically possible or epistemologically justifiable.
Universals which “hold true” but do not “exist”
Rand believes everything in reality is concrete, that, in reality, there is “no such thing” as the universal “manness” which ties together all concrete men, at all times and in all places. This “manness” is rather our organization of concrete men.
She claims that, by properly organizing concrete men, one can thus arrive at a universal “manness” which will hold true for all concrete men, at all times and in all places.
So does the universal does exist mentally but not in reality? Does it “hold true” in reality, and just doesn’t “exist” in reality? There is this odd reluctance to grant the existence of something “in reality”.
Dual aspect metaphysics grants this idea of an “abstract reality”. Some abstraction which holds true in reality, therefore is real. It gives a kind of reifying existence and power to the abstraction, the abstraction is what is metaphysically making it hold true, as opposed to something else making it hold true and the abstraction merely epistemologically “recognizing” that the truth is holding, presumably for some other reason.
It’s kind of an odd question- what is the real thing which is making this universal hold true? There must be something with the force of reality which is making this truth hold- what is that force? Where does that force come from?
Rand asserts that there are no abstractions with this power: only concretes are “really real”. But even some given concrete has to have some abstract nature. Is the material of the concrete supposed to be powering the nature of the concrete? It doesn’t really make any sense when thought about clearly. Only the dual aspect perspective, a la Aristotle’s hylomorphic compounds, actually makes any sense.
Apparently Rand’s perspective is that one cannot say why, but things just “happen” to work universally. That’s just the way the concretes behave- but they don’t behave that way because of some abstract principle of their nature. That form or principle is just a “way man describes” what matter is doing, it only exists in the mind, not in reality itself.
It is bizarre to say that and also hold that induction is possible, as in McCaskey’s article, where he insists that it is possible to have 100% certainty about regularities despite there being no principle of uniformity. How can one have 100% certainty that a regularity will hold, if one denies the reality of some principle to it? There is no way to make a valid inference from any number of observations of a behavior to a universal rule of the behavior. What is to say it won’t change, if it is not a real aspect of the thing’s nature?
McCaskey for example claims:
“If you have good guidelines and follow them, you can be certain that someone absolutely cannot contract cholera unless exposed to the bacterium Vibrio cholerae, certain that all men are mortal, certain that the angles of all planar triangles sum to 180°, and certain that 2+3=5.”
Well no, under this system, none of these is certain. No conclusion science has ever achieved can be described as true, or knowledge, or certain. They simply happen to be true under the concretes previously observed, and one predicts it will continue to be so.
Yes, even with math. Is it certain that 2+3=5? At all times and in all places, universally? How? That may have held up under previous observations, and one may predict that it will continue under similar circumstances, but all of one’s predictions are unjustified and unreliable; one hasn’t observed every single instance that has ever occurred.
From the article: “It’s not that you must presume uniformity in order to classify. It’s that you classify to find uniformities.”
The whole problem with this is that one hasn’t “found” any more uniformity than one had to begin with! One is left in exactly the same position as he agreed with earlier in the article: “The Scholastics lamented (rightly) that unless you had surveyed all magnets or all animals, the inference was not certain”
“Certainty” without proof
A fall back here is to argue that concepts are not universal, but that one can still have a kind of “certainty” which could be mistaken, that it is still “knowledge” which one can hold beyond a reasonable doubt.
If something hasn’t been proven to be true then there is a reasonable doubt that it could be different at some other time or place. After all, it has been asserted that one cannot make justifiable claims for something at all times and all places.
If one is making a universal claim about something at all times and all places, and holds that such claims are invalid, cannot be justified, then how in the world can one feel certainty about them? It has been asserted that one can’t hold such general claims as true, certain, knowledge.
Universal claims are either justified or unjustified. One must choose. If they are unjustified then one cannot claim “certainty” and “the impossibility of doubt”. If universal claims are justifiable, and a given one is proven, then one can claim certainty and the impossibility of doubt.
Either the claim “2+3=5” is unjustified and therefore fallible, or else it is justified and therefore infallible.
It makes absolutely no sense to declare that some claim is unjustifiable, but also true beyond a reasonable doubt.
Conclusion
If one denies the existence of universals metaphysically, then there’s no reason to believe that an abstraction will extend beyond the range of the small set of previously observed concretes to which it currently refers (and certainly not to believe that one has knowledge or certainty about it). In that case these “universal” epistemological abstractions do not provide knowledge, one cannot have certainty about them – and indeed the opinion of a nominalist is that the use of or belief in such “epistemological universals” is foolish and counter-productive, after all, what’s the point in having or believing in some “timeless essential” if it’s not referring to something that is actually timeless and essential in reality? These universal abstractions are actually false and misleading, they distort the view of reality since there are no such things. There are only retrospective categories of reference.
Calling such epistemic categories “concepts” or “universals” is mistaken. None of the positive results that Ayn Rand tries to claim follow, like the ability to have conceptual knowledge, or certainty about reality, or the validity of induction. None of this is really consistent with this view; one is a skeptic about any general statement about reality, a nominalist who believes in categories of convenience, and the epistemic standard (and thus, necessarily, the moral and political standard) is subjective and pragmatic. There are plenty of people who own up to holding exactly this view, nominalists of all kinds own up to this and wear it proudly, declaring that all that is possible to man are pragmatically guided categories and statistical correlations, and that belief in concepts, in universals that actually hold in reality, is akin to a religious fantasy from which one must break away.
If on the other hand one is not truly a skeptic about reality, if deep down there is a belief that it is possible to justifiably know things that are necessary and certain and universal, then there must be a conversation about the metaphysics of universals. Either way there’s an inconsistency in Ayn Rand’s thinking, and one should be clear and honest with themselves on exactly where one stands.
One must choose a side. Either there are universals which actually hold in reality, or else there is no such thing. If there is nothing in reality which holds universally, then one cannot have knowledge of things which hold universally. It is not possible. One either needs to own up to one’s metaphysical stance epistemologically, or own up to one’s epistemological stance metaphysically. It cannot be held both ways. The concept of metaphysical universality cannot be stolen in epistemology while denying it in metaphysics – not if one is being honest with one’s self. Either one has a merely pragmatic stance (i.e. holding this concept as if it were a universal, even if there are no such things, since that seems to work well) in which case one ought to own up to that epistemologically as a nominalist, or else one does believe that universal knowledge is possible but is operating on a stolen metaphysical premise, in which case one ought to own up to that metaphysically as an intrinsicist.

You missed the most important part: because this is preposterous and absurd, we should resolve the apparent contradiction. The theory it is based on must be fixed or a new theory must be proposed. Spoiler alert, they resolve the contradiction in the end. No one in the documentary claimed that the law of identity is invalid. 
Did you stop watching after 15 minutes?

Thanks, Greg.
“Let no one unversed in geometry enter here” is said to have been inscribed over the entryway of Plato’s Academy. Excepting the value-philosophy areas, I think that Euclidean geometry should be a requirement, along with elementary logic, for any beginning philosophy student. I mean anyone beginning to study epistemology and metaphysics. Otherwise, one is missing necessary acquaintance with subject matter and tools of theoretical philosophy.
The synthetic and the analytic of the synthetic-analytic distinction in philosophy, from Leibniz, Hume, Kant, logical positivists, and linguistic analysts, are not the synthetic (e.g. Euclid) and the analytic (e.g. Descartes) of geometry.
On the synthetic-analytic distinction in philosophy, the notions and distinction argued against in Peikoff 1967, there are the recent works Truth in Virtue of Meaning (2008) by Gillian Russell and Analyticity (2010) by Cory Juhl and Eric Loomis. This distinction has continued to be a controversy among philosophers, though most weigh in against soundness of the distinction.
Books on my shelf pertaining to the quite different distinction between synthetic and analytic geometry (all these books still awaiting my full assimilation) are History of Analytic Geometry (1956) by Carl Boyer, The Development of the Foundations of Mathematical Analysis from Euler to Riemann (1970) by Ivor Gratton-Guiness, and Analysis and Synthesis in Mathematics (1997) edited by Michael Otte and Marco Panza.
Greg mentioned that the two distinctions of synthetic-analytic—in philosophy v. in geometry—are not equivalent distinctions. Though the concept of the analytic in geometry is not the same as the analytic in epistemology and although the synthetic in geometry is not the same as the synthetic in epistemology, the distinction from geometry can be of great import for contemporary metaphysics and epistemology. At least this looks so in my present large and main project.*

Soon I will be reading the entirety of the Organon, or more generally all the books within this one about reasoning and demonstration. I say reading group because my intent is to focus on Aristotle's writing without bringing much outside interpretation. I have enough background on Aristotle and Oism to guide reading discussions in a productive way. Not simply to understand what Aristotle said, but to integrate it all with furthering my study of other fields which for me are mostly psychology and neuroscience. If you have a different academic interest, like history or economics, that's even better, because we could apply the ideas to more contexts.
So, post here or send me a private message if you would like to join in. Weekly meetings would go well, we can work out how many chapters to read each time. Probably one hour meetings. I don't expect us all to have the same translation. Sometimes, different translations can be useful.
Secondly, does anyone have suggestions for which translations to use?

II. Analysis (cont.)
Affordance of Ordinal Measures
Recall again Rand's characterization of measurement: identification of "a quantitative relationship established by means of a standard that serves as a unit" (1966, 7). The phrase "a standard that serves as a unit" suggests that Rand's conception of measurement for her measurement-omission analysis of concepts was ratio-scale or interval-scale measurement. These two types possess interval units that can serve as interval standards. They possess interval units that can be meaningfully summed to make measurements. The quantitative relationship established in measurements equipped with interval units entails summation of elementary units. The summation might be simple addition or a more elaborate mathematical combination, and the basis for the summation in concrete reality might be susceptibility to concatenation (for ratio scales) or to composition of ordered difference-intervals (for interval scales).
The measure values required for Rand's theory need not be interval units. As Rand realized, merely ordinal measurement suffices for her measurement-omission scheme (1966, 33). I say that the magnitude structure captured by ordinal measurement is the minimal structure implied for metaphysics if, as I supposed at the outset, all concretes fall under one or more concepts for which Rand's measurement-omission analysis holds. What is the magnitude structure captured by ordinal measurements?
All magnitude structures captured by ratio- or interval-scale measurements contain a linear order relation. A magnitude structure consisting only of such a linear order relation is a structure for which merely ordinal measurement is appropriate. An example is the hardness of a solid. I mean specifically the scratch-hardness, which is measurable using the Mohs hardness scale. Calcite scratches gypsum, but not vice versa; quartz scratches calcite, but not vice versa; therefore, yes, quartz scratches gypsum, but not vice versa. Degrees of hardness have an order that is anti-symmetric and transitive.
Mohs scale assigns the numbers (2, 3, 7) to the degrees of hardness for (gypsum, calcite, quartz). All that is intended by the scale is to be true to the order of the degrees of hardness. That Mohs has chosen these three numbers to be integers is of no significance. They could as well be the rational triple (14.7, 55.3, 56.9) or the irrational triple (√2, π, 1.1π). Unlike the numbers on interval scales, the ratios of difference-intervals between the numbers on these scales are not meant to be of any significance. The hardness degrees (2, 3, 7) are not intended to imply that the hardness of calcite is closer to the hardness of gypsum than it is to the hardness of quartz. For all we know, and for all our ordinal measurements signify, there simply may be no fact of the matter whether the scratch-hardness of calcite is closer to that of gypsum than to that of quartz.
The magnitude structure of hardness (scratch-hardness, not dent-hardness) evidently does not warrant summations or equal subdivisions of some sort of interval unit of hardness. This particular hardness concept is founded analytically on merely ordinal measure. To fall under this concept hardness, an occasion need only present the quality at some measure value on the merely ordinal scale, and that may be any measure value on that scale. Affordance of ordinal measurement is all that Rand's measurement-omission recipe entails for the magnitude structure of all concretes. Her theory does not entail that every attribute of concretes—hardness, for example—must in principle afford ratio- or interval-scale measurements. Her theory does not imply that, were only our knowledge improved enough, it would be possible to make ratio- or interval-scale measurements of scratch-hardness[24].
The magnitude structure affording merely ordinal measurement is a linear order whose automorphisms are the order-automorphisms of (same-order subsets of) the real numbers in their natural order. Such a magnitude structure affords characterization by a lattice (a type of partially ordered set) formed of sets and subsets of possible Dedekind-cuts of its linear order. This linear order might be scattered or dense; ordinal measurement is possible in either case[25].
The magnitude structure affording merely ordinal-scale measurement affords metrics. Each of the three scales adduced above to capture degrees of hardness bears a metric defined by the absolute values of those scales' numerical differences. A magnitude structure affording a (separable) metric belongs to the topological category known as a (separable) uniformity. Topologies that are uniformities in this sense are Hausdorff topologies, but they need not be compact nor (topologically) connected[26]. The topological character of the magnitude structure entailed for all concretes by Rand's measurement-omission theory of concepts is the character of a uniformity.
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 as a uniform topological lattice.
Concerning multidimensional magnitude structures of concept classes, I concluded in the preceding subsection that Rand's theory entails neither affine nor absolute structure. What is entailed: concept classes with a 2D or 3D magnitude structure will have the structure of at least an ordered, distance geometry[27]. Significantly, it is implied that planes and spaces concretely realizable will have at least that much structure.
Superordinates and Similarity Classes
Hardness, fatigue cycle limit, critical buckling stress, shear and bulk moduli, and tensile strength all fall under the superordinate concept strength of a solid. The conceptual common denominator (Rand 1966, 15, 22–25; 1969, 143–45) of these various strengths of solids is that they are all forms of resistance to degradations under stresses. What is the common measure of this resistance the different species of strength have in common? What is the common measure of strength that all specific forms of the strength of solids have in common? The magnitude structure of hardness affords only ordinal-scale measurement. The magnitude structure of tensile strength affords ratio-scale measurement. Only the ordinal aspect of the tensile-strength measure could be common with the hardness measure. Is the ordinal aspect of each specific form of strength a same, single, common measure? No, the ordinal measure of hardness is not the same as the ordinal measure of tensile strength. Resistance to being scratched is not the same as resistance to being pulled apart under tensile stress.
The way in which an ordinal measure of hardness and an ordinal measure of tensile strength can form a common ordinal measure for the general concept strength of a solid is only as substitution units, not as distinct measure values along some common ordinal measurement scale. The some-any locution can be applied to substitution units (e.g., Rand 1966, 25). We sensibly say that strength of a solid in general must have some type of ordinal strength measure but may have any such type. That sort of use of some-any pertains to units as substitution units: there must be some specific form of strength to instantiate the general concept strength of a solid, but it may be any of the specific forms.
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.
Analytically, identity precedes similarity[28]. For purposes of her theory of concepts and concept classes, Rand defined similarity to be "the relationship between two or more existents which possess the same characteristic(s), but in different measure or degree" (1966, 13). I concur. Occasions of scratch-hardness are similar to each other because they are all occasions of scratch-hardness, exhibiting that hardness in various measurable degrees. This much accords with Rand's definition and use of similarity in the theory of concepts.
Occasions of scratch-hardness are also more like each other than they are like occasions of tensile strength. This is a perfectly idle invocation of comparative similarity (comparative likeness). The work that comparative similarity pretends to be doing here can be accomplished fully by simple identity (sameness) without any help from similarity: scratch-hardness is itself and not something else, such as tensile strength.
The shapes of balls are similar to each other because they have principal-curvature measures at various values within certain ranges. Likewise for the shapes of cups (to keep the illustration simple, consider a Chinese teacup, not a cup with a handle). Moreover, ball shapes are more like one another than they are like cup shapes because ball values of principal curvatures are closer to each other than they are to cup values of principal curvatures. Here the invocation of comparative similarity is not idle. To say that ball shapes are more like one another than they are like cup shapes is to say something beyond what is claimed in saying: Shapes that balls have are themselves and not something else, such as shapes that cups have.
The strengths of a solid are of various kinds that are not simply of various values along some common dimension(s). The shapes of a solid are of various kinds, and unlike kinds of strengths, these kinds are of various values along some common dimension(s).
Rand's conception of similarity as sameness of some characteristic, but difference in measure, can be put squarely to work in analyzing comparative similarities of shapes of solids with each other. Then this conception of similarity is a genuine worker, too, in the analysis of the concept shape of a solid, superordinate for the concepts ball-shape and cup-shape. This employment of Rand's conception of similarity in the analysis of comparative similarity, thence in the analysis of superordinates, is just as Rand would have it (1966, 14). But such an employment of Rand's conception of similarity as sameness of some characteristic, but difference in measure, is incorrect in application to the comparative similarities of the various strengths of solids, thence to their superordinate concept strength of a solid.
What will be the proper analysis as we move on up the superordinates? Strengths of a solid are more like strengths of a solid than they are like shapes of a solid. Let us suppose, as Rand supposed, that the reason we can say that strengths of a solid are more like each other than they are like shapes of a solid is because there is some common dimension, the dimension of the conceptual common denominator, between strengths and shapes of a solid. Property of a solid fits the bill for conceptual common denominator. Strengths and shapes of a solid are both properties of a solid. What is the measurable dimension of the concept property of a solid that is common to both strength and shape of a solid? Like the common dimension for strength, it is a dimension consisting of nothing more than various substitution dimensions. The measurable dimension of property of a solid will be the hardness dimension or the tensile-strength dimension or the principal-curvature dimensions or . . . . There is no single, common measure of property of a solid that all specific properties of solids have in common. Rand supposed in error that there were, for she supposed it always the case that there is some same, common measurable dimension supporting the conceptual common denominator for any superordinate concept (1966, 23)[29]. That supposition is here rejected, and measurement-omission analysis of superordinate concepts is here corrected in this respect.
Suppose for a moment, though it be false, that there were some common measurable dimension of property of a solid that was singular, not common merely by substitutions. Then in saying that strengths of a solid are more like each other than they are like shapes of a solid, we could reasonably contend that the values of strengths are closer to each other on the hypothetical common property-of-a-solid dimension than they are to the values of shapes on that common dimension (Rand 1966, 14)[30]. Then the magnitude structure of the common dimension for property of a solid could not be one that affords only ordinal measures. On such measures, there is no telling whether a value between two others is closer to the one than to the other. (Then in an order of values ABCD, one has no measure-basis for clustering B or C with A or D :  B might cluster with A and C cluster with D; or B and C might both cluster with A; or . . . .) The magnitude structure of the common dimension for property of a solid would have to afford additional measurement structure. It would need to afford ratio- or interval-scale measurements. But it is not at all plausible that a measurable dimension common to each instance of property of a solid should have not only ordinal-scale structure, but ratio- or interval-scale structure, when hardness, for instance, has only ordinal structure.
Amended Measure-Definitions of Similarity and Concepts
With possible exception for the most general concepts such as property, Rand supposed that concept classes are always similarity classes (1969, 275–76). This is immediately apparent from comparison of her definition of similarity with her definition of concepts. In the present study, I likewise make Rand's supposition.
Now I have said that a solid's resistance to being scratched is not the same as its resistance to being pulled apart under tensile stress. Nonetheless, these two sorts of strength of a solid are similar. Occasions of hardness are similar to occasions of tensile strength because the same characteristic, limit of resistance to some sort of stress, is possessed by both in different measurable forms. These measurable forms could be merely ordinal, yet in this way be a basis of similarity. Moreover, hardness and tensile strength are more similar to each other than to shape because hardness and tensile strength are two different measurable forms of the same characteristic that is different from the measurable characteristic (pair of principal curvatures) shared by shapes in different degrees.
So I should amend Rand's definition of similarity as follows: Similarity is the relationship between two or more existents possessing the same characteristic(s), but in different measurable degree or in different measurable form.
The corresponding definition of concepts would be: Concepts are mental integrations of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted or with the particular measurable forms of their common distinguishing characteristic(s) omitted.
Every concrete falls under both sorts of concept. Both sorts of conceptual description have application to every concrete. Occasions of hardness fall under the hardness concept by sameness of characteristic in various measures. Those very same occasions of hardness fall under the strength concept by sameness of characteristic varying in measurable form. Occasions of cup-shape fall under the cup-shape concept by sameness of (pairs of) characteristics in various measures. Those very same occasions of cup-shape fall under the spatial property concept by sameness of a characteristic, spatial extension, that has various measurable forms.
Conclusion of Core Task
My amendments to Rand's definitions of concept and concept class (similarity class) do not implicate metaphysical structure beyond what is already implied by Rand's definitions. Where I have spoken of various measurable forms of a characteristic, all of those forms are the same measurable dimensions that are also at work in concept classes based on variation of measure values along a dimension.
What is the magnitude relation under which all concretes must stand such that conceptual rendition of them is possible? They must stand in the relation of a uniform topological lattice, at least one-dimensional. This is the magnitude structure implied for metaphysics, for all existence, by the theory of concepts in Rand's epistemology. The same magnitude structure is implied by that theory with my friendly amendment.
What is the mathematical character of universals, of the collection of potential concept-class members, implicit in Rand's theory of concepts? In Rand's theory, universals are recurrences, repeatable ways that things are or might be. Properties, such as having shape or having hardness, are examples of such ways. That universals are recurrences is a traditional and current mainstay in the theory of universals[31]. In Rand's theory, however, universals are not only recurrences, they are recurrences susceptible to placement on a linear order or they are superordinate-subordinate organizations of recurrences susceptible to placement on such linear orders.
Universals as (abstractions that are) concepts are concept classes with their linear measure values omitted. If the concept is a superordinate, then the linear measurable form might also be omitted, that is, be allowed to vary across acceptable forms. Universals as collections of potential concept-class members are recurrences on a linear order with their measurement values in place[32]. For either sense of the term universals, they are an objective relation between an identifying subject and particulars spanned by those universals (Rand 1966, 7, 29–30, 53-54; 1965, 18; 1957, 1041).
(Continued below.)

II. Analysis
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.
Before entering argumentation for the minimal mathematical structure implied for the metaphysical structure of the world, let us check that we have our proper bearings on objective structure and intrinsic structure. I have ten fingers, eight spaces between those fingers, and two of my fingers are thumbs. That's how many I have of those items. Period. Those numerosities are out there in the world, ready to be counted, and they are what they are whether I count them or not. In our positional notation for expressing and calculating numbers, we choose the number base, but the different base systems designate the same things, the numbers. In base ten, my (fingers, spaces, thumbs) are (10, 8, 2); in base eight (12, 10, 2); and in base two (1010, 1000, 10). The three numbers referred to in all these bases are the same three numbers. In Rand's terminology, the various bases are objective schemes; they are appropriate tools for getting to the intrinsic structure of numbers. But the numbers have intrinsic character—even or odd, whole or fraction, rational or irrational, analytic or transcendental—quite independently of our choices, such as choice of number base.
In asking for the minimal magnitude structure that all concretes must possess if all concretes can be subsumed under concepts for which Rand's measurement-omission analysis holds, we are seeking intrinsic structure, obtaining under every adequate objective expression of that structure. Now we are ready.
Affordance of Ratio or Interval Measures
I have said that the units suspended in the formula "two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted" are units in a double sense: substitution units and measure values. We focus now on units in the latter sense. Rand spoke of measurement as "identification of a relationship in numerical terms" (1966, 39) and as "identification of a relationship—a quantitative relationship established by means of a standard that serves as a unit" (1966, 7; also 33; see further 1969, 188, 199–200). The measure-value sense of unit is the one at work here.
By the expression "a standard that serves as a unit" and by some of her examples of concepts and their measurement bases, one might suppose that Rand's theory of concepts entails that all concretes stand under magnitude relations affording some sort of concatenation measurement. That supposition would be incorrect.
Rand illustrates her theory with the concept length. The pertinent magnitudes of items possessing length are magnitudes of spatial extent in one dimension. Another illustration of Rand's is the concept shape (1966, 11–14; 1969, 184–87). The pertinent magnitudes of items possessing shape, in 3D space, are pairs of linear, spatial magnitudes such as curvature and torsion for shapes of curves or the two principal curvatures for shapes of surfaces[8].
Shapes must possess such pairs of magnitudes in some measure but may possess them in any measure. Observe that Rand's measurement-omission theory does not entail what number of dimensions for the magnitude relations among concretes is appropriate for the concept. Length requires 1D, shape requires 2D. Rand's theory works for any dimensionality and does not entail what the dimensionality must be, except to say that it must be at least 1D. Observe also that the conception of linearity to be applied here to each dimension is not the more particular linearity familiar from analytic coordinate geometry or from abstract vector spaces. It is merely the linearity of a linear order[9].
The magnitude structure of the concretes falling under the concept length affords concatenations. Take as unit of length a sixteenth of an inch. Copies of this unit can be placed end-to-end, in principle, to form any greater length, such as foot, mile, or light-year. This standard concatenation of lengths is properly represented mathematically by simple addition. That is a numerical rule of combination appropriate to concatenations of the concrete magnitude structure in the case of length.
The magnitude structure of the concretes falling under the concept length also affords ratios that are independent of our choice of elementary unit. The ratio of the span of my left hand, thumb-to-pinky, to my height is simply the number it is, regardless of whether we make those two measurements using sixteenths of an inch as elementary unit or millimeters as elementary unit.
Mass is another concept whose concept-class magnitude structure affords simple-addition concatenations and affords ratios of its values that are independent of choice of elementary unit. Because of the latter feature, conversion of pounds to kilograms requires only multiplication by a constant. Such measurement scales are called ratio scales[10]. The mathematical combinations reflecting the concatenations need not be simple addition. This category of scales is somewhat more inclusive than that. It would include the scale for the concept grade (grades of roads, say). Grades can be concatenated, although the proper mathematical reflection of this concatenation is not simple addition[11].
Finest objectivity requires measurement scales appropriate to the magnitude structures to which they are applied. What does appropriate mean in this context? It means that all of the mathematical structure of the measurement scale is needed to capture the concept-class magnitude structure of concretes under consideration. It means as well that all the magnitude structure pertinent to the concept class is describable in terms of the mathematical structure of the measurement scale[12].
What is the magnitude structure of concretes that is appropriately reflected by ratio-scale characterization? It is a magnitude structure whose automorphisms are translations[13]. Translations are transformations of value-points (i.e., points, which may be assigned numerical values) of the magnitude structure (the ordered relational structure of the concept-class concretes) that shift them all by the same amount, altering no intervals between them.
Rand's measurement-omission analysis of concepts and concept classes applies perfectly well to cases in which the measurement scale appropriate to the pertinent magnitude structure of concretes is ratio scale. But Rand's theory does not entail that all concretes afford ratio-scale measures. For Rand's theory does not necessitate that the scale type from which measurements be omitted be ratio scale. Her analysis also works perfectly well for scales having less structure. The magnitude structure entailed for all concretes by Rand's theory is less than the considerable structure that ratio scales reflect.
An analogous conclusion obtains for multidimensional magnitude structures of concept classes. Rand's theory does not entail that all 2D or 3D magnitude structures have both affine structure and absolute structure, as Euclidean geometry has[14]. That is, Rand's theory does not entail that multidimensional magnitude structures of concept classes afford a metric (a measure of the interval between two value-points) definable from a scalar product (a measure of perpendicularity of value-lines)[15].
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]. The magnitude structure underlying the concept class temperature affords only an interval scale of measure. Such magnitude structures do not afford concatenations, unlike the natures of length or mass, but they do afford ordering of differences of degree, and they afford composition of adjacent difference-intervals[17].
Such magnitude structures do not afford ratios of degrees that are independent of choice of unit, but they afford ratios of difference-intervals that are independent of choice of unit and choice of zero-point[18]. Ratio scales have one free parameter, requiring we select the unit, such as yard or meter. These scales are said to be 1-point unique. Interval scales have two free parameters, requiring we select the unit, such as ˚F or ˚C, and requiring we select the zero-point, such as the freezing point of an equally portioned mixture of salt and ice or the freezing point of pure ice. These scales are said to be 2-point unique[19].
The magnitude structure of concretes affording interval-scale characterization is one whose automorphisms are fixed-point collineations, preeminently stretches[20]. Stretches are transformations of the value-points of a magnitude structure such that one point remains fixed and the intervals from that point to all others are altered by a single ratio.
Rand's measurement-omission analysis of concepts and concept classes applies perfectly well to cases in which the measurement scale appropriate to the pertinent magnitude structure of concretes is interval scale. The temperature attribute of a solid or fluid must exist in some measure, but may exist in any measure[21]. But Rand's theory does not entail that all concretes afford interval-scale measures. For Rand's theory does not necessitate that the scale type from which measurements be omitted be interval scale. Her analysis also works perfectly well for a kind of scale having less structure. The magnitude structure entailed for all concretes by Rand's theory is still less than the considerable structure that interval scales reflect.
An analogous conclusion obtains for multidimensional magnitude structures of concept classes. Rand's theory does not entail that all 2D or 3D magnitude structures have not only order structure, but affine structure, as Euclidean and Minkowskian geometry have[22]. That is, Rand's theory does not entail that multidimensional magnitude structures of concept classes afford a metric definable from a norm (a measure on vector structure)[23].
(II. Analysis – continued below)

This essay of mine was first published in V5N2 of The Journal of Ayn Rand Studies – 2004.
Universals and Measurement
I. Orientation
Rand spoke of universals as abstractions that are concepts (1966, 1, 13). Quine spoke in the same vein of "conceptual integration—the integrating of particulars into a universal" (1961, 70). Those uses of universal engage one standard meaning of the term. Another standard meaning is the potential collection to which a concept refers. This is the collection of class members consisting of all the instances falling under the concept[1]. In the present study, the character of universals in the latter sense will be brought into fuller articulation and relief. That vantage will be attained by amplifying the mathematical-metaphysical requirements of Rand's theory of universals as conceptual abstractions.
To begin I situate the topic of the present study within Rand's larger system of metaphysics and epistemology. My core task for the present study will then emerge fully specified.
Rand's system relies on three propositions taken as axioms[2]. (E) Existence exists. (I) Existence is identity. (C ) Consciousness is identification. Rand's set of axioms conveys the fundamental dependence of consciousness on existence. Existence is and is as it is independently of consciousness, whereas consciousness is dependent on existence and characters of existence (Rand 1957, 1015–16; 1966, 29, 55–59; 1969, 228, 240–41, 249–50).
As part of the meaning of (I), Rand contended (Im): All concretes, whether physical or mental, have measurable relations to other concretes (1966, 7–8, 29–33, 39; 1969, 139–40, 189, 199–200, 277–79)[3]. Every concrete thing—whether an entity, attribute, relation, event, motion, locomotion, action, or activity of consciousness—is measurable (Rand 1966, 7, 11–17, 25, 29–33; 1969, 184–87, 223–25).
As part of the meaning of (C ), Rand made the point (Cm): Cognitive systems are measurement systems (1966, 11–15, 21–24; 1969, 140–41, 223–25). Perceptual systems measure[4], and the conceptual faculty measures. 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.
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.

That is arguably the most dangerous issue that we are dealing with as a country. To hear that Trump has been talking about removing the two year limit for a President, was terrifying. Then his believe in complete presidential immunity and the fact that he pardoned people convicted of doing his dirty work. I thought this type of thing only happens in third world fascist systems. When Trump recently tried to remove the position of inspector general, the GOP revolted.
I admit, I was a Trump fan for a generation, watching the Apprentice and trying to learn something from him and yet, from day one of his presidency, claiming his crowd was bigger than Obama's first inauguration, not acknowledging that it is hard to beat an actual first African American President in the History of the United States was shockingly disappointing.
You are assuming that Trump is not a socialist. He wants to create public works like FDR, having the government become the biggest employer. When the Covid first bailout was being negotiated, he wanted the government to have equity in the bailed out companies (even Bernie had not pushed for overt communism).
What will Trump Veto?? He claims that Biden has stolen his economic plan.
Currently he is exacerbating racial tensions, appealing to a silent majority mob, and acting like he fights the evil that he already embodies.
People are worried about a highly unlikely journey to communism with Biden and his Supreme Court nominees, while the worst Americans will tolerate is what it is like in Canada or the UK or Sweden. We are not headed to Communism.
But, we are already a crony capitalist, fascist lite country. Full blown fascism is any easy next step. Trump will not stop it, in fact he is opening the door to it (potentially for the next racist leader that actually likes to kill people). 
The other possibility is civil war. Again, who is best equipped and temperamentally suited to bring factions together?

Basically, if anything goes well, WhyNot will immediately attribute that to Trump enforcing or promoting some good policy and having been right all along, and if anything goes badly, will attribute that to the fact Trump "had to" bow down to consensus. It's a strange way of thinking, because if he wanted to be consistent, and if he thinks that Trump had to bow down, then he should think that Trump is making things worse by not standing by his principles. At which point he'll say your standards are too high. Such is the result of someone using anti-concepts, as you identified. 
For the sake of the entire thread, don't bother engaging him. You won't learn anything, even if you think you might. You'll get dizzy going in circles.
I'm curious about your thoughts on Biden though? 

Then you have destroyed what it means to be quantitative. A range implies that two things can be measured with the same kind of measurement. Sweet and sour cannot be measured with the same kind of measurement. Indeed, they are perceived by the same sense mode, but this is exactly what I mean by qualitative aspects that are not dependent on some prior (implicit) measurement.
This is not precise enough.
I'm going to assume that you already know that cones of the eye are not sensitive to specific colors per se, but sensitive to a range of wavelengths, ranges of wavelengths that we name as colors by convention. But those cones are not sensitive to anything else about color, particularly the impact of luminosity and saturation. That part of perception occurs further in visual processing. For this reason, we can't say that cones are "just" specialized for certain colors. Instead, they are "just" specialized for certain wavelengths. In some way, your body does "care" that the colors are on a spectrum, that's what makes it possible to compare colors in a fully commensurable way!
Not to mention that specialization to "just" different colors would only add further support to what I'm saying: your body is capable of detecting differences of kind (without reliance on a quantitative feature) in addition to differences of quantitative measurement. Color perception is so complex that qualitative differences might be a better picture to the whole story. 
I was confident enough to say that it is a quantity, but I think after reading your post just now, I wouldn't call it a quantity anymore (I like your last sentence). It doesn't alter the point I was getting at though, so everything else I wrote stays the same.

I'm fine with saying that orientation is a relationship, not an attribute - but that's because I agree that orientation must be in respect to something else. How does that support your position though that all qualitative things are epistemological artifacts? After all, you can't think about a length either unless you've compared it to some other entity. Length is intrinsic to an entity, sure, yet omitting particular measurements of length is dealing with something relational because you would be dealing with "long" and "longer". 
Of course orientation can be measured quantitatively, I'm claiming that you can also do qualitative measurement omission. I'd rather say that for a concept to be verifiable, quantitative measurement must be at some level possible. The problem I'm running into is how you would explain coming up with the concept "shadow", or the concept "sour". You could come up with sour by trying a variety of sour things (comparing the blueberry to a lemon for example), but you could also do it by trying a sour thing and then a sweet thing (trying a lemon, then eating chocolate cake). Sweet and sour are comparable as far as both being flavors, but their commensurability ends there. They are not on the same range of intensity; there is sweet and sweeter, and there is sour and more sour. But becoming more sour is not the same as becoming less sweet. Neuroscientifically, this is because each flavor refers to a specific chemical combination. This is in contrast to color, where your eyes detect an intensity of wavelength that is not sharply divided each color. 
Basically, if it can be valid to come up with concepts with qualitative methods, you could then say that some sort of spatial thinking method is as valid as quantitative measurement omission.
Patterns are not concepts, or at least this isn't what this thread would be about (at least as far as Rand's ideas). I've agreed with a lot of what you said, but this looks like a dead end for your position. I see how you mean that it isn't induction now, but even my own experience with pattern recognition, I don't recognize patterns as concepts (you can develop concepts of patterns, though). 

I would like to see a direct quote of Hilbert on that. 
Hilbert did discuss that, in one way, formal systems can be viewed separately from content or meaning. But that does not imply that in another way they cannot be viewed with regard to content or meaning. Indeed, Hilbert was very much concerned with the "contentual" aspect of mathematics. 
Granted, descriptions of Hilbert as viewing mathematics as merely "a pure game of symbols", "without meaning", et. al do occur in literature that simplifies discussion of Hilbert. But for years I have asked people making the claim (here moderated to "reliability") to provide a direct quote from Hilbert. And just looking at Hilbert briefly is enough to see that he was very much concerned with the contentual in mathematics.
I'm simplifying somewhat, but Hilbert distinguished between (1) statements that can be checked by finitistic means and (2) statements that cannot be checked by finitistic means.
Finitistic means are those that can be reduced to finite counting and combination operations - even reducing to finite manipulations of "tokens" (such as stroke marks on paper if we need to concretize). This is unassailable mathematics, even for finitists and constructivists. If one denies finitistic mathematics, then what other mathematics could one possible accept?
On the other hand, mathematics also involves discussion of things such as infinite sequences (try to do even first year calculus without the notion of an infinite sequence). So Hilbert wanted to find a finitistic proof that our axiomatizations of non-finitistic mathematics are consistent. So, there would be unassailable finitistic mathematics (which has clear meaning - that of counting and finite combinatorics) and there would be axiomatized non-finitistic mathematics (of which people may disagree as to whether it has meaning and, if it does have meaning, what that meaning is) that would at least have a finitistic proof of its consistency.
So, of course Hilbert regarded finitistic mathematics as having meaning and being completely reliable. And, I'm pretty sure you will find that Hilbert also understood the scientific application of non-finitistic mathematics (such as calcululs). But he understood that it cannot be checked like finitistic mathematics; so what he wanted was a finititistic (thus utterly reliable) proof that non-finitistic mathematics is at least consistent.
However, Godel (finitistically) proved that Hilbert's hope for a finitistic consistency proof cannot be realized. 
Regarding looking at formal systems separately from content: Imagine you have a formal system such as a computer programming language. We usually regard it to have meaning, such as the actual commands it executes on physical computers or whatever. But also, we can view the mere syntax of it separately, without regard to meaning. One could ask, "Is this page of code in proper syntax? I don't need to know at this moment whether it works to do what I want it to do; I just need to know, for this moment, whether it passes the check for syntax." 
So formal symbol rules can viewed in separation from content, or they can also be viewed with regard to content. Hilbert emphasized, in certain context the separation from content, but in so doing, he did not claim that there is not also a relationship with content.

The system of equations:
2x + 3y = 16
x + 2y = 10
can be placed in matrix form and be pictured with 2-dimensional Cartesian coordinates. I wish I could show the matrix form, but I don't know how to do so here. I omit the picture (graph), too.
Similarly, a system of 3 equations and 3 unknowns can be placed in matrix form and be pictured with 3-dimensional Cartesian coordinates.
On the other hand, a system of higher order, 4 or more, cannot be pictured with spatial coordinates of any kind. Hence, I for one would not describe such a system as "about the world", but rather "about how we can think about the world." Surely, when we start talking about multiplying matrices, we are not talking "about the world", but rather "about how we can think about the world."
Calculus, with its concepts of limits, infinite series, infinitely large and infinitely small, we are not talking "about the world", at least the external world, but rather "about how we can think about the world" and/or methodical thought that takes place in our internal, mental world.

People who say this are usually people who didn't get past undergrad (or have animosity towards people who do get up to the PhD or graduate level). It's a kind of anti-intellectualism. Have you taken graduate courses in philosophy? Have you taken time to understand really complex philosophy, even if you didn't agree? I'm not trying to demean you here - I'm asking that you check your premises that you can and should badmouth people who dare to say something formally in philosophy as would be expected of people in universities who do philosophy of math or logic.
What SK is saying isn't really so scary in all. Oist epistemology is not simple or easy. Sometimes people think it might be, but this is because Rand only wrote an introduction. Adding some formalism doesn't destroy anything. If Oist epistemology can't survive some formal treatment, then it would be a trash epistemology. But the cool thing is that it can. Type theory is probably the absolute best avenue to follow to improve or fairly criticize Oist epistemology from a formal perspective. If you don't like philosophy from a very formal perspective (which this thread is about), that's fine. If you want to participate with it though, you should take the time to understand before criticizing. Rand was generally informal about her philosophy. That doesn't mean it can't get a fair formal treatment.
 
 

I wrote this paper for my own purposes to explain and think more about what makes an object an object. I don't think my idea is incompatible with Objectivism, and it offers new and interesting ideas. Feel free to nit-pick, I edited it to get the exact words I want.
 
Might the universe be an object?
 
I look at something in front of me. I recognize it as a table: four legs, flat surface, wooden. Other things are placed on its surface, like a pencil and a camera. I am comfortable with these identifications. After all, I can touch them, see them, even hear them if I move them in a certain way. Furthermore, I can see edges where one thing ends and the other begins. In other words, these things are entities: things which are bounded and distinct[1]. Through their behavior, they exhibit an identity. Even more, the identity of these entities is independent of my seeing or recognizing them. The entire world in front of me is this way, filled with entities. My perception allows me to see this.
 
Consider that a table is made of parts which are not accessible in a perceptual way. Certainly, I can chop off the legs of the table and then have individual legs, but this is no problem because the legs still remain directly accessible to unaided perception. The parts that I am talking about are not accessible to unaided perception, that is, what the table is decomposable into and is therefore constituted of. The table is constituted of molecules, which cannot be detected just by looking. With a microscope molecules can be detected, and they would have the same distinctness I recognize when I looked at the table without any help. In order to make the distinction between things which I can see without assistance of the things I can't see without assistance, I will consider objects to be both of these, while I will consider entities to be those things that I can see without assistance. Such a distinction is important because I cannot recognize the aspects of a molecule in the same way I can recognize the aspects of a table. A different method is required in order to comprehend a molecule, that is, reliance on tools.
 
A related consideration for the ontology I am sketching out is the objects on the table. To put it simply, the objects share no causal identity to the extent they are distinct and unconnected. The only aspect they all have in common in relation to each other is a spatial characteristic. The camera is on the table, which is as far as the relationship between table and camera extends. I could refer to them as “objects on the table”, and treat them as a set in order to talk about statements like “I knocked over the table, so everything fell off” or “the table is full, I can't put another object on it”. However, there is nothing causal about one object to another in the set of objects on the table. They do not form a system whose constituents operate together.
 
So far, this ontology is nothing radical, and even common sense. Is this all there is to consider though? If I stop here, I may as well say it was sufficient for the Greeks to consider what was immediately and directly available to their perception. As soon as I recognize concrete constituents of an object, and recognize that these constituents are also objects, I need to think about the relation these constituents stand towards other constituents which are not available to unaided perception. Is the relationship only spatial? Or are they directly and causally related, despite my inability to see this possible relationship without a microscope? Intuitively, my answer is that they are causally related exactly because any further distinction I have made is based on having recognized an entity with which I could use a microscope on. The set of objects on the table on the other hand are not reduced from my having seen a larger entity. I did not see an entity and then break it down further. In this way, I have determined that the universe cannot be object. The above view I call perceptual ontology, i.e. ontology bounded and set by perceptual capacities. Important to keep in mind is that objecthood doesn’t depend upon perceptual capacities. Rather, the class of existents (e.g. ideas, concretes, actions) that qualify as objects are specified by what is perceptually detectable without aid and its decomposition.
 
As characterized, perceptual ontology is immediately vulnerable to subjectivity in metaphysics. Indeed, in terms of epistemology, perceptual capacities need not imply a subjectivist epistemology -- there could still be definitive and objective rules to recognizing or knowing that an object is in fact an object. Even more, entities can be considered primary or fundamental to comprehending metaphysics. However, since what qualifies as an object is a direct consequence of a physical reduction from the entity level, I begin to wonder about creatures smaller than humans and what they can see as entities [2]. A microscopic bacteria could detect molecules unaided, that is, molecules would be entities to a microscopic bacteria. Anything too much larger may as well be like the set of objects on the table -- perhaps connected but not complete and entirely contained. Going the other direction, in principle, a massive creature could detect groups of planets like a person detects a dog. The group of planets could plausibly be an entity specifically to that creature. By this reasoning, to a human the group of planets is a set of planets yet not an object because they were not a reduction from the entity level, while the massive creature sees the group of planets on the entity level so the group would qualify as an object. If the very category of existents that qualify as objects in the first place vary based on perceptual differences, the resulting metaphysics would be a direct consequence of the given subject and not a direct consequence of reality.
 
The schema of this problem is easily illustrated:
1. a = {c1, c2, c…, cn}; the constituents of Cx are united as a single abstraction a
2. e = {c1, c2, c…, cn} = {o1, o2, o…, on}; the constituents of Cx are united as entity e. Any element of Cx or unification within Cx is an object.
3. Per(Cx) = e; the function, i.e. the faculty of perception, which recognizes some aspect of the world as bounded and distinct as opposed to an abstraction. The perception of set Cx is sufficient and necessary for set Cx to be an entity and all its elements to be objects.
 
1 is equivalent to the earlier “objects on the table”. Each constituent is an element of the abstraction. 2 is equivalent to a table constituted by molecules and follows a pattern similar to 1. All constituents of entities are objects. 3 means that what qualifies as an entity will be different for any variation of perceptual faculties. 
 
4. The whole set C being an object or not therefore depends on the perceptual capacities of the creature in question. The unity is an object if and only if the constituents are already a division of an entity. Otherwise, it is an abstraction or mental object, which is by definition neither physical nor independent of one’s awareness.
 
One solution to the problem is that efforts to define objects are inherently subjective, that there is in fact no way to objectively state what is or is not an object in any circumstance. More specifically, there is not a multiplicity of objects in reality. Thus, the word “object” ceases to have meaning - there is either exactly one object, or no objects. With Hindu philosophy, there is singular “object” called Brahman, the underlying nature of reality[3]. Any further distinction is considered  “maya”, illusion[4]. At worst, maya is human conceit attempting to satisfy a constant desire to label and categorize, a cause of suffering. At best, it is the world of appearances that the subject acknowledges, which need not determine how the world really is. Brahman is in fact a singularity of all, the only “object” which really counts. So on it goes, towards the denial of one's own ego, towards passive acceptance of existence. Such a consequence is hardly worthwhile.
 
Argument by consequence, however, is not reason to reject a metaphysical claim. If existence is exactly one object, then that’s how it is, for better or worse. The consequence only alters my response to the fact; it may impact the epistemology I develop, or my ethical theories, but disliking the consequences is not a counterargument. The idea of a singularity is wrong because it is parasitic upon more fundamental premises: to speak of a singularity requires having already defined or conceptualized a variety of objects. Denying a multiplicity of objects would just as well deny the means to conceptualize or witness Brahman – denying perception. Ultimately, then, the “solution” is to wipe away a perceiver, such that perceptual faculties are ignored. While metaphysics does not take into account a perceiver for a claim to be valid, coming to understand metaphysical claims takes addressing how one is conscious of reality. It seems that rather than reality being a singularity, there is a threshold on the number of objects one is able to grasp[5]. So, perception’s limits leave me unable to determine which things qualify as objects - besides what I see as an entity, and its constituents.
 
Accepting that there is a threshold on the number of qualifiable objects due to one’s perceptual faculties is no solution, either. This would be taking a stance in favor of maya instead of a singularity. I’d be saying there are an unknown and ungraspable set of objects in reality[6]. If the set of all objects in reality include these “invisible” objects, then all the criteria for an object to fit into the set of all objects are unknown. By this point, it would not be determinable if the known objects really qualify as objects. I would not be able to say if they should be disqualified as objects – for I would be admitting no one will ever find out all the necessary criteria of objecthood. Some currently-known objects may turn out to be non-objects. Unfortunately, no one would ever be able to find out. As a result, no objects in reality are graspable.
 
Imagine the set of all known fruits, then also yet-to-be-discovered fruits. Both sets are graspable, the criteria for qualifying as a fruit can be understood differently in the future, perhaps leading me to recategorize. But if there are a set of extra-dimensional fruits that are unknowable and ungraspable because of the limits to perception, then the set of all fruits would lack any definable criteria. Apples qualify, as do bananas, but I would never know about gooblegorks. If I will never know of gooblegorks, nor why they qualify as fruits, likewise, I won’t know why apples or bananas qualify. So, the entire category of fruit becomes arbitrary or merely nominal - maya. There would be no basis to say what fruits are or are not besides a subjective impression. The same form of reasoning would apply to “invisible” objects.
 
Of course, the above paragraph is a discussion of epistemology. At the same time, any solution to a problem can only be reached by referring to the thinking required. The greater point is to emphasize that all solutions so far are parasitic upon defining objecthood already by means of my awareness and consciousness. A solution requires keeping the idea that entities are distinct and bounded - explaining objecthood any other way is parasitic.
 
The solution I see is to say that objects are also the things entities supercompose into. Just as a table decomposes into molecules, certain entities may supercompose[7] into greater objects. In this way, all things that are objects depend on composability. Entities help with a starting point for qualifying objects; composability as a principle makes use of entities; entities are not a threshold for objecthood. Thus I avoid parasitism issues. I am maintaining premises 1 and 2, the premises pertaining to entities as a basis to qualifying an ontology. Explicitly, I am denying that function 3 expressed as perception (and the tools to extend perception) and decomposability expressed as 2 are sufficient to determine objecthood. Rather, I am proposing that composability is needed – decomposition only works because of composition.
 
Stated generally for any object:
5. Comp(c1 + c2 + c… + cn) = o; the function, i.e. the composing, of a set of constituents. The composibility of set Cx is sufficient and necessary for set Cx to be an object and all its elements to be objects.
 
Stated for any object greater than an entity:
6. SuperComp(e1 + e2 + e… + en) = o; the function, i.e. the supercomposing, of a set of entities. The supercomposibility of set Ex is sufficient and necessary for set Ex to be an object and all its elements to be objects.
 
Why not instead suppose that composition extends infinitely? Because a supercomposition is a combination of entities. If the cardinality of E is 30, then the possible number of supercompositions is 30c30. Each round of compositions will be fewer, and so on until the resulting combination in a single object. A number of the proceeding arguments are why compositions end at a single object as opposed to two or more.
 
If tables decompose into molecules, then molecules compose into tables. In principle, there is no reason to say nothing composes from entities like tables into “supertable”. There is no necessity to stop composition at the level of entity. All that can be said is that determining composition is difficult. Certainly, “supertable” is not an object, because a group of tables have no causal relation, only a spatial relation like objects on a table. Yet if a group of entities have a causal relation, the group is just as much an object as a table composed of molecules. Not just any causal relation will work, though. Two balls bouncing off each other is a direct causal relation, but they still are not singular. They need to be a bounded and distinct unity to be a singular “ballcluster” object as well as two balls. Similarly as an example, two molecules passing through one another doesn’t alone render them into a table. 
 
Three necessary conditions are robust enough for a group of objects to be a composition. Among the group’s elements, there would need to be causal relation strong enough to be called an object as opposed to an abstraction.
 
Systematic
 
The elements in a group of objects operate together simultaneously and affect one another. A loss of one constituent will affect how the group behaves. Taking flour out of a cake recipe will radically alter all aspects of a cake, including texture, shape, baking time, and more. Flour itself will not alter the nature of eggs, but the unity of flour and eggs along with the rest of the ingredients make a specific cake. The nature of the group is different if any element is taken out.
 
Relational
 
A function exists which binds the elements in a group of objects. Being relational makes explicit that the elements are connected. “Next to” is a relation, as is “X > Y” and “the moon orbited the earth”. 
 
Emergence
 
The resulting group possesses one or more attributes which none of the individual elements possess. For example, people are volitional, but their constituents are not volitional – a single neuron is not volitional. Likewise, the process of life and the resulting attribute of being alive don’t make all of a creature’s constituent elements alive. The fact that properly arranging constituents in just the right way (the right mix of carbon molecules, the right external conditions like temperature, etc.) results in a living creature suggests that the constituents form a causal unity with each other.  
 
There are no good examples of a supercomposition aside from science fiction. However, one particular candidate may qualify as a supercomposition: the universe, the unification of all objects that exist. If true, there would be interesting clarifications and ideas regarding identity. First, I need to determine if the universe is an object.
 
The universe is systematic.
All objects that exist make up the nature of the universe. The actions of a planet orbiting a star impacts other stars and other stars’ planets. It is possible to focus on planets as singulars, but the idea is planets and impacted stars, and so on, operate as a system. The more alterations within a system, the more each element will be altered. Taking into account all objects at once is only an expansion of this. Moreover, given that causality never ceases at some ultimate point in time[8], the effects of one object will necessarily continue eternally within the system to the degree the system is complete and bounded. In terms of the universe, it is bounded by all that exists. The universe is the most complete system there is, and its bounds are definite. There is nothing to remove from the universe, and there is nothing to add. Otherwise, the system is incomplete, meaning that it is something different than the universe as defined.
 
The universe is all related.
At minimum, by virtue of being physical, there is a spatial relation between all objects. Two asteroids, or two atoms, placed at opposite ends of the universe bear a spatial relation. The spatial relation makes it possible in principle for any two objects to effect a causal relation. There is always a chronological relation as well, as any group of objects will be acting in some manner.  
 
The universe has emergent attributes.
Exactly the emergent properties of the universe are for cosmologists to discover through science. Cosmology, however, is not the only way to figure out in general attributes unique to the universe. As a complete whole, time holds between all objects at once, which requires a unique time standard. A universal time standard cannot be identical to time standards of more narrow systems. If it were identical, it would be part of an identical system – identical standards would mean using a standard not defined by the context of all objects. So, a time attribute of the universe is emergent, assuming the universe’s systematicity is true. This leads to saying all objects operate together; the actions of one object affect the rest because time applies equally. This is why knowledge of one fact affects how all knowledge is structured.
 
Many more examples of all three are possible. The main idea is that causality spreads in a systematically related way across a system with emergent attributes, or for all constituents of a given object. Applied to the whole universe, causality is eternal and will not cease as long as the universe exists. Eternal causality entails a systematically related universe with at least one emergent attribute.
 
Thus, the universe is an object.
 
A similar idea is that the universe is plenum[9], a continuous substance that connects all things that exist. To be clear, a theory of compositionality is not compatible with universe-as-plenum. Plenum would be a theory that the universe is an object because all objects are directly related spatially - the universe is a Jell-O slab with pieces of fruit suspended inside. But as I argued before, a spatial relation is insufficient for a group of objects to be an object. The added “closeness” of plenum does not help. Instead, compositionality is a theory that the universe is perpetuum[10], a total expanse of all that exists linked by causality.
 
I call it perpetual ontology.
 
<<>>
 
 
[1] This sense of the word entity is intended to be the same as Rand: "The first concepts man forms are concepts of entities—since entities are the only primary existents. (Attributes cannot exist by themselves, they are merely the characteristics of entities; motions are motions of entities; relationships are relationships among entities.)" -ITOE, page 15.
 
[2] I'm reminded of Peikoff's thought experiment on meta-energy puffs. OPAR, pages 45-47.
 
[3] The Brahman is not apprehendable by human means. A yogi may feel being one with the Brahman, but not through their perceptual faculties to see or grasp it. C.f. https://en.wikipedia.org/wiki/Brahman
 
[4] Maya is not just perceptual illusion, but all ways of conceiving of the world whether through perception or cognition. C.f.  http://www.davar.net/EXTRACTS/FICTION/INDIAN.HTM
 
[5] To grasp is used here as a term to cover any form of grasping from mere perceptual awareness to conceptualization. Knowledge is a grasp, as well as perception. Witnessing Brahman would be a grasp, but something distinct from knowledge and perception.
 
[6] Unknown objects are not necessarily ungraspable, just as atoms were not always known. Atoms were always possibly graspable. The issue is proposing that a theoretical, yet-to-be-demonstrated object that cannot ever be grasped.
 
[7]The prefix super- is used to convey that the composition is a composition at or above the level of entity. A supercomposition is not a special form of composition with unique attributes.
 
[8] Time is itself a relation between two actions, so a time which lacks a coinciding action is no time at all. Furthermore, a causal-free point in time implies regions of reality which lack causality. In both ways, the absolute end to any causal chain would be an absolute end to the universe, or at least the universe would be in a frozen state. I’d argue that a frozen state is identical to nothingness, i.e. is nonexistence.
 
[9] Latin - (with genitive, or ablative in later Latin) full (of), filled, plump; https://en.wiktionary.org/wiki/plenus#Latin
 
[10] Latin -  perpetual, continuous, uninterrupted; https://en.wiktionary.org/wiki/perpetuus#Latin

Some of these points have been mentioned, but I'd like to summarize:
 
(1) P -> ~P, as ordinarily understood, is a formula of symbolic logic, which is a context that may differ from the Objectivist notion of logic. One cannot understand such formulas of symbolic logic without first studying the textbook basics of symbolic logic (the Kalish-Montague-Mar book that was mentioned is indeed a fine introductory text). It is not meaningful to discuss such formulas with only a "kinda sorta" vague understanding mixed with Objectivist terminology that might not apply to the specialized terminology of symbolic logic.
 
(2) Symbolic logic has different systems. The usual context of such a formula is what is called 'classical sentential logic' (or 'classical propostional logic'). A less usual context is intuitionistic sentential logic. And there are others. But for purposes of basic discussion, I'll keep to the context of classical sentential logic.
 
(3) Reidy is incorrect that (P -> ~P) -> (P & ~P) is a theorem. What is, for example a theorem, is (P -> ~P) -> ~P. (Also, his mention of the completeness theorem in this context is wrong.)
 
(4) The letter 'P' here is a variable that ranges over "statements" (more precisely 'formulas'). It does not range over other objects. So conflating '->' with 'is' makes no sense.
 
(5) The symbol '->' stands for the Boolean function that maps <P Q> to 0 when P is mapped to 1 and Q is mapped to 0, and maps <P Q> to 1 otherwise. (And '0' can be interpreted as 'false' and '1' as 'true'.) 
 
In this sense, 'P -> Q' is understood as 'if P then Q' (this is called the 'material conditional'). P -> Q is false when P is true and Q is false, and it is true otherwise. 
 
This 'if then' is not claimed to correspond to all other English language meanings of 'if then'. It does correspond usually, but not always, and it is not meant to always correspond. In particular, some people find it wrong, or at least odd, that P -> Q is true when P is false. But in context, it is not intended that this sense of 'if then' corresponds always with certain other ordinary English senses, though it does correspond in a basic way, in the sense of the following analysis:
 
We do not need '->' in symbolic logic. We could take it as a defined symbol in this way:
 
P -> Q
by stipulative definition of our specialized symbol '->' is merely an abbreviation for 
~(P & ~Q).
 
And it is easy to see that, even in virtually all everyday English contexts, ~(P & ~Q) is false when P is true and Q is false, and it is true otherwise, just as we said for P -> Q. 

No, P -> ~P does not just mean "some true proposition entails another true proposition that's not the same as the first proposition."
 
Rather, it means, "If P is true then P is false".
 
And it is not a contradiction. Rather, it boils down to saying "P is false". It does NOT say that there is a statement P such that P is both true and false. '->' is NOT the same as '&'. It says that P implies its own negation, so P itself implies a contradiction since P implies itself (of course) but also it implies its own negation. So P is false since P implies a contradiction. 
 
Again, P -> ~P is not a contradiction. What is a contradiction is to assert both P and P -> ~P.
 
Please, I wish all the people in this thread who are opining about this subject without FIRST learning the basics from a textbook, would indeed first read the chapters of a texbook and then come back to discuss it.

New trolley problem:
The trolley is speeding towards a stack of dynamite that would explode and kill you if you crash into it. If you hit the switch, you will change tracks and be safe. But, you don't own the trolley! Either you get involved the workings of their property, or you die. What do you do?

Come on man, troll a bit better than this! I think it would've been more fun to connect Obama to this.

(1) Those passages don't quote Hilbert or cite any reference to his texts.
(2) The passages are from what I think might be a popularizing book [Godel: A Life In Logic] on the subject. Often such popularizations misleadingly oversimplify the subject. Without having read the book, I won't claim that it does misleadingly oversimplify, but I would caution to look out for possible oversimplifications. That set of passages onto itself might be okay yet it could stand some explanation.
(3) Anyway the passages don't say or even imply that Hilbert took mathematics as entirely a meaningless game of symbols. 
(4) And not only do those passages not say or imply that Hilbert took mathematics as entirely a meaningless game of symbols, but the passages say the OPPOSITE.
/
I am not an authority on this subject; I have read only some of Hilbert's translated writings and none of his writings in German that remain untranslated to English. So my own comments may be too simple or require qualification or sharpening. For a first reference on the Internet, I would suggest:
http://plato.stanford.edu/entries/hilbert-program/
http://plato.stanford.edu/entries/formalism-mathematics/
Moreover, a few years ago, one of the contributors to the Foundations Of Mathematics Forum asked whether anyone knows of any attribution to the writings of Hilbert in which he said that mathematics is only a game of symbols. As I recall, at that time, no one did. (Posters on The Foundations Of Mathematics Forum are almost entirely scholars in the field of mathematical logic and the philosophy of mathematics.)
That said, here are some general points:
(1) Hilbert recognized the role of mathematics in the sciences. He would not regard mathematics as merely a symbol game. 
(2) Hilbert may regard formal systems as subject to being taken, in certain respects, as without meaning. However, I know of no attribution in which Hilbert claimed that mathematics is merely formal systems. Moreover, Hilbert recognized that, while in one aspect formal systems are to be regarded as without meaning, in other aspects, formal systems are to receive interpretation and in interpretation we evaluate meaning. 
The rough idea is that syntax onto itself is without meaning but with semantics we do evaluate meaning.
The syntax includes the formation rules for formulas and the rules for proof steps. Syntax is regarded onto itself as without meaning so that no "subjective", vague, or inexact considerations are allowed in checking whether a symbol string does obey the formation rules for formulas or whether a purported formal proof does indeed use only allowed inference rules. For example, regarding formation rules, when you run a syntax check on lines of computer program code, the syntax checker doesn't care about the "meaning" of your code (say, for example, what it will accomplish for the user of the application or whether the user will like the results, etc.) but only whether the code follows the exact rules of the syntax of the programming language. 
The semantics include the interpretation of the symbols and of the formulas made from the symbols. This is meaning. The interpretation itself can be done either in formal or informal mathematics. For example:
Ax x+0 = x
This is formal string of symbols that in itself has no meaning.
But with a semantics that specifies the domain of natural numbers and interprets 'A' as 'all', 'x' as a "pronoun", '+' as the operation of addition, '0' as the natural number zero, and '=' as identity, we have the interpretation:
zero added to any number is that number
Of course, that example is so simple as to make the method seem silly; with more complicated formulations we see the advantage of the method.
(3) Also, Hilbert distinguished between the contentual and the ideal in mathematics.
Most basicially, the contentual is the the finitary mathematics of "algorithmic" operations on natural numbers. This was later articulated as the formal system PRA (primitive recursive arithmetic), though Hilbert's own earlier work was in a different but akin system. Such operations on natural numbers can be mutually understood as operations on finite strings of symbols. 
The ideal are the infinitary notions of set theory that is used to axiomatize real (number) analysis, as with analysis we regard infinite sequences, etc. 
Hilbertian formalism ("Hilbert's program") is:
The finitary is "safe" and unimpeachable. But, while the ideal may itself be without contentual meaning, it is used as a formal framework for deriving formal theorems (that are later interpreted as generalizations regarding natural numbers and also for real analysis). Then, we wish to know whether the finitary mathematics can prove that the infinitary mathematics is consistent (without formal self-contradiction).
It is Hilbert's hope and expectation of such a finitary proof of the consistentency that was proven by Godel to be unattainable. With regard to Hilbert's program, Godel's second incompleteness theorem reveals that there is no finitary proof of the consistency of the theory of natural numbers (generalizing beyond Godel's particular object theory, say, for example, first order Peano arithmetic) let alone of real analysis.
/
Now let's look at some of the passages from that book:
(1)  "getting at the mathematical truth "
Truth pertains to meaning. If Hilbert was concerned with "getting at the mathematical truth", then he could not have regarded mathematics as merely meaningless symbol manipulating.
(2)  "the statements (symbol strings) should be paradox-free.  In particular, there should be no undecidable propositions "
I don't know all that's intended there, but (un)decidability is a separate (though in certain ways, related) question from consistency (consistency being a formal counterpart to "paradox-free"). 
(3)  "how to interpret the meaningful mathematical objects in terms of meaningful formal ones"
I might say that what are meaningful or not are not objects but instead formulas (or even notions). In any case, again, we see that Hilbert is indeed concerned with meaning. Notions about ideal objects may not be meaningful, but notions about contentual objects are meaningful. PRA has an immediate and "concrete" meaningful interpretation. Then other systems give rise to abstract infinitary notions that don't have such concrete meaning but are "residue" of said formal system that provides theorems regarding generalizations with finitary mathematics and real analysis (which is the theory of the real number calculus used as the basic mathematics of the sciences). Hilbert hoped further that finitary mathematics would prove the consistency of infinitary mathematics - but that's the part proven by Godel not to be possible.
(4)  "Hilbert didn't believe that any Russell-type paradoxes [Set Paradox, Barber Paradox, etc.] lurked in the world of mathematical truths, even though they might exist in the far fuzzier realm of natural language"
Hilbert would have easily known that the Russell paradox can occur even in a formal system (most saliently, Frege's system). Formalization itself does not ensure consistency. 
(5)  "And the way he thought we could prevent them from crossing the border separating ordinary language from mathematics was to formalize the entire universe of mathematical truth.  What Godel showed was that Hilbert was dead wrong."
Hilbert hoped for (indeed, expected) a consistent and complete formal axiomatization of the arithmetic of natural numbers and of analysis. By 'consistent' we mean that there is no formal sentence of this system such that both the sentence and its negation can be proven in the system. By 'complete' we mean that for every formal sentence of this system, either the sentence or its negation can be proven in the system, thus that the sentence is decidable, i.e. that there is an algorithm to decide whether there exists a proof in the system of the sentence (for example, we could keep running proofs until we reach one that either proves the sentence or proves its negation). Godel proved that that expectation was wrong. But this does not entail that formal axiomatizations are not still of great value and interest, as indeed the vast amount of "ordinary" analysis is formalized in any of various formal systems.

Under the Objectivist epistmology, it is a problem to propose a ‘definition’ for an anti-concept. But furthermore, this definition needs some correcting. First, the words is actually used without regard to which political level the redistricting applies to – it could be county, state or federal levels of government. Second, this isn’t a definition of gerrymandering, it is an empirical claim about a result of gerrymandering plus some other political facts. If the Republicans (qua majority party) were to redraw voting districts so that Democrats would most likely become the majority party, that too would in fact be gerrymandering, though it doesn’t satisfy the profferred definition of the word.
I propose that gerrymandering should be simply defined as any redistricting action that serves a political goal other than equal apportionment. If a state has 100 districts and a population of 7,405,743 citizens, then each district shall contain 74,057 citizens (there shall be rounding to accommodate the fact that districts are based on physical residences which can contain various numbers of people, and you can’t have 43% of a person assigned to each district). Any non-random assignment of geographical areas to districts is thus gerrymandering. This covers choices that favor one party over another; it also covers choices intended to increase or decrease the percentage of voters in a district of a certain race, religion, age, occupation, etc.
A computationally-heavy geometry-based approach could be used to choose between SN’s three graphs (but there might also just be three solutions, one of which is selected at random. Because of the population-remainder problem, it is virtually guaranteed that some districts will have 1 more citizen that others. Because (by assumption, open for discussion) the content of a district is a collection of physical addresses and an address can (usually does) contain more than 1 person, addresses need to be included in / excluded from a district in such a way to minimize differences in populations.
However, this does presuppose the principle of geographical representation, largely because it is constitutionally mandated.
 
 

I've threatened to do this for quite some time... so I guess now is as good a time as any.
There's been some discussion on this topic recently, and I'm not opposed to importing quotes -- but for this OP, I'd like to start fresh. I don't have a particular thesis or argument, but I would like to explore the topic of "evasion," and importantly how it intersects with ethics. That is, given "evasion" (however we conceive of it, though obviously that's central to the discussion) what do we do about it? How do we recognize and deal with evasion in others? How do we recognize and deal with evasion in ourselves.
Let me back up for a moment. The first time I ever dealt with evasion, and recognized it as such, was long before reading Rand/discovering Objectivism. I'm sure I didn't use the word "evasion" to describe the phenomenon -- probably something like "denial" would have been quicker to my mind -- but I was debating the Biblical story of the Garden of Eden with a Christian friend of mine, and I wished to make a point by reference to the text of Genesis. I didn't have a copy on-hand, but I was certain that my friend must keep a copy (and we were at his home). I asked him to break out his Bible, so that I could demonstrate the textual basis of my argument -- show that the Bible really did say what I claimed that it did -- and... my friend refused. He did not want to look at the Bible, to see whether I was right or wrong. He didn't want to know.
Now, I know that many people will think that this is besides the point. "Evasion" is an internal phenomenon, a subconscious phenomenon, and so it is. You can't see it happen. I agree. But I have come to believe that evasion often has surface features and effects which may be recognized and addressed. It's kind of like a "tell" in poker: you can't see the other person's cards, but you can see their reaction to their cards, and often people have a characteristic reaction, depending on their hand. That is information, and just like any information, we can try to make sense of it through our best use of reason (bearing in mind the context that we may easily make mistakes in doing so; and sometimes you bet in poker on the basis of what you think you know, and lose).
Usually, this doesn't take the form of someone specifically refusing to look at something -- refusing to look through the proverbial (or literal) telescope -- though sometimes it does. But especially through a long history of debate and conversation, on this forum and elsewhere, what I've found is more often a pronounced reluctance or resistance to specific argument. There are untold arguments where someone has made a claim of, "I will get to that point soon," and then they never, ever do. Not even if it is brought up time and again, or made a point of emphasis. This is not, in itself, proof of anything, let alone "evasion," but especially in context I consider it my best means of determining when a partner in conversation is focused and oriented (in the manner that they would need to be to determine their own error, should I be correct) or otherwise. When examples go unaddressed, when my arguments are paraphrased incorrectly (sometimes wildly so), when questions are asked but never answered, and so forth, it is all information that helps me to see whether someone is engaging with the discussion... or perhaps deflecting it.
And then, in myself, I wonder: how should I know it, if I evade? Because I take it for granted (though perhaps I shouldn't) that a person does not have conscious awareness of his own evasion; if he had conscious awareness, it wouldn't be evasion. That's what makes it so damned troublesome!
What I have found in others, I look for in myself. I look for the effects of evasion, rather than counting on my ability to detect it, as such (or rather than what I fear most do, which is to implicitly assume that I am the only human on earth somehow immune to evasion, of my nature). When I feel reluctant to address some argument or answer some question directly, I try to make it a point of emphasis to do exactly what I am initially disinclined to do. If a question is asked of me, and I fear that my answer will somehow put me at a rhetorical disadvantage, because my instinctive answer somehow "sounds bad" for me or the point I'm trying to make, I consider it doubly important to answer the question directly, and to try to assess whether what I consider a "rhetorical disadvantage" isn't actually just me being wrong about something. I may also choose to answer such a question at length, in an attempt to explain myself more fully, or provide the proper context for interpreting my answer, but I don't let it pass unaddressed because it seems "easier" or feels more comfortable. I fear that those emotional cues, sometimes, may actually be symptoms of an attempt at evasion.
For as I'd recently mentioned elsewhere, I have come to regard evasion as a sort of psychological self-defense mechanism. I think no one is immune. When I try to imagine the extremes of evasion, what I come up with is something like "dissociative identity disorder." To be very honest, I'm not certain whether that's a real phenomenon or not (or the extent, at least, of its "reality"). But suppose that it is. My layman's perspective on it is that it might make sense for a person, in a given context, to "go to war with reality" to some certain extent in order to preserve one's sanity otherwise. To pretend that some outrageous forms of abuse (especially in early childhood) are not truly happening to the self, but "someone else." It is a desperate measure in the face of the truly horrendous, and it portends a lifetime of difficulty and recovery, but in some cases it might still be better than the alternative.
I think that, to lesser or greater extents, evasion is a subconscious means of such self or "ego" preservation.
With my Christian friend (and I sorely wish that I had this level of insight then), it's worth asking: what would he need to defend? What vested interest does a teenager (as we both were) have in the details of the story of bleeding Genesis, such that his emotions would scream at him to avoid looking at his own professed Holy Book? Well, only everything. He'd been raised Christian, in a Christian family, in a Christian community. Though it may not be so simple as this, he regarded his own understanding of the universe -- and his own morality, his own self -- as being based upon his beliefs in the Bible. So... if he were wrong about that, even to the smallest degree, what would that mean for... his beliefs about literally everything else? What would it mean for his regard for his family, for his friends, for himself (in that he had been so thoroughly taken in)? Having been so wrong about this, how could he ever again trust himself going forward?
It's an immensity to consider. And I think that this lies at the heart of the pushback against thought, against evidence, that evasion fundamentally represents. Our survival, our happiness, our lives and all that this represents, depends upon our ability to think, and to be right. And so the possibility of being wrong (and sometimes thoroughly wrong) feels like an attack on our very lives. Evasion, then, is the fear of pain that being wrong, and all that it entails, made manifest at the subconscious level... and then represented at the conscious level by emotional reactions and biases that shade our responses, choices and actions, whether it be something so striking and obvious as an explicit "refusal to look," or something so subtle as an indirect answer to a direct question.
Beyond looking for the "tells" I'd mentioned, resolving to answer questions directly, and etc., what can one do to fight against this tendency? I think that some of my conscious convictions have helped (or at least, so I hope). My conviction, for instance, that being wrong is no moral crime. That it is, in fact, a wondrous joy to discover my own errors -- not a slight against my ego or value, but a tribute to my ability and intelligence. This is how I have come to view debate and argument, not as a contest between enemies, but as a collaboration between allies. I do not feel put off by challenging material; I am drawn to it. (And indeed, I read Rand initially, not because I thought she would agree with me or provide me with some defense of already-held arguments... but because I thought she would disagree with me utterly, and I looked forward to the project of identifying her errors!)
There is an analogy to be made here with my experience of playing games with my daughter. She does not like to lose. Of course. Nobody does. But over the course of my life -- and reflective of what I hope to instill in her early on -- I have come to view losing at games (or "failures" more generally) as being instrumental to the course of improvement... and eventual winning/success. So it is with being wrong. We are all wrong, at times. We are all probably wrong, right now, with respect to some of our beliefs. It is no moral failure to be wrong about things. But the right way of viewing this, in my opinion, is to deeply value the experience of being proved wrong. To then put ourselves in the best position possible to be proved wrong, and to embrace that feeling, embrace the difficult emotions associated with a stern challenge to one's ego, as being part of the true path towards success. And then, also, to look for the concrete manifestations that I have mentioned -- and seek out and discover others, and amend our actions accordingly.
It ain't easy. I'm not always successful, either. But I believe that it's the key to addressing one's own evasion and pushing past it to discover and embrace the truth.

That would be the Pope.

SK,
First off, thank you for kicking off an interesting discussion. It has been most enjoyable.
Now, I don't understand what you say on page 6 paragraph 3. In particular, you say
I do not know what "subly" is. Was this supposed to be "subtly"?
Let me subtly assume the existence of a concept that is tailor-made, etc. There is a concept in mathematics called a field. Examples of fields include the real numbers, complex numbers, finite fields and so on. Would it be fair to say that the concept "field" subsumes these subjects? Now, there is something called a complete field. Examples of complete fields include real numbers and complex numbers, etc. These subjects are subsumed by the concept "complete field". Finally, there is something called a complete ordered field. It seems to me that this is a concept. I have combined well-defined concepts to form a new concept. A priori, one does not know whether there exist any such subjects until once shows that the real numbers do in fact constitute a complete ordered field. Therefore we know that this is not an empty notion. Also, a priori one does not know how many such subjects exist. However, there is a proof that, up to isomorphism, there is only one such subject, the real numbers.
Was "complete ordered field" never a concept? If it was a concept until it was shown that there was only one, at what point did it cease to be a concept? Was it no longer a concept when someone first proved that there was only one such subject? What is a concept in your own mind until someone informed you of the proof that there was only one, making "authority" the determining factor concerning concepts. Or, was it once you read and understood the proof that it ceased to be a concept?
I believe that "complete ordered field" is a concept despite the fact that there is only one. Was this a cheap shot?