Every concrete stands in measurement relations—at least up to ordinal scale—with other concretes (M, MP). If a stands in certain ordinal relations R with b, and b stands in certain ordinal relations R′ with c, then a stands in the composition of ordinal relations R′◦R with c (mathematical composition of morphisms). That does not mean there is necessarily some ordinal relation along a single dimension between a and c. But it means that a stands in at least piece-wise ordinal relations with c, and that yields a relation (R′◦R) in addition to the relation of conjunction.

Now, invoking mathematical induction (ordinal), and perhaps some petite propositions supplementing Rand’s thesis that all concretes stand in (at least ordinal) measurement relations to some other concretes, I might be able to demonstrate that every concrete stands in some piece-wise ordinal relations to every other concrete. That would show a unity to all that exists beyond conjunction.

I do not think that such a result would establish that the concrete existent that is all that exists is an entity, because I conjecture that entities are best conceived as entities only if standing in some measurement relations above ordinal (viz., interval and ratio and multidimensional spaces of these). So I reject the slide from existent to entity and the rationale I offered for it in the third paragraph of Good Points.

~~~~~~~~~~~

Frame for this Research

Six years ago, I composed the study “Universals and Measurement” which appeared in The Journal of Ayn Rand Studies 5(2):271–305. In that work, one of the ways in which I characterized the distinctive magnitude structure for metaphysics implied by Rand’s measurement-omission analysis of concepts was according to mathematical category. I reached this result:

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

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

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

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

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

I noted in Ocean of Concretes that “there remains the question of what various levels of unified wholeness can be identified for all that exists beyond its unity as a concrete existent and its thoroughgoing contrast with a singular nothing.” Demonstrating that every concrete stands in some piece-wise ordinal relations to every other concrete would be a characterization, in terms of minimal measurement-affordance, of the right level of unified wholeness that can be rounded (without physics) from the magnitude structure implied by Rand’s measurement-omission analysis of concepts. A companion characterization of that unified wholeness in terms of the mathematical category of the magnitude structure implied by Rand’s theory also awaits discovery.

In addition to the references I listed in the note Easy and Not, I should have mentioned the following one:

Newman, Andrew. 2002. The Correspondence Theory of Truth. Cambridge.

See especially Chapter 8 "The Correspondence Theory and Complex Propositions" with cross hair on 8.3 "Conjunction as a Function."

~~~~~~~~~~~

Complementary to the work of Newman, is that of Eric Wefald in Truth and Knowledge (TK), which appeared in 1996. In my review of this book, a review titled “Mapping Reality,” I mentioned Wefald’s view of knowledge by direct acquaintance:

“In acquaintance we know not only objects, but relations and structures. That is to say, in acquaintance we can know facts. Wefald interprets Wittgenstein’s dictum ‘The world is everything that is the case’ (T 1) to be asserting not ‘that the world is the set of all facts, but rather is the one total fact of which all other facts are substructures’ (TK 9). A structure consists of objects and relations. . . . Wefald would not identify a relation fundamentally with its occasions or its extension, but with its intension. . . . Intensions are not intentions. Possibilities, relations, and intensions are in the world without mind (contrary to Leibniz).” (pp. 229-30)

The entire review is available at Objectivity Archive. Click on V2N4. The review is on pages 227-33.

At her epistemology seminar (c. 1970), responding to a question from Harry Binswanger (A), Rand remarked that the concept infinity, taken as existing in reality, “means something without identity, something not limited by anything, not definable” (ITOE App., 149)

Within the text of her Introduction to Objectivist Epistemology (1966), Rand writes: “It is only after the child has grasped a number of entities that he can advance to the stage of abstracting attributes from entities and forming separate concepts of attributes” (15). I assembled developmental psychology research supporting this thesis in my 1990 essay “Capturing Concepts” which can be read in V1N1 at Objectivity Archive.

“The fundamental question of how particular men learn concepts and the question of what concepts are, are two different issues. . . .” But as an example of a plausible route, Rand shows how concepts of various objects—such as table, chair, bed, or cabinet—could be integrated, in accord with her measurement analysis of concepts, into the wider concept of furniture. The particular kinds of furniture stand as units within the wider scope of the concept furniture. “The distinguishing characteristics of these units are specified categories of measurement of shape, such as ‘a flat, level surface and support(s)’ in the case of tables” (21, italics added). (See further an exchange between Binswanger and Rand on pp. 146–47.)

The “categories of measurement” refers here not to the various kinds of measurement we have—such as ordinal, hyperordinal, interval, and ratio—rather it refers here to dimensions, such as the dimensions of shape, along which measurements can be made. (See also the exchange between Leonard Peikoff [E] and Rand on pp. 198–200.)

Would it be possible for some existent(s) to have an infinite number of dimensions? The question I here pose is not whether it is possible for a mathematical entity, such as a function, to have an infinite number of dimensions. The question here posed is whether from metaphysics and epistemology one should rule out the possibility of a concrete existent having an infinite number of dimensions.

In our ordinary and scientific experience, we have no indication of there being such a thing. Our hard-won identification of the dimensions of things indicates that for concrete existents there are only a finite number of dimensions had by them. The clock beside me has only a finite number of dimensions even if I don’t know all of them. The number of dimensions whose measure is being marked by the clock is also finite, for it is exactly a single dimension (which also happens to have invariant magnitude, presumably infinitely divisible, in all inertial frames).

Is our knowledge that the number of dimensions of any concrete existent, including the totality that is all that exists, a finite number (indeed, some very large, but particular, positive integer) something we can demonstrate in a proof? If we suppose that either a leaf or a stone has an infinite number of dimensions beyond the ones we know, can it be proven that the identities of leaf and stone would, contrary to fact, not be distinct? So far, I doubt that such a proof can be constructed.

(In the Part 1 of Spinoza’s Ethics [P1–P23], he constructs arguments to the contrary result that the number of dimensions of widest existence must be infinite. I leave the assessment [presumably negative] of Spinoza’s labor to thine own sagacity, gentle reader. Do not neglect his definitions.)

Are you aware of the Objectivist position on the primacy of existence?

If you haven't taken the extended courses on these issues by Peikoff I can understand that there could be misunderstandings about these issues.

The courses on these issues are a revelation one can not, on the other hand, ever turn back the clock on in terms of understanding sense perception and logic's lock-step relationship with truth, and the difference between the metaphysical and the man-made, the perceptual and the conceptual. I really don't regard anyone as being an objectivist even lower case if they haven't taken these courses. They are essential. The rest is just "I read her novel and loved it." Which is great and everything, but it's not the fullness of her philosophy which is set down and forth by Peikoff in his courses. Anyone interested in this must take these courses. You'll be blown further away than you ever thought you were by the books. Critics, especially, must take these courses, or have no credibility with me.

Casey

So all the concrete existents in the room, together, constitute a concrete existent. All of them together in the room during the first second of the last hour constituted a single concrete existent. They constitute a concrete existent among all the concrete existents of the past (right up to and including the initial singularity [an existent], if you want to incorporate the possibility posed by classical general relativity). Then they with all the concretes in all the compartments of space and past time constitute an existent that is all the concrete existents of the past.

All the concrete existents of the past could be an infinite number, yet constitute a single existent. Forget not the divisibility-candidates of one second of duration, an existent. And forget not that all the concrete existents of the present together constitute a concrete existent.

Not an entity, by the reckoning this far, but a concrete existent. Constituent entities of a concrete existent have not been shown to necessarily combine into a single entity. Whether the concrete existent that is all of present and past existence is necessarily an entity remains to be argued from “existence exists” and “existence is identity.”

If we restrict sufficiently the class of existents we want to designate by the term entity, we can say up front that the totality "all that exists" is not an entity. From page 1016 of Atlas, one gets the impression that Rand would call by entity the following things: leaf, stone, man, table, cell, molecule, atom, and the universe. That would be a natural way. But whether one would call all these things entities or restrict that designation so as to exclude the universe, there remains the question of what various levels of unified wholeness can be identified for all that exists beyond its unity as a concrete existent and its thoroughgoing contrast with a singular nothing.

Rand held that fundamentally, “that which exists is concrete” (P-E of Art 23). I imagine Ross, Casey, and Jon hold this view as well. I hold it, you bet.                   

One can hold that fundamentally existents are concrete without also holding any and all of them to be perceivable directly. Some concretes are perceivable directly, some are not. Some concretes directly perceivable are the foundations through which we know what we know. But there are other concretes, not directly perceivable, that we come to know by way of perceivable concretes.

Concrete existence is more than the concretes we directly perceive or could directly perceive. Fundamentally, all the existents here in my library are concrete particulars. Whether they are concrete entities, attributes, actions, or relationships, they are concrete particulars. These are the fundamental existents in the room (fundamental to any abstractions concerning them). All of those concretes are in the room concretely, even if many are not directly perceivable and even if many are not yet known specifically by us at all. And we know that. A proof demonstrating the truth of that knowledge claim might be helpful in discovering proof(s) that there is an existent, fundamentally concrete, that is the totality of all concrete existents.

Meanwhile, I should make explicit the following thesis. I hold that this concrete particular existent and that concrete particular existent are a concrete particular: just this one and that one, conjoined in reference as an existent because coexistent. Coexistence of concretes is a concrete relation, whatever further concrete relations may exist between them and whatever be the abstract relations (sets or concepts) in which we might place them. The coexistence of concrete particulars is a concrete circumstance obtaining without us. Rand and I are evidently together on this pertinent point (ITOE 37, 197).

"The collection of all that exists is a determinate existent whether or not it is finite."

But it's only conceptual, in that it relates to, or embodies, lesser, directly perceivable things. The universe as a whole--your universal existent--is not perceivable on its own. It's only an idea, and as such we use it for convenience, for the purposes of discussion and other things. As such it can't be finite; it's an abstraction and therefore can include infinite concretes. Surely.

The first statement in my previous post was: “Anything more than existence is nonexistent.” In view of Jon’s concerns, I should stress that by my use of more in that statement, I am not prejudging whether the total number of existents is finite or infinite.

There are things that do not exist (all of them the same nothing, really), whether or not the number of existents is finite. The collection of all that exists is a determinate existent whether or not it is finite.

Sometimes the term finite is used simply to mean determinate. That is fine, but as Jon has pointed out, it is incorrect to suppose that finite in that sense entails finite in the sense that the number of one’s fingers is finite.

Beside me is a clock. Its hands tick off the seconds. The ticks are existents, specifically, events. The duration they mark is an existent. To say that this existent is determinate does not prejudice whether the duration of one second consists of: (i) a large finite number (say 10exp25) of indivisible durations, (ii) infinitely many divisible durations, where the infinity coincides with the rational numbers, or (iii) (as we suppose in our physics today) infinitely many divisible durations, where the infinity coincides with the real numbers (rationals and irrationals).

Now, a finer point. I stated in the previous post that everything that exists is bounded by a single nothing. Duration is no exception. Duration is only what it is and not anything else. Duration that is what it is not does not exist. Duration is bounded by that same metaphysical zero. Here is the finer point: Other boundaries of duration not nothing are existents. The boundaries of a particular duration of one second are existents. True, they cannot exist without durations, nevertheless, they exist.

Anything more than existence is nonexistent. Every nonexistent is the same single nothing. The fact that my pencil is a hexagonal prism, a solid material, and contains graphite means that the instrument with which I write is not round, not liquid, and not free of graphite. The nonexistent that is the nonexistent round pencil in my hand is, in reality, not different than the nonexistent liquid pencil in my hand or the nonexistent graphite-less pencil in my hand. Moreover, the never-existing pencils like mine (hexagonal, solid phase, and containing graphite) are the same single nothing. The nonexistent does not progress from the singular to a some more than one nor to an all more than one.

Existence—all that exists—is bounded by a single nothing. In this way, whatever other ways there may be, all that exists constitutes a single whole.

If this much is a sound philosophical argument, what other ways in which all that exists is a single unified whole can be proven by such philosophical demonstrations? I drafted the argument above just this morning, after reading Jon’s latest post, so perhaps my present argument will not stand up even for such a minimal degree of unified wholeness. Criticisms welcome.

Stephen,

You say you are interested in a philosophical (metaphysical,) as opposed to scientific proof of the existence of a single existent that collects and includes all existents. As you know, I hold that no such proof is possible.

The best attempt I have seen goes like this: We know that some things exist, since we perceive entities, their relationships, etc. Further, we know that every entity, relationship among entities, etc. we have ever perceived is finite. So we form the abstraction “the universe” as the collection of all that exists. It’s like a very big basket into which we mentally place every existent. Once everything is inside the basket, and nothing left out, we have a single, very big collection. Its constituents are finite, so the collection is, too. It’s the biggest of big entities. It cannot have any relationship to anything else, because there is nothing else—so nothing, not even relationships, is left out. Like every other entity, it is finite. Therefore, the universe is finite, and, qua collection, it is a single existent that includes all existents.

The problem with the above is its commission of Primacy of Consciousness that Casey refers to (imputing it to the wrong side of the argument.) The problem is in thinking that if the “very big basket” can be imagined as leaving out nothing, then it is true that there is such a collection which leaves out nothing. In other words, the above “proof” ASSUMES that the universe is finite. But what if there is no limit to the entities in existence? Then you could never finish putting things into the basket, because there would always be something else to put in. And if you can’t finish filling the basket, then you can’t say there is an existent (the basket) that includes all existents.

I think we can still speak of “the universe” and be understood. It’s just that we can’t say it is necessarily finite.

You wrote, "I think we can have that much knowledge of the universe before we get the specific dimensions, quantities, structures, and activities of the universe from our exquisite modern science."

The concept of universe is not "knowledge." It's a bucket for putting knowledge into. The concept of "universe" philosophically speaking does not tell us anything about the universe -- it merely provides a folder in which everything can be filed. It's the equivalent of a library that excludes nothing and includes everything. It's not physics, it's philosophy.

Proof of the “exactly one” clause appears easy. I composed this one in a few minutes:

If there is at least one totality existent that includes all existents, there may be one such totality or more than one such totality. If there is more than one, then some existent in one is not in some other. Otherwise, those two totality existents would be one, not two. But all existents must be included in any one totality that includes all existents. Therefore, if there is at least one totality existent that includes all existents, there is only one such totality.

Proof of the antecedent in the preceding proof seems to be more difficult. That would be proof that there is one totality existent that includes all existents, rather than no such totality.

The following books contain work pertaining to the present challenge. The considerable dividends of having the thesis “there is an existent that is the totality of all existents” are set out by David Armstrong (1997, 2004).

Armstrong, David. 1997. A World of States of Affairs (Chap. 13). Cambridge.

--------. 2004. Truth and Truthmakers (§§5.2, 6.3). Cambridge.

van Fraassen, Bas. 2002. The Empirical Stance (Chap. 1 & App. A). Yale.

Lewis, David. 1986. On the Plurality of Worlds. Blackwell.

--------. 1991. Parts of Classes. Blackwell.

Ross and Casey:

From standard modern physics, we know that the universe has a definite total mass-energy and a definite total angular momentum (which we think is zero by observations thus far) and that these two quantities remain constant though time. The parts of the universe are tied together throughout the whole by gravitational attraction and by conservation of angular momentum.

Some might say that from philosophy, without science, we can only speak of “all existents” and leave open whether all existents taken together compose a whole, an existing totality. Then let science investigate and find, as it has: “It turns out that all existents are a definite total, and they have such-and-such specific characteristics totaling to certain definite values which we are tracking down.”

I think philosophy can go a little further even before the light from modern science. Our concept of “all existents, however many they may be and whichever characteristics they may share” is a concept targeting all of existence. It is not only a concept, but has as referent: all that exists. I think we can have that much knowledge of the universe before we get the specific dimensions, quantities, structures, and activities of the universe from our exquisite modern science.

Whatever the stage of scientific knowledge, I think we can know that “there is an existent, exactly one, that is the totality of all existents.” In search of proofs.

The "universe" is a concept, not an "existent." It is all existents however many that may be. There is nothing in reality that specifically ties them together into an "entity" any more than the words "all planets" or "all apples" apply to some kind of unit in reality. The universe refers to everything which ever did, does or ever will exist. Whatever that might comprise.

Let's not start confusing the metaphysical with the manmade!

"Can we deny that “there is an existent, exactly one, that is the totality of all existents” and show a contradiction with “existence exists” or “existence is identity”?"

Well, we can't deny that we have a *concept* for all existents, which we call existence (the universe). But I'm just restating my original reply. It's just a collective noun, isn't it? Just like a herd of cattle is comprised of individual animals. The herd can only be held to exist as long as its constituents (cows) do. If existence is an existent in and of itself, then it wouldn't depend on its constituent existents which, like a herd, it obviously does. Existence without anything actually existing is a contradiction.

When I read your original post--and this may be testament to the power of art--I immediately thought of the final scene in the original Men in Black. As if reversing a film, we pull back from the Earth, past the moon, past the outer planets; our solar system recedes to show the Milky Way, which itself shrinks to reveal millions of such galaxies, and finally our universe is revealed... inside a *marble*, one of many little marbles being played with by an alien, whose hand comes down into frame and scoops the marbles into a cloth bag.

Of course, that stunning sequence doesn't stop us contemplating where the retrogression stops.

Ross and Jon,

Yes, our concept existence is our concept of all existents taken together. Rand once remarked to Allan Gotthelf: “The concept existence, at least the way I use it, is in a certain way close to the concept universe—all that which exists” (ITOE app. 241). The referent of this concept of existence includes entities and their attributes, actions, and relationships. As you will recall, Rand used entity not as an equivalent for existent, but as the primary existents, the existents in which attributes, actions, and relationships have their existence. All of these are existents, and they can all obtain concretely, not only abstractly.

Philosophers often use the term entity to mean any item whatever. That is one customary usage and perfectly alright. Rand wanted to take entity into her technical vocabulary as something more narrow. In her sense, entity is much like Aristotle’s substance, but wearing identity on its sleeve.

I spoke of “an existent that is a whole and not part of some yet larger entity.” That shift from existent to entity is sprung by the phrase “whole and not part of some yet larger.” I am hasty in calling a largest existent comprising all existents an entity. Staying with Rand’s more restrictive sense of entity, I should first speak of “an existent that is a whole and not part of some yet larger existent.” Then I should justify the transition from existent to entity as the last word in that formula. After all, attributes, actions, and relationships can also stand in part-whole relations. But since they stand in part-whole relations inhering always in entities, allow my haste here from existent to entity.

Jon, I distinguish between a collection and a set in the logical/mathematical sense of set. The lamp, the inclined editor’s desk, the pen in my hand, and the sheet of paper on which I write are a collection. They exhibit the finite set that is the number four. But they are a collection, the concrete collection they compose, quite apart from our abstract set-theoretic comprehension of them. (Thanks to modern science, we may say further that this collection has the combined rest mass it has now, available to our measurement if we want to know it.) To be sure, we use set in an everyday way too, by which we simply mean a collection. Perhaps that is all you meant by the term.

(It is only with the logical/mathematical rendition of set that we encounter the set-theoretic paradoxes, which were exiled by the modern axioms.)

The abstraction “all of the apples in the bushel basket” is truly an abstraction. But things arrived at by abstraction can sometimes also be concretes existing without the abstraction. We concur in this: The thesis that all existents taken together do in fact exist as a collection does not entail that their combined mass-energy or their collective spatial extent is finite. That is a good reason for Rand stopping short of identifying “all that which exists” tout court with “the universe”. Perhaps it is best to reserve universe for “all that exists” or “all of existence” only when these are more specifically characterized and informed by common experience and science.

I do think, as Ross writes, that “to claim that existence [all of existence, as a whole] itself was an existent . . . you’d be implying that it was only one of many such [subsidiary] existents.” (Please correct me if my interpolations change what you meant to say, Ross.) That’s neat. I am hoping to unearth the nature of that implication. I hope we can further explicate the implication on which our implying rests in this case. Can we exhibit the implication in a proof, a metaphysical proof? Can we deny that “there is an existent, exactly one, that is the totality of all existents” and show a contradiction with “existence exists” or “existence is identity”?

Ross,

I agree with you. I have a problem with calling the universe an entity. Rather, it is a SET of entities. A bushel of apples is not an apple. “All apples” is an abstraction, not an entity, just as “all entities” (the universe) is not an entity.

In my experience through various discussions over the logical possibility of an infinitely extensive universe, the side arguing for logically necessary finitude trots this out so they can employ the Law of Identity: Identity requires that every entity have a specific, delineated, finite nature— the universe is an entity—therefore, the universe is finite.

Of course, every particular entity has a specific, delineated, finite nature. The question is: how many entities exist? Can we know, a priori, that logic requires there be a limit? They avoid the issue by defining the set as itself an entity.

Existence is a concept, subsumed within which are the actual existents of the universe.

I'd have thought that to claim that existence itself was an existent, that you'd be implying that it was only one of many such existents.