Issue #61 April 2023

Unsaturated Entities and the Unity of Science

Gene Davis, Untitled, (1976).

Why is this thing in front of me a table? What makes it one? What connects it to other things that are also tables, and distinguishes it from yet others that are not? Platonic answers to such questions come by reference to the Form Table, which, most important here, is an object: a perfect specimen of tablehood much as the standard meter kept at the International Bureau of Weights and Measures in Sèvres, France, was from 1889 to 1960 the standard of length, but not to be confused with tablehood itself—not to be identified with that quality. It is not clear, though, how in fact the Form Table helps answer the questions asked about the table in front of me. Clearly, the two are radically different: one is an ideal, atemporal, indeed perfect and unchanging example of what tables are supposed to be; the other is a faulty, ephemeral inhabitant of our spatiotemporal swamp. There must be a way of mediating between the two, of bridging the gap; and what could that be? A third table, maybe—which brings together the ideal and the spatiotemporal ones by being similar to both? And how is that “similarity” going to be cashed out? Perhaps by adding further mediations: a fourth table bringing together the ideal one, the spatiotemporal one, and the first mediation; a fifth table bringing together all of the above…?

This is familiar territory, and leads to an equally familiar, sobering conclusion: a homogeneous ontology of objects is incapable of accounting for the structure of the world. If objects are all we have, then the world will not stick together: it will reduce to a laundry list of loose items, each impossible to describe (i.e., to compare and contrast with others). In addition to objects, we need non-objectual entities, connecting links among objects; and objectifying the links (making them objects) will generate a plethora of third-table (or third-man) arguments.1placeholder

All this nasty mess is addressed by Plato—or buried by him—through invoking the vague notion of participation. When we get to Aristotle, we are offered a more mature, and more radical, approach. For him, being is not a genus: there is an irreparable split at its very origin, which gives rise not to a tree but to a forest of Porphyry. The branches ending in the various categories of being are not to be seen as merging, however asymptotically, in a single superordinate point. And the world can stick together because the different categories fulfill different tasks, so that a quality can attach to an object (in his jargon an ousía, or in Latinate jargon a substance), as tablehood does to the table now in front of me, without needing any mediation, as it is the sort of entity designed to do just that.

But there are no free lunches, in philosophy or anywhere else. The price to pay here is the rising of a different problem: if being is irreparably split, how can there be a single science of it? Surely we can say that objects and qualities and quantities and relations are; but isn’t that just a case of ambiguity (in Aristotle’s jargon, of homonymy)? If there is no supercategory under which all these different senses of being fall, haven’t we just traded the lack of unity of the world for the lack of unity of the discourse on it, and reduced ourselves to having a number of homonymous fields of inquiry—which for the sake of clarity we might want to use different labels to pick out, instead of the confusing, deceptively simple word “being”?

The Aristotelian strategy in facing this problem consists of postulating a peculiar structure of being, organized around what has been called the focal meaning of the word “being”: being as substance. Being is not a genus but is not homonymous either: it has a center, and everything that is (called) being is so because of its relation with the center. Qualities, quantities, and relations are because they constitute qualities, quantities, and relations of substances; their being is parasitical on the being of the latter. And there is a single science of being qua being because it is enough for a science to be one that its subject-matter be made of entities that are, in the way I just explained, related to one another.2placeholder

It is not clear that this solution will work: to establish that it does, one would have to perform a total reconnaissance operation of all senses of being and prove that they come together in the structure Aristotle postulates—that they form the analogue of a Solar system with the being of substances as the Sun and every other kind of being properly revolving around it, chained to an inflexible gravitational force. Elsewhere I argued that there is no reason to think of such a picture as correct, consequently that a more decisive move may be needed: away from Aristotelian, analytic semantic theory and toward a dialectical, Hegelian, narrative one where the meanings of words are given not by collections of traits but by (conceptual) stories.3placeholder Here, however, I want to leave this issue aside, in the background from which it has controlled so much of our tradition of thought, and take up a specific question—a specific instance of the general issue, if you will. For Gottlob Frege’s ontology is Aristotelian. There are no ten categories for him, but only two; still, these fewer resources allow him to explain the unity of the world in the same basic way as Aristotle. The world can stick together because some members of it (some entities) are unsaturated: they come with places to fill, and in those places objects (i.e., saturated entities) fit nicely, thus accounting for the connectedness of the whole.

Frege is no general metaphysician; his contribution to the field has mostly to do with the metaphysics of mathematics—though he did, especially after his program failed, venture into some more encompassing statements. So, unsurprisingly, his model for the contrast between unsaturated and saturated entities comes from mathematical practice: from the contrast between a function and its arguments. A function like + has two places to fill, and when you fill it with the arguments 2 and 3 (which, on the contrary, have no places to fill) you get the value 5 (which also has no places to fill). A function whose values are truth-values is a concept for him: the function is German, for example, comes with one place to fill; and if you fill it with the object Gottlob Frege (if Gottlob Frege is the argument of the function) you get the truth-value T(rue) as a value, whereas if you fill it with the object Bertrand Russell you get the truth-value F(alse)—which makes it a concept. (Notice that concepts and objects are part of ontology, not of language: is German is an entity in the world, unsaturated as it turns out, and so are, though saturated, Gottlob Frege and Bertrand Russell. In order to bring all this into language, we need names like “is German,” “Gottlob Frege,” and “Bertrand Russell.”)

The main point I intend to make here is that Frege’s Aristotelian solution is not immune to the problem that plagued Aristotle’s own: the unity of the world is obtained at the expense of the unity of the discourse on it. As a first step toward making that point, consider one of Frege’s most opaque and controversial statements, contained in the following passage from his 1892 essay “On Concept and Object”:


“It must indeed be recognized that here we are confronted by an awkwardness of language, which I admit cannot be avoided, if we say that the concept horse is not a concept, whereas, e.g., the city of Berlin is a city, and the volcano Vesuvius is a volcano.”4placeholder


The struggle one needs to engage in with ordinary language and the resulting awkwardness into which one is often forced are constant themes in Frege, and elsewhere I provided ample evidence of them.5placeholder Now I focus on this particular instance of awkwardness: what is Frege trying to get across here, through linguistic impropriety?

A function can be applied to arguments, and a concept to objects; correspondingly, the name of a function, or of a concept, can be used, as when I say


(1) 2 + 3 = 5

(2) Gottlob Frege is German.


But a function, and a concept, cannot (mostly) be talked about. If I try to, I typically end up objectifying them: making them into something that is no longer unsaturated. An expression like “the concept horse”—or, for that matter, like “the function +”—are no less names of objects than “Gottlob Frege” is, and in fact I can use them in sentences like


(3) The concept horse is discussed by Gottlob Frege

(4) The function + is known to Gottlob Frege,


where the only unsaturated entities in play are is discussed by and is known to, which get you the value T, respectively, when applied to the concept horse and Gottlob Frege, and to the function + and Gottlob Frege, and the value F, respectively, when applied to Ovid’s Ars amatoria and Gottlob Frege, and to the atomic bomb and Gottlob Frege.6placeholder

The awkwardness has been smoothed out now, allowing the serious question to emerge. How can we have a science of being, unitary or otherwise, if one of two major components of being we (mostly) cannot even talk about?7placeholder

Apparently, there is a way out of this difficulty. For to each concept (from now on, I will only deal with those functions that are concepts) correspond not only one or more names but also a single course-of-value, that specifies what objects give the value T when the concept is applied to them (in traditional terminology, what objects fall under the concept). Restricting ourselves for simplicity to concepts with a single place to fill, we can say that to every concept corresponds a single set of objects; for example, to the concept natural number corresponds the set of natural numbers. So, if we cannot talk about concepts, we can do the next best thing: talk about objects that represent them in a unique way. The nonexistent (because largely ineffable) theory of concepts gives way to set theory.

Gene Davis, Untitled, (ca.1983).

Frege’s program consisted of reducing arithmetic to set theory; more explicitly, of proving the axioms of arithmetic to be theorems of the most natural (“naïve”) formulation of set theory. This program failed, and there have been various attempts over the years of extracting some sound fragments from the edifice that collapsed. I have nothing to say here about these attempts or about the failure itself; what I am interested in are the metaphysical details of the program.

There are two distinct notions of a set. According to the first one, a set is a collection of objects. Georg Cantor, the creator of set theory, provides the following definition:


“By an “aggregate” [Menge, the German word commonly translated into English as “set”; “Mengenlehre” is German for “set theory”] we are to understand any collection into a whole M of definite and separate objects m of our intuition or our thought. These objects are called the “elements” of M.”8placeholder


If I go around my study and collect a bunch of random objects (a book, a paper clip, a printer cartridge, my phone), and add to them a few more random objects I am thinking of (Cantor, my dislike for ice-cream, set theory), I can bring them together into the set (or aggregate, in Jourdain’s translation) whose elements are all and only these very objects. As in mathematics curly brackets are used to refer to sets, the set I just imagined would be named by something like


(5) {book X, paper clip Y, printer cartridge Z, my phone, Cantor, my dislike for ice-cream, set theory}.


This notion of a set, however, does not have any immediate relation to concepts, or to Frege’s metaphysics as I described it so far. What we need for that purpose is the second notion, according to which a set is the extension of a concept (where, by the limitation I decided to accept for the sake of simplicity, we will understand solely one-place concepts). Here the relevance to Frege is obvious: as already indicated above, to the concept yellow corresponds (as its extension) the set of yellow objects; to the concept book in my study corresponds the set of books in my study; and, regrettably, to the concept object that does not belong to itself corresponds the contradictory (Russellian) set of objects that do not belong to themselves. In quasi-mathematical argot, these sets would be named by something like


(6) {x: x is yellow} (read the colon as “such that,” and the whole thing as “the set of all x such that x is yellow”)

(7) {x: x is a book in my study}

(8) {x: x does not belong to itself}.


Frege saw his attempted reduction of arithmetic to set theory as a(n attempted) reduction of arithmetic to logic, and by looking at these two notions of a set we can understand why he would see it that way. Arithmetic is the theory of natural numbers, and natural numbers can be read as collections: the number 5, say, as a collection of five units. So arithmetic, it seems, would have to find its place in a universal theory of collections. Concepts, on the other hand, are the traditional province of logic, and at least since George Boole9placeholder that province was meant to include the extensions of concepts. Therefore, if the notion of a set as a collection can somehow be reduced to that of a set as the extension of a concept, and the theory of collections can thereby be reduced to the theory of the extensions of concepts, not just arithmetic but the whole of mathematics insofar as it can be read as a theory (or theories) of collections10placeholder should be reducible to the extensional logic of concepts. (And indeed, Frege had planned a third volume of his magnum opus Grundgesetze der Arithmetik, where the reduction to logic was to be proved for real, and possibly complex, analysis as well. The collapse of the program meant that that volume never came to pass.) The question now is: can this reduction be carried out? Can we interpret any set issuing from collecting random objects as the extension of a concept?

At first blush, it would seem that the answer is Yes. For any object a, for example my phone, can be matched uniquely with the concept is identical with a (for example, is identical with my phone), and the same holds for pairs of objects, triples of objects, quadruples of objects, and so on. The collection of objects listed in (5) above, for example, can be matched with the concept is identical with book X or with paper clip Y or with printer cartridge Z or with my phone or with Cantor or with my dislike for ice-cream or with set theory. And, based on this matching, set (5) can be seen to be the same as set


(9) {x: x = book X or x = paper clip Y or x = printer cartridge Z or x = my phone or x = Cantor or x = my dislike for ice-cream or x = set theory}.


A collection of random objects is thus identified with the extension of a (however outlandish) concept. Prospects look good for a general theory of collections to find room within logic.

Not so fast. The strategy I sketched cannot be used for infinite collections of random objects. If the objects are not random—if they all obey some law that can be stated in finitely many words—then everything is fine. The collection E of all even natural numbers, for example, can be defined as follows:


(10) E =df {x: x is a natural number and x divided by 2 has remainder 0}.


But if D is an infinite collection of natural numbers generated by throwing dice and adding the outcome to the last member that was generated—a truly random collection—then nothing will make the strategy work. There is no (finitely statable) law that we can use to gather all elements of the collection, and assuming that a, b, c are the first three numbers we generated, and that we start using our strategy to get


(11) D =df {x = a or x = b or x = c or …},


the dots will bring out our predicament: however far we get in the sequence, there is no way of knowing what comes next, hence the “definition” of D will never be completed. This collection cannot be rewritten as the extension of a concept.

If we direct our attention, for a moment, to real numbers (where, as I said, Frege originally also wanted to go) we get in trouble even with finite collections. There are various ways of defining the reals, the easiest being a Dedekind cut: a partition of the rational numbers (ratios of natural numbers—fractions, in more common parlance) into two sets A and B such that all elements of A are smaller than all elements of B and A has no largest element. If B has a smallest element, the cut is a rational number; if it does not, it is an irrational number. For example, √2 is the cut where, intuitively, A contains all rational numbers smaller than √2 and B all those larger than √2 (and, being √2 irrational, B has no smallest element). All of this can be expressed as the extension of a concept, giving us the following definition of the set of real numbers R:


(12) R =df {x: x = <A, B> and A and B are sets of rational numbers and every rational number is a member of either A or B and every element of A is smaller than every element of B and A has no largest element},


where <A, B> is the ordered pair of A and B.

The specific element √2 of R can also be captured by the extensional logic of concepts, by first defining


(13) A =df {x: x is a rational number and x2 is smaller than 2}

(14) B =df {x: x is a rational number and x2 is larger than 2}


and then agreeing that


(15) √2 =df <A, B>


But here we must stop. √2 is an algebraic number, that is: a root of an equation in one variable, with rational coefficients. (Specifically, √2 is a root of the equation


(16) x2 = 2.)


In plain English, we can use the relevant equation to name √2, or any other algebraic number, and to talk about them. Algebraic numbers, however, are a tiny portion of the reals. The great majority are non-algebraic, also called transcendental numbers. There are as many algebraic, or rational, numbers as there are natural numbers 0, 1, 2, …; but the reals are of a larger, uncountable cardinality, equivalent to the power-set of the naturals, and what makes the difference in cardinality is the transcendentals—there are as many of them as there are reals. Some of them we have named, like π and the base of natural logarithms e, though to be quite frank about it we do not exactly know what we are naming since we could only ever get approximations to them and there are no formulas like √2 that uniquely capture them. The largest contingent, at any rate, have no names and could not possibly have them, as it is hard to imagine what a language with uncountably many names would amount to. So, if I imagine a collection, finite or infinite, constituted of transcendental numbers and want to use the usual strategy for converting it into the extension of a concept, I get stuck right away:


(17) {x: x = ? or x = ?? or x = ??? or …}.


The great majority of transcendental numbers are something I cannot talk about, even approximately, so how could I possibly state what concept (17) is the extension of?

Let us be clear about what this says and what it does not say. If you are Frege, you will not be bothered by the conclusion that a lot of your ontology is not expressible in language, or beyond the scope of your knowledge. The concept to be associated with any infinite random collection of natural numbers is anyway, so is the concept to be associated with any collection, finite or infinite, of transcendental numbers, and so are their extensions. That we cannot provide names for them or for their extensions does not detract from the being of the ones or the others. But, if the world is no less unitary for any of that, with the saturated entities fitting nicely in the places left open by the unsaturated ones, and releasing truth-values as a result, the above shows that, in the unending oscillation that began (at least) with Plato and Aristotle between the unity of the world and the unity of the discourse on it, the latter is now to suffer. There are collections of objects—themselves objects—on the one hand, and concepts on the other; and no single science of being can account for both. A science of the extensions of concepts can make do for a science of those concepts that remain (mostly) unspeakable, but it will not cover many collections.

Frege’s attempt was a truly heroic one. Mathematicians, and philosophers of mathematics, will concentrate on what it meant for arithmetic or set theory; full-blown metaphysicians, I have suggested here, should be attracted more to what its success would have meant for the perpetual oscillation I have been discussing. If the attempt had worked, we would have had a world unified by positing an Aristotelian kind of being, divided between the mutually exclusive categories of objects and concepts (or, more generally, functions), as well as a science (a logic) that dealt in a unified way with both objects and—via their extensions—concepts (or, more generally, with functions via their courses-of-values). It did not work, for reasons different from the paradoxes that get all the press; and we are left with the oscillation.

Ermanno Bencivenga is a Distinguished Professor of Philosophy and the Humanities, Emeritus, at the University of California. The author of seventy books in three languages and one hundred scholarly articles, he was the founding editor of the international philosophy journal Topoi (Springer) for thirty years, as well as of the Topoi Library. His most recent book in English is Theories of the Logos (Springer, 2017). His two books on Kant are Kant’s Copernican Revolution (Oxford UP, 1987) and Ethics Vindicated: Kant’s Transcendental Legitimation of Moral Discourse (Oxford UP, 2007).


An infinite regress like the one I suggested is argued for (by the eponymous character) in Plato’s Parmenides 132a-b; but Plato never uses the phrase “third-man argument.” Aristotle, on the other hand, who uses it several times (at Sophistical Refutations 178b and at Metaphysics 990b, 1039a, 1079a), never spells out what the argument is. One gets the impression that it was a well-known difficulty among students of the Academy, and that it had to do with the relation between forms and spatiotemporal objects. Whether it was to be identified with the argument in the Parmenides is not to be known for sure; but it was definitely something of that sort.


See Aristotle’s Metaphysics 1003a-b: “There are many senses in which a thing may be said to ‘be,’ but they are related to one central point, one definite kind of thing, and are not homonymous. Everything which is healthy is related to health, one thing in the sense that it preserves health, another in the sense that it produces it, another in the sense that it is a symptom of health, another because it is capable of it. […] So, too, there are many senses in which a thing is said to be, but all refer to one starting-point; some things are said to be because they are substances, others because they are affections of substance, others because they are a process towards substance, or destructions or privations or qualities of substance, or of things which are relative to substance, or negations of some of these things or of substance itself. […] As, then, there is one science which deals with all healthy things, the same applies in the other cases also. For not only in the case of things which have one common notion does the investigation belong to one science, but also in the case of things which are related to one common nature […]. It is clear then that it is the work of one science also to study all things that are, qua being.” Translation edited by Jonathan Barnes (Princeton, NJ: Princeton University Press, 1984).


See my Hegel’s Dialectical Logic (New York: Oxford University Press, 2000).


Translations from the Philosophical Writings of Gottlob Frege, by Peter Geach and Max Black (Oxford: Basil Blackwell, 1977), p. 46.


In Chapter I of my Logic and Other Nonsense: The Case of Anselm and His God (Princeton, NJ: Princeton University Press, 1993).


Nothing much would change if we dropped “the concept” and said “is German is a concept.” If is German were to retain its conceptual nature, it would have a place to fill and what we said would not be a complete sentence. If, on the other hand, we filled that place by saying “Gottlob Frege is German is a concept,” what we said would be patently false. That means, of course, that when I say here things like “is German is an unsaturated entity” what I am saying (by Frege’s lights) should not be said—it is literally forcing the bounds of grammar. (In “Why Frege Should Not Have Said ‘The Concept Horse Is Not a Concept’,” History of Philosophy Quarterly 4, 1986, 449-465, Terence Parsons discusses not so much what Frege said—and clearly meant—but what, in his view, he should have said. I will not deal with such speculations here.) See also the following footnote.


A qualification is in order, which accounts for the parenthetical “mostly.” Frege does allow for cases where concepts are talked about, as long as what he regards as second-order concepts (like existence and number) are applied to them. We talk about the concept square root of 9, for example, if we say “there exist square roots of 9”, and about the concept satellite of Mars if we say “there are two satellites of Mars.” But a science of concepts that was limited to such statements would be very limited indeed: it could not talk about the concept irrational number by saying “the ancient Greeks did not admit the concept of an irrational number”—or any reasonable rephrasing of it.


Contributions to the Founding of the Theory of Transfinite Numbers, translated by Philip E. B. Jourdain (New York: Dover, 1955), p. 85.


See his The Laws of Thought (Amherst, NY: Prometheus Press, 2003).


Traditionally, that would leave out geometry, which deals with continuous quantities, whereas collections are constituted of discrete (distinct) entities (in Cantor’s definition above, the elements of a set are separate). In contemporary mathematics, the dominant (and, in my view, misguided) tendency is to reinterpret continuous quantities as also being collections of discrete entities. I have dealt with this issue, for example, in Theories of the Logos (Berlin: Springer, 2017).


April 2023


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by Jack Graveney

Unsaturated Entities and the Unity of Science

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Mad Utopias - Using Madness as Political Praxis

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Deleuze's "The Logic of Sense", (Chapter 9, Ninth Series of the Problematic)