The Nothing and the Empty Set (undergraduate paper by David Backer)

Written as a term paper for a class entitled "Phenomenology and Existentialism," submitted and presented at two student conferences in the winter of 2006.


Abstract:
Written off as a continental philosopher by many steeped in the analytic tradition, Martin Heidegger is laughed at for, among other things, flagrant instances of contradiction in his writing. “What is Metaphysics?” might be considered one example of this. Heidegger argues here that metaphysics is concerned primarily with an entity called the nothing, which he defines as “the negation of the totality of beings.” This idea has, at least in my conversations with other philosophers, been met with rolling eyes.
In an attempt to undermine the closed-minded attitudes of both continental and analytic philosophers, I argue in the following that Heidegger’s argument is actually supported by set theory, a mathematical logic. I will first represent Heidegger’s position. Second, I introduce set theory and the concept of the empty set. Finally, I show how this set allows us to talk about Heidegger’s ‘the nothing’ without contradicting ourselves.



‘The Nothing’ and the Empty Set"


‘The Nothing’
Heidegger’s “What is Metaphysics?” is concerned with the “prescientific”1 nature of metaphysics. To begin answering his title question, Heidegger extracts a conception of nothingness by describing the scientific approach as the following:
That to which the relation to the world refers are beings themselves—and nothing besides. That from which every attitude takes its guidance are beings themselves—and nothing further. That with which the scientific confrontation in the irruption occurs are beings themselves—and beyond that nothing…What about this nothing? Is it only a manner of speaking—and nothing besides?2
Because science is concerned with beings in the world (that which is and nothing else) Heidegger finds a niche for metaphysics: a discourse concerned with that which lies beyond the physics, the “beings themselves”— a discourse concerned with that which is not. Heidegger, to further explicate his interpretation of metaphysical discourse, goes on to ask: “How is it with the nothing?” Known throughout the essay as ‘the nothing’, he defines this centerpiece of his argument as the “negation of the totality of beings.”3 This set of ‘negated beings’ is, for Heidegger, the set with which metaphysics should concern itself. But Heidegger notes that
In our asking we posit the nothing in advance as something that ‘is’ such and such; we posit it as a being. But that is exactly what it is distinguished from. Interrogating that nothing—asking what and how it, the nothing is—turns what is interrogated into its opposite. The question deprives itself of its object.4
In other words, the question “how is it with the nothing?” contains the relation ‘…is…’ which contradicts the very idea of ‘the nothing’—which is not by definition. In other words: “…the ‘proper’ nothing itself—is not this the camouflaged but absurd concept of a nothing that is?”5 This problem of contradiction is a serious monkey wrench in Heidegger’s definition of metaphysics: How can we talk about something that is not something by definition? How do we employ our verb ‘to be’ when discussing a thing that necessitates the negation of this verb? Heidegger responds to these questions by citing the “intellectual” necessity of negation. To make his discussion of ‘the nothing’ viable he skirts the problems of logical contradiction by arguing that negation is an activity of the intellect, and that ‘the nothing’ should be considered because of this:
…the proposition that contradiction is to be avoided, universal “logic” itself, lays low this question [concerning ‘the nothing’]. For thinking, which is always essentially thinking about something, must act in a way contrary to its own essence when it thinks of the nothing…Is not the intellect the taskmaster in this question of the nothing?6
While citing negation as an intellectual activity is a clever way to make discussion of ‘the nothing’ possible, Heidegger does not satisfactorily rid his argument of contradiction. If thought is thought of something, how are we to wrap our brains around nothing? While Heidegger is able to continue his discussion of ‘the nothing’ through a series of questions surrounding the origin of negation and ‘the not’, the thorn of contradiction persists. Throughout his discussion one is left wondering: Is there a way to remove it?
2) The Empty Set
In the first few pages of the essay, during his discussion of science and other pursuits, Heidegger writes that
Mathematical knowledge is no more rigorous than philological-historical knowledge. It merely has the character of exactness which does not coincide with rigor.7
We might interpret this to mean that mathematics does not have the capacity to reveal anything about the essence of phenomena, using Husserl’s interpretation of “rigor” as a guide. But while mathematics might lack this phenomenological rigor, its exactness or strict use of deductive proof actually helps Heidegger overcome the aforementioned problems of contradiction in his ‘the nothing’. Set theory, a mathematical logic that “first of all…can be used as a vehicle for communication,”8 provides a way of referring to ‘the nothing’ that follows validly from set-theoretic axioms.
First, what is a set? Machover defines it as “a definite collection, a plurality of objects of any kind, which is itself apprehended as a single object.”9 Potter uses the term “aggregation” to refer to this idea more generally.10 We find aggregation all over the place. Take a gaggle of geese for example. The referent of the word ‘gaggle’ is the group of geese as a singular entity, one thing composed of other individual things (like this or that gosling).11 The term “collection,” as Machover is employing it, refers specifically to a group of things considered as a single object. As Potter remarks, “A collection…does not merely lump several objects together into one: it keeps the things distinct and is a further entity over and above them.”12 Thus ‘gaggle’ refers to one thing composed of many things: the single set (the gaggle) of several geese—that is, the gaggle (like a set) is counted as one thing.
Second, what is the empty set and how does it follow deductively from the axioms of set theory? David Lewis defines the empty set as “the set-theoretical intersection of x and y, where x and y have no members in common.”13 In other words: imagine two gaggles of geese that share no goslings in common. The empty set is the set of the goslings the two gaggles share. To get a bit more technical, Machover defines the empty set as the following:
If n is any natural number and a1…..an are any objects, we put {a1…..an} = df{x: x = x or x = a1… x = an} In particular, for n = 0 we get the empty class { } = {x : x does not equal x} which we denote by O.14
O is a set. Clearly, O is included in any class, and in particular any set…Hence O is included in some set, and by the Axiom of Subset is itself a set.15

In English: the empty class is defined as the group of things that are not themselves ({x: x does not equal x}). And because this group can be found in any collection of objects (that is, there is always room for nothing in any collection), by a set-theoretical axiom we deduce that this collection is itself a set: the set of nothing. Lewis continues to note (with language rarely found in texts on mathematical logic) that the empty set can also be thought of as “a little speck of sheer nothingness, a sort of black hole in the fabric of Reality itself…a special individual with a whiff of nothingness about it.” 16 17
Again, Heidegger’s definition of ‘the nothing’ is “the negation of the totality of beings.” Given the definition of the empty set as the set of things that are not themselves, it is difficult to ignore the similarities between “the set of nothing,” a set-theoretical entity that refers to all things that are not; and “the negation of the totality of beings.” The former, an entity in a logic used for purposes of communication, can be used to refer to the latter. Thus mathematical logic can alleviate the aforementioned worries about Heidegger’s ‘the nothing’.
To head off some initial concerns: this paper does not equate Heidegger’s ontology with that of set theory. In fact set theory, as Potter notes, is a language used to communicate about objects. This is a point in harmony with Heidegger’s comment concerning mathematical knowledge’s lack of rigor—that is, it is difficult to make claims about the metaphysical status of other objects (like geese) using mathematics. Set theory is thus an ideal tool for Heidegger’s arguments because it is a language with which we can logically interpret his idea of ‘the nothing’.
But with such an unusual (and by no means obvious) correlation between a phenomenological account of metaphysics and a branch of mathematical logic, a more careful explanation of arithmetic and how it helps Heidegger is necessary to make this connection clear.
3) Zero, Absence, ‘the Nothing’, and the Empty Set
A simple way of thinking about numbers, such as zero or one or two, is that they are a method of counting. Numbers quantify objects in the world: they account for the presence of things. For instance, take the statement “there are two cumquats in the basket.” ‘Two’ in this sentence tells us how many objects, namely cumquats, there are in the basket. There are also no elephants in the basket. Zero, then, is a number that tells us how many elephants are in the basket: none.
Bringing the discussion back to set theory, one might say that the empty set is the set of all the elephants in the basket. If we wanted to talk about the set of all things that are not in the basket, we would use the same language: the empty set is the set of all things that are not in the basket. Straying now from the basket image entirely, it is clear that the empty set is the set of all things that are not—it is the set of nothing.
This brings us back to Heidegger. As has been noted several times, Heidegger defines ‘the nothing’ as “the negation of the totality of beings.” To alleviate the problems of contradiction in Heidegger’s concept of ‘the nothing’, I think it is possible to adopt a set-theoretical understanding of it.
To restate the problem at hand: how are we to discuss a concept called ‘the nothing’ when thought necessarily thinks of something? If we remember the empty set at this point and conclusion from section two, we should be able to speak logically about the collection of negated things: we can just refer to the ‘the nothing’ with the empty set and continue the discussion without issue. But can the empty set really refer to “the negation of the totality of beings”? Perhaps a better question to address before this one is: Where do Heidegger’s ideas meet the quantificational realm of mathematics, and how does this meeting ground provide a sturdy platform on which Heidegger and set theory can be brought together?
Heidegger writes that
We can of course think the whole of beings in an ‘idea,’ then negate what we have imagined in our thought, and thus ‘think’ it negated. In this way we do attain the formal concept of the imagined nothing…18
If we were to imagine ‘the nothing’ as such, we might picture a cloudy mass with no particular shape or size. Here, again, we find that we are thinking about something that is nothing. This is a problem also present in the concept of the number zero: to what does zero really refer? If it refers to simply nothing, then it still seems to be the case that this nothing is something because a word is used to refer to it and words necessarily refer to things. So this confusion about zero ends up being somewhat short-sighted: zero refers to the absence of things—not just absence. We experience the absence of things all the time, and thus we experience the nothing all the time: “we do know the nothing if only as a word we rattle off everyday.”19 The number of elephants in the basket is one case, but there are countless others. For example, the number of red letters on this page is zero. The number of dogs in my last philosophy lecture was zero. The number of books that I have read by Dan Brown is zero. Indeed, when we ask if are there any red letters on this sheet of paper the answer, given that there are none, is some variation of: there are not any red letters on the paper. So when we negate a thing (not-red letter, not-dog, not-elephant, etc) the linguistic result is that we have no-thing. We refer to a thing’s absence with an absence—the fact that it is not present. Zero is how we deal with this absence in arithmetic. It is here, where zero and negation meet, that Heidegger’s metaphysics meets mathematics.
The most important connection here is between the reference of the word ‘zero’ and the definition of ‘the nothing’ because it is parallel to the connection between ‘the nothing’ and the empty set. If we imagine sets as circles, like Venn diagrams, the empty set refers to the number of things that two non-overlapping circles share: nothing (remember the gaggles of geese sharing no goslings in common). The empty set is the collection of all things that are not quantified as one: all the red letters on this page, the dogs in my last lecture, the number of Dan Brown books I have read, etc. Going back to Machover’s technical definition of the empty set “for n = 0 we get the empty class { } = {x: x does not equal x},” remember that the empty class is at the zero value in a numbered collection of objects. The empty set is thus a singular entity that refers to all things that are absent. In other words: it is a collection whose members are no-things. Now we see more clearly how the empty set refers to Heidegger’s ‘the nothing’: the former refers to a set of things that are not and the latter refers a group of things that are not. Here, it is hard to see how the empty set could not be utilized as a theoretical aid for Heidegger’s account of metaphysics.
But, one might ask, is there a philosophical difference between the presence of a thing and the being of a thing? In other words, can we equate the mathematical concept of presence with Heidegger’s concept of being? Heidegger writes that “the nothing…is nonbeing pure and simple.”20 These questions highlight the differences between the philosophical approaches of a set theorist and a phenomenologist like Heidegger. If mathematical presence (denoted by any number greater than zero) and being (which denotes an extremely complex existential state of affairs) are similar, then zero does in fact refer to the negation of a thing as Heidegger means it when he writes “nonbeing…pure and simple.” But are nonbeing and absence similar in this regard? This is a significant issue that deserves much attention. But a thorough response to these questions would entail an analysis of Heidegger’s conception of being and the mathematical philosopher’s conception of existential quantification. Such an analysis is not within the bounds of this essay, but, again, is deserving of close attention.
While this issue remains up in the air, there are several more concrete problems with the present account. First among them is that Heidegger explicitly says the ‘the nothing’ is not an object: “The nothing is neither an object, nor any being at all.”21 In the definitions provided here of the empty set, all have called the empty set an object or entity that is quantifiable as one thing. This is a problem for the claim that there is a relationship between the empty set and the nothing. However, as mentioned in the beginning of the paper, the ontology of set theory is not identified with that of Heidegger’s here. The claim is merely that set theory provides a logical way of referring to the nothing, which is supposedly an incoherent entity. While it appears to be the case that ‘the nothing’ is something and therefore a dubious topic for philosophical argument, the empty set is an entity (also composed of nothing) that is used to refer to the collection of objects that are not; and it is confidently and extensively written about by logicians of axiomatic set theory. So whether or not the empty set is a being does not necessarily affect its use as term with which we can talk about ‘the nothing’.22
Logicians, however, disagree about how to interpret the empty set; that is, how it fits into the rest of set theory. Lewis, for instance, claims that set theory need only be thought of as “memberlessness” in set theory to do the empty set’s work.23 These differing interpretations might limit the empty set’s helping power. If it is supposed to refer to the nothing, and some of interpretations of the empty set (like Lewis’) have little to do with its characterization as “the collection of things that are not,” then the relationship between the two (that one refers to the other) is less clear. This matter of interpretation is certainly a problem. But consider the Machover quotation cited several pages ago that stipulates nothingness as being present in any grouping: “Clearly, O is included in any class and in particular any set…” That absence should be accounted for in set theory makes some sense. Recall the basket of cumquats. If there are no elephants present in that basket, that must be accounted for. While there are quite a few no-things in the basket (no-pterodactyls, no-Japanese babies, no- stalagmites, the list goes on) it is clear that nothing, to some extent, is present in the basket (because there are no Japanese babies, stalagmites, etc). So it goes with any set or class. And while it is not mandatory according to Lewis that nothing be accounted for in set theory, it certainly makes things much easier. In fact, in set theory, it is possible to derive the entire set of natural numbers (one, two, three, four… all the way to infinity) from nothing. As Lewis himself remarks
You better believe in it [the empty set], and with the utmost confidence; for then you can believe with equal confidence in its singleton, the class of that singleton and the null set, the new singleton of that class, the class of that new singleton and the old singleton and the null set, and so on until have enough modeling clay to make the whole of mathematics.24
As noted above, nothing is always present in collections of things. Given this, along with the very fact that the empty set is interpretable at all, the differing interpretations of the empty set do not present debilitating problems for the present account. So the empty set and ‘the nothing’ can, in fact, still have a working relationship.
4) Conclusions
Heidegger, in his definition of metaphysics, posits the “negation of the totality of beings” despite its seemingly contradictory status. The empty set is follows deductively from set theoretic axioms as an entity in mathematical logic. I have shown in the above that the latter allows us to speak logically about former.
By offering this conclusion I hope to, at the very least, inspire a dialogue between experts in the streams of philosophy at play in it. Too long have the minds of both analytic and continental philosophers been closed in this regard, and too long have they been oblivious to what each might be able to offer the other.