Prelude to Mathematical Neo-Platonism

"Is not being, then, unified number,
and beings number unfolded,
and Intellect number moving in itself,
and the living creature number
embracing everything?" (Plotinus)

This post argues for a Mathematical Neo-Platonism (MNP), where a transcendent source – analogous to the One in historical Neo-Platonism (NP) – is seen as generating the Platonic reality of mathematics, which in turn generates the physical universe in which we find ourselves. First I will discuss some interesting parallels between NP and Zermelo-Fraenkel set theory (including the axiom of Choice, together abbreviated as "ZFC"). Subsequently I will discuss the consequences of Mathematical Monism (MM) in light of the reduction of mathematics to ZFC. MM is the claim that all of reality – including physical reality – consists entirely of mathematical structures. According to physicists like Max Tegmark, MM follows from the success of modern mathematical physics, since the latter describes physical reality entirely in mathematical terms. I will argue that MM leads to MNP when we take into account the reduction of math to ZFC, where the existence of the empty set, designated by "∅", and a small number of other axioms are sufficient to generate the entire universe of pure sets in which the bulk of mathematics fits. Thus, given MM and the reduction of math to ZFC, Leibniz's famous question "Why is there something rather than nothing?" reduces to: Why does ∅ exist? And why do the axioms of ZFC hold? As I will argue in my next post, it is only from a Neo-Platonic perspective that these questions become fully answerable.

The analogy between set theory and Neo-Platonism
Is there any substance to MNP? Or is it no more than a pretentious sounding but ultimately empty combination of words, a mere flatus vocis? Very interesting in this regard is the remarkable role played by axiomatic set theory in contemporary mathematics. As most mathematicians nowadays recognize, axiomatic set theory functions as the foundation for virtually all of mathematics – and some mathematicians would go even further than this, e.g. John Mayberry: "set theory is not really, or not just, a foundation for mathematics. It simply is modern mathematics." (1988: 353) This privileged role played by axiomatic set theory holds in particular for ZFC, which is standardly used in mathematics and mathematical logic. As I will show in the following, ZFC reproduces surprisingly many of the conceptual structures characteristic of NP, notably its hierarchical universe deriving from a single and indeterminate source. Now, suppose that this analogy between ZFC and NP holds up under closer analysis. Wouldn't we then be justified in concluding that ZFC = MNP, since ZFC reproduces NP in the context of mathematics? Let's see how far this analogy goes.

As already noted, ZFC functions as the standardly used axiomatic foundation for virtually all of mathematics. It is important to keep in mind just how remarkable this intellectual achievement is! Especially when you realize that ZFC requires only one existential axiom, namely the existence ∅, and a small number of other axioms stating which operations may be performed on sets in general, in order to generate an endless "cumulative hierarchy" of pure sets (i.e. sets containing nothing but sets), starting from ∅ as the hierarchy's sole existential basis. Amazingly, virtually all of mathematics can be located somewhere in this hierarchy (commonly called V). As the mathematician Enderton explains: "[M]athematical objects (such as numbers and differentiable functions) can be defined to be certain sets. And the theorems of mathematics (such as the fundamental theorem of calculus) then can be viewed as statements about sets. Furthermore, these theorems will be provable from our axioms [i.e. the axioms of ZFC, PS]. Hence our axioms provide a sufficient collection of assumptions for the development of the whole of mathematics – a remarkable fact." (1977: 11) Remarkable indeed!

The hierarchical universes of NP and ZFC
To what extent does ZFC reproduce the core ideas of NP? As already noted, both NP and ZFC present a hierarchically structured universe. Whereas ZFC gives us the set-theoretical hierarchy V deriving from ∅, NP gives us the metaphysical hierarchy One → Intellect → Soul → Nature (where "→" stands for emanation). And this analogy is all the more apt insofar as the Neo-Platonic hierarchy is, like the set-theoretic hierarchy, a graded unfolding of increasing multiplicity. For Plotinus, the One is an utterly undifferentiated unity, which generates the plurality-in-unity of the Intellect, which in turn produces the still more complex multitudes of Soul and Nature, finally terminating in the utter chaos of unordered Matter. Likewise in ZFC, where the hierarchy V starts on the 0th level (called "V0") with the utterly simple unity of ∅, from there on generating ever higher levels of complexity, such that V1={∅}, V2={∅, {∅}}, V3={∅, {∅}, {{∅}}, {∅, {∅}}}, and so on ad infinitum. Thus "in set theory one is always climbing upward" (Devlin 1993: 47). And the higher one goes in this endless hierarchy, the more complexity one encounters, until finally sets are generated which are so mind-bogglingly huge that they escape mathematical understanding altogether. True, the limit of what mathematicians do understand about V is constantly being raised higher, thanks to continuing mathematical research. However, since V is literally endless, it is clear that some limit will always remain for us finite human beings: the limit may be eternally moving upwards, but beyond it will always remain utterly incomprehensible multiplicity. It is tempting to see this as analogous to the way in which the Plotinian hierarchy ultimately dissipates in the incomprehensibility of unordered Matter.

The 'empty unity' at the beginning
It is also at their starting points, however, that the set-theoretic hierarchy V and the metaphysical hierarchy of NP are surprisingly similar. For NP, this starting point is the utterly undifferentiated One. For V, this starting point is ∅, the empty set. The analogy between them is obvious: both ∅ and the Plotinian One are, in a sense, 'empty unities'. Let's take a closer look at this. Exactly why does Plotinus say that the One is an undifferentiated and therefore 'empty' unity? This is actually a very complicated question, but for now the simple answer must do. Plotinus reasons roughly as follows: if the One were a definite something, i.e. if it had well-defined properties, then its properties would in a sense limit its nature (since by having these properties it would not have the contrary properties), and thereby its power to produce would be limited as well, so it would no longer be omnipotent. Thus, in order to be the cause of Everything, the One itself must be (or rather: contain) Nothing. As Plotinus writes about the One: "It is because there is nothing in it that all things come from it." (Enneads, V.2.1.1-5) But, as we have seen, in ZFC the starting point of V is likewise a kind of 'unified nothing', namely, ∅, the set which collects nothing. In a sense, then, ∅ could be seen as the set-theoretic 'equivalent' of the Plotinian One. The crucial question is obviously what this 'equivalence' is worth. Is it no more than a coincidental analogy? Or does it rather reveal a substantial, doctrinal agreement between NP and ZFC?

Before dealing with this question, however, it is interesting to note that this analogy between ∅ and the Plotinian One has been noted before, notably by Rudy Rucker in his widely read book Infinity and the Mind. Rucker (1995: 40) clearly refers to this analogy when he writes: "∅ is the One obtained by taking together... nothing". Although Rucker does not mention Plotinus in this context, the Neo-Platonic overtones of his mention of "the One" are loud and clear. Also because Rucker (in the same section) explicitly states that he inclines towards a Platonic interpretation of set theory (i.e. as describing an ideal reality existing outside of space and time). So if the set-theoretic universe (i.e. hierarchy V) constitutes a Platonic realm of ideal objects, then Rucker's description of ∅ as "the One" is clearly suggestive of a Neo-Platonic view, where ∅ is seen as a kind of transcendent, metaphysical source of V, analogous to the Plotinian One. It would seem, then, that our notion of MNP has already been anticipated by Rucker! Unfortunately, he does not develop this suggestion any further, so at best Rucker remains a 'closet Mathematical Neo-Platonist', hiding under the official cover of MP.

The Pythagorean element in NP
Let's return to the question how far the analogy between ∅ and the Plotinian One extends. Is this indeed a substantial equivalence? One way to discuss this issue is by reversing the approach taken so far. Up till now we have mainly focused on the question to what extent ZFC repeats the core ideas of NP – but we can also turn this around and ask: to what extent did NP anticipate ZFC? There has, after all, always been a mathematical side to NP, partly because mathematics was already dear to Plato himself (who saw in geometry a privileged way to understand the ideal reality of the Forms), but mainly because the development of NP in the first centuries CE coincided with a revival of interest in the mathematical philosophy of Pythagoreanism. The philosophical and religious doctrines of the historical Pythagoras are largely unknown, but the claim that "All is number" was and is widely attributed to him. Thus Pythagoreanism has come to be understood as a form of Mathematical Monism (MM), the claim that all of reality consists of mathematics. However, together with this surprisingly modern doctrine, Pythagoreanism always went hand in hand with ascetic and magical practices, the latter based on a numerological belief in the magical properties of certain numbers. This numerological side obviously conflicts with the scientific aspect of Pythagoreanism. And this also holds for the Neopythagorean revival of the first and second centuries which strongly influenced the development of NP.

For the Neopythagoreans, reality consisted of a hierarchy flowing from God into something they called the original Monad and then into the original Dyad and finally into the numbers that shape physical reality. This scheme clearly anticipated the Plotinian hierarchy of the One unfolding into the multiplicities of Intellect, Soul and Nature (cf. Remes 2008: 15). In fact, Plotinus was seen by some of his contemporaries (notably Longinus) as the principal expositor of Pythagoras's doctrines (cf. Gatti 1996: 12-13). This Pythagorean aspect of NP comes clearly to the fore in the very title of Plotinus's master piece, the Enneads (the 'Nines'), although it should be remembered that this title came not from Plotinus himself but from his editor Porphyry. When Plotinus died in 270, Porphyry took it upon himself to prepare the master's manuscripts for wider circulation and by shifting around Plotinus's original texts (sometimes even breaking them up into smaller sections), Porphyry obtained 54 treatises which he divided into six groups of nine, "a combination of mystical numbers that delighted him" (Wallis 1995: 46).

Given this Pythagorean side of NP (which grew even stronger in later Neo-Platonists, notably Iamblichus), one could say that NP already was a form of MNP right from the start. One could even say, with some goodwill, that the Neo-Platonic hierarchy in its mathematical aspect – i.e. as an unfolding of increasing multiplicity out of an original but empty unity – anticipated the set-theoretic hierarchy of ZFC and other axiomatic set theories. In a sense, given NP's commitment to the Pythagorean claim that "All is number", NP can even be said to have anticipated MM – thus Plotinus, for example, writes: "Is not being, then, unified number, and beings number unfolded, and Intellect number moving in itself, and the living creature [i.e. the World Soul, PS] number embracing everything?" (Enneads, VI.6.4.29-31) But obviously the claim that NP anticipated ZFC and MM ultimately runs afoul of the lack of scientific rigor and the intrusion of numerology into the mathematical aspect of NP, which really did not go beyond vague claims about the One generating the Dyad which in turn generates all numbers that somehow "embrace everything". The how and why of this mathematical unfolding of the One remained shrouded in mystery.

All in all, Plotinus and later Neo-Platonists really lacked the mathematical theories that would have allowed them to precisify the Neopythagorean claim that reality is a mathematical outflow from a single transcendent source. But can't we now say that this situation has changed with the development of axiomatic set theory in combination with the MM of modern, i.e. mathematical physics? Doesn't ZFC+MM allow us to revive the Neoplatonic claim that reality is a mathematical outflow of the One? If MM is true, then the reduction of mathematics to ZFC implies that ∅ stands not only at the origin of V but also at the origin of physical reality, because the latter would then be a substructure of V. In short: MNP=ZFC+MM?

Modern physics and Mathematical Monism
So let's examine MM more closely. What exactly does it say? And is it acceptable? To repeat, MM is the claim that all of reality – including physical reality – is quite literally composed of mathematical objects. And, as already said, although MM is of quite ancient origin, tracing back to the Pythagorean claim that "All is number", it is nowadays making a comeback due to the success of modern, mathematical physics. The basic point behind this comeback is simple enough: since physics describes and explains the whole of physical reality in essentially mathematical terms, the huge experimental success of physics shows that the 'substance' of this reality is ultimately nothing but what these mathematical terms refer to, namely, mathematical objects such as numbers, vectors, functions, fields, topological spaces, etc. One contemporary and very outspoken advocate of MM, the cosmologist Max Tegmark, summarizing the main results from relativity theory and quantum mechanics, describes this thoroughly mathematical nature of physical reality as follows:

"[W]e saw that the very fabric of our physical world, space itself, is a purely mathematical object in the sense that its only intrinsic properties are mathematical properties – numbers such as dimensionality, curvature and topology. [W]e saw that all the "stuff" in our physical world is made of elementary particles, which in turn are purely mathematical objects in the sense that their only intrinsic properties are mathematical properties [...] such as charge, spin and lepton number. [W]e saw that there's something that's arguably even more fundamental than our three-dimensional space and the particles within it: the wave function and the infinite-dimensional Hilbert space where it lives [...] and the wave function and Hilbert space are purely mathematical objects." (Tegmark 2014: 253-254)

It is on the basis of these considerations that Tegmark puts forward his own version of MM, which he calls the "Mathematical Universe Hypothesis" (MUH): "our external physical reality is a mathematical structure" (idem: 319). Similar arguments for MM can be found in scientists and philosophers like John Wheeler, Frank Tipler, Roger Penrose, W.V.O. Quine, James Ladyman, Steve French, Don Ross, Vlatko Vedral, and many, many others. I am inclined to agree with them that modern physics shows MM to be true. However, I can't really argue for that claim here (other than referring to the authority of said scientists and philosophers), simply because of lack of space. So in the following I will simply assume that MM is true, as shown by physics, and then develop the joint consequences of MM and ZFC, to see if this leads anywhere in the direction of MNP.

Three remarks on Mathematical Monism
Before I continue, however, there are three remarks about MM that I would like to make in order to put things in proper perspective. Firstly, it is important to keep in mind that MM does not equate physical reality with the mathematical realm in its entirety. That would simply be incorrect, because the vast majority of topics studied by mathematicians have no connection to physics whatsoever or to any other science apart from pure mathematics. Thus we have to assume that if physical reality is a mathematical structure, then it is a relatively small substructure of a much, much (perhaps infinitely) larger realm of mathematical objects. True, it always remains possible that particular results from pure mathematics find unexpected applications in physics (as happened, for example, with non-Euclidean geometries which were first developed by pure mathematicians but which turned out to find a surprising application in relativity theory). But even if more and more of pure mathematics is 'sucked into' physics, it seems extremely likely that pure mathematics will always dwarf the mathematics needed for physics. This means that MM faces a particular obligation, namely, to explain why certain substructures of mathematics and not others are singled out as constituting physical reality, and exactly what this 'singling out' amounts to.

Secondly, note that MM should be understood as a radicalized form of Mathematical Platonism (MP). Both original Platonism and MP are dualistic in that they make principled distinctions between the spatiotemporal realm of physical reality on the one hand and the ideal realm of Forms / mathematical truths on the other. MM, on the other hand, is monistic in that it reduces all of reality, including physical reality, to mathematics. In that way MM can be seen as a radicalization of MP insofar as it takes the ideal realm of mathematics to 'engulf' the spatiotemporal realm. Tegmark puts this very clearly: "Mathematical structures are eternal and unchanging: they don't exist in space and time – rather space and time exist in (some of) them." (Tegmark 2014: 318) This conception is also known in the context of special relativity as the "block universe", where time is the fourth dimension of the geometrical structure of spacetime, such that the entire universe, from past to future, exists 'at once' as a single, mathematically structured 'block'.

My third remark is a caveat. Although I agree with the argument that the success of modern physics shows that MM is true, I also think it is crucial to note that this argument limits the truth of MM to physical reality, i.e. reality as described by physics. Thus there might be non-physical realities that escape mathematical treatment. Indeed, I think this follows from the famous Hard Problem of Consciousness, which shows that the reduction of reality to mathematics stops short of how we experience reality, insofar as the qualia of that experience refuse direct reduction to non-conscious building blocks, be it the physical structures that form the human brain or the mathematical structures that model the functional organization of the brain (see Chalmers 1996). So here, in the Hard Problem of Consciousness, MM reaches its limit. However, as I will argue in my next post, this fact – that consciousness falls outside of mathematics – is precisely what will make a Neo-Platonic approach to mathematics possible. For now, however, I will abstract from the Hard Problem and simply assume that MM is true tout court. Why? Because this puts in very sharp relief the most fundamental question of ontology...

Why does ∅ exist? Mathematical Nihilism averted
If MM were true tout court, i.e. if all of reality reduces to mathematics, then – given the reduction of mathematics to ZFC – Leibniz's famous question "Why is there something rather than nothing?" would in turn reduce to: Why does ∅ exist? And why do the axioms of ZFC hold? Now it might be thought that this turns Leibniz's question into a 'no-brainer' because – as is sometimes said – "∅ is nothing". So to explain why physical reality exists, we simply have to assume that 'in the beginning there was nothing' et voilà we have ∅, the sole existential assumption needed to derive the whole of mathematics! But, obviously, this approach is nonsensical. First of all, ∅ is not nothing, it is rather {nothing}, i.e. a set with nothing in it. And a set, even if it is empty, is something, not nothing: "It is not the same thing as nothing because it has whatever kind of existence a set has, although it is unlike all other sets." (Gardner 1977: 15). This difference between ∅ and nothing is clarified by a nice example from Enderton': "a man with an empty container is better off than a man with nothing – at least he has the container". (Enderton 1977: 3) Secondly, even if we were warranted in simply assuming the existence of ∅, we would still need to explain why the other axioms of ZFC are valid, because it is only in combination with them that ∅ yields V. Clearly, then, we have to reject the idea that ZFC provides an easy or otherwise attractive solution to Leibniz's question because 'it starts from nothing'.

Nevertheless, this confusion of ∅ with some kind of 'primordial nothingness' explains a lot of the current 'sexiness' of axiomatic set theory, as if the latter were a sort of 'Mathematical Nihilism' giving new content and credibility to the idea of an absolute creatio ex nihilo. Such a view can be found, for example, in Jim Holt's popular book Why does the world exist?. Commenting on the set-theoretic construction of V on the basis of ∅, he writes: "Out of sheer nothingness, a remarkable profusion of entities has come into being." Then, bringing in the hypothesis of MM, i.e. that all of reality reduces to mathematics and thereby to set theory, he writes: "The whole show of reality can be generated out of the empty set – out of Nothing." (Holt 2013: 40) Admittedly, for Holt this is merely one of the many possible solutions to Leibniz's question he examines in his book, so we should perhaps not pin him down on it. The situation is different, however, with the popular French philosopher Alain Badiou, who has developed a wide-ranging set-theoretic ontology (including a political theory) based on a conception of ∅ as a kind of "primordial void" on which all existence is somehow founded, "the nothing from which everything proceeds" (Badiou 2005: 59). Thus he writes: "In Set Theory, the primitive name of Being is the void, the empty set. The whole hierarchy takes root in it. In a certain sense, it alone "is"." (Badiou 2006: 98) As the scare quotes around "is" indicate, Badiou takes ∅ (the sole basis of existence, according to him) to be itself inexistent, i.e. nothing. If this is the foundation of Badiou's ontology, then clearly that ontology is based on a mistake.

Confusion of ∅ with 'nothing' widespread
This confusion is not just confined to foggy philosophers, however. It can also be found in otherwise respectable set theorists and mathematicians, people who really should know better. For example, Keith Devlin (1993: 36) in The Joy of Sets writes about the construction of V: "we commence with nothing, that is to say, the empty set". Mary Tiles (1989: 124) in The Philosophy of Set Theory writes that the universe of sets "is a wholly abstract universe generated, as it were, out of nothing" (admittedly, she qualifies her claim with the phrase "as it were", but still, the damage has been done). John D. Barrow (2000: 167), a prominent theoretical physicist and mathematician, writes about set theory that "it has enabled us to create all of the numbers from literally nothing, the set with no members". But the most curious case of this confusion can be found in Enderton, whose remark about "a man with an empty container" we quoted earlier precisely to dispel the confusion! For if we look at the context of that remark, we see that Enderton himself is confused as well. Here is what he writes:

"Note that {∅} ≠ ∅, because ∅ ∈ {∅} but ∅ ∉ ∅. The fact that {∅} ≠ ∅ is reflected in the fact that a man with an empty container is better off than a man with nothing – at least he has the container." (Enderton 1977: 3)

Clearly, the notion of a container here represents the notion of a set, since a set is a container of sorts (it 'contains' its members). So when Enderton speaks of "an empty container" one naturally supposes he is referring to ∅. But a closer look reveals that this is not the case. Enderton in fact uses the difference between a man with an empty container and a man with no container to clarify the difference between {∅} and ∅. So, in this analogy, {∅} is represented by the empty container, whereas ∅ is represented by no container at all! This clearly indicates that Enderton makes the mistake of identifying ∅ with nothing. If {∅} is represented by the empty container, a container with nothing in it, then this means that ∅ is nothing. Similarly, if ∅ is represented by a man with no container at all ("a man with nothing" as Enderton writes), then this also means that ∅ is nothing. So even if Enderton has in his hands the germ of the insight that ∅ ≠ nothing given his example of the difference between an empty container and no container, he fails to see the true significance of this example. Apparently, then, thinking about the empty set is a tricky affair and its confusion with nothing is always lurking, even for the ablest expositors of set theory.   

Transition to MNP
So ∅ ≠ nothing. This means that Leibniz's question, even in its set-theoretic form ("Why does ∅ exist? And why do the ZFC axioms hold?"), still presents us with a genuine problem, i.e. a problem not solved by simply assuming the 'existence of nothing' (whatever that is supposed to mean). As I will argue in my next post, it is especially in the light of Leibniz's question that the importance of NP will become apparent, because Plotinus was actually the very first philosopher who grappled with this problem in a systematic way and who developed a systematic answer in the form of his notion of the One as causa sui. So, if we return to our analogy between ∅ and the Plotinian One, and we allow ourselves to speak rather loosely, we could say that Plotinus supplies us with a way to understand how ∅ has brought itself into existence! More about this in my next post.

References
-Badiou, Alain (2005), Being and Event. New York: Continuum.
-Badiou, Alain (2006), Briefings on Existence: A Short Treatise on Transitory Ontology. Albany: SUNY Press.
-Barrow, John D. (2000), The Book of Nothing. London: Jonathan Cape.
-Chalmers, David J. (1996), The Conscious Mind: In Search of a Fundamental Theory. Oxford University Press: New York and Oxford.  
-Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory. New York: Springer.
-Enderton, Herbert B. (1977), Elements of Set Theory. New York: Academic Press.
-Gardner, Martin (1977), Mathematical Magic Show. London: Penguin.
-Gatti, M.L. (1996), "Plotinus: The Platonic Tradition and the foundation of Neoplatonism", in: Lloyd P. Gerson (ed.), The Cambridge Companion to Plotinus. Cambridge: Cambridge University Press.
-Holt, Jim (2013), Why Does The World Exist? One Man's Quest for the Big Answer. Profile Books: London. -Mayberry, John (1988), "What are numbers?", in: Philosophical Studies, 54 (3), 317-354.
-Remes, Pauliina (2008), Neoplatonism. Stocksfield: Acumen.
-Rucker, Rudy (1995), Infinity and the Mind: The Science and Philosophy of the Infinite. London: Penguin Books.
-Tegmark, Max (2014), Our Mathematical Universe. New York: Alfred A. Knopf.
-Tiles, Mary (1989), The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. Mineola: Dover.
-Wallis, R.T. (1995), Neoplatonism. London: Gerald Duckworth & Co.

The Inconsistency of Nothing: Objective or Subjective?

In my previous post on this blog I argued that if we want to answer Leibniz' famous question ("Why is there something rather than nothing?") we have no choice but to start with the assumption that nothing at all exists and then investigate how we might derive existence from this state of nothingness. The rationale behind this approach is obvious: as long as we start with some primordial being (e.g. God or the laws of physics) as the cause of all other beings, we will not have truly answered Leibniz' question, since in that case we still have to explain why the supposedly primordial being existed. Why does God exist? Or where did the laws of nature come from? The late Robert Nozick put this problem succinctly as follows: "The question [of Leibniz] appears impossible to answer. Any factor introduced to explain why there is something will itself be part of the something to be explained". (Nozick 1981: 115) Hence, only if we start with the assumption that nothing at all exists will Leibniz' question become answerable.

Gottfried Wilhelm Leibniz

Answering Leibniz: The inconsistency of nothing
Of course, this does by no means imply that there is a true answer to Leibniz' question. Indeed, trying to explain being on the basis of nothing may seem such a logical absurdity that the question may appear as unanswerable as before. The received wisdom, after all, is that ex nihilo nihil fit, from nothing only nothing can come. As William James put it: "from nothing to being there is no logical bridge". (James 1911: 40) However, one of the things I argued in my previous post is that this problem becomes tractable once we focus on the paradoxes involved in the notion of absolute nothingness. These paradoxes have been with us since antiquity, from Parmenides up to modern thinkers like Lewis Carroll and Rudolf Carnap. The latter, for example, famously argued in his polemic against Heidegger that his talk of "the nihilating Nothing" ("das nichtende Nichts") ""involves a contradiction": "For even if it were admissible to introduce "nothing" as a name or description of an entity, still the existence of this entity would be denied in its very definition". (Carnap, 1959 [1931]: p.71) In other words, if we say that 'the nothing' exists, we create a paradox because 'the nothing' is by definition non-existent. From such paradoxes we might conclude that the concept of absolute nothingness is just logically incoherent. And if that is the case, then we appear to have a very simple answer to Leibniz' question, an answer that still starts with the assumption that nothing exists but then goes on to point out the contradiction in this assumption. In short, being is logically necessary because the existence of nothing is logically impossible.

Subjective or objective inconsistency?
In the following I want to investigate a question left hanging in the above answer to Leibniz' question. The question is this: Is the existence of nothing inconsistent in itself? Or is the inconsistency merely to be found in our concept of nothingness? In other words: Is the fact that the existence of nothing is ruled out by logic an objective fact, i.e. a fact that holds independently of us? Or is it a subjective fact, i.e. a fact about the limitations of our cognitive capacities, which are such that we simply cannot think coherently about nothingness? It is clear that the answer to these questions is crucial to how we can go about answering Leibniz' question. If the logical impossibility of nothingness is merely subjective, i.e. merely an effect of our inability to think nothingness, then we can't use this impossibility to answer Leibniz' question. After all, it would be absurd and indeed circular to say that there must be being since we cannot imagine it otherwise. The circularity of such a proposal follows from the fact that we ourselves are part of being, so on this proposal we exist because we cannot imagine ourselves as not existing, which is patently absurd. If the logical impossibility of nothingness is to explain why there is being, then this impossibility must be an objective fact.

Naming nothingness: Badiou's fallacy
The prospect, however, does not look good for the objective interpretation of the impossibility of nothingness. The problem simply seems to be that we cannot think about something without thinking about something. In short: thinking is inherently thematizing and objectifying. When we think about something we automatically turn it into an object, an object referred to by the grammatical subjects of our thoughts and statements. This is more or less unproblematic as long as we think about things that obviously exist. It becomes somewhat more problematic when we think about things that obviously do not exist (e.g. when we think about Pegasus; does it make sense to say that Pegasus itself is the object of our thought?). But it becomes outright paradoxical when we think about what is not anything at all, i.e. when we think about a 'state' where nothing exists. Just by thinking about such absolute nothingness we turn it into a something and thereby contradict its 'nature'. This point is nicely illustrated by some fallacious reasoning by the popular French philosopher Alain Badiou, who has built an entire ontology out of the set-theoretic construction of mathematics based on the empty set. Having developed the axioms of set theory (the Zermelo-Fraenkel system or "ZF" for short), Badiou writes:

Alain Badiou pondering the null set...

"We definitely have the entire material for an ontology here. Save that none of these inaugural statements in which the law of Ideas [i.e. the ZF axioms] is given has yet decided the question: 'Is there something rather than nothing?'... The solution to the problem is quite striking: maintain the position that nothing is delivered by the law of Ideas, but make this nothing be through the assumption of a proper name." (Badiou 2005: 66-67).

In other words: Badiou simply gives "this nothing" a name (namely, "the empty set"), et voilà, here we have our first being, the empty set, on the basis of which all other sets can be created. Now it will be obvious that, as an answer to Leibniz' question, this is totally unsatisfactory. I greatly admire the set-theoretic construction of mathematics out of the empty set. I'm even sympathetic to the idea that this construction may have some real ontological weight to it. But to answer the question "Is there something rather than nothing?" by simply giving a "proper name" to nothingness seems nothing more than a bad joke. Badiou's fallacy illustrates something of importance concerning the paradoxes surrounding the concept of nothingness. As soon as we start using "nothing" as a referring noun, we are in trouble: nothingness becomes a referent, an object. In that case, if we say that nothing exists, we imply that there exists this object called "the nothing", which is contradictory. It is clear that this contradiction is not an objective fact concerning the state where nothing exists. The contradiction is merely an effect of our objectification of this state. Just like Badiou cannot conjure being out of nothingness by giving the latter a proper name, so nothingness cannot be made inconsistent merely by our objectification of it.

Russell's theory of descriptions
But let's not jump to premature conclusions. The above analysis is predicated on the assumption that when we think about a state where nothing exists, we must use the word "nothing" as a noun to refer to this state, thus turning the latter into some mysterious entity. But is this assumption correct? Not according to an influential tradition in analytical philosophy, a tradition stretching back to Bertrand Russell's theory of descriptions. One of the reasons why Russell developed this theory was to solve a logical problem concerning the truth value of statements about non-existent objects. His famous example was the statement: "The present king of France is bald." Obviously, this sentence is false: if we consider all the bald men, the present King of France isn't among them, since there is no present King of France. But if it is false, then -- given the law of the excluded middle -- one would expect that the negation of this statement is true, namely, "It is not the case that the present King of France is bald" (or its logical equivalent: "The present King of France is not bald"). But this sentence is false as well: if we consider all the non-bald men, the present King of France isn't among them either. Thus it seems that the law of the excluded middle does not hold for all propositions! 

Bertrand Russell

Russell proposed to solve this problem (and save the universality of the excluded middle) by means of his theory of descriptions. According to Russell, a definite description like "the present King of France" simply isn't a referring expression at all, although it superficially appears that way. If we analyze the proposition "The present king of France is bald" we arrive at a logical deep structure that crucially involves existential quantification. What that proposition really says, according to Russell, is this: "There exists an object x such that x is the present king of France and x is bald" (formally: ∃x(Kx∧Bx) where K means "is the present king of France" and B means "is bald"). This existential statement has a definite truth value: it is clearly false. And its negation, -∃x(Kx∧Bx),  is clearly true. So problem solved.

Rudolf Carnap

Carnap contra Heidegger
That there is indeed some truth to Russell's theory of definite descriptions becomes especially apparent when we consider ordinary statements figuring "nothing" as the grammatical subject, for example "Nothing was stolen from my house" or "Nothing in this painting has the color green". In these statements we are obviously not referring to some mysterious entity, the nothing, which was stolen from my house and which has the color green in said painting. Clearly these statements must be analyzed, along the lines of Russell's theory, as negative existential statements: "There is no x such that x was stolen from my house" and "There is no x such that x has the color green in this painting". But if this holds for our ordinary use of "nothing" as a grammatical subject, then perhaps it also holds for the metaphysical proposition "Nothing exists". This was precisely Carnap's point in his polemic against Heidegger, where he used Russell's theory of descriptions to debunk Heidegger's talk of the "nihilating Nothing". When we issue the statement "Nothing exists", are we referring to some mysterious entity? No, says Carnap, that statement merely functions as shorthand for the negative existential statement "There is no x such that x exists" (formally: -∃x(Ex) where Ex means "x exists"). This negative existential statement does not commit us to the existence of 'the nothing'. Hence the air of paradox surrounding the claim "Nothing exists" evaporates. Heidegger simply violates the logical deep structure of language when he uses "nothing" as a referring noun. His talk of 'the nothing' is meaningless (Carnap, 1959 [1931]: 70).

Truth supervenes on being
So where does this leave us? It would seem that we can think about 'absolute nothingness' without contradiction after all! But wait a minute... Not all is well in Carnap's logical-positivist, Nothingness-less paradise! Trouble comes from a principle that is broadly accepted in Anglo-American philosophy and that captures a large part of our common sense attitude toward truth. This is the principle that truth supervenes on being (see Jackson 2000: 118). The basic idea is that for every truth there must exist a truthmaker, i.e. an objectively existing state of affairs or fact that makes it true. Thus the statement "It is raining now" is true iff it is an objective fact that it is raining now. Or more formally: "p" is true iff there is a fact that p. Note that this is almost a tautology and thus a logical truth. It is therefore extremely difficult to argue against the idea that truth supervenes on being. It is so deeply ingrained in our common sense mentality that it is virtually impossible to do without it. But it spells trouble for Carnap's logical diffusion of nothingness. Consider the negative existential statement: there is no x such that x exists (-∃x(Ex)). If this statement is true, then it too must supervene on being and so there must exist an objective fact which makes it true. Hence, there still exists something, namely, this fact. The statement -∃x(Ex) is therefore contradictory, even if we are not directly referring to some mysterious entity called "the nothing".

David Lewis

This conclusion, that -∃x(Ex) is contradictory because truth supervenes on being, was – to my knowledge – first drawn by the philosopher David Lewis. Suppose, he says, "that there might have been absolutely nothing at all. It would then have been true that there was nothing. Would there have been a truthmaker for this truth? -- If so, there would have been something, and not rather nothing. Contradiction." (Lewis 1999: 220)

Referring to an empty fact
It appears, then, that the contradiction in thinking about the state where nothing exists is inevitable after all. Moreover, it seems that when we think about this state we cannot shed the supposition that there exists this mysterious entity, the nothing, even if we explicitly try to avoid this through Russell's theory of descriptions. For, so we might ask, what kind of a fact is it that has to exist iff  "-∃x(Ex)" is to be true? It is a fact lacking any positive determination, a fact with no being in it, with virtually no propositional content, an empty fact therefore. What kind of an entity is this utterly empty, being-less fact if not nothingness itself? It is clear that if you accept the existence of facts as truthmakers for propositions, you are committed to the existence of a certain kind of entity, a kind of object even, insofar as facts can be referred to by noun phrases. Thus facts satisfy at least one traditional criterion for object-hood (namely, "must be a possible referent of noun phrases"). For example, nouns referring to facts can be grammatical subjects in statements like "The fact that p is F" (e.g. "The fact that it rains is lamentable / should not deter us / is welcomed by farmers etc."). And if you think that propositional that-clauses ("that p") make for dubious noun phrases, consider the fact that the reference of a that-clause can always be taken over by pronouns through anaphoric reference (e.g. "It is lamentable that it is raining", "I saw that the sun came up and it made me happy"). So if we take the common sense view that only objects can be referred to by noun phrases (which include pronouns), then facts are certainly a kind of object. Indeed, to be precise, facts are those objects which are referred to by true statements. But then the empty fact, the truthmaker for "-∃x(Ex)", becomes a very special kind of object: an empty, being-less, property-less object.

Obviously, this object which is not an object is the nothing if anything is. In conclusion, when thinking about the state where no beings exist, we cannot avoid assuming the existence of the nothing, even if we take the Russell/Carnap approach. Clearly this 'object which is not an object' is a paradoxical entity. This follows from the fact that "-∃x(Ex)" is a contradiction: its claim that no beings exist is contradicted by the existence of its truthmaker. Its truthmaker, then, makes it both true and false simultaneously, which is to say that this 'object' (the nothing) both exists and does not exist. The contradiction in thinking about nothing seems ineluctable.

The objective inconsistency of "-∃x(Ex)"
Let's return to our starting question: Is this contradiction objective or subjective? What has become apparent, I think, is that this issue turns on how we think about the ontological status of truth. If, so to speak, the truth is out there -- objectively, independent of us -- then it's hard to escape the conclusion that the contradiction of "-∃x(Ex)" is an objective one. Take for example elementary truths like "1 + 1 = 2" or "The earth orbits the sun". Isn't it obvious that they are true independently of whether they are conceived or not? Thus one might state as a general principle that if "p" is true, then it is true independently of any observer, and hence it is an objective fact that p.* For "-∃x(Ex)" this means that if it's true, then it's true independently of us, and then the fact that nothing exists is an objective fact, including its contradictory nature. The only way to avoid this conclusion is by saying that truth is not objective, that truths only arise when thinking subjects make assertions. Thus, one could say, the contradiction inherent in "-∃x(Ex)" only arises when it is uttered or thought, so that the contradiction is subjective after all. But this is a very problematical position which ultimately cannot be made coherent. There are, I think, basically two ways in which this position (i.e. that truths are relative to subjects who utter them) can be construed. A first, innocent construal would be to say that truth does not just require a truthmaker but also a truthbearer, i.e. something that is made true by the truthmaker, e.g. a thought or statement. On such a conception, truthbearers need to be produced by thinking subjects: if there are no such subjects, or if they simply fail to produce (i.e. utter or think) truthbearers, then there are no truths. So if no one thinks or utters the claim that -∃x(Ex), there can be no corresponding truth and hence no contradiction. But it is easy to see this solution fails to work. Even if there are no truthbearers, there are still the truthmakers, which exist independently from the truthbearers. So on this conception, even if there were no truth, there would still be objective reality as such, which would make truthbearers either true or false as soon as they are produced. Hence, if "-∃x(Ex)" is true when uttered or thought, then its truthmaker must (pre-)exist objectively and we are still faced with objective contradiction. The only way to avoid this objectivity is to construe the subjective relativity of truth in a second and much more extreme way, namely, by saying that both truthbearers and truthmakers are dependent on the subject who conceives them. But such a vision is tantamount to absolute idealism, where the existence of reality as such is produced by thought. I think we can safely say that in this case the cure is worse than the disease. Ultimately such extreme idealism is incoherent. For if all of existence is the product of thinking, then how did the thinking subject itself come into existence? It would be circular and thus absurd to say that the thinking subject thought itself up... In short, thought always presupposes independent being as its ontological basis. To paraphrase Marx: being does not depend on consciousness, consciousness depends on being. But if that is the case, then the contradiction inherent in "-∃x(Ex)" can only be an objective one. And then our answer to Leibniz' question still stands. That is to say: Why is there something rather than nothing? Because nothing is inconsistent!

Well, nothing does not exist, but whatever...

I would like to end this post with a cautionary note. The above analysis of the logical impossibility of nothingness turns on a great many questions, such as: what exactly is reference? when do we call something an object? wherein does truth consist? what are truthbearers? what is the ontological status of facts? These are issues about which philosophers of language, logic and knowledge have argued and continue to argue endlessly. In other words: there are many different ways to answer these questions. Hence there are also many different ways in which the logical status of "-∃x(Ex)" might be judged. Nevertheless, I think the above conclusion concerning the objectivity of the contradiction inherent  in "-∃x(Ex)" is pretty straightforward and commonsensical. Basically it turns on two presuppositions which are not easily put aside. The first is that truth supervenes one being, so that if a statement is true it is true by virtue of some objective feature of reality. The second is that truth (if not the truthbearer then at least the truthmaker) is independent of the thinking subject, hence that absolute idealism is false. Together these assumptions imply that if "-∃x(Ex)" is true, then there is an objective reality which makes it true. And this is basically all we need to show the objectivity of the contradiction involved.

* With the exception, of course, of subjective facts, i.e. facts concerning conscious experience, like the fact that I feel pain or that I see redness. Such subjective facts are obviously observer-dependent. But clearly we are not talking about subjective facts here. In particular, "-∃x(Ex)" cannot assert a subjective fact since by definition there is no consciousness to perceive it.

References:
-Badiou, Alain (2005), Being and Event. New York: Continuum.
-Carnap, Rudolf (1959 [1931]), "The Elimination of Metaphysics Through Logical Analysis of Language", in: A. J. Ayer (ed.), Logical Positivism. Glencoe, The Free Press, pp. 60-81.
-Jackson, Frank (2000), From Metaphysics to Ethics. Oxford University Press.
-James, William (1911), Some Problems of Philosophy: A Beginning of an Introduction to Philosophy. Longmans, Green, and Co., New York.
-Lewis, David (1999), Papers in Metaphysics and Epistemology. Cambridge University Press.
-Nozick, Robert (1981), Philosophical Explanations. Belknap Press, Cambridge Mass.