Why is there something rather than nothing? Because nothingness is (dialectically) unstable

For a printable version of this text see: Why is there something rather than nothing? From quantum theory to dialectics

"Why is there something rather than nothing?" Ever since Leibniz first raised this question in 1714, philosophers and scientists have been exasperated by it. Some have thought it unanswerable and therefore meaningless. Some have deemed the question trivially wrong, since according to them the world has always existed and will always exist, so no mystery in that department. Still others have been unconvinced by this answer and continue to take Leibniz' question to be the most fundamental of all. Only a few philosophers and scientists, however, have actually grappled with it, getting their hands dirty and stretching the very limits of language and logic in desperate attempts to conceive of the possibility of an answer. "The question cuts so deep," the philosopher Robert Nozick wrote in a famous essay on this matter, "that any approach that stands a chance of yielding an answer will look extremely weird. Someone who proposes a non-strange answer shows he didn't understand this question." (Nozick 1981: 116) 

From quantum theory to dialectics?
In the following I want to examine one possible type of solution which has recently gained considerable currency in the ongoing attempt to answer this question. This type of solution, which definitely does not belong to the non-strange category, has been termed the "something from nothing theory" and its proponents have aptly been called "nothing theorists". It may surprise the unsuspecting reader to learn that these theorists are renown quantum physicists (and the odd chemist) rather than obscure philosophers. Here is how Jim Holt introduces the quantum theory of nothing in his recent book Why Does The World Exist?:

"Perhaps the world arose spontaneously from sheer nothingness. All existence might be chalked up to a random fluctuation in the void, a "quantum tunneling" from nothingness into being. Exactly how this could have happened has become the province of a small but influential group of physicists who are sometimes referred to as "nothing theorists". With a mixture of metaphysical chutzpah and naivete, these physicists – who include Stephen Hawking among their number – think they might be able to resolve a mystery heretofore considered untouchable by science." (Holt 2013: 27-8)

In the following I offer a philosophical analysis of these quantum theories of nothing. Can these theories really answer Leibniz' question? I will argue that ultimately they cannot, even if they are scientifically sound and offer crucial insights into how the universe came into being. The difficulty is that these theories, as belonging to science and physics in particular, still presuppose too much ontological baggage, notably the false vacuum of 'empty' space and/or the laws of quantum mechanics. Thus their 'nothing' is still not the absolute nothingness which is required if we truly want to answer Leibniz' question. Here, I think, philosophy must come to the rescue of physics. 

Near the end of this post, therefore, I will argue for a dialectical conception of nothingness as self-negating. This, according to me, is the necessary philosophical complement to the quantum theory of nothing. A crucially important fact in this regard is the often noted energetic polarity of the physical universe, i.e. its separation into the positive energy captured in matter and the negative energy of the gravitational force. Since the amounts of positive and negative energy in the universe are equal, they ultimately cancel each other out and leave the total energy level of the universe at exactly zero (Hawking 1988: p.129). And since everything in the universe consists in one form or another of energy, this means that the universe is literally made out of nothing, but a nothing split into opposites (Atkins 2011, pp.13-17). A similar conclusion is suggested by the fluctuation of the false vacuum, where particle and antiparticle pairs spontaneously pop into existence out of the fluctuating 'zero' energy level of empty space. As I will argue, these facts clearly point in the direction of a dialectical conception of nothingness as self-negating, since on such a conception polarity is an intrinsic feature of nothingness itself, divided as it is between itself and its negation. Only a dialectics of nothingness, then, is truly able to answer Leibniz' question.  

Preliminary remarks: Why nothingness is unavoidable
Before dealing in more detail with the quantum theories of nothing, however, I want to make some preliminary remarks about how Leibniz' question should be answered. My contention is that this question by itself already forces us to entertain the concept of nothingness as the ultimate answer. This is because any other answer simply leads to a regress or vicious circle. Thus if we answer the question "Why is there anything at all?" by referring to some existing thing as the ultimate cause (say, God), we still have not truly answered our question. For what then explains the existence of that first thing? Why then does God exist? Obviously it might then be answered, as theology has done for centuries, that God is causa sui, his existence is self-caused and hence eternal. Or one might invoke St. Anselm's ontological argument: "God necessarily exists, because as the utmost perfect being his existence is included in his essence." But can these arguments satisfy? The concept of self-causation surely seems viciously circular. And Anselm's ontological argument just seems to define God into existence. But definitions can only yield tautological truths, not synthetic truths that tell us about what is 'really out there'. In short, it seems clear there can't be any magical 'something' the existence of which is self-explanatory and which can then be used to answer Leibniz question. As long as that question is answered by reference to another existing something, the answer runs afoul of a vicious regress or circle. Ayer, the don of logical positivism, put this problem succinctly as follows:

"Supposing you asked a question like 'Where do all things come from?' Now that's a perfectly meaningful question as regards any given event. Asking where it came from is asking for a description of some event prior to it. But if you generalize that question, it becomes meaningless. You're then asking what event is prior to all events. Clearly no event can be prior to all events. Because it's a member of the class of all events it must be included in it, and therefore can't be prior to it." (Ayer quoted in Holt 2013: 24)

As this quote reveals, true to the iconoclastic spirit of logical positivism, Ayer thinks Leibniz' question is nonsensical because it is unanswerable on principle – unanswerable, that is, as long as the range of possible answers is restricted to the domain of existing somethings and events. But what if we leave this domain behind and look for the answer in what does not exist and is not anything at all? What if we look to nothingness as a possible answer? No doubt, the logical positivist Ayer will reject this possibility as nonsensical as well, agreeing with fellow logical positivist Rudolf Carnap that the concept of nothingness is illogical and meaningless since by definition it cannot have a referent. The concept of nothingness, after all, refers to... nothing at all. Thus it is a pseudo concept, or so Carnap argued in his criticism of Heidegger (more about this below).
 
The trouble with infinitism
We should, however, also take note of another possibility, one not mentioned by Ayer but nonetheless often taken as the only possible answer to the question why there is anything at all. According to this line of reasoning, which we may call "infinitism", the cause of the universe is unproblematic, because there simply is no first cause: there is an infinite chain of causes, stretching all the way back into an infinite past and forward into an infinite future. Simply put, the universe exists eternally; it has – in one form or another – always existed and will always exist. There is no mystery, then, as to why the universe exists. This answer has been appealing to many great rational minds: Aristotle, Galileo, Hume, Spinoza, Newton, Einstein... Still, however, I think the infinitist answer misses the point of Leibniz question. Here I fully agree with what Jim Holt writes about the infinitist solution:

"But there's still something missing here. This infinite world is like a railroad train with an infinite number of carriages, each pulling the one behind it – and no locomotive. It can also be likened to a vertical chain with an infinite number of links. Each of these links holds up the link below it. But what holds up the chain as a whole?" (Holt 2013: 86)

The idea of an eternally existing universe – for example in the form of an eternal cycle of Big Bangs – might turn out to be a scientifically legitimate hypothesis. It might even turn out to be true. But it still doesn't answer the question why there is anything at all. It doesn't answer the question why there is this infinite series to begin with. It might be objected that this question makes no sense because in an infinite series of causes there simply is no first cause. But this objection assumes that the ultimate cause of the universe must be temporal, existing in time, like the universe itself. But why can't the ultimate cause be non-temporal? This, indeed, is what contemporary physics suggests about the cause of the Big Bang: since not only space and matter but also time itself only came into existence with the Big Bang, the cause of the Big Bang must be timeless. This notion of a non-temporal cause is also inescapable for the infinitist solution. A temporally infinite series of causes has no first cause in time, but it must have an ultimate cause outside of time, a non-temporal cause. Otherwise the infinite series will itself remain unexplained. Even the infinitist solution, then, must posit some ultimate non-temporal cause. And as long as this cause remains an existing something, we are back in the problem of vicious regresses and circles. Hence, also on the infinitist solution we are driven to an ultimate cause that is not an existing something – that is to say: were are driven to entertain nothingness as the ultimate cause of the universe.

Nothingness and the zero-energy universe

Still, one is inclined to ask, how can something emerge from nothing? Isn’t this plainly impossible? After all, as the ancients said, ex nihilo nihil fit, from nothing only nothing can come. Christianity, in contrast, was able to imagine a creatio ex nihilo, but only by presupposing a God who could perform this magic trick. So even according to Christianity no true creation out of nothing took place, since God pre-existed the creation. And, indeed, doesn’t it seem wildly absurd to suppose that nothing can cause the existence of something? As William James put it: "from nothing to being there is no logical bridge" (James 1911: p.40).

Yet what if we don’t really need such a bridge? What if the entirety of being is after all nothing but… nothing? Strange as it may sound, this indeed seems to be the conclusion of present-day physics. The point is that the physical universe (and is there anything else?) consists of nothing but energy in different forms (matter, light, movement, heat, gravity). On the most elementary level, this total energy of the universe consists in a negative and a positive part: two parts of equal magnitude, which – as opposites – cancel each other out, thus leaving the net energy of the universe at precisely zero! And since the total energy of the universe is zero, there really is – in terms of energy – nothing at all, albeit a nothing split in two opposing parts.

This obviously requires some further explanation. Let’s start with the concept of positive energy. This is the energy invested in matter (including light and antimatter), both in the constitution of matter itself (‘frozen energy’) and in its movement (kinetic energy). Obviously, given the sheer size of the material universe, there is a tremendous lot of positive energy (though no one is quite sure how much). At the same time, however, there is an equal amount of negative energy stored in the gravitational attraction that exists between all pieces of matter. The positive energy of matter is precisely balanced by the negative energy of gravity, so ultimately there is no energy in the universe at all. Here is how Stephen Hawking explains it:

“Two pieces of matter that are close to each other have less [positive] energy than the same two pieces a long way apart, because you have to expend energy to separate them against the gravitational force that is pulling them together.” In other words: since it takes energy to separate the two pieces of matter, gravity must be using an opposed form of energy to pull them together. Thus, as Hawking writes: “the gravitational field has negative energy… this negative gravitational energy exactly cancels the positive energy represented by the matter. So the total energy of the universe is zero.” (Hawking 1988: p.129)

Particles and antiparticles

Nature seems to have a taste for such polarities, such that the opposites ultimately cancel each other out, leaving nothing as their sum total. For not only is there the polarity of positive and negative energy, there is also within the realm of positive energy – to be precise: within the constitution of matter – the polarity of particle and antiparticle (collectively referred to as “fermions”). According to quantum physics, for every type of particle there is a type of antiparticle with opposite properties, such that when they meet they annihilate each other. In fact, particles and antiparticles can only come into existence together, in pairs. Here is what John Gribbin (2007) writes about it:

“The only way you can make a 'new' fermion, such as an electron, out of energy is if, at the same time, you make a mirror-image anti-particle (in this case, a positron). The mirror-image particle has opposite quantum properties (including, in this case, positive electric charge instead of negative electric charge) so the two cancel each other out for the purpose of counting fermions, with one negative and one positive adding up to nothing.” (p. 17) Thus “when a positron meets an electron, both particles disappear in a puff of high-energy photons – gamma rays – as their opposite quantum properties cancel each other out.” (p.62)

Electromagnetic polarity is a prime example of such fermionic polarity in nature. Positrons have positive electric charge, they repel each other but attract the electrons which have negative charge. Since there is a negative charge for every positive charge, all the charges ultimately cancel each other out, so in the final analysis the total electric charge of the universe is precisely zero. It is important to remember, however, that electromagnetic polarity is only one example of fermionic polarity. Even the particles with no electric charge have this fundamental property of being paired to a type of antiparticle. There is an antimatter counterpart for the neutron, for example, even though these particles lack electric charge.

All this, however, does not mean that the physical universe consists of nothing but such polarities. There are indeed many fundamental aspects of the physical universe which apparently do not exhibit polarity. For example, closely connected to the fermions are the bosons, which are not precisely particles, though they have some particle-like properties (e.g. bosons are field quanta). Bosons are the mediators between the fermions, conveying the fundamental forces (or interactions) from one particle to another. Bosons, however, do not exhibit polarity like the fermions: they do not come in pairs of opposites.

A splitting of 0 into 1 and -1?

Nevertheless, polarity does remain a remarkably deep feature of nature at many different levels (positive and negative energy, electromagnetic polarity, fermionic polarity), a feature that still cries out for a general explanation. And, indeed, it is a feature of nature that is very suggestive when it comes to answering Leibniz’ question. For the fact remains that on the most fundamental level – the level of pure energy, the basic ‘stuff’ of physical existence – the universe consists of two opposed magnitudes, positive and negative energy, which in the end cancel each other out. The net amount of energy in the universe is thus strictly speaking zero, so that in an energetic sense the universe is literally nothing. Hence, as the chemist (and famous popularizer of science and atheism) Peter Atkins notes, explaining how the universe ‘popped into being’ out of nothing may turn out to be less of a paradox than was always believed. For if the universe is itself ultimately nothing, then surely it can come out of nothing, since ex nihilo nihil fit. As Atkins writes:

“First, it is important to realize that there probably isn’t anything here anyway… Of course we are part of and surrounded by things; but at a deep level there is nothing… The bottom line, prejudiced with a dash of speculation, is that the initial endowment of energy at the creation was exactly zero, and the total energy has remained fixed at that value for all time… What we see around us is in fact nothing, but Nothing that has been separated into opposites to give, thereby, the appearance of something”. (Atkins 2011, p.13, 17)

What Atkins is suggesting, then, is that the creation of the universe may have been something like “1 + (-1) = 0” in reverse. That is to say: not 1 and -1 coming together to make 0, but rather 0 splitting itself into the polarity of 1 and -1. Analogously, Atkins speculates that the universe emerged out of a primordial nothing because this nothing divided itself into positive and negative energy as well as into particle-antiparticle pairs.

Fluctuation of the false vacuum

However, even if – in terms of energy – the universe is ultimately nothing, the idea of nothing splitting into opposites may still seem wildly speculative and absurd, not to say horribly close to New Age spirituality (Yin and Yang and all that). Nevertheless, quantum physics has revealed that something like this does actually happen. This is the quantum fluctuation of the false vacuum. This is a phenomenon whereby particle-antiparticle pairs (such as electrons and positrons) spontaneously pop in and out of existence in empty space for very short durations.

In quantum mechanics, this is explained by Heisenberg’s uncertainty principle, which – among many other things – says you cannot precisely measure both the value of an energy field and the rate at which it changes. Knowledge of the one implies uncertainty about the other, and vice versa. The point is that this pretty much rules out the possibility of empty space. Empty space, or the vacuum, is by definition a state in which the amount of energy is zero. But Heisenberg’s principle tells us that if the value of a field is precisely known to be zero, its rate of change is completely random and thus can’t be zero. So even in ‘empty’ space, the energy level fluctuates randomly. This is also why the vacuum is better described as a false vacuum, since strictly speaking a real vacuum is impossible, ruled out by the uncertainty principle. In reality, 'empty space' is seething with activity on the quantum scale, with particle-antiparticle pairs popping in and out of existence all the time. Mostly such pairs are extremely short lived, since nearly every particle and antiparticle pair annihilates itself almost immediately after popping into existence. Hence such pairs are generally known as virtual particle-antiparticle pairs. Yet despite their virtuality, they are very real in their consequences, since laboratory experiments have shown that virtual pairs directly affect the energy levels of existing atoms.

“Maybe the universe is a quantum fluctuation!”

So now we have virtual particle and antiparticle pairs spontaneously emerging from the almost nothing of ‘empty’ space… Could this perhaps be the key to how the universe came into existence? The key to how primordial nothingness split into polarities? The first to entertain such an idea seems to have been physicist Ed Tryon who in 1969 – during a talk by a celebrity physicist at Columbia University – suddenly blurted out: “Maybe the universe is a quantum fluctuation!” Reportedly his remark was greeted with derisive laughter from the several Nobel laureates present at that meeting. Nevertheless, Tryon’s idea stuck and was subsequently developed further by Tryon himself and other physicist. Nowadays the idea has bloomed into a serious scientific theory whose proponents include renowned physicists like Stephen Hawking, Alan Guth, Frank Wilczek, Lawrence Krauss, Alexei Filippenko and Jay Pasachoff. What allowed the idea to grow into scientific theory was the fact that it fitted nicely with the inflationary theory about the expansion of the universe right after the Big Bang. I am not going to discuss the inflationary theory here in any detail, since that would take us too far a field. Suffice it to say that together with inflation the occurrence of quantum fluctuations in primordial empty space may quite possibly have been enough to cause the Big Bang. Here is how Filippenko and Pasachoff relate the story in a well-known paper entitled A Universe from Nothing:

“Perhaps many quantum fluctuations occurred before the birth of our universe. Most of them quickly disappeared. But one lived sufficiently long and had the right conditions for inflation to have been initiated. Thereafter, the original tiny volume inflated by an enormous factor, and our macroscopic universe was born.” (Filippenko and Pasachoff 2010)

If this theory is correct, then the emergence of the universe was a matter of sheer chance, a result of the randomness implied by Heisenberg’s uncertainty principle. “In answer to the question of why it happened”, Tryon later commented, “I offer the modest proposal that our universe is simply one of those things which happen from time to time.” In a similar vein Alan Guth has described the universe as the “ultimate free lunch”. Finally, physicist and Nobel laureate Frank Wilczek famously epitomized this theory by answering the question "Why is there something rather than nothing?" with the pithy remark: "Because nothing is unstable." Unstable, that is, insofar as the energy level of 'empty' space fluctuates randomly.

The problem posed by a piece of rubber

But does this really answer Leibniz’ question why there is anything at all? This theory is certainly suggestive about how ‘nothing’ can split itself into opposites, namely, the virtual particle and antiparticle pairs. Nevertheless, it is quite clear that the theory itself is not yet the answer. After all, according to this theory, quite a bit of things must have existed before the Big Bang: there must have been ‘empty’ space, and there must have been the laws of nature (as described by quantum mechanics) in order to facilitate the fluctuations of the vacuum that supposedly caused the Big Bang. About all these things we must still ask why they were there in the first place.

Take, for example, the idea of ‘empty space’. It is clear that this is not absolute nothingness. The space of the quantum vacuum is not really empty. It has a complicated mathematical structure; it bends and flexes like rubber; it is saturated with energy fields and seethes with virtual-particle activity. Why would such a complicated object like the quantum vacuum ever have existed? As Alan Guth has observed (thereby in fact retracting his earlier “ultimate free lunch” remark): “A proposal that the universe was created from empty space seems no more fundamental than a proposal that the universe was spawned by a piece of rubber. It might be true, but one still would want to ask where the piece of rubber came from.” (Guth quoted in Holt 2013: 142)

A quantum tunnel from nothing to something?

The physicist who seems to have come closest to solving this “piece of rubber” problem is Alex Vilenkin. When he talks about the universe as arising from nothing, he literally means nothing. “Nothing is nothing!”, he said during an interview: “Not just no matter. It’s no space. No time. Nothing.” How can he pull of such a feat? Actually Vilenkin cheats a bit. He still defines nothing in spatial terms, admittedly not as empty space, but as a space (or rather spacetime) with zero dimensions. Imagine spacetime as the surface of a sphere. (This is what is called a closed spacetime, which curves back on itself; it is finite even though it has no boundaries.) Now what Vilenkin asks us to do is to imagine this sphere as shrinking, like a balloon losing its air. The radius goes smaller and smaller. Eventually the radius goes all the way down to zero. The surface of the sphere disappears completely and with it spacetime itself. Thus we arrive at a mathematically precise definition of nothingness: a closed spacetime with zero radius. Now with this mathematical definition in hand Vilenkin was able to do an interesting calculation. Using the principles of quantum theory he showed that out of such an initial state of nothingness a tiny bit of false vacuum could spontaneously pop into existence (Vilenkin calls this process “tunneling”). Then, driven by inflation, this tiny bit of vacuum would expand dramatically and turn into the Big Bang.

It is true that – if his calculations are correct – Vilenkin has got rid of the problem of the empty spacetime pre-existing the Big Bang. Yet his primordial nothingness still doesn’t seem to be absolute nothingness, since he is still presupposing the laws of nature. Obviously these laws are not quotidian things like physical objects, but still there is a sense in which they exist or hold true. So we still have to ask why these laws were there in the first place. Why these laws? Why not others? And why any law at all? It would seem that absolute nothingness would also have to be void of law. In fact Vilenkin acknowledges the problem. Here is what he writes:

“The tunneling process is governed by the same fundamental laws that describe the subsequent evolution of the universe. It follows that the laws should be ‘there’ even prior to the universe itself. Does this mean that the laws are not mere descriptions of reality and can have an independent existence of their own? In the absence of space, time, and matter, what tablets could they be written upon? The laws are expressed in the form of mathematical equations. If the medium of mathematics is the mind, does this mean that mind should predate the universe?” (Vilenkin quoted in Holt 2013: p.161)

Asked whose mind this could be, Vilenkin answered: “If you like you can say they [the laws of nature] are in the mind of God.” (Ibid.) Thus with one stroke Vilenkin makes clear we still have not answered Leibniz’ question. Even if the laws of nature are such as to make nothingness impossible, we would still want to know why these laws were there to begin with.

On the boundary of science

So where do we go from here? I think there are two conclusions to be drawn form the above discussion – two conclusions which together will point us in the right direction. The first conclusion follows from the inadequacy of the quantum theories of 'nothing' as answers to Leibniz' question. I am, of course, not saying these theories are false or scientifically unsound: as far as we know, they might very well be true. Nevertheless, they fail to answer Leibniz' question because they still presuppose too much ontological baggage. That is to say: they presuppose either the false vacuum and the laws of physics (Hawking, Tryon, Guth, Wilczek, Krauss e.a.) or just the laws of physics (Vilenkin) as pre-existing the Big Bang. And maybe this is as far as science can go in explaining how something emerged from nothing. Science, and physics in particular, has to proceed through experimental observation and mathematization of the observed results. But you cannot empirically observe absolute nothingness (whether you can mathematize nothingness is an open question; see axiomatic set theory with its foundational empty set). Hence already Vilenkin's theory of how the laws of physics imply the "tunneling" of a false vacuum out of nothingness moves on the very boundary of science, since – given the unobservability of nothingness – the theory doesn't seem to be open to empirical falsification. This indicates the dilemma we are in. If we truly want to answer Leibniz' question, we must somehow develop a solution in terms of absolute nothingness, without even presupposing the laws of physics. But then by the same token we seem to step outside of science, or at least outside of physics, given the unobservability of such a 'thing' as absolute nothingness, if it exists.

A logical transition from nothing to something?

But then again physics is not the only science. And not every science is dependent on empirical testing. Just think of pure mathematics or pure logic. And this brings me to the second conclusion to be drawn from the preceding discussion, namely, that the transition from nothing to something should perhaps primarily be thought of as logical rather than temporal. Consider Vilenkin's scenario, where the transition from nothing to something must have happened outside of time (since time only emerged with the Big Bang) and on the basis of just the laws of physics (laws which are mathematical in nature, as Vilenkin emphasizes). Such a transition seems to be logical or conceptual in nature, insofar as it is ordained by a timeless realm of mathematical truths. Jim Holt puts this very well in his discussion of Vilenkin's theory:

"Since time itself (along with space) is created in the transition from Nothing to Something, this transition can't very well take place in time. It seems to unfold logically rather than temporally. If Vilenkin is right, nothingness never had a chance: the laws of physics eternally ordained that, with some appreciable probability, there would be a universe. But what gives ontological clout to these laws? If they are logically prior to the world, where exactly are they written down?" (Holt 2013: p.144)

The only thing wrong with Vilenkin's theory, as an answer to Leibniz' question, is that it presupposes the laws of physics; this is also what Holt indicates in the quote above. So what if we replace the laws of physics with the 'pure' laws of logic and mathematics? Certainly the elementary truths of logic (such as the principles of identity, non-contradiction, tertium non datur) are much more fundamental than the laws of physics, which as far as we know are only true for our particular universe, whereas these logical truths hold for every possible universe. Thus the logical truths are certainly timeless if any truth is. Whether this timeless validity also holds for mathematics is an open question, although the reducibility of the bulk of mathematics to logic and axiomatic set theory is certainly suggestive here (not least because set theory crucially involves its own version of nothingness in terms of the foundational empty set). But let us for the time being just focus on the 'eternal' truths of logic. What happens if we apply these truths to the concept of pure nothingness? Perhaps – to paraphrase Jim Holt's quote above – nothingness never had a chance given the laws of logic? Perhaps logic forbids nothingness and thus eternally ordains that there is being? Perhaps being is just a logical necessity?

Logical problems with nothingness

Come to think of it, it is very strange that this possibility has not figured more prominently in the academic discussion surrounding Leibniz' question. It is after all obvious that there is a logical problem with the concept of nothingness. The apparent paradox of this concept has, since time immemorial, been the source of countless jokes and puzzles concerning the 'existence' of nothing or absence in general. In Homer's Odyssey, for example, the cunning Ulysses utilizes a version of this paradox by telling the cyclops Polyphemus his name is "Nobody" before piercing the cyclops' eye with a burning stake. Then, when asked by the other cyclopes why he is screaming, Polyphemus replies that "Nobody" is hurting him. Or take the episode in Alice in Wonderland where Alice says "I can see nothing" and the Cheshire Cat replies "My, you must have good eyes".

The paradox, then, turns on what we might call the referentiality of "nothingness". For if we take "nothing" to be a referring expression, referring to a definite object, then paradox immediately arises, since the referent of this term must be... nothing and as such it must be absent or non-existent. But how can this term refer if it has no referent? As a referring expression "nothing" undermines its own referentiality. It is, as philosophers say, performatively inconsistent, since it negates the existence of its referent in the (performative) act of referring to it as "nothing".

In more general terms, the paradox concerns the supposed existence of nothingness. If nothingness can be said to exist, then it must be a being, a thing that exists, an object able to function as the referent of a referring expression. But then again, nothingness is precisely nothingness because it is not any of these things: not a being, not a thing that exists, not an object and not a referent. Hence it seems clear that nothingness can't exist and can't be referred to.

In logical terms, nothingness thus violates the most elementary law of logic, the principle of identity, which states that “each thing is identical with itself and different from another”. For how can nothingness be self-identical if it has no identity to begin with? Nothingness, after all, cannot be referred to by means of the demonstrative "this", which is a precondition for having identity. It makes no sense to speak of "this nothingness" as if it could be distinguished from other "nothingnesses". To suppose nothingness has an identity is to turn it into a something, which it precisely is not. In a similar vein, we can say that nothingness also violates the second-most basic law of logic, the principle of non-contradiction, which states that "either something exists or it does not exist". For, as we have seen, a paradox arises when we say nothingness exists, since then we turn it into a being. The only way for nothingness to exist, then, is by not existing. Or as Jacques Lacan, always a lover of paradox, put it in a somewhat different context: "Nothing exists insofar as it does not exist." (Lacan 1966: p.392) The supposed existence of nothingness, then, is inherently contradictory. A further analysis of the logical impossibility of nothingness can be found here.

From Parmenides to Carnap

In the history of philosophy, these logical paradoxes are well-known. They have motivated a long tradition of philosophers rejecting the logical possibility of talk about nothingness, a tradition ranging from Parmenides to Carnap. In the late 6th century BC, the presocratic philosopher Parmenides of Elea already argued that "you cannot know what is not, for that is impossible - nor can you utter it", concluding from this that thought and being must coincide, since you can only think of what exists. In the 20th century, the logical positivist Rudolf Carnap – though obviously not promoting Eleatic idealism – deployed essentially the same argument to denounce Heidegger's talk of "the Nothing" that "nothings". Such talk, Carnap argued, "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) Carnap, then, basically repeats the argument that ascribing existence to nothingness is contradictory, since by definition it is nothing.

Kantianism, Platonism or Dialectics?
It seems, then, we have found our answer to Leibniz' question! And it seems this answer is infinitely more simple than anything proposed by the quantum theories of nothing. It seem that to the question "Why is there something rather than nothing?" we should simply answer: because the concept of nothingness is inconsistent, ruled out by the timeless truths of logic! Nothingness is logically impossible, hence its negation – the statement that there is something – is logically necessary. So is this the end of the matter? Not quite. For it is still an open question how this logical impossibility of nothingness should be interpreted. Three interpretations seem possible:

1) A subjective or Kantian interpretation: Logic is primarily about the normative structure of human cognition, we don't know if it applies to reality-in-itself. So the fact that nothingness is logically impossible simply means that we can't imagine or think nothingness – being is merely logically necessary for us, not in itself.

2) An objective Platonic interpretation: The laws of logic are in themselves timelessly true, independently of human cognition, they belong to a Platonic realm of ideal truths. The logical impossibility of nothingness, then, means that being is a logical necessity in itself, not just for us. Being is timelessly ordained by objective logic.

3) An objective dialectical interpretation: Nothingness really exists (or really existed), but since its existence is contradictory (i.e. self-negating), nothingness negated itself and thereby produced being. Being is a manifestation of the contradictory nature of nothingness in itself, not just for us.

Which interpretation is the correct one? It seems the subjective interpretation can be ruled out from the start. Logic may be just subjective, being no more than the inherent structure of human thought. But as such it cannot declare the necessity of existence. It is 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, after all, are part of being, so on this proposal we exist because we cannot imagine ourselves as not existing. In such a scenario, then, we would be causa sui, since we would have imagined or thought ourselves into existence. But this is plainly absurd.

Why is there ought anyway?
So the issue comes down to a choice between the Platonic and the dialectical interpretations. Here, I think, we have to admit the dialectical interpretation is the stronger one. Two reasons in particular seem to plead in its favor. First of all, it seems clear that on the Platonic interpretation the logical impossibility of nothingness does not really answer Leibniz' question. For on this interpretation we still have to presuppose the existence of the Platonic realm of ideal truths. Thus we have not genuinely explained how something emerges from nothing. To this it may be objected that "existence" is not the right term to describe this 'obtaining' of the ideal truths, rather they have a normative force or validity: in terms of the familiar is/ought distinction, we should say that the obtaining of such ideal truths is not a matter of "is" but of "ought". This objection, however, seems vacuous to me. For even if it were correct, we could still say that there is such a thing as normativity or ideal validity. And then we would still want to know why there was such a thing in the first place.

Dialectics and the polarity of energy
The second reason for the dialectical interpretation, however, is more decisive. It has to do with the polarities which, as we have seen, are fundamental to the physical universe: the polarity of positive and negative energy, and the fermionic polarity of particle and antiparticle, which includes the polarity of positive and negative electric charge. This proclivity for polarities is an objective feature of nature, which still cries out for a comprehensive explanation. Moreover, since the opposites in these polarities cancel each other out, they ultimately imply that the universe is in a sense nothing at all. The polarity of positive and negative energy seems especially fundamental in this regard. In the physical universe, after all, everything is in one form or another a manifestation of energy. And since the total energy level of the universe is zero (because of the mutual cancellation of positive and negative energy), this implies that the universe is literally made out of nothing, but a nothing split into opposites. Or to repeat an earlier quote from chemist Peter Atkins: "What we see around us is in fact nothing, but Nothing that has been separated into opposites to give, thereby, the appearance of something." (Atkins 2011, p.17) Now isn't it clear that this division of nothing into polarity fits hand in glove with the dialectical conception of nothingness as self-negating? For if nothingness is indeed self-negating, it is by the same token its own opposite, its own negative counterpart, its own 'antiparticle' (or rather 'antibeing') so to speak. Thus, as nothingness negates itself, it necessarily splits in two opposed 'parts', namely, itself and its negation. From a dialectical viewpoint, then, polarity seems to be an inherent attribute of nothingness.

Where does this leave us? As we have seen, the only way to answer Leibniz' question without getting stuck in the regress or circle problem is to presuppose nothing – that is to say: nothing but the concept of nothing and the elementary laws of logic, without which no thought is possible. In the end, only the logical impossibility of nothingness itself can be the ultimate answer to the question why there is something rather nothing. Thus it seems clear to me that the parallel between the energetic polarity of the universe and the dialectical polarity of nothingness can be no mere coincidence.  

Concluding remarks
Obviously there are still loads of questions to be answered. For example, how does a dialectical conception of nothingness as self-negating relate to the laws of quantum mechanics which facilitate the fluctuation of the vacuum or even – on Vilenkin's scenario – the "quantum tunneling" of the false vacuum out of nothingness? It would of course be a pseudo-scientific absurdity to attempt a direct derivation of quantum mechanics from the dialectical logic of nothingness. But what about mathematics? The laws of quantum mechanics are thoroughly mathematical in nature, and perhaps there is a route from pure mathematics to the equations of quantum physics, as mathematical Platonists like Roger Penrose have hypothesized. If so, then the reduction of mathematics to logic and axiomatic set theory does seem to forge an indirect link between quantum physics and the dialectics of nothingness. For axiomatic set theory knows its own version of nothingness in the form of the foundational concept of the empty set. And if nothingness is indeed self-negating, it then seems to have a recursive structure analogous to the recursive procedure by which all higher sets are defined on the basis of the foundational empty set (for this analogy between set theory and the dialectics of self-negation, see Ware 1999: pp.230-238). In short, could it perhaps be the case that the set-theoretic derivation of mathematics is isomorphous to the dialectical structure of self-negating nothingness? And if so, doesn't this imply that mathematics is implicit in that dialectical structure? In that case, the step from the dialectics of nothingness to the mathematical laws of quantum mechanics is perhaps not so daunting as it seems. But, obviously, for now this is all just speculation and hypothesizing. I hope to be able to investigate these issues in the future and publish the results on this blog. However, for further ontological implications of the dialectics of nothingness and its relation to physics, I can already refer the reader to my earlier post: Theses towards a dialectical ontology

References
-Atkins, Peter (2011), On Being: A scientist's exploration of the great questions of existence. Oxford University Press, Oxford.-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.

-Filippenko, Alexei V. and Pasachoff, Jay M. (2010), "A Universe from Nothing" (a lecture for the Astronomical Society of the Pacific): http://www.astrosociety.org/publications/a-universe-from-nothing/

-Gribbin, John (2007), The Universe: A Biography. Allen Lane, London.
-Hawking, Stephen (1988), A Brief History of Time: From the Big Bang to Black Holes. Bantam, New York.
-Holt, Jim (2013), Why Does The World Exist? One Man's Quest for the Big Answer. Profile Books, London.
-James, William (1911), Some Problems of Philosophy: A Beginning of an Introduction to Philosophy. Longmans, Green, and Co., New York.
-Lacan, Jacques (1966), Écrits. Seuil, Paris.
-Nozick, Robert (1981), Philosophical Explanations. Belknap Press, Cambridge Mass.-Ware, Robert Bruce (1999), Hegel: The Logic of Self-Consciousness. Edinburgh University Press, Edinburgh.

Theses towards a dialectical ontology

1. Everything is what it is by differing from what it is not. Hence the identity of indiscernibles (Leibniz): if beings are not different from each other in at least one respect, they are numerically identical.
 
2. The difference between beings is their mutual negativity, that is, their mutual non-identity or 'not-being-the-same-as-each-other'. This negativity determines what they are. In other words: mutual non-identity determines individual identity. Hence there must always be more than one being. For if there were only one being, it would lack all difference with other beings and would as such be completely indeterminate, as good as nothing. Beings must therefore co-exist simultaneously, which is only possible in space. Space is first and foremost the medium of the negativity between beings. 

3. As the origin of determinate beings, the negativity between them must necessarily precede them, just as any cause must precede its effect. Though this precedence need not be temporal, it must at least be ontological, in the sense that the negativity between beings must 'exist' independently from them. Otherwise the origin would presuppose the originated and the process of origination would never get off the ground. We must, therefore, presuppose a pure negativity, a negativity as such or in itself, as the origin of determinate beings.

4. Up till now, however, we have only discussed what beings are, their identities. We have found this 'what-ness' of beings dependent on pure negativity. But to fully grasp the concept of a being, we must also distinguish its 'that-ness', i.e. the fact that beings exist. Logically at least, the existence of a being is not implied in its identity, since we can conceive of determinate identities (e.g. Pegasus, the winged horse) which nevertheless do not exist. Hence the pure negativity between beings, which determines their identities, does not also determine their existence.

5. Starting from proposition 1, we have to define existence as the difference from non-existence. In other words: beings exist because they differ from what doesn't exist. But how is this possible? Where do beings find this non-existence and how can they relate to it? 

6. This non-existence is to be found between beings, in the pure negativity presupposed by their mutual differentiation. After all, what is pure negativity in itself if not an active, nihilating nothingness, "das nichtende Nichts" (Heidegger)? Of such pure negativity in itself, we cannot say that it exists in the same sense that beings exist. We can also put this as follows: Pure negativity is the source of the determinacy of beings. But the source of determinacy cannot itself be determinate, on pain of a vicious circle. Hence it must be completely indeterminate, lacking any distinction. Hence it must be nothing, which is non-existence par excellence.

7. Beings exist because they differ from the pure negativity between them. So even if pure negativity does not determine the existence of beings in the same way as it determines their identities, it is nevertheless presupposed by their existence. In order to exist, beings must stand in a negative relation to this pure negativity.

8. We now reach a remarkable conclusion: there is a contradiction between identity and existence. The determinacy of identity requires negativity between beings, their differentiation. Yet existence requires that beings differ from this negativity. That is to say: existence is difference from difference, hence it is unity. In order to exist, beings must overcome their differences, relinquish their separate identities and form a unity. In the terminology of dialectics we can say that existence is the 'negation of negation', i.e. the negation of the negativity that ensures the separate identities of beings.

9. To better understand this contradiction between identity and existence, we must have recourse to the concepts of force and counter-force, such that we conceive of identity and existence as competing forces. For in reality a contradiction cannot obtain if the contraries are facts: either something is the case or it is not the case, tertium non datur. This is an elementary law of logic, even for dialectics, if it is not to trivialize itself (since ex falso sequitur quodlibet). The only contradictions capable of existing in reality are contradictions of forces or tendencies, where each force contradicts a counter-force. Indeed, such a contradiction is internal to the very concept of force, since a force cannot exert itself unless it is resisted by a counter-force. The contradiction between identity and existence, then, must be a contradiction of forces. On the one hand we must assume the force of negativity, resulting in the differentiation of beings. On the other hand we must assume a counter-force of unification, resulting in existence, where beings are differentiated from the negativity inherent in their mutual differentiation. 

10. Hence there is always a trade-off between identity and existence. The more determinate the identities of beings become, the more intense their mutual negativity, the lesser their level of existence. Conversely, the more beings are differentiated from their mutual negativity, the more intense their existence, the less determinate their identities are.

11. Hence there are levels of existence as well as levels of determinacy. Absolute existence would mean the total erasure of difference, the total unification of all beings into a 'One' lacking all determinacy, which as such would really be a nothing. Absolute determinacy, on the other hand, would be complete lack of difference from difference: beings would be infinitely determined yet lacking in existence. These highest levels can never be reached, however, since the forces of identity and existence must always limit each other. 

12. Since there is always a trade-off between identity and existence, there cannot be a perfect equilibrium between them. Existence always implies a disintegration of identity; identity always involves a lessening of existence. Beings, then, constantly oscillate between existence and identity – constantly fading in and out of existence as they constantly gain and loose their separate identities. In other words: beings constantly change. Hence beings are temporal. Time is this constant oscillation between identity and existence. 

13. Since the force of negativity must exert itself against the counter-force of unification (i.e. existence), it must manifest itself as mutual repulsion of beings, their resistance against unification. In contrast, the force of existence must manifest itself as resistance against this repulsion, i.e. as the mutual attraction of beings. 

14. Attraction forces the mutually repelling forces/beings together into spatial unities. This is why all beings must take the form of point-like particles in a repulsive force field, where the particles are nothing more than compressions of the repulsive force itself. Forced to push each other away and flee from each other, beings seek the maximum distance between them allowed by the force of attraction. This maximum distance between beings is accomplished through their maximum compression into point-like particles.

15. To repeat: the force of attraction forces mutually repelling beings into spatial unities. The only form, however, in which such unities are (relatively) stable, is the core-orbit form, where one being forms a core around which a repelled/repelling being orbits. Repulsion forces beings to move away from each other. This movement-away, then, must be synthesized with their mutual attraction in a unity. The orbiting movement is this synthesis. It allows the orbiting being to constantly flee the core while at the same time forming a unity with it. In the core-orbit form, then, a maximum synthesis of identity and existence is achieved.

16. Hence the empirical fact that the most stable structures in the universe exhibit this core-orbit form: atoms, solar systems, galaxies, and clusters of galaxies. In fact, the life span of atoms is virtually endless. In the universe there are no more stable unities than atoms.

17. We now seem to end up with some sort of cosmic dualism, a dualism of the most fundamental forces that constitute reality as such, namely, repulsion and attraction (= identity and existence = differentiation and unification).

18. Speaking metaphorically (though it is actually more than just a metaphor), we can equate repulsion with strife or war among beings, and attraction with peace or love among beings. In that sense, we can agree with Heraclitus's famous statement that "war is the father of all things" (namely, the origin of their separate identities). But at the same time we must qualify this statement by adding the counter-statement: "but love is the mother of all things" (namely, the origin of their existence). 

19. Yet on closer inspection these two mutually opposed forces (identity and existence = differentiation and unification = repulsion and attraction) are not distinct forces at all but rather two aspects of one and the same 'thing', namely pure negativity. 

20. Let us retrace our steps. We started from the assumption that beings are what they are because of the negativity between them. Thus this negativity must precede them as their origin and must as such be pure negativity. We then argued that for beings to exist, they must stand in a negative relation to this pure negativity, since it is by differing from non-existence that beings exist. Note, however, that as such existence implies a doubling of negativity: for a being to exist, there must be a second negativity that differentiates that being from the first negativity that determines its identity. This second negativity must also be pure, since it must precede the existence of all beings. Here the same reasoning applies as in proposition 3: if it were not independent from all beings, the origin would depend on the originated and the process of origination (in this case: the coming into existence of beings) would not get off the ground. So we have to distinguish two forms of pure negativity: N1 = the pure negativity between beings, the origin of their identities, and N2 = the pure negativity that elevates beings into existence by distinguishing them from N1. 

21. What is the relation between N1 and N2? Are they numerically distinct? Or are they merely different forms or functions of one and the same pure negativity? If the latter is the case, we must assume there is some kind of self-reflexivity involved in pure negativity, since it then relates negatively to itself in order to elevate beings into existence.

22. But let's start with the assumption that N1 and N2 are numerically distinct. Obviously, this does not make a lot of sense. From the standpoint of common sense, the assumption of one pure negativity is bad enough, but to assume the existence of two distinct pure negativities is downright outrageous. It is also at odds with the methodological principle of Occam's Razor (among competing hypotheses, the one with the fewest assumptions should be selected). From a methodological standpoint, then, it is better to start from the hypothesis that there is only one pure negativity involved in both identity and existence. Finally, this hypotheses is also favored by our own starting point (proposition 1) and the identity of indiscernibles that follows from it. N1 and N2 are both instances of pure negativity; as such they cannot be distinguished and it must be assumed they are one and the same. 

23. It seems, then, we must assume that N1 and N2 are merely different forms or functions of pure negativity in itself. As already said in proposition 21, this commits us to the assumption of a self-reflexivity at work in pure negativity, which not only distinguishes beings from each other (thereby establishing their separate identities), but also distinguishes beings from itself (thereby establishing their existence). What arguments can we give in favor of this self-reflexivity of pure negativity?

24. To better understand this self-reflexivity, we must analyse the concept of nothingness. We have already noted how pure negativity turns out to 'be' nothing (proposition 6). As the source of all determinacy (= N1), pure negativity must itself be completely indeterminate and hence must be nothing. Similar reasoning applies to N2, the pure negativity that distinguishes beings from N1 and thereby elevates them into existence. As the source of all existence, N2 cannot itself be said to exist (at least not in the same way beings can be said to exist) on pain of a vicious circle. Hence N2 is non-existent. Given the assumption that N1 = N2, then, we must conclude that pure negativity is a non-existent nothing.

25. What can we assert about this nothing? Apparently nothing, since it lacks any distinction and existence. It is the completely indeterminate. It seems that the only assertion we can make is a completely trivial tautology: "Nothing is... absolutely nothing!" Yet from a dialectical viewpoint, even this tautology is false, or rather it is both true and false at the same time. For this tautology still attributes a property, albeit a minimal property – a minimum of determinacy – to this nothing, namely, the property of self-identity. Saying that nothing is absolutely nothing is after all the same as saying that nothing is itself or identical with itself. Yet even self-identity is still too much determinacy for the completely undetermined. What we must do, then, is radicalize our tautology, follow through its inner dialectical logic and let it turn into its antithesis: Nothing is absolutely nothing to such a degree that it is not even itself! Nothing is not just nothing, it subverts and negates itself. Nothing is a negativity turned inwards, a self-reflexive negativity. Hence the self-reflexivity of pure negativity mentioned in proposition 23.
 
26. This conception of nothingness as a negativity turned inwards also agrees with our earlier arguments concerning pure negativity. Remember: beings are what they are because of the negativity between them. Hence this negativity, as their origin, must precede all beings and must be pure or in-itself. But the negativity between beings is their mutual non-identity or difference, their not-being-the-same-as-each-other. Pure negativity is therefore non-identity or difference absolutized, a not-being-the-same tout court. It stands to reason, however, that absolute non-identity is not even identical with itself, or that absolute difference differs even from itself. Thus the negativity turned inwards, which we arrived at through the dialectical development of the concept of nothing, can also be found in the concept of pure negativity. The dialectical development of the concept of nothing merely mirrors the inherent self-reflexivity of pure negativity.

27. Let us return to the duality of basic forces (i.e. identity and existence, or repulsion and attraction). We can now see that these forces are nothing other than the contradictory aspects of one and the same 'thing', namely, pure negativity in its self-reflexivity. On one hand, pure negativity manifests itself between beings as the origin of their mutual differentiation and determination (repulsion). On the other hand, this very same pure negativity turns inward and negates itself, thus lifting beings into existence by unifying them, i.e. by letting them differ from their difference (attraction).

28. Hence it is not just the contradiction between identity and existence that has to be conceived as an opposition of forces. Both are rather derivatives of a still more basic force, namely, pure negativity. Since a contradiction cannot obtain in reality unless it is a contradiction of forces (proposition 9), we must interpret the contradiction inherent in pure negativity (i.e. its self-negation, logically exemplified by the contradiction inherent in the tautology "nothing is nothing") in terms of force as well. We must, then, conceive of pure negativity as a force that is its own counter-force, a force exerted against itself. It is this self-opposing force that produces reality as such.

29. To arrive at a better understanding of this production of reality, we must take a closer look at the self-reflexivity of pure negativity. In terms of nothingness this self-reflexivity means that nothingness negates itself, that nothingness is not itself. In dialectical terms this is the 'negation of negation'. But negation of negation is affirmation. The self-negation of nothingness is thus the affirmation of something. No-thing becomes some-thing. In its difference from itself, pure negativity produces the positivity of being. This being is the self-produced difference with pure negativity, that is, self-produced by pure negativity itself. Pure negativity, then, does not vanish in its self-negation; rather it produces being as its own contrary. The end result is an opposition between being and pure negativity.

30. This being, produced by the self-negation of pure negativity, is – and is what it is – because of its opposition to pure negativity. In the case of this being, then, existence and identity coincide, in the sense that both consist in a negative relation to the same 'thing'. Yet this coincidence lasts no more than an infinitely small instant. For the self-differentiation of pure negativity (through the production of a determinate being) has the effect of giving identity and hence determinacy to pure negativity itself. This is because only identities can stand in relations of mutual differentiation. By differentiating itself from a something, then, pure negativity 'reifies' itself and becomes a something as well. In that sense, pure negativity does cease to be, since it becomes a being, a negative being, the contrary of the first being. Thus we end up with an opposition between a being and its anti-being, where one is the negation of the other. In mathematical terms, these beings stand to each other as 1 and -1. From now on I will use such numerals to refer to the beings produced by pure negativity. I will also refer to 1 as the "positive being" and to -1 as the "negative being".
 
31. As soon as the production of 1 by pure negativity effects the latter's 'reification' into -1, the coincidence of the identity and existence of 1 (mentioned in proposition 3) comes to an end. For in that case 1 no longer differs from pure negativity itself (a difference necessary for its existence) but only from its anti-being -1 (and this difference can only uphold its identity).

32. Although pure negativity ceases to be when it becomes -1, it does not (and in fact cannot) disappear altogether. For it returns in the difference between 1 and -1. We can put this in mathematical terms by saying that 1 and -1 always remain separated by the zero between them.

33. The self-differentiation of pure negativity can therefore also be seen as its self-division into 1 and -1. This plausible also if we look at it from another angle. The beings produced by pure negativity arise out of nothing. They are in fact nothing more than nothingness as it negates itself. Thus it stands to reason that in the end these beings 'add up' to nothing. Hence they must stand to each other as 1 and -1, since 1 + -1 = 0. 

34. This thesis, that beings can emerge from nothing but only if they stand to each other as 1 and -1, is empirically confirmed by what quantum physicists call the polarization of the vacuum. According to quantum field theory, the vacuum between interacting particles is not simply empty space. Rather, it contains particle-antiparticle pairs which are created out of the vacuum through spontaneous fluctuations of the zero-point energy field. When the particle and antiparticle meet again they annihilate each other.
 
35. Up till now, we have reckoned with just two beings (1 and -1) as resulting from pure negativity's ontological productivity. It is easy to see, however, that this productivity must result in many more beings, indeed, infinitely many. For we have seen how pure negativity returns in the difference between 1 and -1. Thus the process of ontological production as described in propositions 29 and 30 must repeat itself: the pure negativity between 1 and -1 must also divide itself and thus give rise to a second pair of being and anti-being (let's call them 2 and -2). Subsequently, the pure negativity between 2 and -2 divides itself into 3 and -3, and so on ad infininitum. Thus between 1 and -1 there emerges an infinite chain of beings and anti-beings, like this: 1, 2, 3, 4, ... , ∞, 0, -∞, ... , -4, -3, -2, -1. The central zero indicates the pure negativity between 1 and -1, a pure negativity that keeps dividing into being and anti-being pairs, all the way up to infinity.

36. In fact, the infinity of beings doesn't stop here. For as soon as the pure negativity between 1 and -1 has divided itself into 2 and -2 and so on, the necessity arises for 1 to differentiate itself also from 2 and -2, from 3 and -3 and so on (the same holds for -1). Thus pure negativity also returns between 1 and 2, between 1 and -2, between -1 and 2, and between, -1 and -2, and so on. In each case, pure negativity divides itself into a new pair of being and anti-being. It is easy to see that this process never stops.

37. Up till now, we have treated this process of ontological production as if it were a linear process, moving from 1 and -1 to 2 and -2 and then to 3 and -3, up to infinity, unfolding through time. Actually, however, this is incorrect. This semblance of linearity merely resulted from the linearity of the discursive reasoning to which we are confined, where one thought must be developed after the other. In reality, the ontological production of the infinity of beings and anti-beings is instantaneous. The moment when pure negativity splits itself into 1 and -1 is the same moment when pure negativity splits itself in 2 and -2, in 3 and -3 etc. After all, there 'is' only one pure negativity. 

38. This instantaneity of ontological production means that the infinity of beings is actual. Initially this infinity appeared to be merely potential, since we introduced it via the endless enumeration of pairs of being and anti-being (1 and -1, 2 and -2, 3 and -3, and so on...). As such the infinity appeared as the unreachable limit of an ever growing number – a number which as such must always be finite, since the enumeration can never reach the infinity towards which it progresses. Such a potential infinity is never completed. Yet here too the linearity of the enumeration is merely a semblance created by the linearity of discursive reasoning. Given the instantaneity of ontological production, we must rather think of the progression towards infinity as instantaneously completed. Thus the infinity of beings is actual.

39. Since the original infinity of beings and anti-beings is produced instantaneously, we cannot single out one pair of being and anti-being as the primordial pair, call them 1 and -1 and then say that all other beings and anti-beings emerge between 1 and -1 (emerge, that is, from the pure negativity between 1 and -1). There is no such primordial pair; all pairs are produced simultaneously. Although we can say that 2 and -2 emerge (from the pure negativity) between 1 and -1, we can with equal right hold the converse and say that 1 and -1 emerge between 2 and -2, or between 3 and -3, and so on. Thus all beings and anti-beings are in-between each other. They surround each other to such a degree that there is no outer limit where beings cease to be surrounded. But if there is no outer limit, then neither is there a center. Or rather we should say that the true center is everywhere, namely, the pure negativity between all beings and anti-beings. 

40. This thesis, the instantaneous production of infinitely many beings and anti-beings out of the nothingness of pure negativity, is empirically confirmed by the big bang. The implosion of nothingness is the explosion of being.

41. In fact our theory of the ontological productivity of pure negativity is the only available explanation of the big bang. Physicists point out that to explain the big bang, we have to explain how the universe emerged from nothing, since prior to the big bang there literally was nothing: no matter and even no space and time, since both developed along with the universe only after the big bang had taken place. Hence an explanation of the big bang in terms of the polarization of the vacuum will not do, since this already presupposes empty space. We have to explain how the universe emerged from nothing at all. The above account of the ontological productivity of pure negativity is the only such explanation around.

41. Some physicists hope to be able to explain the big bang by presupposing the laws of physics. These physicists assume the laws of physics are timeless. But this is smuggling. For where, then, do the laws of physics come from? Why these laws and not others? This stands in need of explanation. As long as this explanation is lacking, physicists cannot be said to have explained the big bang, even if they can derive the big bang from the laws of physics alone (which as yet they cannot). Here, too, the fruitfulness of the present approach is apparent, since we can derive at least the most basic laws of physics (the forces of attraction and repulsion, and the polarity of being and anti-being) from the dialectical structure of pure negativity.
 
42. Up till now we have considered the ontological production of the infinity of beings merely under the aspect of identity, their mutual differentiation through the self-differentiation of pure negativity. We must now also consider it under the aspect of existence. What guarantees the existence of this original infinity of beings? Remember that existence is difference from difference, that is, unification, such that the unified beings stand in a negative relation to the negativity inherent in their mutual differentiation (see propositions 7 and 8). Thus in the very instant of ontological production, the infinitely many beings must not only differ from each other; they must at the same time form a unity.

43. We have seen that identity is manifested spatially as the mutual repulsion of beings, whereas their existence (unification) is manifested spatially as attraction. The original infinity of beings, then, must find an equilibrium between the forces of repulsion and attraction. 

44. However, we now no longer have just one duality in our ontology, namely, the duality of repulsion and attraction. For in the meantime we have argued for a second duality, namely, that being beings and anti-beings. We now have to superimpose these two dualities.

45. It is tempting to identify the splitting of pure negativity into being and anti-being with repulsion as such, and hence to conclude that repulsion is first and foremost a relation between a being and its anti-being. But this is a mistake. The point is that a being has to differentiate itself not only from its anti-being but also from all other beings and anti-beings. In mathematical terms: 1 must differentiate itself not only from -1 but also from 2 and -2, 3 and -3, and so on (and the same holds for -1). Thus repulsion is active between all beings alike, independently of their being positive or negative.

46. This does not mean, however, that the forces of repulsion and attraction always operate in the same way between all beings. To be precise: between a being and its anti-being (1 and -1) these forces operate differently from how they operate between two 'positive' beings (1 and 1) and between two 'negative' beings (-1 and -1). The point is that the difference between 1 and -1 is intrinsic and hence not in need of repulsion (up to a point; see proposition 47). In contrast, 1 and 1 are intrinsically the same and hence need repulsion for their mutual differentiation (the same holds for -1 and -1). The consequence is that the force of attraction appears to be much stronger between 1 and -1 than it is between 1 and 1 or between -1 and -1. The reason is that the attraction between 1 and -1 is not counteracted by repulsion as it is in the case of 1 and 1 or -1 and -1. This is empirically confirmed by the attraction of opposite charges in nature.

47. Still, this does not mean that repulsion is entirely absent between 1 and -1. Their mutual attraction must not be unlimited. Otherwise 1 and -1 will annihilate each other in the moment of their unification. Existence, after all, is the rationale behind attraction (= unification). Yet this purpose is defeated if 1 and -1 are allowed to meet. Hence the attraction between 1 and -1 turns into its opposite (i.e. into repulsion) when they come 'too close' to each other, that is, when their mutual annihilation becomes an imminent possibility. 

48. Hence the only unity possible for 1 and -1 is the core-orbit form. Attraction draws them together until repulsion sets in, forcing one in orbit around the other. 

49. This explains the empirical fact that although the protons and electrons in an atom attract each other, they are not both allowed in the atom's core, since that would mean their annihilation and hence the annihilation of the atom itself. Outside the confines of the atom, the attraction between proton and electron is not counteracted by repulsion. Inside the atom, however, their repulsion is activated. 

50. The standard model of quantum mechanics really has no satisfying explanation of why the electron, attracted as it is to the proton, does not crash into the atom's core. It tries to explain this by pointing out that the electron is not really a discrete particle in orbit around the core at all, rather it forms a wave function or probabilistic cloud around the core. The model then simply postulates that this cloud can never intersect with the protons at the core: the nuclear proton and the electron cloud around it are said to repel each other. But this is postulation, not explanation. The question is precisely where this repelling force comes from. Simply describing the electron as a wave function or a probabilistic cloud does not cancel the attraction between proton and electron, hence it cannot explain this sudden repulsion when proton and electron threaten to collide. As we have seen, however, we can explain this fact on the basis of the insight that attraction equals existence (propositions 48 and 49). This obviously does not mean that the standard model is wrong and that the bound electron cannot be described as a probabilistic cloud around the core. It does mean, however, that this model is explanatory incomplete and needs to be supplemented by the present analysis of repulsion and attraction as the forces of identity and existence respectively.

51. The very first unities to emerge from the primordial chaos must have been atoms. This follows from two facts. Firstly, the even distribution of beings and anti-beings in the original infinity (proposition 39). Secondly, the fact that the force of attraction is greater between 1 and -1 than it is between 1 and 1 or between -1 and -1 (proposition 46). Together these two facts imply that unities of 1 and -1 must have formed faster and hence earlier than other possible unities. And as we have seen, the only possible unity of 1 and -1 is the core-orbit structure, such that orbit and core have opposite charge. Empirically speaking, such unities are the atoms. And once beings formed atoms, they could no longer form other kinds of unities, with the exception of combinations of atoms. Atoms, then, are the basic building 'blocks' of determinate existence. The atomic unity is the primary guarantor of the universe's existence, that is, its difference from difference.