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.
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.
