The mathematical understanding of symmetry is conspicuously at odds with human intuition, which is prescribed by our linguistic understanding of similarity and difference; the relation between Same and Other. Symmetry has a specific meaning in mathematics that is therefore somewhat counter-intuitive. The symmetry of an object is confirmed by that fact that any rotation, reflection or translation leaves it looking exactly the same. We are perhaps most familiar with bilateral symmetry or mirror reflection, which we commonly encounter in faces and the physical symmetry of bio-organisms. However, a less familiar type of symmetry is bland uniformity; the white flat-painted wall continuing to infinity in all directions. Such a wall would have a huge amount of symmetry in that it appears the same whatever transformation you apply[i]. No amount of successive transformations will reveal any distinguishing features and we’d be unable to discern any difference in the flat featureless plane. This kind of extensive symmetry is so ubiquitous it is hidden from us, we are habituated to it and accordingly, it exceeds our attentional focus. Not only is it spatially indistinguishable, it also has a continuity of temporal similarity and as such we can fail to notice such ubiquitous sameness until the symmetry of the plane is broken[ii].
This interruption of symmetry is much like the lightning flash that Heidegger speaks of in The Heraclitus Seminar (1967); the dissymmetry or lightning flash that breaks the abyssal, undifferentiated symmetry of the night sky, a rupture in the continuum of sameness.[iii] Only when the lightning cracks the bland uniformity of the night does it reveal the continuous symmetry of the darkness; the illumination of lightning is the bringing-forth-to-appearance, or phenomenal presencing of the undifferentiated symmetrical ground. It is a wisdom-like ‘eureka-flash’ evoked by dissymmetry that ‘tears open the dark night, and in its gleam, it lights up and lets all individual things be seen’ [HS, p.8]. Accordingly, the forked lightning flash is identified as a temporal dynamic: ‘…at the same time it is also a mobile power [and] this movement passes into the movement of things’ [HS, p.8]. Here Heidegger explains the identity of lightning, which we could more accurately say is the identity of identities that brings-together disparate identities. It is ‘the unifying One of lightning’ or that which Heraclitus calls Zeus! The mobile power here described is akin to the dialectical movement that illuminates the identity of all things. The lightning flash for Heidegger is specifically technology, the singular world-forming event that ruptures homogenous symmetry and illuminates the realm of differences. Or as Heraclitus would have it: ‘lightning steers the universe’ and brings unity to opposites.
Since the dissymmetry of the lightning is nominated as the identity of identities – a heterogeneous difference that illuminates homogeneity of the same – this leads us to ask the necessary question: What is Identity in philosophy? In logic the identity relation is normally defined as the relation that holds only between a thing and itself. That is, identity is the two-place predicate, ‘=’, such that for all x and y, ‘x = y’ is true if x is the same thing as y. Accordingly, identity is a temporally transitive, symmetric, and reflexive relation. Therefore it is an axiom of most normal modal logics that for all x, if x = y then necessarily y = x. In such cases, ‘=’ stands for identity itself, an empty signifier for identity and the equality of its self-similarity. However, the question of identity can be further deepened in the distinction between qualitative and quantative identity, which requires a slightly stricter definition.
If we encounter any two arbitrary objects which are duplicates they can be said to be qualitatively identical, that is to say that the twin objects a and b share identical common qualities. For instance, if two green Volkswagen Beatles were built on the same assembly line in the same year and were both exactly the same production model we’d say that they had qualitative identity according to a relaxed standard of similarity. If we were to set a stricter standard of exact similarity we might apply it to two carbon atoms in different locations, which would share a much higher degree of qualitative similarity. Alternatively, two objects could be said to be quantatively or numerically identical if objects a and b are one and the same thing (that is if a = b, yet there is only one thing that is called ‘a’ and ‘b’ on differing occasions). For instance, a classic example of quantative identity would be to say that Dr. Jekyll is numerically identical to Mr. Hyde, in that they are one and the same person, who are both identified as different people on differing occasions. By simultaneously being the same person (i.e. Dr. Jekyll = Mr. Hyde) they are counted as numerically one self-similar entity in a way that two qualitatively identical carbon atoms in two separate locations are not.
However, in contrast to this, positivist modal logic follows Leibniz’s Law of the excluded middle, by which: every judgment is either true or false, not both or neither (either x or not x). Bertrand Russell explains this further as having developed via the preceding axioms:
- Law of identity: ‘Whatever is, is.’
- Law of non-contradiction: ‘Nothing can both be and not be.’
By which Leibniz arrives at the law of the excluded middle: ‘Everything must either be or not be.’ This takes the axiomatic form (x)[f(x) ⊕ ~f(x)]. So when applied to our example, if they were indeed two separate people as Dr. Jekyll and Mr. Hyde then they would not equal each other as per the Law of excluded middle. It is then either Jeykll or Hyde, not both or neither, therefore decidedly one entity or the other. Either the proposition is true or its negation is. So here our definition of identity becomes slightly more precarious and our problem deepens. Will the real Slim Shady please stand up?
In contrast to this, in the canon of Western philosophical thought, Georg W. F. Hegel is perhaps the most preeminent thinker of identity as defined by its negative. In the often inconsistent terminology of Hegelese, Identity is synonymous with the Same. So within Hegelian thought, what does it mean for something to be the same as itself, in other words, to have self-similarity? Here again we are confronted with the temporal dilemna of transitive identity, which like the rotting apple changes over time (Is applet the same as applet+1?). As the Hegelian philosopher Alexandre Kojève explains, describing the first chapter of Hegel’s Phenomenology:
Look at your watch […] and note that it is, let us say, noon. Say it and you will have enunciated a truth. Now write this on a piece of paper: “It is now noon”. At this point Hegel remarks that a truth cannot cease to be because of being formulated in writing. And now look at your watch again and reread the sentence you have written. You will see that the truth has been transformed into error, for it is now five minutes past noon.
What can be said, except that real being can transform a human truth into error – at least so far as the real is temporal and Time has an identity. [IRH, p. 186]
It is therefore important for the Hegelian system to track transitive identity through a series of movements, in order to index the identity of identities that Heidegger attributes to the lightning flash. This is the question first asked by Heraclitus, who notes that: we never step in the same river twice. This observation highlights a paradox present in the question of identity relations: is an object the same object if all of its components are replaced? Is Spock the same Spock if all of his molecules are beamed across space and reassembled?[iv]
Leibniz maintained that x can only equal y if every predicate true of x is true of y as well, in keeping with logical positivism. However, Hegel maintained that identity is possible through revealing the inherent self-contradiction of the thing (i.e. the negative ‘mark its own identity’ or what it is not). Hegel inherits this from Heraclitus for whom all identities are connected by temporal change, denoted by a tension of opposites and contradictions. “There is no sentence of Heraclitus’ that I have not taken into my Logik,” Hegel confessed. Consequently within the Hegelian system, like the Heraclitan flux of the river, every identity changes and nothing remains still. And it is from this that Hegel comes to develop the negative dialectic, in order to track transitive identity and although he never explicitly explains his methodology, it is nonetheless consistently present throughout his logic. The negative dialectic is based on the Aristotlean form of a deductive argument known as a syllogism. The syllogism follows a classical form of reasoning that assumes the principle of logical identity: a = a or a is not non-a.
In contrast, the dialectic assumes a tripartite structure that incorporates ‘the identity of identity and non-identity’, finding a higher unity of synthesis of the two. Here we detect the famous triadic inner structure internal to dialectical thought (thesis-antithesis-synthesis) that preserves the two prior identities through its inner dynamic. This mobile structure in which truths can only be relative opposes Aristotle’s notion of static identities that are unambiguous facts. By the term negation or contradiction we mean the incorporation into identity of its antithesis. By contrasting the essence of a thing with that which it is not – a ‘fight’ between opposing irreductive essential entities – Hegel imputes a wide variety of relations such as difference, opposition, reflection or relation. The antithesis is not simply the polar opposite of its thesis, it can indicate the mere insufficiency of a category or its incoherence (that which Lacan would refer to as its lack). Most dramatically, categories are sometimes shown to be self-contradictory, which forms the basis of Hegel’s critique of the arbitrary nature of the Kantian categories by which all objects can be judged. However, Frederic Jameson, in The Hegel Variations cautions against the ‘vulgarization’ of the Hegelian method, since to characterize Hegel as only deploying the tripartite formula is to ‘fall into the temptation’ of reifying Hegelian thought. The error of construing it as merely systemic process is to lapse into reified thinking (Verstand), the sense-certainty that Hegelian Reason (Vernuft) seeks to overcome.[v]
Through the movement of the dialectic through thesis, antithesis to higher synthesis, we come to see Hegel’s logic is a fractal architecture of dialectical movements that overcome opposing identities – therefore constituting a form of differential calculus – contracting on the identity of the Whole. In this manner Hegel assumes that it is possible for self-sufficient thought to become commensurate with reality, perhaps the most grandiose of absolute idealism’s claims. Nothing is lost or destroyed by the dialectic, but each stage raised-up and preserved as in a logarithmic spiral. We could think of it as the opening of a fern[vi] (the circinate vernation of the frond) or the helical chambers of a nautilus shell, an auto-affectation by which thought unfolds out of itself to become adequate to reality[vii]. This is the teleological process by which Hegel, the Wise Man, can think the totality of the Whole. It is therefore an organic rather than mechanical logic. Hegel’s special term for the overcoming of contradictions, whilst at the same time preserving, is Aufhebung; sometimes translated as ‘sublation’. As Kojève states, with regard to the passive contemplation of the Wise Man, which constitutes the dialectical and ‘scientific’ process of the philosopher:
His role is that of perfectly flat and indefinitely extended mirror: he does not reflect on the Real; it is the Real that reflects on him, is reflected in his own consciousness, and is revealed in his own dialectical structure by the discourse of the Wise Man who describes it without deforming it. [IRH, p.192]
However, it is exactly this movement from abstract thought to concrete reality that Marx is suspicious of and inverts with his dialectical materialism (which moves instead from the concrete to the abstract). As Deleuze deftly quips, bearded Marx is Hegel shorn of his idealism.
Deleuze, in contrast to Hegel is the thinker of multiplicity or difference, an aristocratic thinker of ‘crowned anarchy’. Deleuze is anti-Hegelian exactly because Hegel doesn’t give difference its own concept and he treats as essential that which individuals have in common. For Deleuze, like Nietzsche, it is only the extreme that returns from selection – pure or maximal difference/the becoming of thought itself – which cannot be contained by the limits of Identity or the Same. The eternal return does not bring back the Same, but the identity of difference itself. Thought is therefore a selective test, an affirmation of the extremis.[viii] Deleuze also uses a form of differential calculus (dialectical), which he then supplements with a radicalized version of the eternal return (‘The wheel of the eternal return is at once both the production of repetition on the basis is difference and the selection of difference on the basis of repetition’) [D&R, p. 42]. This constitutes an affirmation of the extreme that undermines identity (Hegel) and the conservative nature of representational judgement (Kant). Inequality is therefore a motor of distribution, perhaps best exemplified by productive dissymmetry, the process of development found in the morphology of embryos. Yet, whilst Deleuze provides an account of how the biological subject comes to be differentiated in thought and how intensive difference pours out into the world[ix], he leaves little room for a material account of the individuation of objects in general. This has the effect of undermining a scientific account of the primacy of matter, in favour of the virtual.
In contrast, Badiou is also a thinker of the multiple, and like Deleuze opposed to the One, yet his critique of Hegel allows him to put a certain amount of philosophical distance between Deleuze and himself – although not quite as much distance as he has occasion to sometimes claim. Badiou remarks when examining Identity in Hegel:
‘Something’ – a pure presented term – is determinate for Hegel only in so far that it can be thought as other than other: ‘The exteriority of being other is the proper interiority of something’. [B&E, p.162]
The essence of the thing can therefore only be determined by its polar alterity of the opposite (i.e. the Other). For example: Being’s negative contradiction is Non-Being, on which its necessary being is predicated. Here we encounter the problem of infinity, since being’s identity must somehow transcend the limit (Grenze) of its own finitude in order to be realized as One or Whole. This has the effect of causing a torsion or Möbius strip in logic. ‘Infinity becomes an internal reason of the finite itself’ (B&E, p163). Therefore finite being (or identity) has to be simultaneously infinite under Hegel’s understanding. Consequently, Hegel is unable resolve the paradox of qualitative and quantitative identity, without creating a torsion in thought that represents an impasse to thinking the One and Many. Badiou’s is an argument against thinking the totality, in which the fact that everything belongs to the Whole constitutes an obstacle to the Whole.
This rejoining of the disjunction is what Badiou calls a ‘fragile verbal footbridge’[x] or suture at the heart of Hegel’s account of identity; a point of precariousness that unifies his system, in which he seeks to fuse the distinction between qualitative and quantatitve infinity. And this for Badiou is an attempt to span the appearance of the void (Ø or empty set) in the Hegelian Whole, the gap that Hegel bridges by assimilating qualitative infinity and quantitative infinity. This gap is the abyssal rabbit-hole that Hegel refers to as ‘bad infinity’[xi]. As Badiou states with reference to Hegel:
There is no symmetry between the same and the other, between proliferation and identification. However heroic the effort, it is interrupted de facto by the exteriority itself of the pure multiple. Mathematics occurs here as discontinuity within the dialectic. It is this lesson that Hegel wishes to mask by suturing under the same term – infinity – two disjoint discursive orders. (B&E, p. 169).
Badiou is principally a thinker of the actual (i.e. dilating quantitative infinity or co-extensive multiplicity) and his identification of this asymmetry between the proliferation of quantative infinity and the contraction of qualitative identity points to a discontinuity that he claims only axiomatics can index[xii]. However, this diagnosis has the curious effect of making Deleuze’s account of individuation in Difference and Repetition appear somewhat Hegelian, given that his account also deals with the contraction of individuating differences, which in turn leads him to privilege the intensive virtual (thought) at the expense of undermining the extensive actual (matter). It should perhaps be emphasized that Deleuze subscribes to difference being a positive and productive desiring power that is not defined by lack and Hegel is a thinker of the negative. However, we find a similarity in that they over-emphasize quality over quantity, for which Badiou attempts to hold them to account.
Badiou, A. (2007) Being and Event, trans. O. Feltham (Continuum).
Brassier, R. (2007) Nihil Unbound: Enlightenment and Extinction (Palgrave Macmillan).
Deleuze, G. (1994) Difference and Repetition, trans. P. Patterson (New York: Columbia University Press).
Heiddegger, M. & E. Fink. (1967) The Heraclitus Seminar, trans. Charles H. Seibert (Northwestern University Press)
Hegel, G.W.F. (1969) Hegel’s Science of Logic, trans. A. V. Miller (London: Allen and Unwin).
Jameson, F (2010) The Hegel Variations, (Verso)
Kojève A. (1980) Introduction to the Reading of Hegel: Lectures on the Phenomenology of Spirit (Ithaca: Cornell University Press)
Metzinger, T. (2004) Being No One: The Self-Model Theory of Subjectivity (MIT Press).
Mullarkey, J. (2007) Post-Continental Philosophy: An Outline (Contiuum).
[i] A 360° sphere is perhaps a better example of a bland uniform symmetrical surface that continues monotonously in all directions.
[ii] Thomas Metzinger claims that sameness (transpemporal identity) is a relation in the subject. As human beings we can intuit differences, similarities and nuances, but the limitations of perceptual memory make reidentification of a similar state, based on concepts impossible: “the phenomenal experience of sameness may functionally be based on introspective indistinguishability, and not on a reliable form of identifying reference” [BNO, p.78] Therefore, identity judgements with regard to similarity and difference, as they appear to us phenomenally, cannot constitute knowledge.
[iii] Heidegger’s ‘lightning flash’ is perhaps best visualized as a Litchenburg Figure – also referred to as an electron or lightning tree – a fractal electrical discharge pattern that extends down to molecular level. These self-similar fractal patterns are found in fulgarites, the natural hollow glass tubes formed in silica, or soil by lightning strikes that when excavated give the appearance of petrified lightning.
[iv] This common analogy, derived from the Star Trek novel Spock Must Die, is similar to Donald Davidson’s ‘Swampman’ hypothesis, a thought experiment often deployed in debates around mind-matter identity, subjectivity and consciousness.
[v] “For even if the tripartite rhythm happens to do justice to this or that Hegelian insight, it still reifies that insight in advance and translates its language into purely systemic terms…[The] tripartite formula is calculated to mislead and confuse the reader who seeks to process this material in a series of three steps: something utterly impossible to complete in the structurally far more complex play of oppositions in the chapter on secular absolutism; and alarmingly rebuked by Hegel himself in the famous chapter at the end of the greater Logic in which Hegel allows that ‘three’ might be ‘four’ after all.”[HV, p. 19]
[vi] The fern leaf analogy is the exact aborescent-schema of thought that Deleuze and Guatarri object to, instead favouring of root-like rhizoid structure, the elongated cells that anchor the plant.
[vii] The imposing strength of the Hegelian system is precisely because it consolidates and preserves that which it has overcome through the articulation of its own movements. To use an example from art, Tatlin’s unbuilt Monument to the Third International is a dramatic manifestation of the dynamic teleological structure of the dialectic.
[viii] Deleuze uses Bergson’s example of a converging lens that directs differing colours (concepts) which then participate in a single point of pure white light: “White Light is still a universal, but concrete universal which enables us to understand the particular, because it itself is at the extreme of the particular. Just as things have become nuances or degrees of the concept itself, the concept itself has become a thing. It is a universal thing we could say, since the objects sketched therein are so many degrees, but a concrete thing, not a kind or generality. Strictly speaking there are no longer several objects with the same concept, as the concept is identical to the thing itself, it is the difference between objects related to it, not their resemblance. Such is their internal difference: the concept becomes concept of difference.” [D&R, xix].
[ix] This is precisely where Deleuze, according to Ray Brassier, oversteps what is scientifically feasible, by proposing in a counterfactual manner that thought changes matter (i.e. reality as a correlate of thought).
[x] This rejoining of qualitative and quantitative infinity is explained by Edward Willatts here: http://edwardwillatt.yolasite.com/blog/-fragile-verbal-footbridges-in-badiou-s-history-of-philosophy
[xi] Hegel draws a sharp distinction between echt and schlect Unendlichkeit, ‘true’ and ‘bad’ infinity. True infinity, or the Whole, like the infinitude of an eternal God, is unending. “The infinite as thus posited over and against the finite, in a relation wherein they are as qualitatively distinct others, is to be called the bad infinite, the infinite of the understanding, for which it has the value of the highest, the absolute truth.” [SL, 139] Hegel favours a completable infinity, a circle that begins and ends with Being. In contrast to the completeable continuum of the circle, the twisted Möbius strip is mathematically distinct, in that it has no end, the inside and outside surfaces are continuous, yet the torsion simultaneously constitutes a discontinuity.
[xii] According to Brassier, Badiou encounters a problem at the limit of mathematics, a shortcoming of axiom set theory itself. Badiou’s peculiarly structured philosophy seems to straddle the ontic and ontological divide – in that it tends to the philosophical whilst being conditioned by the ‘extra-logical truths’ of mathematics – spanning both regimes. This creates a dichotomy that contradicts the possibility of scientific realism. Consequently, his philosophy is unverifiable within its own conceptual apparatus, due to the unpresentable nature of its anti-phenomenological structure (i.e. inconsistency); foundering on the very dualism he sets up between his formal mathematical schema and its material content. This unverifiablity is eminently evident in Badiou’s designation that proper name of ‘the void’ or Ø is merely the representation of the uncountable of the count [BE, p.90].