Derrida Contra Austin – Irreducible Polysemy…

The position of Austin seems to relegate writing vis-à-vis speech, even if he maintains that certain aspects of speech are imperfectly captured by writing. Even Searle joins his mentor in admitting the implicit context of speech when compared with the explicit context of writing. Another thematic insistence between speech and writing in Austin is an utterance that is tied to origin. When an utterance is not in the present indicative active, then the utterer is not typically referred to by name or personal pronoun ‘I’ but by the fact that it is he who is speaking and thus the origin of the utterance, and when he happens to be absent and does not use his name or the personal pronoun ‘I’, he will often indicate in the written document that it is he who is the origin by signing it with his name. Derrida voices his criticism on this position for the speaker’s intended meaning isn’t any more unequivocal, if he is present, than, if he would have written. This point is cogently argued by Derrida because for him the presence of the speaker is analogous to the one who signs. He says in Signature, Event, Context,

The signature also marks and retains [the writer’s] having-been present in a past now or present which will remain a future now or present…general maintenance is in some way inscribed, pinpointed in the always evident and singular present punctuality of the form of the signature…in order for the tethering to the source to occur, what must be retained is an absolute singularity of a signature-event and a signature-form: the pure reproducibility of a pure event.

One conclusion that could be favorably drawn from Derrida’s reading of Austin according to the above quote is that for the latter, a permanence is given to the signature that identifies the signer and his presence with/within the text. This also implies at the same time the reproducibility of the mark of signature to deduce that it is recognized as¹ his signature , thus proving not only originality of signature but also its iterability. Derrida’s general criticism of Austin rest upon the latter’s failing to acknowledge the graphematic nature of locutions in addition to performative/constative and serious/parasitic distinctions not being able to fit in, when applied to locutions. This is deducible by arguments that run against the notion of proper contexts thereby hindering the discernment between speech acts that qualify as normal or parasitic and happy or unhappy. A careful reading of Austin’s How To Do Things With Words establishes a thematic rule of classifying and/or categorizing speech acts that are resistant to being unambiguously accounted for one way rather than other, or, in other words, the book’s primary aim is to root out the thesis that context is absolutely determinable, even if there is a recognition of serious and non-serious speech acts with the cautionary treatment of leaving out the non-serious acts during the examination of the serious ones. Derrida, on the contrary gives a lot of seriousness to the “non-serious/non-literal”’ linguistic use, as for him, they are determinate of meaning. This stand of Derrida goes opposite to Austin’s, for who, speech acts, even if they harbor felicities and infelicities, could only be investigated about within ordinary circumstances. In an amazing reading² of Austin, Derrida claims non-serious citations of utterances qua citations, are nothing but instances of the iteration of the utterances that help determine its identity. Moreover, Derrida claims graphematic root of citationality as responsible for why Austin is unable to provide an exhaustive list of criteria to distinguish performatives with constatives, and also because Austin fails to take account of the structure of locution as already entailing predicates that blur the oppositions which are in turn unsuccessfully attempted to be established. Also, failure to recognize the necessity of impure performatives on Austin’s part made Derrida’s criticism more cogent, as for the latter, “impurities” are not just confined to performatives having a constative dimension, or constatives having a performative dimension, but, even normal and parasitic acts weren’t immune anymore to “impurities”. This criticism gains authority, since for Derrida, impurities are necessary and not any accidental facts, and in the absence of proper contexts, “hosts” maybe parasitic on “parasites” implying further that “normal” utterances are relatively normal and “parasitic” utterances are relatively parasitic, since the criterion invoked to differentiate them is the difference in contexts that is somehow missing or blurred in Austin. So, if the constative/performative distinction is an impure distinction in itself for Austin, then he is not successful in legitimizing the normal/parasitic distinction. Derrida claims that Austin’s work shows that the possibility of failure, or infelicity, is a permanent structural and/or necessary possibility of performative utterances, but Austin excludes the risk of such failures as accidental. In other words, Austin shows that performatives are characterized by an essential risk of failure and yet treats that risk as if it were accidental, which Derrida characterizes as a necessary impurity of performatives and constatives. Furthermore, Austin’s investigations of infelicities and total speech situations point to the fact that speakers and hearers can exercise control over speech situations in order to avoid infelicity and secure uptake, which meets its counter- argument in Derridean dissemination or irreducible polysemy by the establishing of locutions as graphematic, thus losing out on any such possibility of securing control on the speech act by either the speaker or the hearer.

In a nutshell, it is safe to say that Austin’s total speech act revolved around a dual notion of a possible elucidation within the total speech situations that left room for a generalized accountability for a formulation to comprehend parasitic deviations from the norm and speech acts construed as an exercise in exposing the lack of distinctions like parasitic/normal involved therein. Therefore, even if in his speech act theory, it is impossible for an utterance to take hold of normal and parasitic tones, it does not rule out the contingency of such distinctions from coming into being. The impossibility of distinctions for utterances in Austin’s case is what moves Searle away from his mentor, as for the latter, utterances could be tagged normal or parasitic due to his literal/utterance-meaning and representation/communication distinctions (his notion of intentionality achieves prominence here with the speaker-writer determining if her utterance is normal or parasitic). For Searle, sentences are loaded with literal ambiguities, since the possibilities of speaking literally or non-literally exist in some sort of a double bind, and this take of his has some parallels in Derrida’s citationality, iterability and dissemination. There is a difference though, in that, Derrida gives credence to the irreducible polysemy and parasitism and unhappiness as permanent and structural, that should in no way be counted as indeterminate or free play, but rather as mired in ambiguities, whereas Searle never thinks of all utterances as polysemic…

1 It should be noted that the ideal signature is one which can only be repeated by one individual, and for Derrida, it is the impossible ideal of something original that remains so even when it undergoes repetition. Effects of signature are the most common thing in the world, with the conditions of its possibility as simultaneously the conditions of its impossibility of a rigorous purity. Functionality is driven, when the signature enjoys repeatable, iterable and imitable forms, which is made possible, when a signature gets detached from its singular, intended production, or in other words, it is sameness which, by corrupting its identity and its singularity, divides its seal. 

2 This reading is evident in the quote (Derrida),

“..ultimately, isn’t it true that what Austin excludes as anomaly, exception, ‘non-serious’ citation (on stage, in a poem, or a soliloquy) is the determined modification of a general citationality- or rather, a general iterability – without which there would not even be a ‘successful’ performative? So that- a paradoxical but unavoidable conclusion – a successful performative is ‘necessarily’ an impure performative, to adopt the word advanced later on by Austin when he acknowledges that there is no pure performative.”

Ruminations on Philosophy of Science: A Case of Volume Measure Respecting Orientation

Let M be an n–dimensional manifold (n ≥ 1). An s-form on M (s ≥ 1) is a covariant field α_b1…bs that is anti-symmetric (i.e., anti-symmetric in each pair of indices). The case where s = n is of special interest.

Let α_b1…bn be an n-form on M. Further, let ξ^ib (i = 1,…,n) be a basis for the tangent space at a point in M with dual basis ηⁱ_b (i=1,…,n). Then α_b1…bn can be expressed there in the form

α_b1…bn = k n! η¹[b₁…ηⁿb_n] —– (1)

where

k = α_b1…bnξ^1b₁…ξ^nb_n

(To see this, observe that the two sides of equation (1) have the same action on any collection of n vectors from the set {ξ^1b, . . . , ξ^nb}.) It follows that if α_b1…bn and β_b1…bn are any two smooth, non-vanishing n-forms on M, then

β_b1…bn = f α_b1…bn

for some smooth non-vanishing scalar field f. Smooth, non-vanishing n-forms always exist locally on M. (Suppose (U, φ) is a chart with coordinate vector fields (γ⃗₁)a, . . . , (γ⃗_n)a, and suppose ηⁱ_b(i = 1, . . . , n) are dual fields. Then η¹[b₁…ηⁿb_n] qualifies as a smooth, non-vanishing n-form on U.) But they do not necessarily exist globally. Suppose, for example, that M is the two-dimensional Möbius strip, and α_ab is any smooth two-form on M. We see that α_ab must vanish somewhere as follows.

A 2-form α_ab on the Möbius strip determines a “positive direction of rotation” at every point where it is non-zero. So there cannot be a smooth, non-vanishing 2-form on the Möbius strip.

Let p be any point on M at which α_ab ≠ 0, and let ξ^a be any non-zero vector at p. Consider the number α_ab ξ^a ρ^b as ρ^b rotates though the vectors in M_p. If ρ^b = ±ξ^b, the number is zero. If ρ^b ≠ ±ξ^b, the number is non-zero. Therefore, as ρ^b rotates between ξ^a and −ξ^a, it is always positive or always negative. Thus α_ab determines a “positive direction of rotation” away from ξ^a on M_p. α^ab must vanish somewhere because one cannot continuously choose positive rotation directions over the entire Möbius strip.

M is said to be orientable if it admits a (globally defined) smooth, non- vanishing n-form. So far we have made no mention of metric structure. Suppose now that our manifold M is endowed with a metric g_ab of signature (n⁺, n⁻). We take a volume element on M (with respect to g_ab) to be a smooth n-form ε_b1…bn that satisfies the normalization condition

ε^b₁…b_n ε_b1…bn = (−1)ⁿ⁻n! —– (2)

Suppose ε_b1…bn is a volume element on M, and ξ^{i b} (i = 1,…,n) is an orthonormal basis for the tangent space at a point in M. Then at that point we have, by equation (1),

ε_b1…bn = k n! ξ¹_[b1 …ξ_bn] —– (3)

where

k = ε_b1…bn ξ^1b₁…ξ^nb_n

Hence, by the normalization condition (2),

(−1)ⁿ⁻n! = (k n! ξ¹_[b1 …ξ_bn]) (k n! ξ^1[b₁ …ξ_^bn])

= k² n!² 1/n! (ξ¹_b1 ξ^1b₁) … (ξⁿ_bn ξ^nb_n) = k² (−1)ⁿ⁻

So k² = 1 and, therefore, equation (3) yields

ε_b1…bn ξ^1b₁…ξ^nb_n = ±1 —– (4)

Clearly, if ε_b1…bn is a volume element on M, then so is −ε_b1…bn. It follows from the normalization condition (4) that there cannot be any others. Thus, there are only two possibilities. Either (M, g_ab) admits no volume elements (at all) or it admits exactly two, and these agree up to sign.

Condition (4) also suggests where the term “volume element” comes from. Given arbitrary vectors γ^1a , . . . , γ^na at a point, we can think of ε_b1…bn γ^1b₁ … γ^nb_n as the volume of the (possibly degenerate) parallelepiped determined by the vectors. Notice that, up to sign, ε_b1…bn is characterized by three properties.

(VE1) It is linear in each index.

(VE2) It is anti-symmetric.

(VE3) It assigns a volume V with |V | = 1 to each orthonormal parallelepiped.

These are conditions we would demand of any would-be volume measure (with respect to g_ab). If the length of one edge of a parallelepiped is multiplied by a factor k, then its volume should increase by that factor. And if a parallelepiped is sliced into two parts, with the slice parallel to one face, then its volume should be equal to the sum of the volumes of the parts. This leads to (VE1). Furthermore, if any two edges of the parallelepiped are coalligned (i.e., if it is a degenerate parallelepiped), then its volume should be zero. This leads to (VE2). (If for all vectors ξ^a, ε_b1…bn ξ^b₁ ξ^b₂ = 0, then it must be the case that ε_{b1 …bn} is anti-symmetric in indices (b₁, b₂). And similarly for all other pairs of indices.) Finally, if the edges of a parallelepiped are orthogonal, then its volume should be equal to the product of the lengths of the edges. This leads to (VE3). The only unusual thing about ε_b1…bn as a volume measure is that it respects orientation. If it assigns V to the ordered sequence γ^1a , . . . , γ^na, then it assigns (−V) to γ^2a, γ^1a, γ^3a,…,γ^na, and so forth.

Metric. Part 1.

A (semi-Riemannian) metric on a manifold M is a smooth field g_ab on M that is symmetric and invertible; i.e., there exists an (inverse) field g^bc on M such that g_abg^bc = δ_a^c.

The inverse field g^bc of a metric g_ab is symmetric and unique. It is symmetric since

g^cb = g^nb δⁿ_c = g^nb(g_nm g^mc) = (g_mn g^nb)g^mc = δ_m^b g^mc = g^bc

(Here we use the symmetry of g^nm for the third equality.) It is unique because if g^′bc is also an inverse field, then

g^′bc = g^′nc δ_n^b = g^′nc(g_nm g^mb) = (g_mn g^′nc) g^mb = δ_m^c g^mb = g^cb = g^bc

(Here again we use the symmetry of g_nm for the third equality; and we use the symmetry of g^cb for the final equality.) The inverse field g^bc of a metric g_ab is smooth. This follows, essentially, because given any invertible square matrix A (over R), the components of the inverse matrix A⁻¹ depend smoothly on the components of A.

The requirement that a metric be invertible can be given a second formulation. Indeed, given any field g_ab on the manifold M (not necessarily symmetric and not necessarily smooth), the following conditions are equivalent.

(1) There is a tensor field g^bc on M such that g_abg^bc = δ_a^c.

(2) ∀ p in M, and all vectors ξ^a at p, if g_abξ^a = 0, then ξ^a =0.

(When the conditions obtain, we say that g_ab is non-degenerate.) To see this, assume first that (1) holds. Then given any vector ξ^a at any point p, if g_ab ξ^a = 0, it follows that ξ^c = δ_a^c ξ^a = g^bc g_ab ξ^a = 0. Conversely, suppose that (2) holds. Then at any point p, the map from (M_p)^a to (M_p)^b defined by ξ^a → g_ab ξ^a is an injective linear map. Since (M_p)^a and (M_p)^b have the same dimension, it must be surjective as well. So the map must have an inverse g^bc defined by g^bc(g_ab ξ^a) = ξ^c or g^bc g_ab = δ_a^c.

In the presence of a metric g_ab, it is customary to adopt a notation convention for “lowering and raising indices.” Consider first the case of vectors. Given a contravariant vector ξ^a at some point, we write g_ab ξ^a as ξ^b; and given a covariant vector η_b, we write g^bc η_b as η^c. The notation is evidently consistent in the sense that first lowering and then raising the index of a vector (or vice versa) leaves the vector intact.

One would like to extend this notational convention to tensors with more complex index structure. But now one confronts a problem. Given a tensor α_c^ab at a point, for example, how should we write g^mc α_c^ab? As α^mab? Or as α^amb? Or as α^abm? In general, these three tensors will not be equal. To get around the problem, we introduce a new convention. In any context where we may want to lower or raise indices, we shall write indices, whether contravariant or covariant, in a particular sequence. So, for example, we shall write α^ab_c or α^a_c^b or α_c^ab. (These tensors may be equal – they belong to the same vector space – but they need not be.) Clearly this convention solves our problem. We write g^mc α^abc as α^abm; g^mc α^a_c^b as α^amb; and so forth. No ambiguity arises. (And it is still the case that if we first lower an index on a tensor and then raise it (or vice versa), the result is to leave the tensor intact.)

We claimed in the preceding paragraph that the tensors α^ab_c and α^a_c^b (at some point) need not be equal. Here is an example. Suppose ξ1^a, ξ2^a, … , ξn^a is a basis for the tangent space at a point p. Further suppose α^abc = ξi^a ξj^b ξk^c at the point. Then α^acb = ξi^a ξj^c ξk^b. Hence, lowering indices, we have α^ab_c =ξi^a ξj^b ξk^c but α^a_c^b =ξi^a ξj_c ξi^b at p. These two will not be equal unless j = k.

We have reserved special notation for two tensor fields: the index substiution field δ_b^a and the Riemann curvature field R^a_bcd (associated with some derivative operator). Our convention will be to write these as δ^a_b and R^a_bcd – i.e., with contravariant indices before covariant ones. As it turns out, the order does not matter in the case of the first since δ^a_b = δ_b^a. (It does matter with the second.) To verify the equality, it suffices to observe that the two fields have the same action on an arbitrary field α^b:

δ_b^aα^b = (g_bng^amδⁿ_m)α^b = g_bng^anα^b = g_bng^naα^b = δ^a_bα^b

Now suppose g_ab is a metric on the n-dimensional manifold M and p is a point in M. Then there exists an m, with 0 ≤ m ≤ n, and a basis ξ1^a, ξ2^a,…, ξn^a for the tangent space at p such that

g_abξi^a ξi^b = +1 if 1≤i≤m

g_abξi^aξi^b = −1 if m<i≤n

g_abξi^aξj^b = 0 if i ≠ j

Such a basis is called orthonormal. Orthonormal bases at p are not unique, but all have the same associated number m. We call the pair (m, n − m) the signature of g_ab at p. (The existence of orthonormal bases and the invariance of the associated number m are basic facts of linear algebraic life.) A simple continuity argument shows that any connected manifold must have the same signature at each point. We shall henceforth restrict attention to connected manifolds and refer simply to the “signature of g_ab”

A metric with signature (n, 0) is said to be positive definite. With signature (0, n), it is said to be negative definite. With any other signature it is said to be indefinite. A Lorentzian metric is a metric with signature (1, n − 1). The mathematics of relativity theory is, to some degree, just a chapter in the theory of four-dimensional manifolds with Lorentzian metrics.

Suppose g_ab has signature (m, n − m), and ξ1^a, ξ2^a, . . . , ξn^a is an orthonormal basis at a point. Further, suppose μ^a and ν^a are vectors there. If

μ^a = ∑ⁿ_i=1 μⁱ ξi^a and ν^a = ∑ⁿ_i=1 νⁱ ξi^a, then it follows from the linearity of g_ab that

g_abμ^a ν^b = μ1ν1 +…+ μmνm − μ(m+1)ν(m+1) −…−μnνn.

In the special case where the metric is positive definite, this comes to

g_abμ^aν^b = μ1ν1 +…+ μnνn

And where it is Lorentzian,

g_ab μ^aν^b = μ1ν1 − μ2ν2 −…− μnνn

Metrics and derivative operators are not just independent objects, but, in a quite natural sense, a metric determines a unique derivative operator.

Suppose g_ab and ∇ are both defined on the manifold M. Further suppose

γ : I → M is a smooth curve on M with tangent field ξ^a and λ^a is a smooth field on γ. Both ∇ and g_ab determine a criterion of “constancy” for λ^a. λ^a is constant with respect to ∇ if ξⁿ∇_nλ^a = 0 and is constant with respect to g_ab if g_ab λ^a λ^b is constant along γ – i.e., if ξⁿ ∇_n (g_ab λ^a λ^b = 0. It seems natural to consider pairs g_ab and ∇ for which the first condition of constancy implies the second. Let us say that ∇ is compatible with g_ab if, for all γ and λ^a as above, λ^a is constant w.r.t. g_ab whenever it is constant with respect to ∇.