Morphism of Complexes Induces Corresponding Morphisms on Cohomology Objects – Thought of the Day 146.0

Let A = Mod(R) be an abelian category. A complex in A is a sequence of objects and morphisms in A

… → Mi-1 →di-1 Mi →di → Mi+1 → …

such that di ◦ di-1 = 0 ∀ i. We denote such a complex by M.

A morphism of complexes f : M → N is a sequence of morphisms fi : Mi → Ni in A, making the following diagram commute, where diM, diN denote the respective differentials:

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We let C(A) denote the category whose objects are complexes in A and whose morphisms are morphisms of complexes.

Given a complex M of objects of A, the ith cohomology object is the quotient

Hi(M) = ker(di)/im(di−1)

This operation of taking cohomology at the ith place defines a functor

Hi(−) : C(A) → A,

since a morphism of complexes induces corresponding morphisms on cohomology objects.

Put another way, an object of C(A) is a Z-graded object

M = ⊕i Mi

of A, equipped with a differential, in other words an endomorphism d: M → M satisfying d2 = 0. The occurrence of differential graded objects in physics is well-known. In mathematics they are also extremely common. In topology one associates to a space X a complex of free abelian groups whose cohomology objects are the cohomology groups of X. In algebra it is often convenient to replace a module over a ring by resolutions of various kinds.

A topological space X may have many triangulations and these lead to different chain complexes. Associating to X a unique equivalence class of complexes, resolutions of a fixed module of a given type will not usually be unique and one would like to consider all these resolutions on an equal footing.

A morphism of complexes f: M → N is a quasi-isomorphism if the induced morphisms on cohomology

Hi(f): Hi(M) → Hi(N) are isomorphisms ∀ i.

Two complexes M and N are said to be quasi-isomorphic if they are related by a chain of quasi-isomorphisms. In fact, it is sufficient to consider chains of length one, so that two complexes M and N are quasi-isomorphic iff there are quasi-isomorphisms

M ← P → N

For example, the chain complex of a topological space is well-defined up to quasi-isomorphism because any two triangulations have a common resolution. Similarly, all possible resolutions of a given module are quasi-isomorphic. Indeed, if

0 → S →f M0 →d0 M1 →d1 M2 → …

is a resolution of a module S, then by definition the morphism of complexes

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is a quasi-isomorphism.

The objects of the derived category D(A) of our abelian category A will just be complexes of objects of A, but morphisms will be such that quasi-isomorphic complexes become isomorphic in D(A). In fact we can formally invert the quasi-isomorphisms in C(A) as follows:

There is a category D(A) and a functor Q: C(A) → D(A)

with the following two properties:

(a) Q inverts quasi-isomorphisms: if s: a → b is a quasi-isomorphism, then Q(s): Q(a) → Q(b) is an isomorphism.

(b) Q is universal with this property: if Q′ : C(A) → D′ is another functor which inverts quasi-isomorphisms, then there is a functor F : D(A) → D′ and an isomorphism of functors Q′ ≅ F ◦ Q.

First, consider the category C(A) as an oriented graph Γ, with the objects lying at the vertices and the morphisms being directed edges. Let Γ∗ be the graph obtained from Γ by adding in one extra edge s−1: b → a for each quasi-isomorphism s: a → b. Thus a finite path in Γ∗ is a sequence of the form f1 · f2 ·· · ·· fr−1 · fr where each fi is either a morphism of C(A), or is of the form s−1 for some quasi-isomorphism s of C(A). There is a unique minimal equivalence relation ∼ on the set of finite paths in Γ∗ generated by the following relations:

(a) s · s−1 ∼ idb and s−1 · s ∼ ida for each quasi-isomorphism s: a → b in C(A).

(b) g · f ∼ g ◦ f for composable morphisms f: a → b and g: b → c of C(A).

Define D(A) to be the category whose objects are the vertices of Γ∗ (these are the same as the objects of C(A)) and whose morphisms are given by equivalence classes of finite paths in Γ∗. Define a functor Q: C(A) → D(A) by using the identity morphism on objects, and by sending a morphism f of C(A) to the length one path in Γ∗ defined by f. The resulting functor Q satisfies the conditions of the above lemma.

The second property ensures that the category D(A) of the Lemma is unique up to equivalence of categories. We define the derived category of A to be any of these equivalent categories. The functor Q: C(A) → D(A) is called the localisation functor. Observe that there is a fully faithful functor

J: A → C(A)

which sends an object M to the trivial complex with M in the zeroth position, and a morphism F: M → N to the morphism of complexes

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Composing with Q we obtain a functor A → D(A) which we denote by J. This functor J is fully faithful, and so defines an embedding A → D(A). By definition the functor Hi(−): C(A) → A inverts quasi-isomorphisms and so descends to a functor

Hi(−): D(A) → A

establishing that composite functor H0(−) ◦ J is isomorphic to the identity functor on A.

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Modal Structuralism. Thought of the Day 106.0

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Structuralism holds that mathematics is ultimately about the shared structures that may be instantiated by particular systems of objects. Eliminative structuralists, such as Geoffrey Hellman (Mathematics Without Numbers Towards a Modal-Structural Interpretation), try to develop this insight in a way that does not assume the existence of abstract structures over and above any instances. But since not all mathematical theories have concrete instances, this brings a modal element to this kind of structuralist view: mathematical theories are viewed as being concerned with what would be the case in any system of objects satisfying their axioms. In Hellman’s version of the view, this leads to a reinterpretation of ordinary mathematical utterances made within the context of a theory. A mathematical utterance of the sentence S, made against the context of a system of axioms expressed as a conjunction AX, becomes interpreted as the claim that the axioms are logically consistent and that they logically imply S (so that, were we to find an interpretation of those axioms, S would be true in that interpretation). Formally, an utterance of the sentence S becomes interpreted as the claim:

◊ AX & □ (AX ⊃ S)

Here, in order to preserve standard mathematics (and to avoid infinitary conjunctions of axioms), AX is usually a conjunction of second-order axioms for a theory. The operators “◊” and “□” are modal operators on sentences, interpreted as “it is logically consistent that”, and “it is logically necessary that”, respectively.

This view clearly shares aspects of the core of algebraic approaches to mathematics. According to modal structuralism what makes a mathematical theory good is that it is logically consistent. Pure mathematical activity becomes inquiry into the consistency of axioms, and into the consequences of axioms that are taken to be consistent. As a result, we need not view a theory as applying to any particular objects, so certainly not to one particular system of objects. Since mathematical utterances so construed do not refer to any objects, we do not get into difficulties with deciding on the unique referent for apparent singular terms in mathematics. The number 2 in mathematical contexts refers to no object, though if there were a system of objects satisfying the second-order Peano axioms, whatever mathematical theorems we have about the number 2 would apply to whatever the interpretation of 2 is in that system. And since our mathematical utterances are made true by modal facts, about what does and does not follow from consistent axioms, we no longer need to answer Benacerraf’s question of how we can have knowledge of a realm of abstract objects, but must instead consider how we know these (hopefully more accessible) facts about consistency and logical consequence.

Stewart Shapiro’s (Philosophy of Mathematics Structure and Ontology) non-eliminative version of structuralism, by contrast, accepts the existence of structures over and above systems of objects instantiating those structures. Specifically, according to Shapiro’s ante rem view, every logically consistent theory correctly describes a structure. Shapiro uses the terminology “coherent” rather than “logically consistent” in making this claim, as he reserves the term “consistent” for deductively consistent, a notion which, in the case of second-order theories, falls short of coherence (i.e., logical consistency), and wishes also to separate coherence from the model-theoretic notion of satisfiability, which, though plausibly coextensive with the notion of coherence, could not be used in his theory of structure existence on pain of circularity. Like Hellman, Shapiro thinks that many of our most interesting mathematical structures are described by second-order theories (first-order axiomatizations of sufficiently complex theories fail to pin down a unique structure up to isomorphism). Mathematical theories are then interpreted as bodies of truths about structures, which may be instantiated in many different systems of objects. Mathematical singular terms refer to the positions or offices in these structures, positions which may be occupied in instantiations of the structures by many different officeholders.

While this account provides a standard (referential) semantics for mathematical claims, the kinds of objects (offices, rather than officeholders) that mathematical singular terms are held to refer to are quite different from ordinary objects. Indeed, it is usually simply a category mistake to ask of the various possible officeholders that could fill the number 2 position in the natural number structure whether this or that officeholder is the number 2 (i.e., the office). Independent of any particular instantiation of a structure, the referent of the number 2 is the number 2 office or position. And this office/position is completely characterized by the axioms of the theory in question: if the axioms provide no answer to a question about the number 2 office, then within the context of the pure mathematical theory, this question simply has no answer.

Elements of the algebraic approach can be seen here in the emphasis on logical consistency as the criterion for the existence of a structure, and on the identification of the truths about the positions in a structure as being exhausted by what does and does not follow from a theory’s axioms. As such, this version of structuralism can also respond to Benacerraf’s problems. The question of which instantiation of a theoretical structure one is referring to when one utters a sentence in the context of a mathematical theory is dismissed as a category mistake. And, so long as the basic principle of structure-existence, according to which every logically consistent axiomatic theory truly describes a structure, is correct, we can explain our knowledge of mathematical truths simply by appeal to our knowledge of consistency.

Functoriality in Low Dimensions. Note Quote.

Let CW be the category of CW-complexes and cellular maps, let CW0 be the full subcategory of path connected CW-complexes and let CW1 be the full subcategory of simply connected CW-complexes. Let HoCW denote the category of CW-complexes and homotopy classes of cellular maps. Let HoCWn denote the category of CW-complexes and rel n-skeleton homotopy classes of cellular maps. Dimension n = 1: It is straightforward to define a covariant truncation functor

t<n = t<1 : CW0 → HoCW together with a natural transformation

emb1 : t<1 → t<∞,

where t<∞ : CW0 → HoCW is the natural “inclusion-followed-by-quotient” functor given by t<∞(K) = K for objects K and t<∞(f) = [f] for morphisms f, such that for all objects K, emb1∗ : H0(t<1K) → H0(t<∞K) is an isomorphism and Hr(t<1K) = 0 for r ≥ 1. The details are as follows: For a path connected CW-complex K, set t<1(K) = k0, where k0 is a 0-cell of K. Let emb1(K) : t<1(K) = k0 → t<∞(K) = K be the inclusion of k0 in K. Then emb1∗ is an isomorphism on H0 as K is path connected. Clearly Hr(t<1K) = 0 for r ≥ 1. Let f : K → L be a cellular map between objects of CW0. The morphism t<1(f) : t<1(K) = k0 → l0 = t<1(L) is the homotopy class of the unique map from a point to a point. In particular, t<1(idK) = [idk0] and for a cellular map g : L → P we have t<1(gf) = t<1(g) ◦ t<1(f), so that t<1 is indeed a functor. To show that emb1 is a natural transformation, we need to see that

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that is

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commutes in HoCW. This is where we need the functor t<1 to have values only in HoCW, not in CW, because the square need certainly not commute in CW. (The points k0 and l0 do not know anything about f, so l0 need not be the image of k0 under f.) Since L is path connected, there is a path ω : I → L from l0 = ω(0) to f (k0) = ω(1). Then H : {k0} × I → L, H(k0, t) = ω(t), defines a homotopy from

k0 → l0 → L to k0 → K →f L.

Dimension n = 2: We will define a covariant truncation functor t<n = t<2 : CW1 → HoCW

together with a natural transformation
emb2 : t<2 → t<∞,

where t<∞ : CW1 → HoCW is as above (only restricted to simply connected spaces), such that for all objects K, emb2∗ : Hr(t<2K) → Hr(t<∞K) is an isomorphism for r = 0, 1, and Hr(t<2K) = 0 for r ≥ 2. For a simply connected CW-complex K, set t<2(K) = k0, where k0 is a 0-cell of K. Let emb2(K) : t<2(K) = k0 → t<∞(K) = K be the inclusion as in the case n = 1. It follows that emb2∗ is an isomorphism both on H0 as K is path connected and on H1 as H1(k0) = 0 = H1(K), while trivially Hr(t<2K) = 0 for r ≥ 2. On a cellular map f, t<2(f) is defined as in the case n = 1. As in the case n = 1, this yields a functor and emb2 is a natural transformation.

Metaphysics of the Semantics of HoTT. Thought of the Day 73.0

PMquw

Types and tokens are interpreted as concepts (rather than spaces, as in the homotopy interpretation). In particular, a type is interpreted as a general mathematical concept, while a token of a given type is interpreted as a more specific mathematical concept qua instance of the general concept. This accords with the fact that each token belongs to exactly one type. Since ‘concept’ is a pre-mathematical notion, this interpretation is admissible as part of an autonomous foundation for mathematics.

Expressions in the language are the names of types and tokens. Those naming types correspond to propositions. A proposition is ‘true’ just if the corresponding type is inhabited (i.e. there is a token of that type, which we call a ‘certificate’ to the proposition). There is no way in the language of HoTT to express the absence or non-existence of a token. The negation of a proposition P is represented by the type P → 0, where P is the type corresponding to proposition P and 0 is a type that by definition has no token constructors (corresponding to a contradiction). The logic of HoTT is not bivalent, since the inability to construct a token of P does not guarantee that a token of P → 0 can be constructed, and vice versa.

The rules governing the formation of types are understood as ways of composing concepts to form more complex concepts, or as ways of combining propositions to form more complex propositions. They follow from the Curry-Howard correspondence between logical operations and operations on types. However, we depart slightly from the standard presentation of the Curry-Howard correspondence, in that the tokens of types are not to be thought of as ‘proofs’ of the corresponding propositions but rather as certificates to their truth. A proof of a proposition is the construction of a certificate to that proposition by a sequence of applications of the token construction rules. Two different such processes can result in construction of the same token, and so proofs and tokens are not in one-to-one correspondence.

When we work formally in HoTT we construct expressions in the language according to the formal rules. These expressions are taken to be the names of tokens and types of the theory. The rules are chosen such that if a construction process begins with non-contradictory expressions that all name tokens (i.e. none of the expressions are ‘empty names’) then the result will also name a token (i.e. the rules preserve non-emptiness of names).

Since we interpret tokens and types as concepts, the only metaphysical commitment required is to the existence of concepts. That human thought involves concepts is an uncontroversial position, and our interpretation does not require that concepts have any greater metaphysical status than is commonly attributed to them. Just as the existence of a concept such as ‘unicorn’ does not require the existence of actual unicorns, likewise our interpretation of tokens and types as mathematical concepts does not require the existence of mathematical objects. However, it is compatible with such beliefs. Thus a Platonist can take the concept, say, ‘equilateral triangle’ to be the concept corresponding to the abstract equilateral triangle (after filling in some account of how we come to know about these abstract objects in a way that lets us form the corresponding concepts). Even without invoking mathematical objects to be the ‘targets’ of mathematical concepts, one could still maintain that concepts have a mind-independent status, i.e. that the concept ‘triangle’ continues to exist even while no-one is thinking about triangles, and that the concept ‘elliptic curve’ did not come into existence at the moment someone first gave the definition. However, this is not a necessary part of the interpretation, and we could instead take concepts to be mind-dependent, with corresponding implications for the status of mathematics itself.

Wittgenstein’s Form is the Possibility of Structure

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For given two arbitrary objects x and y they can be understood as arguments for a basic ontological connection which, in turn, is either positive or negative. A priori there exist just four cases: the case of positive connection – MP, the case of negative connection – MI, the case that connection is both positive and negative, hence incoherent, denoted – MPI, and the most popular in combinatorial ontology the case of mutual neutrality – N( , ). The first case is taken here to be fundamental.

Explication for σ

Now we can offer the following, rather natural explication for a powerful, nearly omnipotent, synthesizer: y is synthetizable from x iff it is be made possible from x:

σ(x) = {y : MP(x,y)}

Notice that the above explication connects the second approach (operator one) with the third (internal) approach to a general theory of analysis and synthesis.

Quoting one of the most mysterious theses of Wittgenstein’s Tractatus:

(2.033) Form is the possibility of structure.

Ask now what the possibility means? It has been pointed out by Frank Ramsey in his famous review of the Tractatus that it cannot be read as a logical modality (i. e., form cannot be treated as an alternative structure), for this reading would immediately make Tractatus inconsistent.

But, rather ‘Form of x is what makes the structure of y possible’.

Formalization: MP(Form(x), Str(y)), hence – through suitable generalization – MP(x, y).

Wittgensteinian and Leibnizian clues make the nature of MP more clear: form of x is determined by its substance, whereas structurality of y means that y is a complex built up in such and such way. Using syntactical categorization of Lésniewski and Ajdukiewicz we obtain therefore that MP has the category of quantifier: s/n, s – which, as is easy to see, is of higher order and deeply modal.

Therefore M P is a modal quantifier, characterized after Wittgenstein’s clue by

MP(x, y) ↔ MP(S(x), y)

Rhizomatic Topology and Global Politics. A Flirtatious Relationship.

 

rhizome

Deleuze and Guattari see concepts as rhizomes, biological entities endowed with unique properties. They see concepts as spatially representable, where the representation contains principles of connection and heterogeneity: any point of a rhizome must be connected to any other. Deleuze and Guattari list the possible benefits of spatial representation of concepts, including the ability to represent complex multiplicity, the potential to free a concept from foundationalism, and the ability to show both breadth and depth. In this view, geometric interpretations move away from the insidious understanding of the world in terms of dualisms, dichotomies, and lines, to understand conceptual relations in terms of space and shapes. The ontology of concepts is thus, in their view, appropriately geometric, a multiplicity defined not by its elements, nor by a center of unification and comprehension and instead measured by its dimensionality and its heterogeneity. The conceptual multiplicity, is already composed of heterogeneous terms in symbiosis, and is continually transforming itself such that it is possible to follow, and map, not only the relationships between ideas but how they change over time. In fact, the authors claim that there are further benefits to geometric interpretations of understanding concepts which are unavailable in other frames of reference. They outline the unique contribution of geometric models to the understanding of contingent structure:

Principle of cartography and decalcomania: a rhizome is not amenable to any structural or generative model. It is a stranger to any idea of genetic axis or deep structure. A genetic axis is like an objective pivotal unity upon which successive stages are organized; deep structure is more like a base sequence that can be broken down into immediate constituents, while the unity of the product passes into another, transformational and subjective, dimension. (Deleuze and Guattari)

The word that Deleuze and Guattari use for ‘multiplicities’ can also be translated to the topological term ‘manifold.’ If we thought about their multiplicities as manifolds, there are a virtually unlimited number of things one could come to know, in geometric terms, about (and with) our object of study, abstractly speaking. Among those unlimited things we could learn are properties of groups (homological, cohomological, and homeomorphic), complex directionality (maps, morphisms, isomorphisms, and orientability), dimensionality (codimensionality, structure, embeddedness), partiality (differentiation, commutativity, simultaneity), and shifting representation (factorization, ideal classes, reciprocity). Each of these functions allows for a different, creative, and potentially critical representation of global political concepts, events, groupings, and relationships. This is how concepts are to be looked at: as manifolds. With such a dimensional understanding of concept-formation, it is possible to deal with complex interactions of like entities, and interactions of unlike entities. Critical theorists have emphasized the importance of such complexity in representation a number of times, speaking about it in terms compatible with mathematical methods if not mathematically. For example, Foucault’s declaration that: practicing criticism is a matter of making facile gestures difficult both reflects and is reflected in many critical theorists projects of revealing the complexity in (apparently simple) concepts deployed both in global politics.  This leads to a shift in the concept of danger as well, where danger is not an objective condition but “an effect of interpretation”. Critical thinking about how-possible questions reveals a complexity to the concept of the state which is often overlooked in traditional analyses, sending a wave of added complexity through other concepts as well. This work seeking complexity serves one of the major underlying functions of critical theorizing: finding invisible injustices in (modernist, linear, structuralist) givens in the operation and analysis of global politics.

In a geometric sense, this complexity could be thought about as multidimensional mapping. In theoretical geometry, the process of mapping conceptual spaces is not primarily empirical, but for the purpose of representing and reading the relationships between information, including identification, similarity, differentiation, and distance. The reason for defining topological spaces in math, the essence of the definition, is that there is no absolute scale for describing the distance or relation between certain points, yet it makes sense to say that an (infinite) sequence of points approaches some other (but again, no way to describe how quickly or from what direction one might be approaching). This seemingly weak relationship, which is defined purely ‘locally’, i.e., in a small locale around each point, is often surprisingly powerful: using only the relationship of approaching parts, one can distinguish between, say, a balloon, a sheet of paper, a circle, and a dot.

To each delineated concept, one should distinguish and associate a topological space, in a (necessarily) non-explicit yet definite manner. Whenever one has a relationship between concepts (here we think of the primary relationship as being that of constitution, but not restrictively, we ‘specify’ a function (or inclusion, or relation) between the topological spaces associated to the concepts). In these terms, a conceptual space is in essence a multidimensional space in which the dimensions represent qualities or features of that which is being represented. Such an approach can be leveraged for thinking about conceptual components, dimensionality, and structure. In these terms, dimensions can be thought of as properties or qualities, each with their own (often-multidimensional) properties or qualities. A key goal of the modeling of conceptual space being representation means that a key (mathematical and theoretical) goal of concept space mapping is

associationism, where associations between different kinds of information elements carry the main burden of representation. (Conceptual_Spaces_as_a_Framework_for_Knowledge_Representation)

To this end,

objects in conceptual space are represented by points, in each domain, that characterize their dimensional values. A concept geometry for conceptual spaces

These dimensional values can be arranged in relation to each other, as Gardenfors explains that

distances represent degrees of similarity between objects represented in space and therefore conceptual spaces are “suitable for representing different kinds of similarity relation. Concept

These similarity relationships can be explored across ideas of a concept and across contexts, but also over time, since “with the aid of a topological structure, we can speak about continuity, e.g., a continuous change” a possibility which can be found only in treating concepts as topological structures and not in linguistic descriptions or set theoretic representations.

Revisiting Twistors

In twistor theory, α-planes are the building blocks of classical field theory in complexified compactified Minkowski space-time. The α-planes are totally null two-surfaces S in that, if p is any point on S, and if v and w are any two null tangent vectors at p ∈ S, the complexified Minkowski metric η satisfies the identity η(v,w) = vawa = 0. By definition, their null tangent vectors have the two-component spinor form λAπA, where λA is varying and πA is fixed. Therefore, the induced metric vanishes identically since η(v,w) = λAπA μAπA = 0 = η(v, v) = λAπA λAπA . One thus obtains a conformally invariant characterization of flat space-times. This definition can be generalized to complex or real Riemannian space-times with non-vanishing curvature, provided the Weyl curvature is anti-self-dual. One then finds that the curved metric g is such that g(v,w) = 0 on S, and the spinor field πA is covariantly constant on S. The corresponding holomorphic two-surfaces are called α-surfaces, and they form a three-complex-dimensional family. Twistor space is the space of all α-surfaces, and depends only on the conformal structure of complex space-time.

Projective twistor space PT is isomorphic to complex projective space CP3. The correspondence between flat space-time and twistor space shows that complex α-planes correspond to points in PT, and real null geodesics to points in PN, i.e. the space of null twistors. Moreover, a complex space-time point corresponds to a sphere in PT, and a real space-time point to a sphere in PN. Remarkably, the points x and y are null-separated iff the corresponding spheres in PT intersect. This is the twistor description of the light-cone structure of Minkowski space-time.

A conformally invariant isomorphism exists between the complex vector space of holomorphic solutions of  ◻φ = 0 on the forward tube of flat space-time, and the complex vector space of arbitrary complex-analytic functions of three variables, not subject to any differential equation. Moreover, when curvature is non-vanishing, there is a one-to-one correspondence between complex space-times with anti-self-dual Weyl curvature and scalar curvature R = 24Λ, and sufficiently small deformations of flat projective twistor space PT which preserve a one-form τ homogeneous of degree 2 and a three-form ρ homogeneous of degree 4, with τ ∧ dτ = 2Λρ. Thus, to solve the anti-self-dual Einstein equations, one has to study a geometric problem, i.e. finding the holomorphic curves in deformed projective twistor space.

Distributed Representation Revisited

Figure-132-The-distributed-representation-of-language-meaning-in-neural-networks

If the conventional symbolic model mandates a creation of theory that is sought to address the issues pertaining to the problem, this mandatory theory construction is bypassed in case of distributed representational systems, since the latter is characterized by a large number of interactions occurring in a nonlinear fashion. No such attempts at theoretical construction are to be made in distributed representational systems for fear of high end abstraction, thereby sucking off the nutrient that is the hallmark of the model. Distributed representation is likely to encounter onerous issues if the size of the network inflates, but the issue is addressed through what is commonly known as redundancy technique, whereby, a simultaneous encoding of information generated by numerous interactions take place, thus ameliorating the adequacy of presenting the information to the network. In the words of Paul Cilliers, this is an important point, for,

the network used for the model of a complex system will have to have the same level of complexity as the system itself….However, if the system is truly complex, a network of equal complexity may be the simplest adequate model of such a system, which means that it would be just as difficult to analyze as the system itself.

Following, he also presents a caveat,

This has serious methodological implications for the scientists working with complex systems. A model which reduces the complexity may be easier to implement, and may even provide a number of economical descriptions of the system, but the price paid for this should be considered carefully.

One of the outstanding qualities of distributed representational systems is their adaptability. Adaptability, in the sense of reusing the network to be applicable to other problems to offer solutions. Exactly, what this connotes is, the learning process the network has undergone for a problem ‘A’, could be shared for problem ‘B’, since many of the input neurons are bounded by information learned through ‘A’ that could be applicable to ‘B’. In other words, the weights are the dictators for solving or resolving issues, no matter, when and for which problem the learning took place. There is a slight hitch here, and that being this quality of generalizing solutions could suffer, if the level of abstraction starts to shoot up. This itself could be arrested, if in the initial stages, the right kind of framework is decided upon, thus obscuring the hitch to almost non-affective and non-existence impacting factor. The very notion of weights is considered here by Sterelny as a problematic, and he takes it to attack distributed representation in general and connectionsim as a whole in particular. In an analogically witty paragraph, Sterelny says,

There is no distinction drawable, even in principle, between functional and non- functional connections. A positive linkage between two nodes in a distributed network might mean a constitutive link (eg. Catlike, in a network for tiger); a nomic one (carnivore, in the same network), or a merely associative one (in my case, a particular football team that play in black and orange.

It should be noted that this criticism on weights is derived, since for Sterelny, relationship between distributed representations and the micro-features that compose them is deeply problematic. If such is the criticism, then no doubt, Sterelny still seems to be ensconced within the conventional semantic/symbolic model. And since, all weights can take part in information processing, there is some sort of a democratic liberty that is accorded to the weights within a distributed representation, and hence any talk of constitutive, nomic, or even for that matter associative is mere humbug. Even if there is a disagreement prevailing that a large pattern of weights are not convincing enough for an explanation, as they tend to complicate matters, the distributed representational systems work consistently enough as compared to an alternative system that offers explanation through reasoning, and thereby, it is quite foolhardy to jettison the distributed representation by the sheer force of criticism. If the neural network can be adapted to produce the correct answer for a number of training cases that is large compared with the size of the network, it can be trusted to respond correctly to the previously unseen cases provided they are drawn from the same population using the same distribution as the training cases, thus undermining the commonly held idea that explanations are the necessary feature of the trustworthy systems (Baum and Haussler). Another objection that distributed representation faces is that, if representations are distributed, then the probability of two representations of the same thing as different from one another cannot be ruled out. So, one of them is the true representation, while the other is only an approximation of the representation.(1) This is a criticism of merit and is attributed to Fodor, in his influential book titled Psychosemantics.(2) For, if there is only one representation, Fodor would not shy from saying that this is the yucky solution, folks project believe in. But, since connectionism believes in the plausibility of indeterminate representations, the question of flexibility scores well and high over the conventional semantic/symbolic models, and is it not common sense to encounter flexibility in daily lives? The other response to this objection comes from post-structuralist theories (Baudrillard is quite important here. See the first footnote below). The objection of true representation, and which is a copy of the true representation meets its pharmacy in post-structuralism, where meaning is constituted by synchronic as well as diachronic contextualities, and thereby supplementing the distributed representation with a no-need-for concept and context, as they are inherent in the idea of such a representation itself. Sterelny, still seems to ride on his obstinacy, and in a vitriolic tone poses his demand to know as to why distributed representation should be regarded as states of the system at all. Moreover, he says,

It is not clear that a distributed representation is a representation for the connectionist system at all…given that the influence of node on node is local, given that there is no processor that looks at groups of nodes as a whole, it seems that seeing a distributed representation in a network is just an outsider’s perspective on the system.

This is moving around in circles, if nothing more. Or maybe, he was anticipating what G. F. Marcus would write and echo to some extent in his book The Algebraic Mind. In the words of Marcus,

…I agree with Stemberger(3) that connectionism can make a valuable contribution to cognitive science. The only place, we differ is that, first, he thinks that the contribution will be made by providing a way of eliminating symbols, whereas I think that connectionism will make its greatest contribution by accepting the importance of symbols, seeking ways of supplementing symbolic theories and seeking ways of explaining how symbols could be implemented in the brain. Second, Stemberger feels that symbols may play no role in cognition; I think that they do.

Whatever Sterelny claims, after most of the claims and counter-claims that have been taken into account, the only conclusion for the time being is that distributive representation has been undermined, his adamant position to be notwithstanding.

(1) This notion finds its parallel in Baudrillard’s Simulation. And subsequently, the notion would be invoked in studying the parallel nature. Of special interest is the order of simulacra in the period of post-modernity, where the simulacrum precedes the original, and the distinction between reality and representation vanishes. There is only the simulacrum and the originality becomes a totally meaningless concept.

(2) This book is known for putting folk psychology firmly on the theoretical ground by rejecting any external, holist and existential threat to its position.

(3) Joseph Paul Stemberger is a professor in the Department of Linguistics at The University of British Columbia in Vancouver, British Columbia, Canada, with primary interests in phonology, morphology, and their interactions. My theoretical orientations are towards Optimality Theory, employing our own version of the theory, and towards connectionist models.