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.

Category Theory as Structuralist. Part Metaphysic, Part Mathematic. (1)

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What are categories good for? Elementary category theory is mostly concerned with universal properties. These define certain patterns of morphisms that uniquely characterize (up to isomorphism) a certain mathematical structure. An example that we will be concerned with is the notion of a ‘terminal object’. Given a category C, a terminal object is an object I such that, for any object A in C, there is a unique morphism of type f : A → I. So for instance, on Set the singleton {∗} is the terminal object, and so we obtain a characterization of the singleton set in terms of the morphisms in Set. Other standard constructions, e.g. the cartesian product, disjoint union etc. can be characterized as universal.

Just as morphisms in a category preserve the structure of the objects, we can also define maps between categories that preserve the composition law. Let C and D be categories. A functor F : C → D is a mapping that:

(i) assigns an object F (A) in D to each object A in C; and

(ii) assigns a morphism F(f) : F(A) → F(B) to each morphism f : A → B in C, subject to the conditions F(g ◦ f) = F(g) ◦ F(f) and F(1A) = 1F(A) for all A in C.

Examples abound: we can define a powerset functor P : Set → Set that assigns the powerset P (X ) to each set X , and assigns the function P(f)::X →f[X] to each function f :X →Y, where f[X] ⊆ Y is the image of f.

Functors ‘compare’ categories, and we can once again increase the level of abstraction: we can compare functors as follows. Let F : C → D and G : C → D be a pair of functors. A natural transformation η : F ⇒ G is a family of functions {ηA : F (A) → G(A)} A ∈|C| indexed by the objects in C, such that for all morphisms f : A → B in C we have ηB ◦ F (f ) = G(f ) ◦ ηA.

Category theory can be thought of as ‘structuralist’ in the following simple sense: it de-emphasizes the role played by the objects of a category, and tries to spell out as many statements as possible in terms of the morphisms between those objects. These formal features of category theory have developed into a vision of how to do mathematics. This has for instance been explicitly articulated by Awodey, who says that a category-theoretic ‘structuralist’ perspective of mathematics, is based on specifying :

…for a given theorem or theory only the required or relevant degree of information or structure, the essential features of a given situation, for the purpose at hand, without assuming some ultimate knowledge, specification, or “determination” of the objects involved.

Awodey presents one reasonable methodological sense of ‘category-theoretic struc- turalism’: a view about how to do mathematics that is guided by the features of category theory.

Let us now contrast Awodey’s sense of structuralism with a position in the philosophy of science known as Ontic Structural Realism (OSR). Roughly speaking, OSR is the view that the ontology of the theory under consideration is given only by structures and not by objects (where ‘object’ is here being used in a metaphysical, and not a purely mathematical sense). Indeed, some OSR-ers would claim that:

(Objectless) It is coherent to have an ontology of (physical) relations without admitting an ontology of (physical) relata between which these rela- tions hold.

On the face of it, the ‘simple structuralism’ that is evident in the practice of category theory is very different from that envisaged by OSR; and in particular, it is hardly obvious how this simple structuralism could be applied to yield (Objectless). On the other hand, one might venture that applying (some form of) this simple structuralism to physical theories will serve the purposes of OSR.

Let us consider which forms of OSR have an interest in such an category-theoretic argument for (Objectless). According to Frigg and Votsis’ detailed taxonomy of structural realist positions, the most radical form of OSR insists on an extensional (in the logical sense of being ‘uninterpreted’) treatment of physical relations, i.e. physical relations are nothing but relations defined as sets of ordered tuples on appropriate formal objects. This view is faced not only with the problem of defending (Objectless) but with the further implausibility of implying that the concrete physical world is nothing but a structured set.

More plausible is a slightly weaker form of ontic structural realism, which Frigg calls Eliminative OSR (EOSR). Like OSR, EOSR maintains that relations are ontologically fundamental, but unlike OSR, it allows for relations that have intensions. Defenders of EOSR have typically responded to the charge of (Objectless)’s incoherence in various ways. For example, some claim that our ontology is ‘structure all the way down’ without a fundamental level (Ladyman and Ross)

, or that the EOSR position should be interpreted as reconceptualizing objects as bundles of relations.

Honey-Trap Catalysis or Why Chemistry Mechanizes Complexity? Note Quote.

Was browsing through Yuri Tarnopolsky’s Pattern Chemistry and its affect on/from humanities. Tarnopolsky’s states “chemistry” + “humanities” connectivity ideas thusly:

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Practically all comments to the folk tales in my collection contained references to a book by the Russian ethnographer Vladimir Propp, who systematized Russian folk tales as ‘molecules‘ consisting of the same ‘atoms‘ of plot arranged in different ways, and even wrote their formulas. His book was published in the 30’s, when Claude Levi-Strauss, the founder of what became known as structuralism, was studying another kind of “molecules:” the structures of kinship in tribes of Brazil. Remarkably, this time a promise of a generalized and unifying vision of the world was coming from a source in humanities. What later happened to structuralism, however, is a different story, but the opportunity to build a bridge between sciences and humanities was missed. The competitive and pugnacious humanities could be a rough terrain.

I believed that chemistry carried a universal message about changes in systems that could be described in terms of elements and bonds between them. Chemistry was a particular branch of a much more general science about breaking and establishing bonds. It was not just about molecules: a small minority of hothead human ‘molecules’ drove a society toward change. A nation could be hot or cold. A child playing with Lego and a poet looking for a word to combine with others were in the company of a chemist synthesizing a drug.

Further on, Tarnopolsky, following his chemistry then thermodynamics leads, then found the pattern theory work of Swedish chemist Ulf Grenander, which he describes as follows:

In 1979 I heard about a mathematician who tried to list everything in the world. I easily found in a bookstore the first volume of Pattern Theory (1976) by Ulf Grenander, translated into Russian. As soon as I had opened the book, I saw that it was exactly what I was looking for and what I called ‘meta-chemistry’, i.e., something more general than chemistry, which included chemistry as an application, together with many other applications. I can never forget the physical sensation of a great intellectual power that gushed into my face from the pages of that book.

Although the mathematics in the book was well above my level, Grenander’s basic idea was clear. He described the world in terms of structures built of abstract ‘atoms’ possessing bonds to be selectively linked with each other. Body movements, society, pattern of a fabric, chemical compounds, and scientific hypothesis—everything could be described in the atomistic way that had always been considered indigenous for chemistry. Grenander called his ‘atoms of everything’ generators, which tells something to those who are familiar with group theory, but for the rest of us could be a good little metaphor for generating complexity from simplicity. Generators had affinities to each other and could form bonds of various strength. Atomism is a millennia old idea. In the next striking step so much appealing to a chemist, Ulf Grenander outlined the foundation of a universal physical chemistry able to approach not only fixed structures but also “reactions” they could undergo.

The two major means of control in chemistry and organic life: thermodynamic control (shift of equilibrium) and kinetic control (selective change of speed). People might not be aware that the same mechanisms are employed in social and political control, as well as in large historical events out of control, for example, the great global migration of people and jobs in our time or just the one-way flow of people across the US-Mexican border!!! Thus, with an awful degree of simplification, the intensification of a hunt for illegal immigrants looks like thermodynamic control by a honey trap, while the punishment for illegal employers is typical negative catalysis, although both may lead to a less stable and more stressed state. In both cases, new equilibrium will be established, different equilibria housed upon different sets of conditions.

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Should I treat people as molecules, unless I am from the Andromeda Galaxy. Complex-systems never come to global equilibrium, although local equilibrium can exist for some time. They can be in the state of homeostasis, which, again, is not the same as steady state in physics and chemistry. Homeostasis is the global complement of the classical local Darwinism of mutation and selection.

Taking other examples, the immigration discrimination in favor of educated or wealthy professionals is also a catalysis of affirmative action type. It speeds up the drive to equilibrium. Attractive salary for rare specialists is an equilibrium shift (honey trap) because it does not discriminate between competitors. Ideally, neither does exploitation of foreign labor. Bureaucracy is a global thermodynamic freeze that can be selectively overcome by 100% catalytic connections and bribes. Severe punishment for bribe is thermodynamic control. The use of undercover agents looks like a local catalyst: you can wait for the crook to make a mistake or you can speed it up. Tax incentive or burden is a shift of equilibrium. Preferred (or discouraging) treatment of competitors is catalysis (or inhibition).

There is no catalysis without selectivity and no selectivity without competition. Equilibrium, however, is not selective: it applies globally to the fluid enough system. Organic life, society, and economy operate by both equilibrium shift and catalysis. More examples: by manipulating the interest rate, the RBI employs thermodynamic control; by tax cuts for efficient use of energy, the government employs kinetic control, until saturation comes. Thermodynamic and kinetic factors are necessary for understanding Complex-systems, although only professionals can talk about them reasonably, but they are not sufficient. History is not chemistry because organic life and human society develop by design patterns, so to speak, or archetypal abstract devices, which do not follow from any physical laws. They all, together with René Thom morphologies, have roots not in thermodynamics but in topology. Anything that cannot be presented in terms of points, lines, and interactions between the points is far from chemistry. Topology is blind to metrics, but if Pattern Theory were not metrical, it would be just a version of graph theory.

¿..Structuralism..?

This is a puerile dig from the archives. Just wanted to park it here to rest and rust and then probably forgotten, until a new post on structuralism as a philosophy of mathematics comes up, which, it shall soon. In the meanwhile, this could largely be skipped.

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Structuralism is an umbrella term involving a wide range of disciplines that came to fruition with the work of Swiss linguist Ferdinand de Saussure. The basic idea revolves round the study of underlying structures of significations that are meaningfully derived from ‘texts’. A ‘text’ is anything that owes its existence to a document or anything that has the potential of getting documented. The analyses for the discovery of structures underlying all these significations and texts and the conditions of possibilities for the existence of these significations and texts is what structuralism purportedly does. Saussure’s ‘Course in General Linguistics’, published posthumously, and seeing the light of the day because of his students’ note taking influenced ‘Structural Linguistics’, thereby explaining the adequacy of language for describing things concrete and abstract and in the process expanding the applicability of what language could do.

The starting point of Saussure’s analysis is Semiology, a science that undertakes the study of signs in society. These signs that express ideas build up the system of language for him. Signs are comprised of langue (language) and parole (speech). Langue is an abstract homogeneous system of language that is internalized by a given speech community, whereas parole is a concrete heterogeneous act of putting language into practice. In Saussurian jargon, Langue describes the social, impersonal phenomenon of language as a system of signs, while parole describes the individual, personal phenomenon of language as a series of speech acts made by a linguist subject. Signs attain their iconic status for Saussure due to meaning production when they enter into relationships with their referents.(1) Every sign is composed of a pair, a couple viz, signifier and signified, where signifier is a sound image (psychologically considered rather than materially), and signified is a concept. Signifier is the sensible part of the sign. A signified on the other hand is a connotation, an attachment that the signifier carries, a meaning, or a mental image of an entity that somehow misses out manifesting in the proximity. In other words, the signified of a signifier is not itself a sensible part of the sign. Signifier without the signified and vice versa strips a sign of its essence and therefore any meaning whatsoever (metaphysics ruled out for the moment!), and meaningfulness of signs in any discourse is derived from internal systemic relations of difference. This is precisely what is meant when Saussure says that language is a system of differences without positive terms (for the record: this is accepted even in Derrida’s post-structuralist critique of Saussure). The positivity of terms needs deliberation here. We recognize language, or more generally the marks inhabiting the language by virtue of how each and every mark is distinct/difference from each and every other mark inhabiting the same language. This distinction or difference is neither a resident with the sensible part of the sign, or signifier, nor with the mental/insensible part of the sign, or signified. Now, if the signifier and the signified are separated somehow, then language as guided by differences connoting negativity is legitimate. But, as has been mentioned; a sign is meaningful only when the signifier and the signified are coupled together, the meaning attaches itself a positive value. This only means that language is governed by differences. In the words of Saussure,

Whether we take signified or the signifier, language has neither ideas nor sounds that existed before the linguistic system, but only conceptual and phonic differences that have issued from the system. The idea or a phonic substance that a sign contains is of less importance than the other signs that surround it. Proof of this is that the value of a term may be modified without either its meaning or sound being affected, solely because a neighboring term has been modified.

Signs were value laden, for only then would linguistics become an actual science, and for this realization to manifest, signs in any language system were determined by other signs in the same language system that helped delimiting meaning and a possible bracketed range of usage rather than a confinement to internal sound-pattern and concept. A couple of ramifications follow for Saussure from here on viz, signs cannot exist in isolation, but emanate from the system in which they are to be analyzed (this also means that the system cannot be built upon isolated signs), and grammatical facts are consolidated by taking recourse to syntagmatic and paradigmatic analyses. The former is based on the syntactic or surface structure in semiotics, whereas the latter is operative on the syntagms by means of identifying its paradigms. The syntagmatic and paradigmatic analyses were what made Saussure assert the primacy of relations of difference that made any language operate. Syntagms particularly belong to speech, and thereby direct the linguist in identifying the frequency of its usage before being incorporated into language, whereas, paradigms relationalize associatively thus building up clusters of signs in the mind before finally imposing themselves on syntagms for the efficient functionality of the language.

So, fundamentally structuralism is concerned with signifiers and relations between signifiers, and requires a diligent effort to make visible what is imperceptible and at the same time responsible for the whole phenomenon to exist, and that being the absent signified. The specialty of absent signified is to carry out the efficacy of structuralism as a phenomenon, without itself sliding into just another singifier, and this is where Derrida with his critique of structuralism comes in, in what is known as post-structuralism. But, before heading into the said territory, what is required is an attempt to polish structuralism by viewing it under some lenses, albeit very briefly.

Structural anthropology as devised by Claude Lévi-Strauss in his Structural Anthropology (1 and 2) studied certain unobservable social structures that nonetheless generated observable social phenomenon. Lévi-Strauss imported most of his ideas from the structuralist school of Saussure, and paralleled Saussure’s view on the unknow-ability of grammar usage while conversing, with the unknow-ability of the workings of the social structures in day-to-day life. Thought as such is motivated by various patterns and structures that show proclivities towards redundancy in these very various situations. This means that the meaning or the signified is derived from a decision that somehow happens to have taken place in the past, and hence already decided. And the very construction of thoughts, experience is what structural anthropology purports to do, but with beginnings that were oblivious to social/cultural systems and wedded to objectivity of scientific perspective. Although criticized for the lack of foundations of a complete scientific account and ignorant towards an integration of cultural anthropology and neuroscience, the structural anthropology remains embraced amongst anthropologists.

Other important political variant of structuralism is attributed to Louis Althusser, who coined the idea of structural Marxism as against humanistic Marxism by emphasizing on Marxism as a science that has ‘studying’ objective structures as its goal, as against the prison house of pre-scientific humanistic ideology embraced by humanistic Marxism. The major tenet of this school of Marxism lay in its scathing critique of the instrumentalist version that argued for the institutions of the state as directly under the control of those capitalist powers, and instead sought out to clarify the functionality of these institutions in order to reproduce the capitalist society as a whole.

After these brief remarks on structural anthropology and structural Marxism, it is time for a turn to examine the critiques of structuralism in order to pave a smooth slide into post-structuralism. The important reaction against structuralism is its apparent reductionist tendency, wherein deterministic structural forces are pitted over the capacities of people to act, thus anthropologically weakening. Within the anthropological camp itself, Kuper had this to say,

Structuralism came to have something of the momentum of the millennial movement and some of its adherents thought that they formed a secret society of a seeing in a world of the blind. Conversion was not just a matter of accepting a new paradigm. It was, almost, a question of salvation.

Another closely allied criticism is confining to biological explanations for cultural constructions, and therefore ignoring the social constructions in the process. This critique is also attached with the Saussurian version, for it was considered as too closed off to social change. This critique could not have been ameliorated for the presence of Voloshinov, who thematized dialectical struggles within words to argue for the language to happen primarily through a ‘clash of social forces’ between people who use words, and thereby concluding that to study changes in signs and to chart those changes mandates the study of class struggles within society.

(1) Back in the 19th Century an important figure for semiotics, the pragmatic philosopher Charles Sanders Peirce, isolated three different types of sign: The symbolic sign is like a word in so far as it refers by symbolising its referent. It neither has to look like it nor have any natural relation to it at all. Thus the word cat has no relation to that ginger monster that wails all night outside my apartment. But its owner knows what I’m talking about when I say “your cat kept me awake all night.” A poetic symbol like the sun (which may stand for enlightenment and truth) has an obviously symbolic relation to what it means. But how do such relationships come about? Saussure has an explanation. The indexical sign is like a signpost or a finger pointing in a certain direction. An arrow may accompany the signpost to San Francisco or to “Departures.” The index of a book will have a list of alphabetically ordered words with page numbers after each of them. These signs play an indexical function (in this instance, as soon as you’ve looked one up you’ll be back in the symbolic again). The iconic sign refers to its object by actually resembling it and is thus more likely to be like a picture (as with a road sign like that one with the courteous workman apologising for the disruption).