Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }$$\end{document} be a set of outcomes, and let I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} on XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }^\mathcal{I }$$\end{document} admits an additive representation. That is: there exists a linearly ordered abelian group (...) R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document} and a ‘utility function’ u:X⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$\end{document} such that, for any x,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document} which differ in only finitely many coordinates, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∑i∈Iu-u≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i\in \mathcal{I }} \left[u-u\right]\ge 0$$\end{document}. Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} also satisfies a weak continuity condition, then the paper shows that, for anyx,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document}, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∗∑i∈Iu≥∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u\ge {}^*\!\sum _{i\in \mathcal{I }}u$$\end{document}. Here, ∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u$$\end{document} represents a nonstandard sum, taking values in a linearly ordered abelian group ∗R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\mathcal{R }$$\end{document}, which is an ultrapower extension of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document}. The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces. (shrink)

Introduction: contemporary conditions and diagrammatic trajectory -- From joy to the gap: the accessing of the infinite by the finite (Spinoza, Nietzsche, Bergson) -- The care of the self versus the ethics of desire: two diagrams of the production of subjectivity (and of the subject's relation to truth) (Foucault versus Lacan) -- The aesthetic paradigm: from the folding of the finite-infinite relation to schizoanalytic metamodelisation (to biopolitics) (Guattari) -- The strange temporality of the subject: life in-between the (...) class='Hi'>infinite and the finite (Deleuze contra Badiou) -- Desiring-machines, chaoids, probeheads: towards a speculative production of subjectivity (Deleuze and Guattari) -- Conclusion: composite diagram and relations of adjacency. (shrink)

Our concern is the axiomatisation problem for modal and algebraic logics that correspond to various fragments of two-variable first-order logic with counting quantifiers. In particular, we consider modal products with Diff, the propositional unimodal logic of the difference operator. We show that the two-dimensional product logic $Diff \times Diff$ is non-finitely axiomatisable, but can be axiomatised by infinitely many Sahlqvist axioms. We also show that its ‘square’ version (the modal counterpart of the substitution and equality free fragment of two-variable (...) first-order logic with counting to two) is non-finitely axiomatisable over $Diff \times Diff$ , but can be axiomatised by adding infinitely many Sahlqvist axioms. These are the first examples of products of finitely axiomatisable modal logics that are not finitely axiomatisable, but axiomatisable by explicit infinite sets of canonical axioms. (shrink)

We propose a canonization scheme for smooth equivalence relations on Rω modulo restriction to E0-large infinite products. It shows that, given a pair of Borel smooth equivalence relations E, F on Rω, there is an infinite E0-large perfect product P⊆Rω such that either F⊆E on P, or, for some ℓ<ω, the following is true for all x,y∈P: xEy implies x(ℓ)=y(ℓ), and x↾(ω∖{ℓ})=y↾(ω∖{ℓ}) implies xFy.

Drawing upon the practice of caregiving and the insights of feminist care ethics, I offer a phenomenology of caregiving through the work of Eva Feder Kittay and Emmanuel Lévinas. I argue that caregiving is a material dialectic of embodied response involving moments of leveling, attention, and interruption. In this light, the Levinasian opposition between responding to another's singularity and leveling it via parity-based principles is belied in the experience of care. Contra much of response ethics’ and care ethics’ respective literatures, (...) this dialectic suggests that they are complementary in ways that productively illuminate themes of each. I conclude by suggesting that when response and care ethics are thought together through the experience of caregiving, such labors produce finite responsibility with infinite hope. (shrink)

The present article seeks to inquire the nature of Novalis’ difficult relation with philosophy. By founding this relation in a primordial spiritual conflict, wherein philosophy is the cause and the solution for the latter, as well as in an inescapable mythical understanding, wherein philosophy is at the same time necessary, and yet necessarily expendable, we intend to prove Novalis’ own seemingly contradictory, yet infinitely productive concept of philosophizing: one where philosophizing is a ductile, organic way of dealing with the problem (...) between the human being and the absolute. Furthermore, we wish to prove how this is most visibly considered in the sub-problem of the origin of philosophy, which sets the tone for Novalis’ remaining philosophical edifice and inscribes him as one of the leading opposers of a systematic philosophy – and hence, as an advocator of the impossibility of attaining the absolute through philosophy. (shrink)

The infinite judgement has long been forgotten and yet, as I am about to demonstrate, it may be urgent to revive it for its critical and productive potential. An infinite judgement is neither analytic nor synthetic; it does not produce logical truths, nor true representations, but it establishes the genetic conditions of real objects and the concepts appropriate to them. It is through infinite judgements that we reach the principle of transcendental logic, in the depths of which (...) all reality can emerge in its material and sensible singularity, making possible all generalization and formal abstraction. (shrink)

I argue that Kant's claim in the “Transcendental Aesthetic” of the Critique of Pure Reason that space and time are immediately given in intuition as infinite magnitudes is undercut by his general theory of mathematical knowledge. On this general theory, pure intuition does not give objects of any determinate magnitude at all, but only forms of possible objects. Specifically, what pure intuition itself yields is the recognition that any determinate space or time is part of a larger one, but (...) it requires an inference of reason to go from that to the claim that space and time are infinite. I further argue that this result is consistent with Kant's claim in the second-edition “Transcendental Deduction” that the unity of space and time are the products of synthesis, but also means that the unity of space and time as objects cannot be used a premise in the Deduction but can only be regarded as a conclusion of the deduction and the following “System of Principles.”. (shrink)

We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the (...) natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the L ζ -stables. It also implies that the machines devised are "Σ 2 Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. (shrink)

We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down $\zeta$, the least ordinal not the length of any eventual output of an Infinite Time Turing machine ; using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals (...) associated with the jump operator satisfy a Spector criterion, and correspond to the L$_\zeta$-stables. It also implies that the machines devised are "$\Sigma_2$ Complete" amongst all such other possible machines. It is shown that least upper bounds of an "eventual jump" hierarchy exist on an initial segment. (shrink)

The single greatest challenge facing managers in the developed countries of the world is to raise the productivity of knowledge and service workers. This challenge, which will dominate the management agenda for the next several decades, will ultimately determine the competitive performance of companies. Even more important, it will determine the very fabric of society and the quality of life of every industrialized nation. … Unless this challenge is met, the developed world will face increasing social tensions, increasing polarization, increasing (...) radicalization, possibly even class war. (shrink)

The aim of this paper is to shed light on our understanding of large scale properties of infinite strings. We say that one string $\alpha $ has weaker large scale geometry than that of $\beta $ if there is color preserving bi-Lipschitz map from $\alpha $ into $\beta $ with small distortion. This definition allows us to define a partially ordered set of large scale geometries on the classes of all infinite strings. This partial order compares large scale (...) geometries of infinite strings. As such, it presents an algebraic tool for classification of global patterns. We study properties of this partial order. We prove, for instance, that this partial order has a greatest element and also possess infinite chains and antichains. We also investigate the sets of large scale geometries of strings accepted by finite state machines such as Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This connects the work with the complexity theory. We also prove that the quasi-isometry problem is a $\Sigma _2^0$ -complete set, thus providing a bridge with computability theory. Finally, we build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings, thus connecting the topic with algorithmic randomness. (shrink)

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics asK4,S4,S4.1,K4.3,GL, orGrzare undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4, K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, ifl1andl2are classes of transitive frames such that their (...) depth cannot be bounded by any fixedn< ω, then the logic of the class {5ℑ1× ℑ2∣ ℑ1∈l1, ℑ2, ∈l2} is undecidable. (On the contrary, the product of, say,K4and the logic of all transitive Kripke frames of depth ≤n, for some fixedn< ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π11-complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics. (shrink)

We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra’s temporal flow semantics for Hájek’s basic logic, and Lewis-Smith, Oliva, and Robinson’s semantics for intuitionistic Łukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a (...) uniform fashion, and extend them to infinitely-many other substructural logics. (shrink)

Leading economist Daniel Cohen provides a whirlwind tour of the history of economic growth, from the early days of civilization to modern times, underscoring what is so unsettling today. The new digital economy is establishing a "zero-cost" production model, inexpensive software is taking over basic tasks, and years of exploiting the natural world have begun to backfire with deadly consequences. Working hard no longer guarantees social inclusion or income. Drawing on economics, anthropology, and psychology, and thinkers ranging from Rousseau to (...) Keynes and Easterlin, Cohen examines how a future less dependent on material gain might be considered and, how, in a culture of competition, individual desires might be better attuned to the greater needs of society."--Publisher's description. (shrink)

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4,K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if.

We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, (...) 1] I which consists of functions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰. (shrink)

We present an abstract social aggregation theorem. Society, and each individual, has a preorder that may be interpreted as expressing values or beliefs. The preorders are allowed to violate both completeness and continuity, and the population is allowed to be infinite. The preorders are only assumed to be represented by functions with values in partially ordered vector spaces, and whose product has convex range. This includes all preorders that satisfy strong independence. Any Pareto indifferent social preorder is then shown (...) to be represented by a linear transformation of the representations of the individual preorders. Further Pareto conditions on the social preorder correspond to positivity conditions on the transformation. When all the Pareto conditions hold and the population is finite, the social preorder is represented by a sum of individual preorder representations. We provide two applications. The first yields an extremely general version of Harsanyi's social aggregation theorem. The second generalizes a classic result about linear opinion pooling. (shrink)

Following Lauwers and Van Liedekerke (1995), this paper explores in a model-theoretic framework the relation between Arrovian aggregation rules and ultraproducts, in order to investigate a source of impossibility results for the case of an infinite number of individuals and an aggregation rule based on a free ultrafilter of decisive coalitions.

Louveau, A., S. Shelah and B. Velikovi, Borel partitions of infinite subtrees of a perfect tree, Annals of Pure and Applied Logic 63 271–281. We define a notion of type of a perfect tree and show that, for any given type τ, if the set of all subtrees of a given perfect tree T which have type τ is partitioned into two Borel classes then there is a perfect subtree S of T such that all subtrees of S of (...) type τ belong to the same class. This result simultaneously generalizes the partition theorems of Galvin-Prikry and Galvin-Blass. The key ingredient of the proof is the theorem of Halpern-Laüchli on partitions of products of perfect trees. (shrink)

Winner of the 1974 National Book Award The product of many years of reflection on phenomenology, this book is a comprehensive and creative introduction to the philosophy of Edmund Husserl. Natanson uses Husserl's later work as a clue to the meaning of his entire intellectual career, showing how his earlier methodological work evolved into the search for transcendental roots and developed into a philosophy of the life-world. Phenomenology, for Natanson, emerges as a philosophy of origin, a transcendental discipline concerned with (...) consciousness, history, and world rather than with introspection and traditional metaphysical warfare. (shrink)

_Winner of the 1974 National Book Award_ The product of many years of reflection on phenomenology, this book is a comprehensive and creative introduction to the philosophy of Edmund Husserl. Natanson uses Husserl's later work as a clue to the meaning of his entire intellectual career, showing how his earlier methodological work evolved into the search for transcendental roots and developed into a philosophy of the life-world. Phenomenology, for Natanson, emerges as a philosophy of origin, a transcendental discipline concerned with (...) consciousness, history, and world rather than with introspection and traditional metaphysical warfare. (shrink)

The creation-of-matter hypothesis of the Bondi-Gold-Hoyle steady-state cosmology requires that in an infinite time to which the first transfinite number may be assigned the number of atoms of matter produced would be equal to the cardinal number of the set of mathematical points in the continuum. The existence of a set of finite atoms with that cardinal number is physically unacceptable. The argument for the production of a non-denumerable set of atoms, in infinite time, is given in terms (...) of a model which is shown to be isomorphic with the original Cantor "diagonal" proof for the existence of a non-denumerable infinity. An alternative model which meets the requirements of the steady-state theory is presented; in this model, the number of atoms is explicitly no greater than countably infinite, and remains countably infinite as long as the past time of the universe is restricted to the unlimited set of finite unit-time intervals. If the origin of the steady-state universe is taken as being within that infinite set, expressed by the negative natural numbers, the contradiction of an atom at every mathematical point does not arise. The contradiction does arise if the origin is not within the set of finite numbers, and accordingly there is a restriction as to which concept of infinite past may properly be maintained in the steady-state theory. (shrink)

In this paper we present a method to reduce the decision problem of several infinite-valued propositional logics to their finite-valued counterparts. We apply our method to Łukasiewicz, Gödel and Product logics and to some of their combinations. As a byproduct we define sequent calculi for all these infinite-valued logics and we give an alternative proof that their tautology problems are in co-NP.

In his interesting paper on babeo and aueo published in CQ 66 , 388–98, Dr. A.S. Gratwick raised a number of questions bearing on my own discussion of the origin and development of the babeo+infinitive construction in CQ 65 , 215–31. First the collapse of the earlier future-tense system. As I said, this was ‘the product of a number of different linguistic events’, phonetic, grammatical, and semantic, which were summarized and illustrated on pp. 220–1 of my paper. Even so Dr. (...) Gratwick believes that I assigned too much weight to ‘phonetic attrition of the future simple’. (shrink)

This paper examines the distribution of thematic infinitive endings in early Greek epic in the context of the long-standing debate about the transmission and development of Homeric epic diction. There are no aorist infinitives in - in Homer which would scan as -before a consonant or caesura (for example *). It is argued that this artificially ending - should be viewed as an actual analogical innovation of the poetic language, resulting from a proportional analogy to the futures. The total absence (...) of aoristic - in Hesiod is unlikely to be coincidental: the analogical form must have been the product of a specifically East Ionic Kunstsprache, and so could have been simply unknown in some other Ionian school of epic poetry where Hesiod was trained. Finally, the striking avoidance of anapaestic aorist infinitives in - is argued to be explained better under the approach to the Aeolic elements in Homeric diction than under the theory. (shrink)

This article addresses “Creativity after Computation” by looking into the concept of artificial imagination, namely the machine’s ability to produce images that challenge artmaking and surprise human beings with the aid of machine learning algorithms. What is at stake is not only art and creativity but also the tension between the determination of machines and the freedom of human beings. This opposition restages Kant’s third antinomy in the contemporary technological condition. By referring to the debate on the question of imagination (...) in Kant, Heidegger, and Stiegler, the article suggests that imagination is always already artificial and that it is more productive to develop an organology of artificial imagination. It clarifies the notion of artificial imagination and offers an organological reading through a reinterpretation of Leibniz’s monadology, Kant’s sublime, and Schiller’s aesthetic education against the backdrop of recursive algorithms. (shrink)

It is well known that if the nonempty player of the Banach–Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box product topology. The converse of this implication may also be true: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.

Based on a recently published essay by Jeremy Gwiazda, I argue that the possibility that the present state of the universe is the product of an actually infinite series of causally-ordered prior events is impossible in principle, and thus that a major criticism of the Secunda Via of St. Thomas is baseless after all.

Results of Sierpiski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is ‘narrow’ in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set ω × ω is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite set. (...) Such partitions or coverings can exist only when the sets forming the product are of limited size.This paper considers such coverings for products of infinitely many sets . In this case, a covering of the product by narrow sets, one for each coordinate direction, will exist no matter how large the factor sets are. But if one restricts the sets used in the covering , then the existence of narrow coverings is related to a number of large cardinal properties: partition cardinals, the free subset problem, nonregular ultrafilters, and so on.One result given here is a relative consistency proof for a hypothesis used by S. Mrówka to produce a counterexample in the dimension theory of metric spaces. (shrink)

Amidst the surge in AI-oriented commercial ventures, music is a site of intensive efforts to innovate. A number of companies are seeking to apply AI to music production and consumption, and amongst them several are seeking to reinvent the music listening experience as adaptive, interactive, functional and infinitely generative. These are bold objectives, having no clear roadmap for what designs, technologies and use cases, if any, will be successful. Thus each company relies on speculative product visions. Through four case studies (...) of such companies, I consider how product visions must carefully provide a clear plan for developers and investors, whilst also remaining open to agile user-centred product development strategies, which I discuss in terms of the ‘blind search’ nature of innovation. I suggest that innovation in this area needs to be understood in terms of technological emergence, which is neither technologically determinist nor driven entirely by the visions of founders, but through a complex of interacting forces. I also consider, through these cases, how, through the accumulation of residual value, all such start-up work risks being exapted for more familiar extractive capitalist agendas under the general process that Doctorow calls “enshittification”. Lastly, I consider a number of other more specific ways in which these projects, if their growth is achieved, could influence music culture more broadly. (shrink)

The infinite-valued logic of ukasiewicz was originally defined by means of an infinite-valued matrix. ukasiewicz took special forms of negation and implication as basic connectives and proposed an axiom system that he conjectured would be sufficient to derive the valid formulas of the logic; this was eventually verified by M. Wajsberg. The algebraic counterparts of this logic have become know as Wajsberg algebras. In this paper we show that a Wajsberg algebra is complete and atomic (as a lattice) (...) if and only if it is a direct product of finite Wajsberg chains. The classical characterization of complete and atomic Boolean algebras as fields of sets is a particular case of this result. (shrink)

The physical process of observation is considered from a specific information theoretical viewpoint. Using the modified concept of an information based on infinite alternatives, a formalism is derived describing the elementary transfer of one bit of information. This bit of information is produced on a virtual (nonreal) sub-quantum level of physical description. The interpretation of the formalism yields the following, complementary points: (i) the effect of spatiotemporal delocalization on the sub-quantum level, and (ii) a possible access to the concept (...) of chaos as an intrinsic property of quantum systems. As a brief example, elementary information transfer is illustrated in a cosmological context. Finally, a formal approach to information production on the sub-quantum level is sketched on the basis of complex time. (shrink)

Throughout the twentieth century, critical art history often chose to ally itself with a restrictive brand of formalism. As a result, representation- and ideology-critical analyses regularly reduced the artwork to the bare bones (Hegel) of the material signifier in its social use. By contrast, in the texts assembled here, elements of a critical materialism are combined with an effort to reevaluate the meta-physical implications of modern abstraction and art since the 1960s. Taking Gilles Deleuze s readings of Spinoza, Nietzsche, and (...) Bergson as his starting point, the author delineates a topic in which the artwork s capacity for resistance is grounded in its relationship to an immanent infinity: the Spinozian substance, Nietzsche s Becoming, Bergson s duree. Against the backdrop of a critical rereading of Heidegger, this infinite dimension is interpreted in temporal and ontological terms as the vertical past of production, which can only be grasped in broken and technically encrypted form in the present shape and materiality of the artwork. Hence the notion of an aesthetics of production does not imply a nostalgia for the artisanal or for the artwork s singularity. The concept of production developed in this book aims at a realm that lies beyond finite representation but is still understood in materialist terms, and that threatens the circulation of positive, conceptually standardized knowledge. In case studies on Piet Mondrian, Marcel Duchamp, Thomas Hirschhorn, and Michael Asher and in framing essays on Kant and Nietzsche as well as Heidegger and Spinoza, this book articulates a concept of the artwork in the long modern era which takes account of the twentieth century s critique of metaphysics but without surrendering the truth claim of art and philosophy in favor of a culturalist and sociological relativism. ". (shrink)