The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Categorical Ontology of Complex Systems, Meta-Systems and Theory of Levels: The Emergence of Life, Human Consciousness and Society

Single cell interactomics in simpler organisms, as well as somatic cell interactomics in multicellular organisms, involve biomolecular interactions in complex signalling pathways that were recently represented in modular terms by quantum automata with ‘reversible behavior’ representing normal cell cycling and division. Other implications of such quantum automata, modular modeling of signaling pathways and cell differentiation during development are in the fields of neural plasticity and brain development leading to quantum-weave dynamic patterns and specific molecular processes underlying extensive memory, learning, anticipation mechanisms and the emergence of human consciousness during the early brain development in children. Cell interactomics is here represented for the first time as a mixture of ‘classical’ states that determine molecular dynamics subject to Boltzmann statistics and ‘steady-state’, metabolic (multi-stable) manifolds, together with ‘configuration’ spaces of metastable quantum states emerging from complex quantum dynamics of interacting networks of biomolecules, such as proteins and nucleic acids that are now collectively defined as quantum interactomics. On the other hand, the time dependent evolution over several generations of cancer cells --that are generally known to undergo frequent and extensive genetic mutations and, indeed, suffer genomic transformations at the chromosome level (such as extensive chromosomal aberrations found in many colon cancers)-- cannot be correctly represented in the ‘standard’ terms of quantum automaton modules, as the normal somatic cells can. This significant difference at the cancer cell genomic level is therefore reflected in major changes in cancer cell interactomics often from one cancer cell ‘cycle’ to the next, and thus it requires substantial changes in the modeling strategies, mathematical tools and experimental designs aimed at understanding cancer mechanisms. Novel solutions to this important problem in carcinogenesis are proposed and experimental validation procedures are suggested. From a medical research and clinical standpoint, this approach has important consequences for addressing and preventing the development of cancer resistance to medical therapy in ongoing clinical trials involving stage III cancer patients, as well as improving the designs of future clinical trials for cancer treatments.\ud \ud \ud KEYWORDS: Emergence of Life and Human Consciousness;\ud Proteomics; Artificial Intelligence; Complex Systems Dynamics; Quantum Automata models and Quantum Interactomics; quantum-weave dynamic patterns underlying human consciousness; specific molecular processes underlying extensive memory, learning, anticipation mechanisms and human consciousness; emergence of human consciousness during the early brain development in children; Cancer cell ‘cycling’; interacting networks of proteins and nucleic acids; genetic mutations and chromosomal aberrations in cancers, such as colon cancer; development of cancer resistance to therapy; ongoing clinical trials involving stage III cancer patients’ possible improvements of the designs for future clinical trials and cancer treatments. \ud \u

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes

Quantum Ecologies in Cosmological Infrastructures: A Critical Holographers Encounters with the Meta/Physics of Landscape-Laboratories

Quantum Ecologies interrogates the role of physics in the construction of an indifferent and disenchanted universe. It explores conceptual resonances within and between new materialism, Indigenous philosophy of place, science fiction, and art. Quantum Ecologies recognizes that the world is alive and wise and considers relevant modes of responsible address within and as the Earth. Through theoretical and historical analysis, site based research and a/v installation Quantum Ecologies has developed the heuristic of the ‘holographic’ as a way to attend to the multi-temporal, co-present, and multi-scalar pluralities and layers of knowing, agency, and landscape. This feminist, anti-colonial art-science framework for critically engaging (physics) sites and philosophies addresses the scientific cosmology of the West that (inadvertently) legitimates the exploitation, dispossession, and extraction of Earthly beings and bodies. Holography as critical interferometry is applied to experimental sites and assemblages known as ‘landscape-laboratories’ as a mode of both reading and (re)writing them. My field/work has taken place in remote environmentally protected sites that are entangled and instrumentalized as cosmological sensing arrays, experimental nuclear fusion energy, or dark matter particle physics laboratories in Russia, France, the UK, Germany, and Canada. By thinking through the strangeness of these planetary quantum assemblages alongside sciences inheritances and genealogies in magic, alchemy, and mysticism I argue for the necessity of ‘another science’ that is situated, compassionate, and responsible. Quantum Ecologies proposes a plural, poly-perspectival assessment of place, where accounting for the promiscuous more-than of materials, sites, forces, and energies is a necessary and continuous (re)configuring of meta/physics and respectful anti-colonial engagement with Land

Quasi-Linear Size Zero Knowledge from Linear-Algebraic PCPs

The seminal result that every language having an interactive proof also has a zero-knowledge interactive proof assumes the existence of one-way functions. Ostrovsky and Wigderson (ISTCS 1993) proved that this assumption is necessary: if one-way functions do not exist, then only languages in BPP have zero-knowledge interactive proofs. Ben-Or et al. (STOC 1988) proved that, nevertheless, every language having a multi-prover interactive proof also has a zero-knowledge multi-prover interactive proof, unconditionally. Their work led to, among many other things, a line of work studying zero knowledge without intractability assumptions. In this line of work, Kilian, Petrank, and Tardos (STOC 1997) defined and constructed zero-knowledge probabilistically checkable proofs (PCPs). While PCPs with quasilinear-size proof length, but without zero knowledge, are known, no such result is known for zero knowledge PCPs. In this work, we show how to construct ``2-round\u27\u27 PCPs that are zero knowledge and of length \tilde{O}(K) where K is the number of queries made by a malicious polynomial time verifier. Previous solutions required PCPs of length at least K^6 to maintain zero knowledge. In this model, which we call *duplex PCP* (DPCP), the verifier first receives an oracle string from the prover, then replies with a message, and then receives another oracle string from the prover; a malicious verifier can make up to K queries in total to both oracles. Deviating from previous works, our constructions do not invoke the PCP Theorem as a blackbox but instead rely on certain algebraic properties of a specific family of PCPs. We show that if the PCP has a certain linear algebraic structure --- which many central constructions can be shown to possess, including [BFLS91,ALMSS98,BS08] --- we can add the zero knowledge property at virtually no cost (up to additive lower order terms) while introducing only minor modifications in the algorithms of the prover and verifier. We believe that our linear-algebraic characterization of PCPs may be of independent interest, as it gives a simplified way to view previous well-studied PCP constructions

Culture, worldview and transformative philosophy of mathematics education in Nepal: a cultural-philosophical inquiry

This thesis portrays my multifaceted and emergent inquiry into the protracted problem of culturally decontextualised mathematics education faced by students of Nepal, a culturally diverse country of south Asia with more than 90 language groups. I generated initial research questions on the basis of my history as a student of primary, secondary and university levels of education in Nepal, my Master’s research project, and my professional experiences as a teacher educator working in a university of Nepal between 2004 and 2006. Through an autobiographical excavation of my experiences of culturally decontextualised mathematics education, I came up with several emergent research questions, leading to six key themes of this inquiry: (i) hegemony of the unidimensional nature of mathematics as a body of pure knowledge, (ii) unhelpful dualisms in mathematics education, (iii) disempowering reductionisms in curricular and pedagogical aspects, (iv) narrowly conceived ‘logics’ that do not account for meaningful lifeworld-oriented thinking in mathematics teaching and learning, (v) uncritical attitudes towards the image of curriculum as a thing or object, and (vi) narrowly conceived notions of globalisation, foundationalism and mathematical language that give rise to a decontextualised mathematics teacher education program.With these research themes at my disposal my aim in this research was twofold. Primarily, I intended to explore, explain and interpret problems, issues and dilemmas arising from and embedded in the research questions. Such an epistemic activity of articulation was followed by envisioning, an act of imagining futures together with reflexivity, perspectival language and inclusive vision logics.In order to carry out both epistemic activities – articulating and envisioning – I employed a multi-paradigmatic research design space, taking on board mainly the paradigms of criticalism, postmodernism, interpretivism and integralism. The critical paradigm offered a critical outlook needed to identify the research problem, to reflect upon my experiences as a mathematics teacher and teacher educator, and to make my lifetime’s subjectivities transparent to readers, whereas the paradigm of postmodernism enabled me to construct multiple genres for cultivating different aspects of my experiences of culturally decontextualised mathematics education. The paradigm of interpretivism enabled me to employ emergence as the hallmark of my inquiry, and the paradigm of integralism acted as an inclusive meta-theory of the multi-paradigmatic design space for portraying my vision of an inclusive mathematics education in Nepal.Within this multi-paradigmatic design space, I chose autoethnography and small p philosophical inquiry as my methodological referents. Autoethnography helped generate the research text of my cultural-professional contexts, whereas small p philosophical inquiry enabled me to generate new knowledge via a host of innovative epistemologies that have the goal of deepening understanding of normal educational practices by examining them critically, identifying underpinning assumptions, and reconstructing them through scholarly interpretations and envisioning. Visions cultivated through this research include: (i) an inclusive and multidimensional image of the nature of mathematics as an im/pure knowledge system, (ii) the metaphors of thirdspace and dissolution for conceiving an inclusive mathematics education, (iii) a multilogical perspective for morphing the hegemony of reductionism-inspired mathematics education, (iv) an inclusive image of mathematics curriculum as montage that provides a basis for incorporating different knowledge systems in mathematics education, and (v) perspectives of glocalisation, healthy scepticism and multilevel contextualisation for constructing an inclusive mathematics teacher education program