Extensional realizability

AbstractTwo straightforward “extensionalisations” of Kleene's realizability are considered; denoted re and e. It is shown that these realizabilities are not equivalent. While the re-notion is (as a relation between numbers and sentences) a subset of Kleene's realizability, the e-notion is not. The problem of an axiomatization of e-realizability is attacked and one arrives at an axiomatization over a conservative extension of arithmetic, in a language with variables for finite sets. A derived rule for arithmetic is obtained by the use of a q-variant of e-realizability; this rule subsumes the well-known Extended Church's Rule. The second part of the paper focuses on toposes for these realizabilities. By a relaxation of the notion of partial combinatory algebra, a new class of realizability toposes emerges. Relationships between the various realizability toposes are given, and results analogous to Robinson and Rosolini's characterization of the effective topos, are obtained for a topos generalizing e-realizability

Formalizing Constructive Analysis: A comparison of minimal systems and a study of uniqueness principles.

Αυτή η διατριβή εξετάζει ορισμένες πλευρές της τυποποίησης και της αξιωματικοποίησης της κατασκευαστικής ανάλυσης. Η έρευνα στους κλάδους της κατασκευαστικής ανάλυσης που αντιστοιχούν στις διάφορες εκδοχές κατασκευαστικότητας διεξάγεται σε μια πλειάδα τυπικών ή όχι συστημάτων, των οποίων οι σχέσεις είναι ασαφείς. Αυτό το πρόβλημα αποβαίνει κρίσιμο για την ανάπτυξη της σχετικά νέας περιοχής των κατασκευαστικών ανάστροφων μαθηματικών. Η εργασία αυτή συμβάλλει σε μια πιο καθαρή εικόνα. Το Μέρος 1 περιέχει μία ακριβή σύγκριση των δύο ευρύτερα χρησιμοποιούμενων συστημάτων που τυποποιούν τον κοινό πυρήνα της κατασκευαστικής, της ενορατικής, της αναδρομικής και της κλασικής ανάλυσης, των Μ και EL, των Kleene και Troelstra, αντιστοίχως. Αποδεικνύεται ότι το EL είναι ασθενέστερο από το M και ότι η διαφορά τους αποτυπώνεται από μια αρχή η οποία εγγυάται την ύπαρξη χαρακτηριστικής συνάρτησης για κάθε αποκρίσιμο κατηγόρημα φυσικών αριθμών. Με παρόμοια επιχειρήματα προκύπτουν συγκρίσεις για τα περισσότερα από τα χρησιμοποιούμενα ελαχιστικά συστήματα. Στην κατασκευαστική ανάλυση χρησιμοποιούνται διάφορες αρχές επιλογής, συνέχειας και άλλες. Στο Μέρος 2, μελετώνται σχέσεις μεταξύ πολλών από αυτές, στις εκδοχές τους με μία συνθήκη μοναδικότητας, ένα χαρακτηριστικό από το οποίο απορρέουν ενδιαφέρουσες ιδιότητες, καθώς και σχέσεις μεταξύ αυτών των αρχών και μη κατασκευαστικών λογικών αρχών, στο πνεύμα των ανάστροφων μαθηματικών.This dissertation investigates certain aspects of the formalization and axiomatization of constructive analysis. The research in the branches of constructive analysis corresponding to the various forms of constructivism is carried out in a multitude of formal or informal systems, whose relations are unclear. This problem becomes quite crucial for the development of the relatively new field of constructive reverse mathematics. This work contributes to a clearer picture. Part 1 contains a precise comparison of the two most widely used systems which formalize the common core of constructive, intuitionistic, recursive and classical analysis, namely Kleene's M and Troelstra's EL. It is shown that EL is weaker than M and that their difference is captured by a function existence principle asserting that every decidable predicate of natural numbers has a characteristic function. Applying similar arguments, comparisons of most of the used minimal systems are obtained. In constructive analysis, various forms of choice principles, continuity principles and many others are used. Part 2 studies relations between many of them, in their versions having a uniqueness condition, a feature from which interesting properties follow, as well as relations between these principles and non-constructive logical principles, in the spirit of reverse mathematics

Stepping Beyond the Newtonian Paradigm in Biology. Towards an Integrable Model of Life: Accelerating Discovery in the Biological Foundations of Science

The INBIOSA project brings together a group of experts across many disciplines who believe that science requires a revolutionary transformative step in order to address many of the vexing challenges presented by the world. It is INBIOSA’s purpose to enable the focused collaboration of an interdisciplinary community of original thinkers. This paper sets out the case for support for this effort. The focus of the transformative research program proposal is biology-centric. We admit that biology to date has been more fact-oriented and less theoretical than physics. However, the key leverageable idea is that careful extension of the science of living systems can be more effectively applied to some of our most vexing modern problems than the prevailing scheme, derived from abstractions in physics. While these have some universal application and demonstrate computational advantages, they are not theoretically mandated for the living. A new set of mathematical abstractions derived from biology can now be similarly extended. This is made possible by leveraging new formal tools to understand abstraction and enable computability. [The latter has a much expanded meaning in our context from the one known and used in computer science and biology today, that is "by rote algorithmic means", since it is not known if a living system is computable in this sense (Mossio et al., 2009).] Two major challenges constitute the effort. The first challenge is to design an original general system of abstractions within the biological domain. The initial issue is descriptive leading to the explanatory. There has not yet been a serious formal examination of the abstractions of the biological domain. What is used today is an amalgam; much is inherited from physics (via the bridging abstractions of chemistry) and there are many new abstractions from advances in mathematics (incentivized by the need for more capable computational analyses). Interspersed are abstractions, concepts and underlying assumptions “native” to biology and distinct from the mechanical language of physics and computation as we know them. A pressing agenda should be to single out the most concrete and at the same time the most fundamental process-units in biology and to recruit them into the descriptive domain. Therefore, the first challenge is to build a coherent formal system of abstractions and operations that is truly native to living systems. Nothing will be thrown away, but many common methods will be philosophically recast, just as in physics relativity subsumed and reinterpreted Newtonian mechanics. This step is required because we need a comprehensible, formal system to apply in many domains. Emphasis should be placed on the distinction between multi-perspective analysis and synthesis and on what could be the basic terms or tools needed. The second challenge is relatively simple: the actual application of this set of biology-centric ways and means to cross-disciplinary problems. In its early stages, this will seem to be a “new science”. This White Paper sets out the case of continuing support of Information and Communication Technology (ICT) for transformative research in biology and information processing centered on paradigm changes in the epistemological, ontological, mathematical and computational bases of the science of living systems. Today, curiously, living systems cannot be said to be anything more than dissipative structures organized internally by genetic information. There is not anything substantially different from abiotic systems other than the empirical nature of their robustness. We believe that there are other new and unique properties and patterns comprehensible at this bio-logical level. The report lays out a fundamental set of approaches to articulate these properties and patterns, and is composed as follows. Sections 1 through 4 (preamble, introduction, motivation and major biomathematical problems) are incipient. Section 5 describes the issues affecting Integral Biomathics and Section 6 -- the aspects of the Grand Challenge we face with this project. Section 7 contemplates the effort to formalize a General Theory of Living Systems (GTLS) from what we have today. The goal is to have a formal system, equivalent to that which exists in the physics community. Here we define how to perceive the role of time in biology. Section 8 describes the initial efforts to apply this general theory of living systems in many domains, with special emphasis on crossdisciplinary problems and multiple domains spanning both “hard” and “soft” sciences. The expected result is a coherent collection of integrated mathematical techniques. Section 9 discusses the first two test cases, project proposals, of our approach. They are designed to demonstrate the ability of our approach to address “wicked problems” which span across physics, chemistry, biology, societies and societal dynamics. The solutions require integrated measurable results at multiple levels known as “grand challenges” to existing methods. Finally, Section 10 adheres to an appeal for action, advocating the necessity for further long-term support of the INBIOSA program. The report is concluded with preliminary non-exclusive list of challenging research themes to address, as well as required administrative actions. The efforts described in the ten sections of this White Paper will proceed concurrently. Collectively, they describe a program that can be managed and measured as it progresses

Learning, realizability and games in classical arithmetic

PhDAbstract. In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel modi ed realizability to a classical fragment of rst order Arithmetic, Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to 0 1 formulas). We introduce a new realizability semantics we call \Interactive Learning-Based Realizability". Our realizers are self-correcting programs, which learn from their errors and evolve through time, thanks to their ability of perpetually questioning, testing and extending their knowledge. Remarkably, that capability is entirely due to classical principles when they are applied on top of intuitionistic logic. Secondly, we extend the class of learning based realizers to a classical version PCFClass of PCF and, then, compare the resulting notion of realizability with Coquand game semantics and prove a full soundness and completeness result. In particular, we show there is a one-to-one correspondence between realizers and recursive winning strategies in the 1-Backtracking version of Tarski games. Third, we provide a complete and fully detailed constructive analysis of learning as it arises in learning based realizability for HA+EM1, Avigad's update procedures and epsilon substitution method for Peano Arithmetic PA. We present new constructive techniques to bound the length of learning processes and we apply them to reprove - by means of our theory - the classic result of G odel that provably total functions of PA can be represented in G odel's system T. Last, we give an axiomatization of the kind of learning that is needed to computationally interpret Predicative classical second order Arithmetic. Our work is an extension of Avigad's and generalizes the concept of update procedure to the trans nite case. Trans- nite update procedures have to learn values of trans nite sequences of non computable functions in order to extract witnesses from classical proofs

Cognition Without Construction: Kant, Maimon, and the Transcendental Philosophy of Mathematics

In the Critique of Pure Reason, Immanuel Kant takes the ostensive constructions characteristic of Euclidean-style demonstrations to be the paradigm of both mathematical proofs and synthetic a priori cognition in general. However, the development of calculus included a number of techniques for representing infinite series of sums or differences, which could not be represented with the direct geometrical demonstrations of the past. Salomon Maimon’s Essay on Transcendental Philosophy addresses precisely this disparity. Maimon, owing much to G. W. Leibniz, proposes that differentials of sensation achieve what Kantian constructions could not. More importantly, Maimon develops a kind of symbolic cognition that is not delimited by the constraint of the pure forms of intuition. The mind does not construct its objects, but constructs itself through inquiry into the real objects of thought

Toward an Analysis of the Abductive Moral Argument for God’s Existence: Assessing the Evidential Quality of Moral Phenomena and the Evidential Virtuosity of Christian Theological Models

The moral argument for God’s existence is perhaps the oldest and most salient of the arguments from natural theology. In contemporary literature, there has been a focus on the abductive version of the moral argument. Although the mode of reasoning, abduction, has been articulated, there has not been a robust articulation of the individual components of the argument. Such an articulation would include the data quality of moral phenomena, the theoretical virtuosity of theological models that explain the moral phenomena, and how both contribute to the likelihood of moral arguments. The goal of this paper is to provide such an articulation. Our method is to catalog the phenomena, sort them by their location on the emergent hierarchy of sciences, then describe how the ecumenical Christian theological model exemplifies evidential virtues in explaining them. Our results show that moral arguments are neither of the highest or lowest quality yet can be assented to on a principled level of investigation, especially given existential considerations