Title: Some philosophical debates around distributed computing.

Abstract: The notion of computation is important to philosophers of mind and cognitive scientists because computer is supposed to work as a model of mind. TM type computation proceeds through manipulation of symbols having language-like structure. But in distributed brain-style computing, units of computing though representational are not language-like. So the first debate that I am going to give an outline of is: is distributed computing as effective a way of mind-modeling as classical computing? The second debate centers round the question whether distributed computing necessarily leads to elimination of propositional states and thus undermines the case for both common-sense psychology and logic?

Title: Expert agreement in peer review and the epistemic exceptionality of mathematics.

Abstract: It is seen as one of the strengths of modern science that its findings are intersubjectively stable: two independent referees who check a scientific fact are expected to reach the same conclusion. This general principle translates into a highly normative belief that in scientific practice, experts should be expected to agree. While this agreement will not always be complete, especially in the experimental sciences, mathematics has been seen by some as an "epistemic exception" where complete agreement is possible.

As a consequence, epert agreement has been used as a proxy for the intersubjectivity of judgments about scientific facts. In a famous paper with the exasperated subtitle "Is agreement between reviewers any greater than would be expected by chance alone?", Rothwell & Martyn (2000) studied this agreement in the case of peer review. Alas, the answer to their question was negative. Geist, Loewe, & Van Kerkhove (2010) did a similar analysis for mathematics and theoretical computer science and obtained slightly higher agreement values than those found by Rothwell & Martyn, but not the very high values that would be expected if mathematics was indeed an epistemic exception.

In this talk, we explore whether it is reasonable to expect that peer review agreement can be used as a proxy for intersubjectivity of agreement about scientific facts.

Title: Non-distributive logics: from semantics to meaning.

Abstract: The term ‘non-distributive logics’ refers to the wide family of non-classical propositional logics in which the distributive laws α ∧(β ∨γ) ⊢ (α ∧β)∨(α ∧γ) and (α ∨β)∧(α ∨γ) ⊢ α ∨(β ∧γ) do not need to be valid. Since the rise of very well known instances such as quantum logic, interest in non-distributive logics has been building steadily over the years, motivated by insights from a range of fields in logic and neighbouring disciplines. Techniques and ideas have come from pure mathematical areas such as lattice theory, duality and representation, and areas in mathematical logic such as algebraic proof theory, but also from the philosophical and formal foundations of quantum physics, philosophical logic, theoretical computer science, and formal linguistics.

We will discuss an ongoing line of research in the relational (non topological) semantics of non-distributive logics, which is technically rooted in duality and (generalized) correspondence theory. Not dissimilarly from the conceptual contribution of Kripke frames to the intuitive understanding of modal logics in various signatures, the relational semantics of non-distributive logics can help to illuminate the intuitive meaning of non-distributive logics at a more fundamental and conceptual level.

We discuss the application of the dual characterization methodology to introduce two relational semantic frameworks for non-distributive logics: the polarity-based frames and the graph-based frames. Despite their common root, polarity-based and graph-based semantics give rise to two radically different intuitive interpretations of non-distributive logics: namely, the polarity-based semantics supports the interpretation of non-distributive logics as logics of categories and formal concepts; the graph-based semantics supports a view of non-distributive logics as hyper-constructivist logics, i.e. logics in which the principle of excluded middle fails at the meta-linguistic level (in the sense that, at states in graph-based models, formulas can be satisfied, refuted or neither), and hence their propositional base generalizes intuitionistic logic in the same way in which intuitionistic logic generalizes classical logic. Consequently, we will argue that graph-based semantics supports the interpretation of non-distributive logics as logics of evidential reasoning.

Title: Towards a Philosophical Theory of Computation.

Abstract: The question, "What is an algorithm?", has important philosophical dimensions: ontological, epistemological and ethical. Mathematicians and physicists are used to talking of “in principle” solutions, "leaving it to implementations" to work out the details. But what if there is another principle that limits implementations as well? Complexity and resource consciousness are modern concerns, both in the physical world and in the world of ideas.

In this talk we discuss some questions of immense philosophical relevance, where computability and complexity play an important role, as also notions like the nature of time, knowledge, communication, agency etc. that are relevant in the context of computing agents and formal logics.

Title: Intuitionistic and constructive set theories: An appetizer.

Abstract: There are many reasons for studying set theories based on intuitionistic or semi-intuitionistic logic. To mention just a few:

(a) If one conceives of the universe of sets as an evolving realm of objects, the old distinction between the actual and the potential, going back to Aristotle, is very pertinent. It has been argued that the logical rules governing the quantifiers in a potential universe ought to be those of intuitionistic logic.

Another place where intuitionistic logic has been proposed as the germane logic is the philosophical and mathematical literature on vagueness. It is concerned with objects that are inherently vague, namely objects whose identity conditions are objectively indeterminate.

(b) Intuitionistic set theory is also very desirable from a computational and proof mining point of view in that one can extract algorithmic information from proofs in such theories.

(c) In mathematics (e.g. algebraic geometry) large categories such as topoi play an important role. The latter can be viewed as models of set theory, but crucially the internal logic of these worlds is generally inuitionistic rather than classical logic.

Mathematical logic is a great tool for furnishing models of all kinds of delicate and elusive objects, e.g., nonstandard analysis for justifying infinitesimals and set theory with the antifoundation axiom for modelling all kinds of circular phenomena. With intuitionistic logic one can go a step further in that one can even model phenomena that are classically impossible. To name just a few, there are (realizability) models of intuitionistic set theories (even with large cardinals) in which all functions from naturals to naturals are computable or all functions from the reals to the reals are continuous.

A further remarkable development in this vein is smooth infinitesimal analysis, featuring nilpotent infinitesimals. In it one can recover a great deal of scientifically applicable classical analysis without having to deploy limit procedures.