Adjoint exactness

Plato's ideas and Aristotle's real types from the classical age, Nominalism and Realism of the mediaeval period and Whitehead's modern view of the world as pro- cess all come together in the formal representation by category theory of exactness in adjointness (a). Concepts of exactness and co-exactness arise naturally from ad- jointness and are needed in current global problems of science. If a right co-exact valued left-adjoint functor ( ) in a cartesian closed category has a right-adjoint left- exact functor ( ), then physical stability is satis ed if itself is also a right co-exact left-adjoint functor for the right-adjoint left exact functor ( ): a a . These concepts are discussed here with examples in nuclear fusion, in database interroga- tion and in the cosmological ne structure constant by the Frederick construction

On a Contrastive Criterion of Testability II: The Material Inadequacy of Contrastive Testability

Elliott Sober has suggested his contrastive criterion of testability as an improvement over previous criteria of empirical significance like falsifiability and the standard Bayesian criterion of empirical significance. I argue that the criterion fails to meet four of the conditions of adequacy for a criterion of empirical significance that follow from Sober’s position or are presumed in his arguments. I suggest to define empirical significance as empirical non-equivalence to a tautology, because this definition does meet the conditions of adequacy. Specifically, it is equivalent to the standard Bayesian criterion of empirical significance whenever all probabilities are defined and contains falsifiability as a special case. This latter feature is important because those conditions of adequacy that apply to criteria of deductive empirical significance single out falsifiability

Takeuti's proof theory in the context of the Kyoto School

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project

Epistemology of String Theory in Quantum Gravity

In quantum field theory, the main obstacle is the occurrence of the untreatable infinities in the interactions of the particles due to the possibility of arbitrary distances between the point particles. Strings, as extended objects, provide a better framework, which allows finite calculations. String theory is part of a research program in which point particles in particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate through space and interact with one another. The purpose of string theory was to replace elementary particles with one-dimensional strings in order to unify quantum physics and gravity. DOI: 10.13140/RG.2.2.18894.8224

Infinite Probabilistic Databases

Probabilistic databases (PDBs) are used to model uncertainty in data in a quantitative way. In the standard formal framework, PDBs are finite probability spaces over relational database instances. It has been argued convincingly that this is not compatible with an open-world semantics (Ceylan et al., KR 2016) and with application scenarios that are modeled by continuous probability distributions (Dalvi et al., CACM 2009). We recently introduced a model of PDBs as infinite probability spaces that addresses these issues (Grohe and Lindner, PODS 2019). While that work was mainly concerned with countably infinite probability spaces, our focus here is on uncountable spaces. Such an extension is necessary to model typical continuous probability distributions that appear in many applications. However, an extension beyond countable probability spaces raises nontrivial foundational issues concerned with the measurability of events and queries and ultimately with the question whether queries have a well-defined semantics. It turns out that so-called finite point processes are the appropriate model from probability theory for dealing with probabilistic databases. This model allows us to construct suitable (uncountable) probability spaces of database instances in a systematic way. Our main technical results are measurability statements for relational algebra queries as well as aggregate queries and Datalog queries

Eventology versus contemporary theories of uncertainty

The development of probability theory together with the Bayesian approach in the three last centuries is caused by two factors: the variability of the physical phenomena and partial ignorance about them. As now it is standard to believe [Dubois, 2007], the nature of these key factors is so various, that their descriptions are required special uncertainty theories, which differ from the probability theory and the Bayesian credo, and provide a better account of the various facets of uncertainty by putting together probabilistic and set-valued representations of information to catch a distinction between variability and ignorance. Eventology [Vorobyev, 2007], a new direction of probability theory and philosophy, offers the original event approach to the description of variability and ignorance, entering an agent, together with his/her beliefs, directly in the frameworks of scientific research in the form of eventological distribution of his/her own events. This allows eventology, by putting together probabilistic and set-event representation of information and philosophical concept of event as co-being [Bakhtin, 1920], to provide a unified strong account of various aspects of uncertainty catching distinction between variability and ignorance and opening an opportunity to define imprecise probability as a probability of imprecise event in the mathematical frameworks of Kolmogorov's probability theory [Kolmogorov, 1933].uncertainty, probability, event, co-being, eventology, imprecise event