Related papers: Mathematical Interpretation of Plato's Third Man A…
Turing machines and spin models share a notion of universality according to which some simulate all others. Is there a theory of universality that captures this notion? We set up a categorical framework for universality which includes as…
In La G\'eom\'etrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation.…
Nonmonotonic logics are usually characterized by the presence of some notion of 'conditional' that fails monotonicity. Research on nonmonotonic logics is therefore largely concerned with the defeasibility of argument forms and the…
The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the…
In his proof of Fermat's Last Theorem, Wiles deployed a commutative algebra technique, namely a numerical criterion for detecting isomorphisms of rings. In our recent work we pick up on Wiles' work and generalize the numerical criterion to…
We study Tao's finitary viewpoint of convergence in metric spaces, as captured by the notion of metastability. We adopt the perspective of continuous model theory. We show that, in essence, metastable convergence with a given rate is the…
In this paper, I present a critical discussion of mathematical arguments employed in the philosophy of event of Alain Badiou. On the basis of "Being and Event" as well as his other writings, I analyze the main notions of his philosophy such…
Tarski gave a general semantics for deductive reasoning: a formula a may be deduced from a set A of formulas iff a holds in all models in which each of the elements of A holds. A more liberal semantics has been considered: a formula a may…
Mathematics is usually regarded as a kind of language. The essential behavior of physical phenomena can be expressed by mathematical laws, providing descriptions and predictions. In the present essay I argue that, although mathematics can…
One of the main claims of the paper is that Dirac's calculus and broader theories of physics can be treated as theories written in the language of Continuous Logic. Establishing its true interpretation (model) is a model theory problem. The…
It was shown by Visser that Peano Arithmetic has the property that any two bi-interpretable extensions of it (in the same language) are equivalent. Enayat proposed to refer to this property of a theory as tightness and to carry out a more…
A new mathematics, the constructive one, characterizes a singular limit as undecidable. Hence, a singular limit between two theories actually represents a difference between two different kinds of mathematics. This particular situation…
This text summarizes and expands the content of a general audience talk given in 2018 at the University of Mainz. Motivated by recent developments in dependent type theory and infinity category theory, it presents a history of ideas around…
We show that a special case of the Feferman-Vaught composition theorem gives rise to a natural notion of automata for finite words over an infinite alphabet, with good closure and decidability properties, as well as several logical…
In the present paper, the existence and multiplicity problems of extensions are addressed. The focus is on extension of the stable type. The main result of the paper is an elegant characterization of the existence and multiplicity of…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
In recent publications in physics and mathematics, concerns have been raised about the use of real numbers to describe quantities in physics, and in particular about the usual assumption that physical quantities are infinitely precise. In…
It is generally accepted that the incompleteness of first-order number theory (PA) is established by an application of Godel's proof. This paper shows that the arithmetization of the syntax of PA implies that the hypothesised class of PA…
In many situations humans have to reason with inconsistent knowledge. These inconsistencies may occur due to not fully reliable sources of information. In order to reason with inconsistent knowledge, it is not possible to view a set of…
Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T formulated in a finite language such that T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an…