1,721,553 research outputs found
Naturalness in Mathematics
No full textIn mathematical literature, it is quite common to make reference to an informal notion of naturalness: axioms or definitions may be defined as “natural,” and part of a proof may deserve the same label (i.e., “in a natural way...”). Our aim is to provide a philosophical account of these occurrences. The paper is divided in two parts. In the first part, some statistical evidence is considered, in order to show that the use of the word “natural,” within the mathematical discourse, largely increased in the last decades. Then, we attempt to develop a philosophical framework in order to encompass such an evidence. In doing so, we outline a general method apt to deal with this kind of vague notions – such as naturalness – emerging in mathematical practice. In the second part, we mainly tackle the following question: is naturalness a static or a dynamic notion? Thanks to the study of a couple of case studies, taken from set theory and computability theory, we answer that the notion of naturalness – as it is used in mathematics – is a dynamic one, in which normativity plays a fundamental role
Preservation of Suslin trees and side conditions
No full textWe show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto
Hilbert between the formal and the informal side of mathematics
Full text linkIn this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution
Cantor e l'infinito
No full textIn this article we review Cantor’s contribution to the construction of a theory of infinity. We will present and discuss the technical achievements and the philosophical ideas that brought Cantor to the creation of set theory and to its justification
Progress on fatigue characterization of ITER primary first wall mock-ups
No full textIn 2001, EFDA has assigned to ENEA a contract for the thermomechanical testing of six mock-ups of the ITER primary wall module. These small scale mock-ups, reproducing representative portions of the reference ITER primary wall panels, were fabricated during ITER EDA phase by solid hot isostatic pressing (HIPping) of an AISI 316L stainless steel back structure to a alumina dispersion strengthened (DS)-Cu alloy heat sink armored with beryllium tiles. The experimental program, carried-out at ENEA Brasimone CEF 1-2 thermal hydraulic facility, was focused on the thermal mechanical testing of these mock-ups aiming at verifying which tile geometry and manufacturing procedure assures the required reliability of the beryllium/DS-Cu alloy/SS joints at high incident heat flux (<0.8 MW/m2) both at steady state and under thermal fatigue tests. The paper presents the progress in the experimental activity of the first test campaign and the main thermomechanical FEM analyses after the detachment and rupture of Beryllium tiles from one mock-up
Thermal-mechanical test on ITER primary first wall mock-ups
No full textIn 1998, in the frame of the ITER EDA phase, an European R & D Programme for the Blanket Design was implemented for developing and selecting the materials and the relevant fabrication procedures for manufacturing the shielding modules of the ITER Primary Wall. The fabrication of several Beryllium armored small scale mock-ups, reproducing representative portions of a Primary Wall panels, was also launched (Fusion Technol. (1998) 195). Further experimental activities were also programmed for investigating the thermal-mechanical behavior of these mock-ups at high heat flux and under thermal fatigue tests. In 2001, the ITER European Home Team decided to assign to ENEA a contract for the thermal fatigue testing of six mock-ups aiming at verifying the reliability of the Beryllium/Dispersion Strengthened Copper alloy/Stainless Steel and Beryllium/Precipitation hardened Copper alloy/Stainless Steel joints manufactured by solid Hot Isostatic Pressing (HIP) procedure (Technical Specification for the Thermal Fatigue Tests of Be protected EDA Mock-ups). The paper presents the results of the FEM thermal-mechanical analyses performed by ANSYS code and the progress of the first test campaign
Hilbert, Completeness and Geometry
Full text linkThis paper aims to show how the mathematical content of Hilbert’s Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring out how Hilbert’s conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play inHilbert’s foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert’s position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert’s for the completeness of set theory.
Genericity and Arbitrariness
Full text linkWe compare the notions of genericity and arbitrariness on the basis of the realist import of the method of forcing. We argue that Cohen's Theorem, similarly to Cantor's Theorem, can be considered a meta-theoretical argument in favor of the existence of uncountable collections. Then we discuss the effects of this meta-theoretical perspective on Skolem's Paradox. We conclude discussing how the connection between arbitrariness and genericity can offer arguments in favor of Forcing Axioms
A note on the introduction of Hilbert’s Grundlagen der Geometrie
Full text linkWe present and discuss a change in the introduction of Hilbert's Grundlagen der Geometrie between the first and the subsequent editions: the disappearance of the reference to the independence of the axioms. We briefly outline the theoretical relevance of the notion of independence in Hilbert's work and we suggest that a possible reason for this disappearance is the discovery that Hilbert's axioms were not, in fact, independent. In the end we show how this change gives textual evidence for the connection between the notions of independence and simplicity
Hilbert, completeness and geometry
Full text linkThis paper aims to show how the mathematical content of Hilbert\u27s Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring out how Hilbert\u27s conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play in Hilbert\u27s foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert\u27s position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert\u27s for the completeness of set theory
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