scholarly journals Relevant Arithmetic and Mathematical Pluralism

2021 ◽  
Vol 18 (5) ◽  
pp. 569-596
Author(s):  
Zach Weber
Keyword(s):  
Mathematical Practice ◽  
Worked Examples ◽  
Incompleteness Theorem ◽  
Classical Counterpart ◽  
The One ◽  
Mathematical Pluralism

In The Consistency of Arithmetic and elsewhere, Meyer claims to “repeal” Goedel’s second incompleteness theorem. In this paper, I review his argument, and then consider two ways of understanding it: from the perspective of mathematical pluralism and monism, respectively. Is relevant arithmetic just another legitimate practice among many, or is it a rival of its classical counterpart—a corrective to Goedel, setting us back on the path to the (One) True Arithmetic? To help answer, I sketch a few worked examples from relevant mathematics, to see what a non-classical (re)formulation of mathematics might look like in practice. I conclude that, while it is unlikely that relevant arithmetic describes past and present mathematical practice, and so might be most acceptable as a pluralist enterprise, it may yet prescribe a more monistic future venture.

scholarly journals Critical Plural Logic†

2020 ◽  
Vol 28 (2) ◽  
pp. 172-203
Author(s):  
Salvatore Florio ◽  
Øystein Linnebo
Keyword(s):  
Mathematical Practice ◽  
The Other ◽  
Plural Logic ◽  
Traditional Form ◽  
Other Hand ◽  
The One

Abstract What is the relation between some things and the set of these things? Mathematical practice does not provide a univocal answer. On the one hand, it relies on ordinary plural talk, which is implicitly committed to a traditional form of plural logic. On the other hand, mathematical practice favors a liberal view of definitions which entails that traditional plural logic must be restricted. We explore this predicament and develop a “critical” alternative to traditional plural logic.

Innovations in Fluid Mechanics Education at the Technical University of Berlin

1982 ◽  
Author(s):  
H. Schade ◽  
H.-H. Fernholz
Keyword(s):  
New York ◽  
Fluid Mechanics ◽  
Engineering Students ◽  
Worked Examples ◽  
Advantages And Disadvantages ◽  
University Of Berlin ◽  
Technical University ◽  
One Year ◽  
The One ◽  
Mechanics Education

At the Technical University of Berlin, the usual Fluid Mechanics Course for engineering students has been offered since 1977 in two different ways simultaneously, between which the students may choose. One is the traditional scheme of lectures with additional example classes; students can obtain a typewritten summary of the course without the examples. As an alternative the students are provided with a complete text of the course arranged for self-study together with worked examples. This text has now been published as a book (H. Schade, E. Kunz: Strömungslehre. Berlin, New York: de Gruyter 1980). In the alternative course the students meet in groups of about five with a professor or a teaching assistant once a week to discuss the section of the text which they had been given as a reading assignment in the previous week. All students, whether they have attended the traditional or the alternative course, must pass the same written and oral examinations at the end of the one-year course. The advantages and disadvantages of both methods will be discussed and an evaluation will be given.

Coarse Geometry of Topological Groups

2021 ◽  
Author(s):  
Christian Rosendal
Keyword(s):  
Geometric Structure ◽  
Metric Spaces ◽  
Automorphism Groups ◽  
Mathematical Practice ◽  
Worked Examples ◽  
Topological Groups ◽  
Diffeomorphism Groups ◽  
Locally Compact ◽  
Open Problems ◽  
Topological Transformation

This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

Nucleation of Disorder in Fe3Al Alloys

1967 ◽  
Vol 25 ◽  
pp. 312-313
Author(s):  
P. R. Swann ◽  
W. R. Duff ◽  
R. M. Fisher
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
Annealing Temperature ◽  
Antiphase Domain ◽  
Effect Of Temperature ◽  
Domain Structures ◽  
Transmission Electron ◽  
Domain Boundaries ◽  
Higher Temperature ◽  
The One

Recently we have investigated the phase equilibria and antiphase domain structures of Fe-Al alloys containing from 18 to 50 at.% Al by transmission electron microscopy and Mössbauer techniques. This study has revealed that none of the published phase diagrams are correct, although the one proposed by Rimlinger agrees most closely with our results to be published separately. In this paper observations by transmission electron microscopy relating to the nucleation of disorder in Fe-24% Al will be described. Figure 1 shows the structure after heating this alloy to 776.6°C and quenching. The white areas are B2 micro-domains corresponding to regions of disorder which form at the annealing temperature and re-order during the quench. By examining specimens heated in a temperature gradient of 2°C/cm it is possible to determine the effect of temperature on the disordering reaction very precisely. It was found that disorder begins at existing antiphase domain boundaries but that at a slightly higher temperature (1°C) it also occurs by homogeneous nucleation within the domains. A small (∼ .01°C) further increase in temperature caused these micro-domains to completely fill the specimen.

Fundamental Research into Thin Films in a Developing Country

1972 ◽  
Vol 30 ◽  
pp. 500-501
Author(s):  
J.A. Eades ◽  
E. Grünbaum
Keyword(s):  
Thin Films ◽  
Growth Process ◽  
Empirical Process ◽  
Nucleation And Growth ◽  
Research Groups ◽  
Film Research ◽  
Fundamental Research ◽  
Controlled Deposition ◽  
The One ◽  
The University

In the last decade and a half, thin film research, particularly research into problems associated with epitaxy, has developed from a simple empirical process of determining the conditions for epitaxy into a complex analytical and experimental study of the nucleation and growth process on the one hand and a technology of very great importance on the other. During this period the thin films group of the University of Chile has studied the epitaxy of metals on metal and insulating substrates. The development of the group, one of the first research groups in physics to be established in the country, has parallelled the increasing complexity of the field.The elaborate techniques and equipment now needed for research into thin films may be illustrated by considering the plant and facilities of this group as characteristic of a good system for the controlled deposition and study of thin films.

STM studies of crystal growth

1991 ◽  
Vol 49 ◽  
pp. 374-375
Author(s):  
M. G. Lagally
Keyword(s):  
Crystal Growth ◽  
Molecular Beam ◽  
Chemical Vapor ◽  
Sputter Deposition ◽  
Field Of View ◽  
Scanning Tunneling ◽  
Wide Field ◽  
Tunneling Microscopy ◽  
Wide Field Of View ◽  
The One

It has been recognized since the earliest days of crystal growth that kinetic processes of all Kinds control the nature of the growth. As the technology of crystal growth has become ever more refined, with the advent of such atomistic processes as molecular beam epitaxy, chemical vapor deposition, sputter deposition, and plasma enhanced techniques for the creation of “crystals” as little as one or a few atomic layers thick, multilayer structures, and novel materials combinations, the need to understand the mechanisms controlling the growth process is becoming more critical. Unfortunately, available techniques have not lent themselves well to obtaining a truly microscopic picture of such processes. Because of its atomic resolution on the one hand, and the achievable wide field of view on the other (of the order of micrometers) scanning tunneling microscopy (STM) gives us this opportunity. In this talk, we briefly review the types of growth kinetics measurements that can be made using STM. The use of STM for studies of kinetics is one of the more recent applications of what is itself still a very young field.

Preparation of Electron Transparent Metal-Matrix Composites

1968 ◽  
Vol 26 ◽  
pp. 334-335 ◽  
Author(s):  
M. R. Pinnel ◽  
A. Lawley
Keyword(s):  
Stainless Steel ◽  
Load Transfer ◽  
Volume Fraction ◽  
Steel Wire ◽  
Initial Step ◽  
Interfacial Region ◽  
Matrix Composites ◽  
Specific Test ◽  
The Matrix ◽  
The One

Numerous phenomenological descriptions of the mechanical behavior of composite materials have been developed. There is now an urgent need to study and interpret deformation behavior, load transfer, and strain distribution, in terms of micromechanisms at the atomic level. One approach is to characterize dislocation substructure resulting from specific test conditions by the various techniques of transmission electron microscopy. The present paper describes a technique for the preparation of electron transparent composites of aluminum-stainless steel, such that examination of the matrix-fiber (wire), or interfacial region is possible. Dislocation substructures are currently under examination following tensile, compressive, and creep loading. The technique complements and extends the one other study in this area by Hancock.The composite examined was hot-pressed (argon atmosphere) 99.99% aluminum reinforced with 15% volume fraction stainless steel wire (0.006″ dia.).Foils were prepared so that the stainless steel wires run longitudinally in the plane of the specimen i.e. the electron beam is perpendicular to the axes of the wires. The initial step involves cutting slices ∼0.040″ in thickness on a diamond slitting wheel.

Grain boundary segregation in metals

1992 ◽  
Vol 50 (2) ◽  
pp. 1204-1205
Author(s):  
C.L. Briant
Keyword(s):  
Fracture Surface ◽  
Grain Boundary ◽  
Grain Boundaries ◽  
Material Properties ◽  
Electron Spectroscopy ◽  
High Vacuum ◽  
Boundary Segregation ◽  
Grain Boundary Segregation ◽  
Local Concentration ◽  
The One

Grain boundary segregation is the process by which solute elements in a material diffuse to the grain boundaries, become trapped there, and increase their local concentration at the boundary over that in the bulk. As a result of this process this local concentration of the segregant at the grain boundary can be many orders of magnitude greater than the bulk concentration of the segregant. The importance of this problem lies in the fact that grain boundary segregation can affect many material properties such as fracture, corrosion, and grain growth.One of the best ways to study grain boundary segregation is with Auger electron spectroscopy. This spectroscopy is an extremely surface sensitive technique. When it is used to study grain boundary segregation the sample must first be fractured intergranularly in the high vacuum spectrometer. This fracture surface is then the one that is analyzed. The development of scanning Auger spectrometers have allowed researchers to first image the fracture surface that is created and then to perform analyses on individual grain boundaries.

Towards quantitative simulations of inelastic electron diffraction patterns and images

1992 ◽  
Vol 50 (2) ◽  
pp. 1170-1171
Author(s):  
Z. L. Wang
Keyword(s):  
Phonon Scattering ◽  
Incident Beam ◽  
Dynamical Theory ◽  
Inelastic Electron ◽  
Additional Advantage ◽  
Coupled Equations ◽  
Atomic Vibrations ◽  
Diffraction Patterns ◽  
Primary Advantage ◽  
The One

A new dynamical theory has been developed based on Yoshioka's coupled equations for describing inelastic electron scattering in thin crystals. Compared to existing theories, the primary advantage of this theory is that the incoherent summation of the diffracted intensities contributed by electrons after exciting vast numbers of different excited states has been evaluated before any numerical calculation. An additional advantage is that the phase correlations of atomic vibrations are considered, so that full lattice dynamics can be combined in the phonon scattering calculation. The new theory has been proven to be equivalent to the inelastic multislice theory, and has been applied to calculate energy-filtered diffraction patterns and images formed by phonon, single electron and valence scattered electrons.A calculated diffraction pattern of elastic and phonon scattered electrons for a parallel incident beam case is in agreement with the one observed (Fig. 1), showing thermal diffuse scattering (TDS) streaks and Kikuchi pattern.

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