Diagrammatisches Denken bei Euklid

Sollen wir Euklids Vorgehen in den Elementen als ein axiomatisches System verstehen—oder als ein System des natürlichen Schließens, in dem die Regeln und Prinzipien, denen wir in unserem Schließen folgen, dargelegt werden? Im Folgenden werde ich darstellen, wie Kenneth Manders, Danielle Macbeth, Marco Panza und andere in jüngster Zeit diese letztere Sicht als eine alternative Lesart von Euklids Elementen dargestellt haben. Insbesondere werde ich versuchen zu zeigen, dass wir in dieser Lesart Euklids eine Art der Argumentation vorfinden, die nicht bloß auf Diagrammen basiert (wie dies auch in anderen diagrammatischen Darstellungen der Fall ist), sondern dass die Euklidische Geometrie hier als wesentlich diagrammatisch aufgefasst wird: Wir argumentieren und denken hier in dem Diagramm, und nicht bloß auf seiner Grundlage

Siegener Beiträge zur Geschichte und Philosophie der Mathematik 2022

Die im nunmehr vorliegenden sechzehnten Band von SieB - Siegener Beiträge zur Geschichte und Philosophie der Mathematik - vereinten Aufsätze dokumentieren jene Pluralität von Themen, Perspektiven und Methoden das große Oberthema Geschichte und Philosophie der Mathematik betreffend, die in den vorangehenden Bänden bereits ein Anliegen der Reihe war. Die Siegener Beiträge bieten ein Forum für den Diskurs im Bereich von Philosophie und Geschichte der Mathematik. Dabei stehen die folgenden inhaltlichen Aspekte im Zentrum: 1. Philosophie und Geschichte der Mathematik sollen einander wechselseitig fruchtbar irritieren: Ohne Bezug auf die real existierende Mathematik und ihre Geschichte läuft das philosophische Fragen nach der Mathematik leer, ohne Bezug auf die systematische Reflexion über Mathematik wird ein Bemühen um die Mathematikgeschichte blind. 2. Geschichte ermöglicht ein Kontingenzbewusstsein, philosophische Reflexion fordert Kontextualisierungen heraus. Damit stellen sich u. a. Fragen nach der Rolle der Mathematik für die Wissenschaftsgeschichte, aber auch nach einer gesellschaftlichen Rolle der Mathematik und deren historischer Bedingtheit.Inhaltsverzeichnis: Harald Boehme: Von Theodoros bis Speusippos. Zur Entdeckung des Inkommensurablen sowie der Seiten- und Diagonalzahlen Jasmin Özel: Diagrammatisches Denken bei Euklid Christian Hugo Hoffmann: Der Hauptsatz in der Ars conjectandi: Interpretationen von Bernoullis Beiträgen zu den Anfängen der mathematischen Wahrscheinlichkeitstheorie Jens Lemanski: Schopenhauers Logikdiagramme in den Mathematiklehrbüchern Adolph Diesterwegs Dolf Rami: Frege über Merkmale von Begriffen Daniel Koenig: Der Raum als Reihenbegriff – Ernst Cassirers Deutung der Geometrieentwicklung des 19. Jahrhunderts Renate Tobies: Zum 100-jährigen Jubiläum des Ernst Abbe-Gedächtnispreises Štefan Porubský: Štefan Schwarz und die Entstehung der Halbgruppentheorie Stephan Berendonk: Ein dialektischer Weg zur Summe der Kubikzahlen Felicitas Pielsticker & Ingo Witzke: Devilish prime factorization – fundamental theorem of arithmeti

First Very Long Baseline Interferometry Detections at 870 μ m

The first very long baseline interferometry (VLBI) detections at 870 μm wavelength (345 GHz frequency) are reported, achieving the highest diffraction-limited angular resolution yet obtained from the surface of the Earth and the highest-frequency example of the VLBI technique to date. These include strong detections for multiple sources observed on intercontinental baselines between telescopes in Chile, Hawaii, and Spain, obtained during observations in 2018 October. The longest-baseline detections approach 11 Gλ, corresponding to an angular resolution, or fringe spacing, of 19 μas. The Allan deviation of the visibility phase at 870 μm is comparable to that at 1.3 mm on the relevant integration timescales between 2 and 100 s. The detections confirm that the sensitivity and signal chain stability of stations in the Event Horizon Telescope (EHT) array are suitable for VLBI observations at 870 μm. Operation at this short wavelength, combined with anticipated enhancements of the EHT, will lead to a unique high angular resolution instrument for black hole studies, capable of resolving the event horizons of supermassive black holes in both space and time

First M87 Event Horizon Telescope Results. IX. Detection of Near-horizon Circular Polarization

Event Horizon Telescope (EHT) observations have revealed a bright ring of emission around the supermassive black hole at the center of the M87 galaxy. EHT images in linear polarization have further identified a coherent spiral pattern around the black hole, produced from ordered magnetic fields threading the emitting plasma. Here we present the first analysis of circular polarization using EHT data, acquired in 2017, which can potentially provide additional insights into the magnetic fields and plasma composition near the black hole. Interferometric closure quantities provide convincing evidence for the presence of circularly polarized emission on event-horizon scales. We produce images of the circular polarization using both traditional and newly developed methods. All methods find a moderate level of resolved circular polarization across the image (〈∣v∣〉 < 3.7%), consistent with the low image-integrated circular polarization fraction measured by the Atacama Large Millimeter/submillimeter Array (∣v int∣ < 1%). Despite this broad agreement, the methods show substantial variation in the morphology of the circularly polarized emission, indicating that our conclusions are strongly dependent on the imaging assumptions because of the limited baseline coverage, uncertain telescope gain calibration, and weakly polarized signal. We include this upper limit in an updated comparison to general relativistic magnetohydrodynamic simulation models. This analysis reinforces the previously reported preference for magnetically arrested accretion flow models. We find that most simulations naturally produce a low level of circular polarization consistent with our upper limit and that Faraday conversion is likely the dominant production mechanism for circular polarization at 230 GHz in M87*

First Sagittarius A* Event Horizon Telescope Results. VII. Polarization of the Ring

The Event Horizon Telescope observed the horizon-scale synchrotron emission region around the Galactic center supermassive black hole, Sagittarius A* (Sgr A*), in 2017. These observations revealed a bright, thick ring morphology with a diameter of 51.8 ± 2.3 μas and modest azimuthal brightness asymmetry, consistent with the expected appearance of a black hole with mass M ≈ 4 × 106 M ⊙. From these observations, we present the first resolved linear and circular polarimetric images of Sgr A*. The linear polarization images demonstrate that the emission ring is highly polarized, exhibiting a prominent spiral electric vector polarization angle pattern with a peak fractional polarization of ∼40% in the western portion of the ring. The circular polarization images feature a modestly (∼5%–10%) polarized dipole structure along the emission ring, with negative circular polarization in the western region and positive circular polarization in the eastern region, although our methods exhibit stronger disagreement than for linear polarization. We analyze the data using multiple independent imaging and modeling methods, each of which is validated using a standardized suite of synthetic data sets. While the detailed spatial distribution of the linear polarization along the ring remains uncertain owing to the intrinsic variability of the source, the spiraling polarization structure is robust to methodological choices. The degree and orientation of the linear polarization provide stringent constraints for the black hole and its surrounding magnetic fields, which we discuss in an accompanying publication

Polarimetric Geometric Modeling for mm-VLBI Observations of Black Holes

The Event Horizon Telescope (EHT) is a millimeter very long baseline interferometry (VLBI) array that has imaged the apparent shadows of the supermassive black holes M87* and Sagittarius A*. Polarimetric data from these observations contain a wealth of information on the black hole and accretion flow properties. In this work, we develop polarimetric geometric modeling methods for mm-VLBI data, focusing on approaches that fit data products with differing degrees of invariance to broad classes of calibration errors. We establish a fitting procedure using a polarimetric “m-ring” model to approximate the image structure near a black hole. By fitting this model to synthetic EHT data from general relativistic magnetohydrodynamic models, we show that the linear and circular polarization structure can be successfully approximated with relatively few model parameters. We then fit this model to EHT observations of M87* taken in 2017. In total intensity and linear polarization, the m-ring fits are consistent with previous results from imaging methods. In circular polarization, the m-ring fits indicate the presence of event-horizon-scale circular polarization structure, with a persistent dipolar asymmetry and orientation across several days. The same structure was recovered independently of observing band, used data products, and model assumptions. Despite this broad agreement, imaging methods do not produce similarly consistent results. Our circular polarization results, which imposed additional assumptions on the source structure, should thus be interpreted with some caution. Polarimetric geometric modeling provides a useful and powerful method to constrain the properties of horizon-scale polarized emission, particularly for sparse arrays like the EHT

Mid-Range Science Objectives for the Event Horizon Telescope

International audienceThe first images of the black holes in Sagittarius A* and M87* have created a wide range of new scientific opportunities in gravitational physics, compact objects, and relativistic astrophysics. We discuss here the scientific opportunities that arise from the rich data sets that have already been obtained and the new data sets that will be obtained, exploiting a wide range of technical advances, including observational agility, receiver upgrades, and the addition of new stations. This document provides a 5-year framework for Event Horizon Telescope (EHT) science structured around four fundamental questions that are used to prioritize the analysis of existing data, guide technical upgrades, and determine the optimal use of future observational opportunities with EHT, ALMA, and multi-wavelength facilities. Through enhancements over this period, the EHT will create the first movie of M87* connecting black hole and jet physics, provide detailed studies of the structure and dynamics of Sgr A*, characterize the magnetospheres of both systems through polarimetric imaging, and explore the spacetime properties of black holes with greater precision and range

Mid-Range Science Objectives for the Event Horizon Telescope

International audienceThe first images of the black holes in Sagittarius A* and M87* have created a wide range of new scientific opportunities in gravitational physics, compact objects, and relativistic astrophysics. We discuss here the scientific opportunities that arise from the rich data sets that have already been obtained and the new data sets that will be obtained, exploiting a wide range of technical advances, including observational agility, receiver upgrades, and the addition of new stations. This document provides a 5-year framework for Event Horizon Telescope (EHT) science structured around four fundamental questions that are used to prioritize the analysis of existing data, guide technical upgrades, and determine the optimal use of future observational opportunities with EHT, ALMA, and multi-wavelength facilities. Through enhancements over this period, the EHT will create the first movie of M87* connecting black hole and jet physics, provide detailed studies of the structure and dynamics of Sgr A*, characterize the magnetospheres of both systems through polarimetric imaging, and explore the spacetime properties of black holes with greater precision and range

First Very Long Baseline Interferometry Detections at 870μm

International audienceThe first very long baseline interferometry (VLBI) detections at 870$\mu$m wavelength (345$\,$GHz frequency) are reported, achieving the highest diffraction-limited angular resolution yet obtained from the surface of the Earth, and the highest-frequency example of the VLBI technique to date. These include strong detections for multiple sources observed on inter-continental baselines between telescopes in Chile, Hawaii, and Spain, obtained during observations in October 2018. The longest-baseline detections approach 11$\,$G$\lambda$ corresponding to an angular resolution, or fringe spacing, of 19$\mu$as. The Allan deviation of the visibility phase at 870$\mu$m is comparable to that at 1.3$\,$mm on the relevant integration time scales between 2 and 100$\,$s. The detections confirm that the sensitivity and signal chain stability of stations in the Event Horizon Telescope (EHT) array are suitable for VLBI observations at 870$\mu$m. Operation at this short wavelength, combined with anticipated enhancements of the EHT, will lead to a unique high angular resolution instrument for black hole studies, capable of resolving the event horizons of supermassive black holes in both space and time