scholarly journals Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

2021 ◽  
Author(s):  
Viktor Blåsjö
Keyword(s):  
Good Reason ◽  
Diagrammatic Reasoning ◽  
Conic Section ◽  
Conic Sections ◽  
Ancient Greek ◽  
Classical Mathematics ◽  
And Mathematics ◽  
Reading Key ◽  
Made In ◽  
Greek Geometry

AbstractI present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid (implicit diagrammatic reasoning, superposition) can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised.

Newton and the Origin of Civilization

2012 ◽  
Author(s):  
Jed Z. Buchwald ◽  
Mordechai Feingold
Keyword(s):  
Good Reason ◽  
History Of Mathematics ◽  
Primary Sources ◽  
Ancient History ◽  
Isaac Newton ◽  
History Of ◽  
Ancient Civilizations ◽  
And Mathematics ◽  
One Year ◽  
Radical Ideas

Isaac Newton’s Chronology of Ancient Kingdoms Amended, published in 1728, one year after the great man’s death, unleashed a storm of controversy. And for good reason. The book presents a drastically revised timeline for ancient civilizations, contracting Greek history by five hundred years and Egypt’s by a millennium. This book tells the story of how one of the most celebrated figures in the history of mathematics, optics, and mechanics came to apply his unique ways of thinking to problems of history, theology, and mythology, and of how his radical ideas produced an uproar that reverberated in Europe’s learned circles throughout the eighteenth century and beyond. The book reveals the manner in which Newton strove for nearly half a century to rectify universal history by reading ancient texts through the lens of astronomy, and to create a tight theoretical system for interpreting the evolution of civilization on the basis of population dynamics. It was during Newton’s earliest years at Cambridge that he developed the core of his singular method for generating and working with trustworthy knowledge, which he applied to his study of the past with the same rigor he brought to his work in physics and mathematics. Drawing extensively on Newton’s unpublished papers and a host of other primary sources, the book reconciles Isaac Newton the rational scientist with Newton the natural philosopher, alchemist, theologian, and chronologist of ancient history.

Oosterweel – from an underground and even underwater road infrastructure plan to an urban transformation project

2021 ◽  
Author(s):  
Luc Hellemans
Keyword(s):  
Traffic Flow ◽  
Large Scale ◽  
Good Reason ◽  
Urban Transformation ◽  
Construction Site ◽  
Successful Completion ◽  
Road Infrastructure ◽  
Living Space ◽  
The City ◽  
Made In

<p>For the coming ten years, the heart of Europe will turn into a gigantic construction site for works on one of the largest hubs of the continent: Antwerp. The Oosterweel Link is the project whereby the motorway ring around Antwerp is undergoing a metamorphosis to reinvigorate traffic flow and add living space to the City. The project had come to a standstill for several years as a result of protests by assertive citizens, but was given a second lease of life following a large-scale participation project.</p><p>To ensure its successful completion, unparalleled efforts are being made in the field and in the area of digitization. It is therefore with good reason that in Belgium the project is referred to as “the construction site of the century”.</p>

scholarly journals PRECIOUS ITEMS FROM THE SHUMEIKO BARROW

2019 ◽  
Vol 33 (4) ◽  
pp. 322-339
Author(s):  
Yu. B. Polidovych
Keyword(s):  
River Basin ◽  
Preliminary Conclusion ◽  
Ancient Greek ◽  
Original Shape ◽  
Gold Plate ◽  
Gold Tip ◽  
Made In

The paper deals to the finds from the barrow near the Shumeiko farm in the Sula river basin (now Sumy region of Ukraine) which was excavated by Sergei Mazaraki in 1899. Objects of Scythian culture were found in the mound: weapons, horse bridles, and vessels. Mikhail Rostovtsev mistakenly attributed to these finds the fragment of ancient Greek kylix of the end of the 6th century BC. Modern researchers date the barrow assemblage near the Shumeiko farm to the first half of the 6th century BC (Igor Bruyako, Denis Grechko, Denis Topal, Oleksandr Shelekhan). Sergey Polin attributes it to Early Scythian time. In the paper three precious items from the barrow are described in detail. This is a sword, the handle of which is plaqued with gold. The ancient craftsman used the granulation technique for decoration. Not only the ancient Greek jewelers used this technique. The masters of Urartu applied it as well. It was used in the decoration of the sword from the Kelermes barrow in the Kuban region, as well as on various adornments. The iron sword has an original shape and belongs to the Shumeiko type (according to Denis Topal, Oleksandr Shelekhan). Such swords were most common in the first half of the 6th century BC. The scabbard was decorated by the gold plate with images of animals and the gold tip. The analysis shows that the images of wild goats and predators are made in the early Scythian animal style. The sheath tip also corresponds to the early Scythian tradition and finds analogies in the Pre-Scythian time. On the contrary, at a later time (the end of the 6th — beginning of the 5th century BC), according to other principles (barrow No 6 near the Oleksandrivka village, Gostra Mogyla near the Tomakovka village) the tips of the scabbard were made. Near the sword the gold plate in the form of a running hare was found. It was made in the Scythian animal style. This plate was probably part of the sheath decor and adorned a side leather ledge that helped to attach the scabbard to the belt. A preliminary conclusion is made about the belonging of precious items from the Shumeiko barrow to the Kelermes horizon of antiquities of the Early Scythian culture.

Advanced Math

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Riemann Hypothesis ◽  
Physical Theory ◽  
A Priori ◽  
Advanced Mathematics ◽  
Ancient Greek ◽  
Mathematical Objects ◽  
Inductive Methods ◽  
Modern Mathematics ◽  
The Continuum ◽  
Greek Geometry

This chapter proposes the idea that advanced mathematics is based on hypotheses—that far from being a priori, it is based on hypothetical assumptions. The concept of quasi-empiricism is often linked with the view that inductive methods are at play when the hypotheses are established. The presence of hypotheses at the very heart of mathematics establishes an important similitude with physical theory and undermines the simple distinction between “formal” and “empirical” sciences. The chapter first elaborates on a hypothetical conception of mathematics before discussing the ideas (and ideals) of certainty and objectivity in mathematics. It then considers the modern problems of the continuum that exist in ancient Greek geometry, along with the so-called methodological platonism of modern mathematics and its focus on mathematical objects. Finally, it describes the Axiom of Completeness and the Riemann Hypothesis.

Fuzzy Set Theory Applied to Accounting Sciences

2020 ◽  
Vol 16 (01) ◽  
pp. 1-16
Author(s):  
Carmen Lozano ◽  
Enriqueta Mancilla-Rendón
Keyword(s):  
Social Sciences ◽  
Fuzzy Logic ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Hamming Distance ◽  
Great Acceptance ◽  
Classical Mathematics ◽  
Triangular Numbers ◽  
And Mathematics

Fuzzy set theory and fuzzy logic have been successfully developed in engineering and mathematics. However, these concepts have found great acceptance in social sciences in recent years since they provide an answer to those problems in the real world that cannot be modeled using classical mathematics. In this paper, we propose a new methodology for accounting science based on fuzzy triangular numbers. The methodology uses Hamming distance between fuzzy triangular numbers and arithmetic operations to evaluate corporate governance of multinational public stock corporations (PSCs) in the telecommunications sector.

scholarly journals Preface

1994 ◽  
Vol 347 (1684) ◽  
pp. 449-449 ◽  
Keyword(s):  
Stock Prices ◽  
Large Scale ◽  
18Th Century ◽  
Telecommunication Networks ◽  
Financial Industry ◽  
Interacting Systems ◽  
Mathematical Sciences ◽  
And Mathematics ◽  
The City ◽  
Made In

There is money to be made in the financial industry. Academics, under pressure to exhibit relevance, are happy to point to their consultancies in the City as evidence of their value in the market, and the industry has shown a notable ability to recruit the brightest and best from our Universities. These observations should not obscure the profound scientific challenges posed by the area of finance. The area has both stimulated and benefited from advances in a range of mathematical sciences, most obviously probability, differential equations, optimization, statistics and numerical analysis. One thinks, for example, of Bernoulli’s resolution, in the 18th century, of the St Petersburg Problem through his introduction of a logarithmic utility, of Bachelier’s description, at the turn of this century, of the stochastic process we now call brownian motion, of Kendall’s investigation, forty years ago, of the statistical unpredictability of stock prices, and of the current enormously fertile interaction between economics and mathematics centred around martingale representations. Looking to the future, some of the mathematical ideas originally motivated by statistical mechanics, and since used to model the large-scale telecommunication networks upon which the financial industry relies, may also provide insight into the very difficult problems that arise in economics concerning interacting systems of rational agents.

scholarly journals Some Observations on the Interdigitation of Advances in Medical Science and Mathematics

2013 ◽  
Vol 79 (12) ◽  
pp. 1231-1234 ◽  
Author(s):  
Michael James Glamore ◽  
James L. West ◽  
James Patrick O'leary
Keyword(s):  
Medical Science ◽  
Written Word ◽  
And Mathematics ◽  
Made In

The immense advancement of our understanding of disease processes has not been a uniform progression related to the passage of time. Advances have been made in “lurches” and “catches” since the advent of the written word. There has been a remarkable interdependency between such advances in medicine and advances in mathematics that has proved beneficial to both. This work explores some of these critical relationships and documents how the individuals involved contributed to advances in each.

scholarly journals The Works of Pergamon and their Influence

1886 ◽  
Vol 7 ◽  
pp. 251-274
Author(s):  
L. R. Farnell
Keyword(s):  
Good Reason ◽  
Right Wing ◽  
Left Wing ◽  
The South ◽  
Hellenic Journal ◽  
The West ◽  
West Side ◽  
South West ◽  
The Right ◽  
Made In

The questions concerning the art of Pergamon, its characteristics and later influence, depend partly for their solution on the reconstruction and explanation of the fragments in Berlin. Much progress has been made in the work during the last year. The discovery which decided what was the breadth of the staircase, and what were the figures which adorned the left wing and the left staircase wall, has been already mentioned in the Hellenic Journal. It is now officially stated that the staircase was on the west side of the altar, although Bohn, in his survey of the site, at first conceived that this was impossible. Assuming that this point is now settled, we may note what is certain, or probable, or what is merely conjectural, in the placing of the groups. We know that the wing on the left of the staircase, and the left staircase-wall, were occupied by the deities of the sea and their antagonists: by Triton, Amphitrite, Nereus, and others which we cannot name. Among them, also, we may perhaps discern the figure of Hephaestos, and in their vicinity we must suppose Poseidon. On the right wing of the staircase, and around the south-west corner, we have good reason for placing Dionysos, with Cybele and her attendant goddesses, although the order of the slabs on which these latter are found is not the same as was formerly supposed.

Demonstration of Conic Sections and Skew Curves with String Models

1946 ◽  
Vol 39 (6) ◽  
pp. 284-287
Author(s):  
H. von Baravalle
Keyword(s):  
High Schools ◽  
Solid Surface ◽  
String Model ◽  
Conic Section ◽  
Slide Projector ◽  
Conic Sections ◽  
Black Background ◽  
Glass Slides ◽  
String Models

For the demonstration of conic sections in high schools and colleges, we generally use the w ooden model of a cone with detachable parts, showing the four kinds of plane sections, a circle, an ellipse, a parabola, and a hyperbola. The wooden model is naturally limited to fixed positions of the intersecting plane and the conic sections appear as four separate facts. As one of the most stimulating parts of a study of conic sections is the realization of how each one of the four curves changes with the position of the intersecting plane and how one kind of conic section can turn into another, a more flexible medium of demonstration seems desirable. It can be found through applying two changes to the usual demonstrations. The first one is to replace the actual cutting of a solid wooden model by projecting a cut on a larger cone, thus achieving an easy flexibility, as either the projector or the model can be moved during the demonstration. The second change is to replace a solid surface by one formed by strings, which not only makes a larger model considerably lighter, but also allows the conic sections to be seen all around the cone. The alternative to string models, models with transparent surfaces, produce too many disturbing light reflections. Figure 1 shows the string model of a cone and Figure 2 its application in the demonstration of ellipses. The system of parallel ellipses in Figure 2 is produced by parallel light planes which are obtained from an ordinary slide projector with slides showing transparent lines on black background. Instead of glass slides also rectangular pieces of stronger drawing paper can be used into which the lines have been cut. Moving the projector slightly to the side makes the ellipses in Figure 2 go up or down the cone and each ellipse widens or contracts during the motion. If one increases the angle of the planes the form of all the ellipses change until they turn into parabolas and finally hyperbolas.

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