Extended Abstract of Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry

2006 ◽  
pp. 127-129 ◽  
Author(s):  
Nathaniel Miller
Keyword(s):  
Twentieth Century ◽  
Euclidean Geometry

Historical Introduction

2019 ◽  
pp. 1-25
Author(s):  
John Stillwell
Keyword(s):  
Twentieth Century ◽  
Mathematical Logic ◽  
Set Theory ◽  
Euclidean Geometry ◽  
Reverse Mathematics ◽  
Pythagorean Theorem ◽  
Base Theory ◽  
The Axiom Of Choice ◽  
The Right ◽  
Main Ideas

This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.

Non-Euclidean Geometry in Russian Art History: On a Little-Known Application of a Scientific Theory

Leonardo ◽  
2020 ◽  
Vol 53 (3) ◽  
pp. 293-298
Author(s):  
Clemena Antonova
Keyword(s):  
Twentieth Century ◽  
Art History ◽  
Scientific Theory ◽  
Euclidean Geometry ◽  
Human Vision ◽  
Russian Mathematician ◽  
Pictorial Space ◽  
Russian Art ◽  
Non Euclidean Geometry ◽  
The One

The author has previously proposed that there are at least six different definitions of “reverse” or “inverse” perspective, i.e. the principle of organizing pictorial space in the icon. Reverse perspective is still a largely unresolved art historical problem. The author focuses on one of the six defi nitions, the one least familiar to Western scholars—namely, the view, common in Russian art-historical writing at the beginning of the twentieth century, that space in the icon is a visual analogue of non-Euclidean geometry. Russian mathematician-turned-theologian and priest Pavel Florensky claimed that the space of the icon is that of non-Euclidean geometry and truer to the way human vision functions. The author considers the scientifi c validity of Florensky's claim.

Toward a Formal Interpretation of Kant's Analogies of Experience

2007 ◽  
Vol 28 (1-2) ◽  
pp. 107-120
Author(s):  
Johan Blok
Keyword(s):  
Twentieth Century ◽  
Natural Science ◽  
Euclidean Geometry ◽  
Scientific Progress ◽  
Specific Form ◽  
Pure Form ◽  
Logical Positivism ◽  
Theory Of Relativity ◽  
Transcendental Logic ◽  
Non Euclidean Geometry

Very often, the rise of non-Euclidean geometry and Einstein's theory of relativity are seen as the decisive defeat of Kant's theoretical philosophy. Scientific progress seems to render Kant's philosophy obsolete. This view became dominant during the first decades of the twentieth century, when the movement of logical positivism arose. Despite extensive criticism of basic tenets of this movement later in the twentieth century, its view of Kant's philosophy is still common. Although it is not my intention to defend Kant infinitely, I think that this view is rather unsatisfactory and even misleading.Let us consider the first factor: non-Euclidean geometry. If one reads the first Critique carefully, it becomes clear that the claims of transcendental logic do not imply Euclidean geometry. Kant's notion of space, as explained in the aesthetics chapter, is rather limited: it does neither entail nor presuppose a specific form of geometry (Cf. B37-B57). None of his statements about the form of space is specific enough to imply or support Euclidean geometry. Although Kant uses several examples, Euclidean geometry does not play any systematic role; only the pure form of space is at issue in the aesthetics chapter. In my view, the same holds in the case of Newton's physics: it is neither presupposed nor entailed by Kant's transcendental logic. The justification of Newton's physics requires further specialisation and application of the transcendental framework to empirical concepts like matter and motion. Kant took this step in his Metaphysical Foundations of Natural Science.

The power of a point for some real algebraic curves

2008 ◽  
Vol 92 (523) ◽  
pp. 22-28 ◽  
Author(s):  
Bogdan D. Suceavă ◽  
Adrian Vajiac ◽  
Mihaela B. Vajiac
Keyword(s):  
Twentieth Century ◽  
Computational Geometry ◽  
Current Literature ◽  
Euclidean Geometry ◽  
Algebraic Curves ◽  
Real Algebraic Curves

According to various sources (e.g. [1, p. 102]), the terminology of thepower of a pointwith respect to a circle is due to Steiner. His definition appears in most classical and contemporary geometry textbooks (to mention just a few references, see [2, 3, 4, 5]). The concept of the power of a point has been revisited not only in advanced Euclidean geometry, but also in computational geometry and other areas of mathematics.In the current literature there are two different definitions of the power of a point with respect to a circle, which we study in detail in section 2. In the first half of the twentieth century there have been published several attempts to generalise the concept of power of the point to real algebraic curves.

Aleksandr Drevin, Nadezhda Udal'tsova: An Exhibition That Never Was

Leonardo ◽  
2002 ◽  
Vol 35 (3) ◽  
pp. 263-269
Author(s):  
Kirill Sokolov
Keyword(s):  
Twentieth Century ◽  
Euclidean Geometry ◽  
Flat Surface ◽  
Western Thought ◽  
Artistic Practice ◽  
Russian Art ◽  
Early Twentieth Century ◽  
Non Euclidean Geometry ◽  
Husband And Wife

This juxtaposition of autobiographical statements written in 1933 by Aleksandr Drevin and Nadezhda Udal'tsova, together with an introduction to their artistic careers and a select chronology designed to place them in the context of their times, is intended to show how early twentieth-century Russian art evolved in parallel to Western thought and artistic practice, taking into account contemporary developments in non-Euclidean geometry, physics, mathematics, the laws of perspective and the awareness of the impossibility of “realistically” representing spatial forms on a flat surface, which, at the time, were exercising many minds. The artists, though from very different backgrounds, were closely involved with one another, as husband and wife and as close colleagues in art. Their artistic course is traced through and beyond the experimental 1910s and 1920s.

Some Polish Composers of To-Day

Tempo ◽  
1948 ◽  
pp. 25-28
Author(s):  
Andrzej Panufnik
Keyword(s):  
Twentieth Century ◽  
Fertile Soil ◽  
Early Twentieth Century ◽  
Progressive Group ◽  
Karol Szymanowski

It is ten years since KAROL SZYMANOWSKI died at fifty-four. He was the most prominent representative of the “radical progressive” group of early twentieth century composers, which we call “Young Poland.” In their manysided and pioneering efforts they prepared the fertile soil on which Poland's present day's music thrives.

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