Euclid Vindicated from Every Blemish

Saccheri, Gerolamo; Risi, Vincenzo De

doi:10.1007/978-3-319-05966-2

Cited by 7 publications

(1 citation statement)

References 46 publications

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“…Other translators have chosen Euclid Freed of Every Flaw. The title above is the one chosen by de Risi [33].…”

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Euclid After Computer Proof-checking

Beeson

2021

Preprint

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Euclid pioneered the concept of a mathematical theory developed from axioms by a series of justified proof steps. From the outset there were critics and improvers. In this century the use of computers to check proofs for correctness sets a new standard of rigor. How does Euclid stand up under such an examination? And what does the exercise have to teach us about geometry, mathematical foundations, and the relation of logic to truth? 4 See [32] (Prologue). But by 1902, when Russell was thirty, he had changed his mind, and published an article pointing out errors in Euclid I. 1, I.4, I.7, and I.16 [31].

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“…Other translators have chosen Euclid Freed of Every Flaw. The title above is the one chosen by de Risi [33].…”

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confidence: 99%

Euclid After Computer Proof-checking

Beeson

2021

Preprint

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From Euclid to Riemann and Beyond: How to Describe the Shape of the Universe

Sunada

2019

Geometry in History

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The purpose of this essay is to trace the historical development of geometry while focusing on how we acquired mathematical tools for describing the "shape of the universe." More specifically, our aim is to consider, without a claim to completeness, the origin of Riemannian geometry, which is indispensable to the description of the space of the universe as a "generalized curved space."But what is the meaning of "shape of the universe" ? The reader who has never encountered such an issue might say that this is a pointless question. It is surely hard to conceive of the universe as a geometric figure such as a plane or a sphere sitting in space for which we have vocabulary to describe its shape. For instance, we usually say that a plane is "flat" and "infinite," and a sphere is "round" and "finite." But in what way is it possible to make use of such phrases for the universe ? Behind this inescapable question is the fact that the universe is not necessarily the ordinary 3D (3-dimensional) space where the traditional synthetic geometry-based on a property of parallels which turns out to underly the "flatness" of space-is practiced. Indeed, as Einstein's theory of general relativity (1915) claims, the universe is possibly "curved" by gravitational effects. (To be exact, we need to handle 4D curved space-times ; but for simplicity we do not take the "time" into consideration, and hence treat the "static" universe or the universe at any instant of time unless otherwise stated. We shall also disregard possible "singularities" caused by "black holes.")An obvious problem still remains to be grappled with, however. Even if we assent to the view that the universe is a sort of geometric figure, it is impossible for us to look out over the universe all at once because we are strictly confined in it. How can we tell the shape of the universe despite that ? Before Albert Einstein (1879-1955) created his theory, mathematics had already climbed such a height as to be capable to attack this issue. In this respect, Gauss and Riemann are the names we must, first and foremost, refer to as mathematicians who intensively investigated curved surfaces and spaces with the grand vision that their observations have opened up an entirely new horizon to cosmology. In particular, Riemann's work, which completely recast three thousand years of geometry executed in "space as an a priori entity," played an absolutely decisive role when Einstein established the theory of general relativity.Gauss and Riemann were, of course, not the first who were involved in cosmology. Throughout history, especially from ancient Greece to Renaissance Europe, mathematicians were, more often than not, astronomers at the same time, and hence the links between mathematics and cosmology are ancient, if not in the

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A Diagram of Choice: The Curious Case of Wallis’s Attempted Proof of the Parallel Postulate and the Axiom of Choice

Therrien

2020

Lecture Notes in Computer Science

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Euclid Vindicated from Every Blemish

Cited by 7 publications

References 46 publications

Euclid After Computer Proof-checking

Euclid After Computer Proof-checking

From Euclid to Riemann and Beyond: How to Describe the Shape of the Universe

A Diagram of Choice: The Curious Case of Wallis’s Attempted Proof of the Parallel Postulate and the Axiom of Choice

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