275 research outputs found
Constructive Geometry and the Parallel Postulate
Euclidean geometry consists of straightedge-and-compass constructions and
reasoning about the results of those constructions. We show that Euclidean
geometry can be developed using only intuitionistic logic. We consider three
versions of Euclid's parallel postulate: Euclid's own formulation in his
Postulate 5; Playfair's 1795 version, and a new version we call the strong
parallel postulate. These differ in that Euclid's version and the new version
both assert the existence of a point where two lines meet, while Playfair's
version makes no existence assertion. Classically, the models of Euclidean
(straightedge-and-compass) geometry are planes over Euclidean fields. We prove
a similar theorem for constructive Euclidean geometry, by showing how to define
addition and multiplication without a case distinction about the sign of the
arguments. With intuitionistic logic, there are two possible definitions of
Euclidean fields, which turn out to correspond to the different versions of the
parallel axiom. In this paper, we completely settle the questions about
implications between the three versions of the parallel postulate: the strong
parallel postulate easily implies Euclid 5, and in fact Euclid 5 also implies
the strong parallel postulate, although the proof is lengthy, depending on the
verification that Euclid 5 suffices to define multiplication geometrically. We
show that Playfair does not imply Euclid 5, and we also give some other
independence results. Our independence proofs are given without discussing the
exact choice of the other axioms of geometry; all we need is that one can
interpret the geometric axioms in Euclidean field theory. The proofs use Kripke
models of Euclidean field theories based on carefully constructed rings of
real-valued functions.Comment: 114 pages, 39 figure
Did Lobachevsky Have A Model Of His "imaginary Geometry"?
The invention of non-Euclidean geometries is often seen through the optics of
Hilbertian formal axiomatic method developed later in the 19th century. However
such an anachronistic approach fails to provide a sound reading of
Lobachevsky's geometrical works. Although the modern notion of model of a given
theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's
geometrical theory turns to be very unusual. Lobachevsky doesn't consider
various models of Hyperbolic geometry, as the modern reader would expect, but
uses a non-standard model of Euclidean plane (as a particular surface in the
Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction,
and show how it can be better analyzed within an alternative non-Hilbertian
foundational framework, which relates the history of geometry of the 19th
century to some recent developments in the field.Comment: 31 pages, 8 figure
The Steiner-Lehmus theorem and "triangles with congruent medians are isosceles" hold in weak geometries
We prove that (i) a generalization of the Steiner-Lehmus theorem due to A.
Henderson holds in Bachmann's standard ordered metric planes, (ii) that a
variant of Steiner-Lehmus holds in all metric planes, and (iii) that the fact
that a triangle with two congruent medians is isosceles holds in Hjelmslev
planes without double incidences of characteristic
Vittorio Checcucci and his contributions to mathematics education: a historical overview
This study deals with Vittorio Checcucci’s ideas and proposals as to mathematics education. The scopes of this work are twofold: 1) the first scope is historical: my aim is to reconstruct Checcucci’s thought. This is a novelty because almost no contribution dedicated to Checcucci exists. The few existing contributions are brief articles whose aim is not to provide a general picture of his ideas; 2) the second scope is connected to mathematics education in the 21st century. A series of argumentations will be proposed to prove that many Checcucci’s ideas could be fruitfully exploited nowadays. For the first time, the thought of this mathematician is exposed to non-Italian readers because his ideas are worthy to be known, rethought and discussed in an international context
Proof-checking Euclid
We used computer proof-checking methods to verify the correctness of our
proofs of the propositions in Euclid Book I. We used axioms as close as
possible to those of Euclid, in a language closely related to that used in
Tarski's formal geometry. We used proofs as close as possible to those given by
Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48
propositions, we proved 235 theorems. The extras were partly "Book Zero",
preliminaries of a very fundamental nature, partly propositions that Euclid
omitted but were used implicitly, partly advanced theorems that we found
necessary to fill Euclid's gaps, and partly just variants of Euclid's
propositions. We wrote these proofs in a simple fragment of first-order logic
corresponding to Euclid's logic, debugged them using a custom software tool,
and then checked them in the well-known and trusted proof checkers HOL Light
and Coq.Comment: 53 page
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