5,640 research outputs found
Herbrand's theorem and non-Euclidean geometry
We use Herbrand's theorem to give a new proof that Euclid's parallel axiom is
not derivable from the other axioms of first-order Euclidean geometry. Previous
proofs involve constructing models of non-Euclidean geometry. This proof uses a
very old and basic theorem of logic together with some simple properties of
ruler-and-compass constructions to give a short, simple, and intuitively
appealing proof.Comment: 12 pages, 5 figure
Doing and Showing
The persisting gap between the formal and the informal mathematics is due to
an inadequate notion of mathematical theory behind the current formalization
techniques. I mean the (informal) notion of axiomatic theory according to which
a mathematical theory consists of a set of axioms and further theorems deduced
from these axioms according to certain rules of logical inference. Thus the
usual notion of axiomatic method is inadequate and needs a replacement.Comment: 54 pages, 2 figure
Marriages of Mathematics and Physics: A Challenge for Biology
The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences
Cabri's role in the task of proving within the activity of building part of an axiomatic system
We want to show how we use the software Cabri, in a Geometry class for preservice mathematics teachers, in the process of building part of an axiomatic system of Euclidean Geometry. We will illustrate the type of tasks that engage students to discover the relationship between the steps of a geometric construction and the steps of a formal justification of the related geometric fact to understand the logical development of a proof; understand dependency relationships between properties; generate ideas that can be useful for a proof; produce conjectures that correspond to theorems of the system; and participate in the deductive organization of a set of statements obtained as solution to open-ended problems
Aligning Mathematics Curriculum to Create Potential for Active Learning in Pre-K Through Eighth Grade Teacher Education
In this paper, we consider some issues surrounding the teaching of mathematics to pre-service teachers. In particular. we look at the possibilities for teaching elementary mathematics from an advanced standpoint and alignments of curriculum that have the capacity to enhance student involvement in the making of the mathematics.The particulars of the James Madison University curriculum are used to illustrate many of the points
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
A Science of Reasoning
This paper addresses the question of how we can understand reasoning in general and mathematical proofs in particular. It argues the need for a high-level understanding of proofs to complement the low-level understanding provided by Logic. It proposes a role for computation in providing this high-level understanding, namely by the association of proof plans with proofs. Proof plans are defined and examples are given for two families of proofs. Criteria are given for assessing the association of a proof plan with a proof. 1 Motivation: the understanding of mathematical proofs The understanding of reasoning has interested researchers since, at least, Aristotle. Logic has been proposed by Aristotle, Boole, Frege and others as a way of formalising arguments and understanding their structure. There have also been psychological studies of how people and animals actually do reason. The work on Logic has been especially influential in the automation of reasoning. For instance, resolution..
The Angle Trisection Solution (A Compass-Straightedge (Ruler) Construction)
This paper is devoted to exposition of a provable classical solution for the ancient Greeks classical geometric problem of angle trisection [3]. (Pierre Laurent Wantzel, 1837),presented an algebraic proof based on ideas from Galois field showing that, the angle trisection solution correspond to an implicit solution of the cubic equation; , which he stated as geometrically irreducible [23]. The primary objective of this novel work is to show the possibility to solve the trisection of an arbitrary angle using the traditional Greeks tools of geometry, and refutethe presented proof of angle trisection impossibility statement. The exposedproof of the solution is theorem , which is based on the classical rules of Euclidean geometry, contrary to the Archimedes proposition of usinga marked straightedge construction [4], [11]
Hilbert's Error?
It is well known that a center of a given circle cannot be constructed using
only a straightedge and that this was proven by David Hilbert. Still it is not
so clear what kind of object is proven to be non-existing. We analyze different
attempts to define a geometric construction appearing in the literature and
observe that none of them is really satisfactory and that Hilbert's proof needs
to be corrected (as noted by Akopyan and Fedorov
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