518 research outputs found
Note di Matematica 26
Abstract. We point out the geometric significance of a part of the theorem regarding the maximality of the orthogonal group in the equiaffine group proved in Keywords: Erlanger Programm, definability, Lω 1 ω -logic MSC 2000 classification: 03C40, 14L35, 51F25, 51A99 A. Schleiermacher and K. Strambach [12] proved a very interesting result regarding the maximaility of the group of orthogonal transformations and of that of Euclidean similarities inside certain groups of affine transformations. Although similar results have been proved earlier, this is the first time that the base field for the groups in question was not the field of real numbers, but an arbitrary Pythagorean field which admits only Archimedean orderings. They also state, as geometric significance of the result regarding the maximality of the group of Euclidean motions in the unimodular group over the reals, that there is "no geometry between the classical Euclidean and the affine geometry". The aim of this note is to point out the exact geometric meaning of the positive part of the 2-dimensional part their theorem, in the case in which the underlying field is an Archimedean ordered Euclidean field. In this case their theorem states that: (1) the group G 1 of Euclidean isometries is maximal in the group H 1 of equiaffinities (affine transformations that preserve non-directed area), and that (2) the group G 2 of Euclidean similarities is maximal in the group H 2 of affine transformations. The restriction to the 2-dimensional case is not essential but simplifies the presentation. The geometric counterpart of group-theoretic results in the spirit of the Erlanger Programm is given by Beth's theorem, as was emphasized by Büch
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
On Constructive Axiomatic Method
In this last version of the paper one may find a critical overview of some
recent philosophical literature on Axiomatic Method and Genetic Method.Comment: 25 pages, no figure
A Galois connection between classical and intuitionistic logics. I: Syntax
In a 1985 commentary to his collected works, Kolmogorov remarked that his
1932 paper "was written in hope that with time, the logic of solution of
problems [i.e., intuitionistic logic] will become a permanent part of a
[standard] course of logic. A unified logical apparatus was intended to be
created, which would deal with objects of two types - propositions and
problems." We construct such a formal system QHC, which is a conservative
extension of both the intuitionistic predicate calculus QH and the classical
predicate calculus QC.
The only new connectives ? and ! of QHC induce a Galois connection (i.e., a
pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying
posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation
translation of propositions into problems extends to a retraction of QHC onto
QH; whereas Goedel's provability translation of problems into modal
propositions extends to a retraction of QHC onto its QC+(?!) fragment,
identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic
modal logic, whose modality !? is a strict lax modality in the sense of Aczel -
and thus resembles the squash/bracket operation in intuitionistic type
theories.
The axioms of QHC attempt to give a fuller formalization (with respect to the
axioms of intuitionistic logic) to the two best known contentual
interpretations of intiuitionistic logic: Kolmogorov's problem interpretation
(incorporating standard refinements by Heyting and Kreisel) and the proof
interpretation by Orlov and Heyting (as clarified by G\"odel). While these two
interpretations are often conflated, from the viewpoint of the axioms of QHC
neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a
simplified version of Isabelle's meta-logic
Some new results on decidability for elementary algebra and geometry
We carry out a systematic study of decidability for theories of (a) real
vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces,
Banach spaces and metric spaces, all formalised using a 2-sorted first-order
language. The theories for list (a) turn out to be decidable while the theories
for list (b) are not even arithmetical: the theory of 2-dimensional Banach
spaces, for example, has the same many-one degree as the set of truths of
second-order arithmetic.
We find that the purely universal and purely existential fragments of the
theory of normed spaces are decidable, as is the AE fragment of the theory of
metric spaces. These results are sharp of their type: reductions of Hilbert's
10th problem show that the EA fragments for metric and normed spaces and the AE
fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs
of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3
On Jordan's measurements
The Jordan measure, the Jordan curve theorem, as well as the other generic
references to Camille Jordan's (1838-1922) achievements highlight that the
latter can hardly be reduced to the "great algebraist" whose masterpiece, the
Trait\'e des substitutions et des equations alg\'ebriques, unfolded the
group-theoretical content of \'Evariste Galois's work. The present paper
appeals to the database of the reviews of the Jahrbuch \"uber die Fortschritte
der Mathematik (1868-1942) for providing an overview of Jordan's works. On the
one hand, we shall especially investigate the collective dimensions in which
Jordan himself inscribed his works (1860-1922). On the other hand, we shall
address the issue of the collectives in which Jordan's works have circulated
(1860-1940). Moreover, the time-period during which Jordan has been publishing
his works, i.e., 1860-1922, provides an opportunity to investigate some
collective organizations of knowledge that pre-existed the development of
object-oriented disciplines such as group theory (Jordan-H\"older theorem),
linear algebra (Jordan's canonical form), topology (Jordan's curve), integral
theory (Jordan's measure), etc. At the time when Jordan was defending his
thesis in 1860, it was common to appeal to transversal organizations of
knowledge, such as what the latter designated as the "theory of order." When
Jordan died in 1922, it was however more and more common to point to
object-oriented disciplines as identifying both a corpus of specialized
knowledge and the institutionalized practices of transmissions of a group of
professional specialists
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