¿Cuándo una demostración es más perspicua que otra?

Proofs contribute to mathematical knowledge in a richer way than through exclusively of their results. Then, a philosophically relevant task is to inquire how diverse demonstrations of the same result concretize that contribution. This essay compares (following a recent work by John Dawson) various demonstrations of an elementary result of number theory, regarding a specific relation: “. . . is more perspicuous than . . . ”. The main conclusion of this work aims to highlight (in the cases considered) the relevance of the analysis of the strategic and expressive contrasts and its peculiar dynamics, in the understanding of the relationship of perspicuity between proofs.

On countable fractions from an elementary class

The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].Let L be a countable first-order language. Let L′ be the relational language having, for each formula φ of L and each sequence υ1,…,υn of variables including the free variables of φ, an n-ary relation symbol Pφ. For any L-structure and any formula Ψ(υ) of L, we define the Ψ-fraction of to be the L′-structure Ψ whose universe consists of those elements of satisfying Ψ(υ) and whose relations {Rφ}φϵL are defined by letting a1,…,an satisfy Rφ in Ψ if, and only if, a1,…, an satisfy φ in .An L-elementary class means the class of all L-structures satisfying each of some set of sentences of L. The countable part of an L-elementary class K means the class of all countable L-structures from K.Theorem. Let K be an L-elementary class and let Ψ(υ) be a formula of L. Then the class of countable Ψ-fractions of structures in K is the countable part of some L′-elementary class.Comment. By the downward Löwenheim-Skolem theorem, the countable Ψ-fractions of structures in K are the same as the Ψ-fractions of countable structures in K.Proof. We give a set Σ′ of L′-sentences whose countable models are exactly the countable Ψ-fractions of structures in K.

Algebraic ideals of semiprime Banach algebras

If A is a semiprime Banach algebra, soc A, rad A the socle and radical of A, then Soc A ∩ rad A = (0). This elementary result enables us to prove some results concerning algebraic ideal and algebraic elements modulo the socle of A. We also deduce several conditions for A equivalent to the condition dim A <+∞.

THE SPHEROMAK AS AN AXISYMMETRIC SPHERICAL DYNAMO: CLARIFICATION OF AN OLD AMBIGUITY

An elementary result in magnetostatics clarifies a fundamental ambiguity in the theory of the Spheromak (which is a compact toroidal plasma without the external poloidal windings of a Tokamak). A free Spheromak (without an external equilibrium field) is shown to possess intrinsic rotational self-forces that amplify its toroidal current, and hence its poloidal magnetic field. This result explains the experimentally observed increase of toroidal current in free compact toroidal plasmas. The possibility of an axisymmetric spherical dynamo that generates its own dipole-like self-field is mentioned.

Groups with many characteristically simple subgroups

1. A group G is called characteristically simple if it has no proper non-trivial subgroups which are invariant under all automorphisms of G. It is known that if G is characteristically simple then each countable subgroup lies in a countable characteristically simple subgroup of G. A similar assertion holds for simple groups. These results were proved by Philip Hall in lectures in 1966, and further proofs appear in (4) and (6). For simple groups there is a well known and elementary result in the other direction: if every two-generator subgroup of a group G lies in a simple subgroup, then G is simple. These considerations prompt the question (first raised, I believe, by Philip Hall) whether a group G is necessarily characteristically simple if each countable subgroup lies in a characteristically simple subgroup.