Examining the method of proofs and refutations in pre-service teachers education

There is some evidence in the mathematics education literature that Lakatos’ proofs and refutation methods can be useful to examining students’ conjecture production and proof construction process. The purpose of this study was to determine how the Lakatos method goes and which steps of the method works in the teacher education program. The population sample for this study consists of 24 senior pre-service teachers in elementary mathematics education in Turkey (16 women and 8 men). Pre-service teachers were given a problem in which they examined the relation between perimeter and area of a rectangle. Data was collected with a camera, field notes, and groups’ written solutions and analyzed on the basis of framework included in Larsen and Zandieh’s (2008) study. The finding revealed that Lakatos’ method was usable in the teacher education program. But some steps of the method described in Lakatos’ (1976) historical case study were not provided in the real classroom environment

The Rational Higher Structure of M-theory

We review how core structures of string/M-theory emerge as higher structures in super homotopy theory; namely from systematic analysis of the brane bouquet of universal invariant higher central extensions growing out of the superpoint. Since super homotopy theory is immensely rich, to start with we consider this in the rational/infinitesimal approximation which ignores torsion-subgroups in brane charges and focuses on tangent spaces of super space-time. Already at this level, super homotopy theory discovers all super $p$-brane species, their intersection laws, their M/IIA-, T- and S-duality relations, their black brane avatars at ADE-singularities, including their instanton contributions, and, last not least, Dirac charge quantization: for the D-branes it recovers twisted K-theory, rationally, but for the M-branes it gives cohomotopy cohomology theory. We close with an outlook on the lift of these results beyond the rational/infinitesimal approximation to a candidate formalization of microscopic M-theory in super homotopy theory.Comment: 32 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 201

Spurious, Emergent Laws in Number Worlds

We study some aspects of the emergence of logos from chaos on a basal model of the universe using methods and techniques from algorithmic information and Ramsey theories. Thereby an intrinsic and unusual mixture of meaningful and spurious, emerging laws surfaces. The spurious, emergent laws abound, they can be found almost everywhere. In accord with the ancient Greek theogony one could say that logos, the Gods and the laws of the universe, originate from "the void," or from chaos, a picture which supports the unresolvable/irreducible lawless hypothesis. The analysis presented in this paper suggests that the "laws" discovered in science correspond merely to syntactical correlations, are local and not universal.Comment: 24 pages, invited contribution to "Contemporary Natural Philosophy and Philosophies - Part 2" - Special Issue of the journal Philosophie

Harmonious architecture and kinetic linear energy

This is a chapter in a book based upon the work from an Erasmus International Project which took place in Athens in July 2012