Obeying a rule : Ludwig Wittgenstein and the foundations of Set Theory

In this paper we propose some reflections on Wittgenstein’s ideas about grammar and rules; then we shall consider some consequences of these for the foundations of set theory and, in particular, for the introduction of major concepts of set theory in education. For instance, a community of practice can decide to follow a particular rule that forbids the derivation of arbitrary sentences from a contradiction: since, according to Radford’s perspective, knowledge is the result of thinking, and thinking is a cognitive social praxis, the mentioned choice can be considered as a form of real and effective knowledge

"Obeying a rule": Ludwig Wittgenstein and the foundations of Set Theory

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Numbers and Polynomials- 50 years since the publication of Wittgenstein\u27s Bemerkungen über die Grundlagen der Mathematik (1956): Mathematical and Educational reflections

According to L. Wittgenstein, the meaning of a mathematical object is to be grounded upon its use. In this paper we consider Robinson theory Q, the subtheory of firstorder Peano Arithmetic PA; some theorems and conjectures can be interpreted over one model of Q given by a universe of polynomials; with respect to nonconstant polynomials some proofs by elementary methods are given and compared with corresponding results in the standard model of PA. We conclude that the creative power of the language can be pointed out in how the language itself is embedded into the rest of human activities, and this is an important track to follow for researchers in mathematics education

Quench energy studies in ITER conductors for different magnetic field perturbations with Jackpot and THEA combined models

The electromagnetic-thermal models for the cable-in-conduit conductors (CICCs) JackPot-ACDC and THEA (thermal, hydraulic and electric analysis of superconducting cables) are combined to predict the stability of ITER central solenoid conductors. The combination of both models allows the prediction of the effect of any type of magnetic field perturbation in time, relevant for the magnet coils during the plasma operation scenario of the reactor. At present, there is no experiment for testing the stability of the ITER Nb3Sn conductors under such conditions. Only limited experimental data on minimum quench energy (MQE), defining the conductor stability, are available but the time and magnetic field amplitude settings are completely different from the actual ITER operating conditions. Nevertheless, such tests are useful as a basis to calibrate and benchmark the codes. The JackPot-THEA combination allows us to determine the MQE for any magnetic field change in time and to fully describe the involved electromagnetic phenomena in strand-level detail in terms of local power dissipation and (peak) electric field along all strands. Thermally, the computation is still on a global scale for identifying the quench initiation and propagation. The predictions from the combined codes are in good agreement with the experimental results and provide a solid basis for extrapolative scaling of the stability of CICCs under plasma operating conditions

Molecular dynamics simulations show how the FMRP Ile304Asn mutation destabilizes the KH2 domain structure and affects its function

Mutations or deletions of FMRP, involved in the regulation of mRNA metabolism in brain, lead to the Fragile X syndrome (FXS), the most frequent form of inherited intellectual disability. A severe manifestation of the disease has been associated with the Ile304Asn mutation, located on the KH2 domain of the protein. Several hypotheses have been proposed to explain the possible molecular mechanism responsible for the drastic effect of this mutation in humans. Here, we performed a molecular dynamics simulation and show that the Ile304Asn mutation destabilizes the hydrophobic core producing a partial unfolding of two α-helices and a displacement of a third one. The affected regions show increased residue flexibility and motion. Molecular docking analysis revealed strongly reduced binding to a model single-stranded nucleic acid in agreement with known data that the two partially unfolded helices form the RNA-binding surface. The third helix, which we show here to be also affected, is involved in the PAK1 protein interaction. These two functional binding sites on the KH2 domain do not overlap spatially, and therefore, they can simultaneously bind their targets. Since the Ile304Asn mutation affects both binding sites, this may justify the severe clinical manifestation observed in the patient in which both mRNA metabolism activity and cytoskeleton remodeling would be affected

Conceptually driven and visually rich tasks in texts and teaching practice: the case of infinite series

The study we report here examines parts of what Chevallard calls the institutional dimension of the students’ learning experience of a relatively under-researched, yet crucial, concept in Analysis, the concept of infinite series. In particular, we examine how the concept is introduced to students in texts and in teaching practice. To this purpose, we employ Duval's Theory of Registers of Semiotic Representation towards the analysis of 22 texts used in Canada and UK post-compulsory courses. We also draw on interviews with in-service teachers and university lecturers in order to discuss briefly teaching practice and some of their teaching suggestions. Our analysis of the texts highlights that the presentation of the concept is largely a-historical, with few graphical representations, few opportunities to work across different registers (algebraic, graphical, verbal), few applications or intra-mathematical references to the concept's significance and few conceptually driven tasks that go beyond practising with the application of convergence tests and prepare students for the complex topics in which the concept of series is implicated. Our preliminary analysis of the teacher interviews suggests that pedagogical practice often reflects the tendencies in the texts. Furthermore, the interviews with the university lecturers point at the pedagogical potential of: illustrative examples and evocative visual representations in teaching; and, student engagement with systematic guesswork and writing explanatory accounts of their choices and applications of convergence tests

Absence of RNA-binding protein FXR2P prevents prolonged phase of kainate-induced seizures.

Status epilepticus (SE) is a condition in which seizures are not self-terminating and thereby pose a serious threat to the patient's life. The molecular mechanisms underlying SE are likely heterogeneous and not well understood. Here, we reveal a role for the RNA-binding protein Fragile X-Related Protein 2 (FXR2P) in SE. Fxr2 KO mice display reduced sensitivity specifically to kainic acid-induced SE. Immunoprecipitation of FXR2P coupled to next-generation sequencing of associated mRNAs shows that FXR2P targets are enriched in genes that encode glutamatergic post-synaptic components. Of note, the FXR2P target transcriptome has a significant overlap with epilepsy and SE risk genes. In addition, Fxr2 KO mice fail to show sustained ERK1/2 phosphorylation induced by KA and present reduced burst activity in the hippocampus. Taken together, our findings show that the absence of FXR2P decreases the expression of glutamatergic proteins, and this decrease might prevent self-sustained seizures

Nuclear accumulation of mRNAs underlies G4C2-repeat-induced translational repression in a cellular model of C9orf72 ALS

A common feature of non-coding repeat expansion disorders is the accumulation of RNA repeats as RNA foci in the nucleus and/or cytoplasm of affected cells. These RNA foci can be toxic because they sequester RNA-binding proteins, thus affecting various steps of post-transcriptional gene regulation. However, the precise step that is affected by C9orf72 GGGGCC (G4C2) repeat expansion, the major genetic cause of amyotrophic lateral sclerosis (ALS), is still poorly defined. In this work, we set out to characterise these mechanisms by identifying proteins that bind to C9orf72 RNA. Sequestration of some of these factors into RNA foci was observed when a (G4C2)31 repeat was expressed in NSC34 and HeLa cells. Most notably, (G4C2)31 repeats widely affected the distribution of Pur-alpha and its binding partner fragile X mental retardation protein 1 (FMRP, also known as FMR1), which accumulate in intra-cytosolic granules that are positive for stress granules markers. Accordingly, translational repression is induced. Interestingly, this effect is associated with a marked accumulation of poly(A) mRNAs in cell nuclei. Thus, defective trafficking of mRNA, as a consequence of impaired nuclear mRNA export, might affect translation efficiency and contribute to the pathogenesis of C9orf72 ALS