Explicit mathematics and operational set theory: some ontological comparisons

We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.

Stratification of high-risk prostate cancer into prognostic categories: a European multi-institutional study

BACKGROUND High-risk prostate cancer (PCa) is an extremely heterogeneous disease. A clear definition of prognostic subgroups is mandatory. OBJECTIVE To develop a pretreatment prognostic model for PCa-specific survival (PCSS) in high-risk PCa based on combinations of unfavorable risk factors. DESIGN, SETTING, AND PARTICIPANTS We conducted a retrospective multicenter cohort study including 1360 consecutive patients with high-risk PCa treated at eight European high-volume centers. INTERVENTION Retropubic radical prostatectomy with pelvic lymphadenectomy. OUTCOME MEASUREMENTS AND STATISTICAL ANALYSIS Two Cox multivariable regression models were constructed to predict PCSS as a function of dichotomization of clinical stage (< cT3 vs cT3-4), Gleason score (GS) (2-7 vs 8-10), and prostate-specific antigen (PSA; ≤ 20 ng/ml vs > 20 ng/ml). The first "extended" model includes all seven possible combinations; the second "simplified" model includes three subgroups: a good prognosis subgroup (one single high-risk factor); an intermediate prognosis subgroup (PSA >20 ng/ml and stage cT3-4); and a poor prognosis subgroup (GS 8-10 in combination with at least one other high-risk factor). The predictive accuracy of the models was summarized and compared. Survival estimates and clinical and pathologic outcomes were compared between the three subgroups. RESULTS AND LIMITATIONS The simplified model yielded an R(2) of 33% with a 5-yr area under the curve (AUC) of 0.70 with no significant loss of predictive accuracy compared with the extended model (R(2): 34%; AUC: 0.71). The 5- and 10-yr PCSS rates were 98.7% and 95.4%, 96.5% and 88.3%, 88.8% and 79.7%, for the good, intermediate, and poor prognosis subgroups, respectively (p = 0.0003). Overall survival, clinical progression-free survival, and histopathologic outcomes significantly worsened in a stepwise fashion from the good to the poor prognosis subgroups. Limitations of the study are the retrospective design and the long study period. CONCLUSIONS This study presents an intuitive and easy-to-use stratification of high-risk PCa into three prognostic subgroups. The model is useful for counseling and decision making in the pretreatment setting.

A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0

We partially solve a long-standing problem in the proof theory of explicit mathematics or the proof theory in general. Namely, we give a lower bound of Feferman’s system T0 of explicit mathematics (but only when formulated on classical logic) with a concrete interpretat ion of the subsystem Σ12-AC+ (BI) of second order arithmetic inside T0. Whereas a lower bound proof in the sense of proof-theoretic reducibility or of ordinalanalysis was already given in 80s, the lower bound in the sense of interpretability we give here is new. We apply the new interpretation method developed by the author and Zumbrunnen (2015), which can be seen as the third kind of model construction method for classical theories, after Cohen’s forcing and Krivine’s classical realizability. It gives us an interpretation between classical theories, by composing interpretations between intuitionistic theories.

How gender fair are German schoolbooks in the twenty-first century? An analysis of language and illustrations in schoolbooks for mathematics and German

Schoolbooks convey not only school-relevant knowledge; they also influence the development of stereotypes about different social groups. Particularly during the 1970s and 1980s, many studies analysed schoolbooks and criticised the overall predominance of male persons and of traditional role allocations. Since that time, women’s and men’s occupations and social functions have changed considerably. The present research investigated gender portrayals in schoolbooks for German and mathematics that were recently published in Germany. We examined the proportions of female and male persons in pictures and texts and categorized their activities, occupational and parental roles. Going beyond previous studies, we added two criteria: the use of gender-fair language and the spatial arrangements of persons in pictures. Our results show that schoolbooks for German contained almost balanced depictions of girls and boys, whereas women were less frequently shown than men. In mathematics books, males outnumbered females in general. Across both types of books, female and male persons were engaged in many different activities, not only gendertyped ones; however, male persons were more often described via their profession than females. Use of gender-fair language has found its way into schoolbooks but is not used consistently. Books for German were more gender fair in terms of linguistic forms than books for mathematics. For spatial arrangements, we found no indication for gender biases. The results are discussed with a focus on how schoolbooks can be optimized to contribute to gender equality.

Using space to represent categories: Insights from gaze position

We investigated the boundaries among imagery, memory, and perception by measuring gaze during retrieved versus imagined visual information. Eye fixations during recall were bound to the location at which a specific stimulus was encoded. However, eye position information generalized to novel objects of the same category that had not been seen before. For example, encoding an image of a dog in a specific location enhanced the likelihood of looking at the same location during subsequent mental imagery of other mammals. The results suggest that eye movements can also be launched by abstract representa- tions of categories and not exclusively by a single episode or a specific visual exemplar.