A new foundational crisis in mathematics, is it really happening?

The article reconsiders the position of the foundations of mathematics after the discovery of HoTT. Discussion that this discovery has generated in the community of mathematicians, philosophers and computer scientists might indicate a new crisis in the foundation of mathematics. By examining the mathematical facts behind HoTT and their relation with the existing foundations, we conclude that the present crisis is not one. We reiterate a pluralist vision of the foundations of mathematics. The article contains a short survey of the mathematical and historical background needed to understand the main tenets of the foundational issues.Comment: Final versio

The AKNS-hierarchy

We present here an overview for the Encyclopaedia of Mathematics of the various forms and properties of this system of equations together with its geometric and Lie algebraic background

Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

The Roma/Non-Roma Test Score Gap in Hungary

This paper documents and decomposes the test score gap between Roma and non-Roma 8th graders in Hungary in 2006. Our data connect national standardized test scores to an individual panel survey with detailed data on ethnicity and family background. The test score gap is approximately one standard deviation for both reading and mathematics, which is similar to the gap between African-American and White students of the same age group in the US in the 1980s. After accounting for on health, parenting, school fixed effects and family background, the gap disappears in reading and drops to 0.15 standard deviation in mathematics.

Mathematics Course Placement Using Holistic Measures: Possibilities for Community College Students.

Background/Context: Most community colleges across the country use a placement test to determine students’ readiness for college-level coursework, yet these tests are admittedly imperfect instruments. Researchers have documented significant problems stemming from overreliance on placement testing, including placement error and misdiagnosis of remediation needs. They have also described significant consequences of misplacement, which can hinder the educational progression and attainment of community college students. Purpose/Objective/Research Question/Focus of Study: We explore possibilities for placing community college students in mathematics courses using a holistic approach that considers measures beyond placement test scores. This includes academic background measures, such as high school GPA and math courses taken, and indicators of noncognitive constructs, such as motivation, time use, and social support. Setting: The study draws upon administrative data from a large urban community college district in California that serves over 100,000 students each semester. The data enable us to link students’ placement testing results, survey data, background information, and transcript records. Research Design: We first use the supplemental survey data gathered during routine placement testing to conduct predictive exercises that identify severe placement errors under existing placement practices. We then move beyond prediction and evaluate student outcomes in two colleges where noncognitive indicators were directly factored into placement algorithms. Findings/Results: Using high school background information and noncognitive indicators to predict success reveals as many as one quarter of students may be misassigned to their math courses by status quo practices. In our subsequent analysis we find that students placed under a holistic approach that considered noncognitive indicators in addition to placement test scores performed no differently from higher scoring peers in the same course. Conclusions/Recommendations: The findings suggest a holistic approach to mathematics course placement may improve placement accuracy and provide access to higher level mathematics courses for community college students without compromising their likelihood of success