Experimental philosophy and the fruitfulness of normative concepts

Philosophical Studies - This paper provides a new argument for the relevance of empirical research to moral and political philosophy and a novel defense of the positive program in experimental...     

Inconsistency in mathematics and the mathematics of inconsistency

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity.     

Modernizing the philosophy of mathematics

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematical theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content.     

Mathematical forms and forms of mathematics: leaving the shores of extensional mathematics

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according to some speculative research programs.     

Animated diagrams in teaching statistics

In this study, we investigated whether computer-animated graphics are more effective than static graphics in teaching statistics. Four statistical concepts were presented and explained to students in class. The presentations included graphics either in static or in animated form. The concepts explained were the multiplication of two matrices, the covariance of two random variables, the method of least squares in linear regression, α error, β error, and strength of effect. A comprehension test was immediately administered following the presentation. Test results showed a significant advantage for the animated graphics on retention and understanding of the concepts presented.     

李淳风数学功绩简论

一、道教学者:李淳风李淳风是唐朝著名的天文学家和数学家,他是道士之子,是中国古代著名的道教学者。他的父亲李播曾出家为道士,据《新唐书·列传第一百二十九方技》记载:“李淳风,岐州雍人。父播,仕隋高唐尉,弃官为道士,号黄冠子,以论撰自见”。这么说来,“李淳风出身于道士之     

Epistemic injustice in mathematics

We investigate how epistemic injustice can manifest itself in mathematical practices. We do this as both a social epistemological and virtue-theoretic investigation of mathematical practices. We delineate the concept both positively—we show that a certain type of folk theorem can be a source of epistemic injustice in mathematics—and negatively by exploring cases where the obstacles to participation in a mathematical practice do not amount to epistemic injustice. Having explored what epistemic injustice in mathematics can amount to, we use the concept to highlight a potential danger of intellectual enculturation.

     

Syntax-directed discovery in mathematics

It is shown how mathematical discoveries such as De Moivre's theorem can result from patterns among the symbols of existing formulae and that significant mathematical analogies are often syntactic rather than semantic, for the good reason that mathematical proofs are always syntactic, in the sense of employing only formal operations on symbols. This radically extends the Lakatos approach to mathematical discovery by allowing proof-directed concepts to generate new theorems from scratch instead of just as evolutionary modifications to some existing theorem. The emphasis upon syntax and proof permits discoveries to go beyond the limits of any prevailing semantics. It also helps explain the shortcomings of inductive AI systems of mathematics learning such as Lenat's AM, in which proof has played no part in the formation of concepts and conjectures.     

Counterfactuals and the applications of mathematics

Conclusion It has been argued that the attempt to meet indispensability arguments for realism in mathematics, by appeal to counterfactual statements, presupposes a view of mathematical modality according to which even though mathematical entities do not exist, they might have existed. But I have sought to defend this controversial view of mathematical modality from various objections derived from the fact that the existence or nonexistence of mathematical objects makes no difference to the arrangement of concrete objects. This defense of the controversial view of mathematical modality obviously falls far short of a full endorsement of the counterfactual approach, but I hope my remarks may serve to help keep such an approach a live option.     

Discussion on the foundation of mathematics

This article provides an English translation of a historic discussion on the foundations of mathematics, during which Kurt GÖdel first announced his (first) incompleteness theorem to the mathematical world. The text of the discussion is preceded by brief background remarks and commentary.     

A longitudinal assessment of early acceleration of students in mathematics on growth in mathematics achievement

Early acceleration of students in mathematics (in the form of early access to formal abstract algebra) has been a controversial educational issue. The current study examined the rate of growth in mathematics achievement of accelerated gifted, honors, and regular students across the entire secondary years (Grades 7–12), in comparison to their non-accelerated counterparts. Using data from the Longitudinal Study of American Youth, hierarchical linear models showed that early acceleration had little advantage among gifted students, small advantage among honors students, but large advantage among regular students. Equity issues, especially gender, racial, and socioeconomic equities, are not a concern once regular students were accelerated, but there are serious concerns about racial gaps among honors students and both gender and racial gaps among gifted students once they were accelerated. Schools played an important role in early acceleration, with school context rather than school climate affecting accelerated students. Students, particularly regular students, having high achievement and attending schools with high average achievement were advantageous in early acceleration. Overall, early acceleration of students in mathematics benefits regular students significantly in terms of growth in mathematics achievement.