This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present
two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case
a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important
to mathematical practice is the relation that exists between the structure and the set. In the second case, from algebraic
topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object
in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but
that structures may not be all there is to mathematics.
I wish to thank Colin McLarty as well as the anonymous referees for helpful comments on earlier versions of this paper. 相似文献
In this paper I present an argument for belief in inconsistent objects. The argument relies on a particular, plausible version of scientific realism, and the fact that often our best scientific theories are inconsistent. It is not clear what to make of this argument. Is it a reductio of the version of scientific realism under consideration? If it is, what are the alternatives? Should we just accept the conclusion? I will argue (rather tentatively and suitably qualified) for a positive answer to the last question: there are times when it is legitimate to believe in inconsistent objects. 相似文献
This paper presents Automath encodings (which are also valid in LF/λP) of various kinds of foundations of mathematics. Then it compares these encodings according to their size, to find out which foundation is the simplest.
The systems analyzed in this way are two kinds of set theory (ZFC and NF), two systems based on Church's higher order logic (Isabelle/Pure and HOL), three kinds of type theory (the calculus of constructions, Luo's extended calculus of constructions, and Martin-Löf's predicative type theory) and one foundation based on category theory.
The conclusions of this paper are that the simplest system is type theory (the calculus of constructions), but that type theories that know about serious mathematics are not simple at all. In that case the set theories are the simplest. If one looks at the number of concepts needed to explain such a system, then higher order logic is the simplest, with twenty-five concepts. On the other side of the scale, category theory is relatively complex, as is Martin-Löf's type theory.
(The full Automath sources of the contexts described in this paper are one the web at http://www.cs.ru.nl/~freek/zfc-etc/.) 相似文献
The purpose of this study was to develop and validate a Japanese version of the Achievement Emotions Questionnaire – Elementary School (AEQ-ES), which assesses enjoyment, anxiety, and boredom experienced by elementary school students within the settings of attending class, doing homework, and taking tests. Japanese elementary school students (n = 863 for the first survey; n = 332 for the second survey) participated in the questionnaire survey. The results showed that the psychometric properties of the Japanese AEQ-ES were comparable to those of the original version. Moreover, the results showed that students' achievement emotions were associated with their control and value appraisals, as well as their academic motivation, learning strategies, academic performance, and support from teachers. These results indicated that the Japanese version of the AEQ-ES was a good measure of elementary school students' emotions and supported the propositions of the control–value theory of achievement emotions. 相似文献
ABSTRACT— Twin studies comparing identical and fraternal twins consistently show substantial genetic influence on individual differences in learning abilities such as reading and mathematics, as well as in other cognitive abilities such as spatial ability and memory. Multivariate genetic research has shown that the same set of genes is largely responsible for genetic influence on these diverse cognitive areas. We call these "generalist genes." What differentiates these abilities is largely the environment, especially nonshared environments that make children growing up in the same family different from one another. These multivariate genetic findings of generalist genes and specialist environments have far-reaching implications for diagnosis and treatment of learning disabilities and for understanding the brain mechanisms that mediate these effects. 相似文献
Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development
of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a
structuralist alternative to set-theoretical foundations. Against this tendency, criticisms have been made that category theory
depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are
dispensable. The goal of this paper is to show that these criticisms are misguided by arguing that category theory is entirely
autonomous from set theory. 相似文献
We examined the effects of teaching overt precurrent behaviors on the current operant of solving multiplication and division word problems. Two students were taught four precurrent behaviors (identification of label, operation, larger numbers, and smaller numbers) in a different order, in the context of a multiple baseline design. After meeting criterion on three of the four precurrent skills, the students demonstrated the current operant of correct problem solutions. These skills generalized to novel problems. Correct current operant responses (solutions that matched answers revealed by coloring over the space with a special marker) maintained the precurrent behaviors in the absence of any other programmed reinforcement. 相似文献