Mathematical object

Abstract: Set theory is the most fundamental part of mathematics. The definition of almost any kind of mathematical object (a group, a ring, a vector space, a topological space, a Hilbert space...) begins: a consists of a set, together with some extra structure in the form of operations, relations, subsets, sets of subsets, functions to the real numbers, or whatever. Also, as we will see, these operations, relations, etc. are themselves special kinds of sets.

Abstract: This chapter attempts to set a perspective on where interactive technologies have taken us and where they seem to be headed. After briefly reviewing their impact in different mathematical domains, including arithmetic, algebra, geometry, statistics, and calculus, we examine what we believe to be the sources of technology’s power, which we feel is primarily epistemological. While technology’s impact on daily practice has yet to match expectations from two or three decades ago, its epistemological impact is deeper than expected. This impact is based in a reification of mathematical objects and relations that students can use to act more directly on these objects and relations than ever before. This new mathematical realism, when coupled with the fact that the computer becomes a new partner in the didactical contract, forces us to extend the didactical transposition of mathematics to a computational transposition. This new realism also drives ever deeper changes in the curriculum, and it challenges widely held assumptions about what mathematics is learnable by which students, and when they may learn it. We also examine the limits of Artificial Intelligence and microworlds and how these may be changing. We close by considering the newer possibilities offered by the Internet and its dramatic impact on connections among learners, teachers, and the immense resources that are becoming available to both. Our conclusion is that we are very early in the technological transformation and that we desperately need research in all aspects of teaching and learning with technology.

Abstract: A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about mathematical objects by simply explaininghow they can do this. My argument is based upon the adoption of a particular version of platonism — full-blooded platonism — which asserts that any mathematical object which possiblycould exist actuallydoes exist.