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The Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the "maverick" trend in the philosophy of mathematics. Several points made by its main representatives are mentioned - from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the new approach questions Hilbert's Thesis, according to which a correct mathematical proof is in principle reducible to a formal proof, based on explicit axioms and logic. [ABSTRACT FROM AUTHOR]

The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, but because these real justifications are distributed in the written archive of mathematics, proofs remain surveyable, hence good. [ABSTRACT FROM AUTHOR]

Kant discovered a philosophical problem with mathematical proof. Despite being a priori, its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals why these proofs—written in the vernacular to this very day—succeed in conforming to those norms. [ABSTRACT FROM AUTHOR]

Abstract: This is a study of mathematics students working in small groups. Our research methodology allows us to examine how individual ideas develop in a social context. The research perspective used in this study is based on a co-constructive view of learning. Groups of three or four undergraduate mathematics majors, with prior experience writing mathematical proofs together, were asked to prove three statements. Computer software, such as Geometers Sketchpad, was available. Group work sessions were videotaped. Later, individuals viewed segments of the group video and were asked to reflect on group activities. Students in some groups did not share a common conception of proof, which seemed to hamper their collaboration. We observed interactions that fit with the co-constructive theory, with bidirectional interactions that shaped both group and individual conceptions of the tasks. These changes in understanding may result from parallel and successive internalization and externalization of ideas by individuals in a social context. [Copyright &y& Elsevier]

We describe how elementary Linear Algebra can be taught successfully while introducing students to the concept and practice of `mathematical proof'. This is done badly with a sophisticated Definition–Lemma–Proof–Theorem–Proof–Corollary (DLPTPC) approach; badly – since students in elementary Linear Algebra courses have very little experience with proofs and mathematical rigor. Instead, the subjects and concepts of Linear Algebra can be introduced in an exploratory and fundamentally reasoned way. One seemingly successful way to do this is to explore the concept of solvability of linear systems first via the row echelon form (REF). Solvability questions lead to row and column criteria for a REF that can be used repeatedly to: compute subspaces, settle linear (in)dependence, find inverses, perform basis change, compute determinants, analyze eigensystems etc. If these subjects are explained heuristically from the first principles of linear transformations, linear equations, and the REF, students experience the power of a concept–built approach and reap the benefit of deep math understanding. Moreover, early `salient point' proofs lead to an intuitive understanding of `math proof'. Once the basic concept of `proof' is ingrained in students, more abstract proofs, even DLPTPC style expositions, on normal matrices, the SVD etc. become accessible and understandable to sophomore students. With the help of this gentle early approach, the concept and construct of a `math proof' becomes firmly embedded in the students' minds and helps with future math courses and general scientific reasoning. [ABSTRACT FROM AUTHOR]