"Logicism is the school of thought that regards pure mathematics as a part of logic. […] If all mathematics can be expressed in purely logical terms and proved from logical principles alone, then the certainty of mathematical knowledge can be reduced to that of logic. Logic was considered to provide a certain foundation for truth." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)
"Mathematical truth ultimately depends on an irreducible set of assumptions, which are adopted without demonstration. But to qualify as true knowledge, the assumptions require a warrant for their assertion. There is no valid warrant for mathematical knowledge other than demonstration or proof. Therefore the assumptions are beliefs, not knowledge, and remain open to challenge, and thus to doubt." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)
"The absolutist view of mathematical knowledge is that it consists of certain and unchallengeable truths. According to this view, mathematical knowledge is made up of absolute truths, and represents the unique realm of certain knowledge, apart from logic and statements true by virtue of the meanings of terms […]" (Paul Ernest, "The Philosophy of Mathematics Education", 1991)
"This absolutist view of mathematical knowledge is based on two types of assumptions: those of mathematics, concerning the assumption of axioms and definitions, and those of logic concerning the assumption of axioms, rules of inference and the formal language and its syntax. These are local or micro-assumptions. There is also the possibility of global or macro-assumptions, such as whether logical deduction suffices to establish all mathematical truths." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)
"The philosophy of mathematics is the branch of philosophy whose task is to reflect on, and account for the nature of mathematics. [...] The role of the philosophy of mathematics is to provide a systematic and absolutely secure foundation for mathematical knowledge, that is for mathematical truth." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)
"Despite being partly familiar to all, because of these contradictory aspects, mathematics remains an enigma and a mystery at the heart of human culture. It is both the language of the everyday world of commercial life and that of an unseen and perfect virtual reality. It includes both free-ranging ethereal speculation and rock-hard certainty. How can this mystery be explained? How can it be unraveled? The philosophy of mathematics is meant to cast some light on this mystery: to explain the nature and character of mathematics. However this philosophy can be purely technical, a product of the academic love of technique expressed in the foundations of mathematics or in philosophical virtuosity. Too often the outcome of philosophical inquiry is to provide detailed answers to the how questions of mathematical certainty and existence, taking for granted the received ideology of mathematics, but with too little attention to the deeper why questions." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)
"It is often the case in mathematics that the definition of truth is assumed to be clear-cut, unambiguous, and unproblematic. While this is often justifiable as a simplifying assumption, the fact is that it is incorrect and that the meaning of the concept of truth in mathematics has changed significantly over time." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)
"The philosophy of mathematics is neither mathematics nor a subset of mathematics. It is a field of study which reflects on mathematics from the outside. It is one of a number of metatheories of mathematics, which also include the sociology, history, psychology, and anthropology of mathematics as well as mathematics education." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)
"The social constructivist thesis is that mathematics is a social construction, a cultural product, fallible like any other branch of knowledge." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)
No comments:
Post a Comment