An Empirical Basis For Arithmetical Knowledge

In the traditional view,mathematical knowledge is regarded as "absolute truth"which is necessary,apriori and analytic.Especially the arithmetic proposition in mathematics is recognized as an example of necessary truth and apriori knowledge.The knowledge of arithmetic and general empirical science(such as physics and chemistry)is different in kind,not only in degree,so it is necessary for us to provide a special set of knowledge theory for arithmetic knowledge.Intuitively,we form three opinions about arithmetic knowledge:(1)Realism:arithmetic proposition is real,that is to say,its existence does not depend on whether we recognize it or not,and arithmetic knowledge is independent of human mind.(2)Empiricism:The arithmetic proposition is directly related to the empirical world,and we need to understand the arithmetic proposition through empirical activities.(3)Apriorism:arithmetic propositions are apriori,that is to say,we do not need any specific experience to defend arithmetic knowledge.From the traditional point of view,when we take the position of realism,realism requires that the proposition of arithmetic be independent of the existence of the mind;when we take the position of empiricism,empiricism requires that all propositions independent of the existence of the mind ultimately depend on experience.Those two propositions imply the view that arithmetic propositions ultimately depend on experience.However,when we stand in the position of apriorism,apriorism requires that arithmetic propositions do not depend on experience.Therefore,contradictions arise.However,we strongly hope that we can retain the three intuitions of realism,empiricism and apriorism about arithmetic knowledge at the same time.Therefore,we need to analyze and expand these three concepts more elaborately so that these viewpoints can be reconciled.This article will focus on Jenkins' solutions and her epistemology of arithmetic.She reconciled realism and anti-realism,empiricism and apriorism so that we can retain three intuitions in arithmetic knowledge at the same time,therefore she laid the theoretical foundation for her epistemology of arithmetic knowledge.In mathematical knowledge,the most typical dispute between realism and anti-realism is Platonism and constructivism.Jenkins holds that when we use the traditional "modal-independence" to define the concept of independence from the mind,we will inevitably fall into the opposite state of realism and anti-realism.Mathematical Platonism must be a realist position and constructivism must be an anti-realist position.But when we use Jenkins's "essential-independence" to define the concept of independence from the mind,as long as the mathematical proof is not part of the mathematical proposition,we can retain constructivism as realism.Therefore,we have reached the original intention that we hope to retain the position of mathematical realism.A classical problem in epistemology is how to understand the role of experience in knowledge(especially apriori knowledge).In Kant's view,there is a clear contradiction between apriorism and experience.Kant believes that universal knowledge,taking mathematical knowledge as an example,experience can not explain the source of universal inevitability.He called universal and necessary knowledge a priori knowledge,believing that these knowledge is independent of all experience and can not be refuted by any experience.Inspired by Williamson,Jenkins enlarges the function of experience.She believes that experience plays a fundamental role in the formation of concepts.It is not only initiative,but also encoding information into our concepts and waiting to be tested.We can acquire apriori knowledge by examining the concept itself and its association with other concepts.Hence,Jenkins reconciled Transcendentalism with empiricism and retained both intuitions.The core of Jenkins,epistemology is to explain what knowledge is by satisfying an illustrative condition.This new type of epistemology can be exempted from the criticism of the Gattier problem.She applied this epistemology to arithmetic knowledge to defend arithmetic truth.