mathematics, philosophy of
The philosophy of mathematics attempts to explain both the nature of mathematical facts and entities, and the way in which we have our knowledge of both. Modern philosophy of mathematics began with the foundational studies of Cantor, R. Dedekind, and K. T. W. Weierstrass in the late 19th century. It received its fundamental impulse from the work of Frege and Russell on the relations between numbers, sets, and logic. The logicist programme in the philosophy of mathematics, culminating in Russell and Whitehead's great Principia Mathematica of 1910–13, attempted to reduce mathematics to logic, in the sense of proving that all of mathematics could be represented in a system whose axioms were simply axioms of pure logic. It is generally thought that the programme failed over the need to admit sets as entities subject to their own particular axioms, whose logical status was not at all obvious. At the beginning of the 20th century the place of set theory in mathematics was well established, but the contradictions of naïve set theory, and the sheer scale of the full transfinite hierarchy, divided mathematicians into several camps, reflecting different philosophies of mathematics. Full-scale realism or Platonism takes mathematical entities to be real, independent objects of study about which discoveries (and mistakes) can be made. Constructivism takes it that we ourselves construct what we talk about. Formalism assimilates mathematics to the purely syntactic process of following proof procedures, without any question being raised about the interpretation of the theorems proved. The Achilles heel of formalism has always been the application of mathematics in our best descriptions of the world, so realism and constructivism have emerged as the main contrasting ideologies. The division has mathematical consequences. For the realist there is no problem in admitting completed infinite collections, whether or not we have an effective method of specifying their members. For the constructivist this will be inadmissible.
Quite apart from this issue modern debates on the epistemology of mathematics have been infected by the general flight from the category of a priori knowledge. Suggestions taking its place include the general conventionalism associated with Wittgenstein, the view that abstract objects such as sets are not so frightening after all, and indeed are in fact perceptible, and the view that the general success of science, of which mathematics is an indispensable part, affords an argument for the truth of mathematics.

Philosophy dictionary. . 2011.

Look at other dictionaries:

  • mathematics, philosophy of — Branch of philosophy concerned with the epistemology and ontology of mathematics. Early in the 20th century, three main schools of thought called logicism, formalism, and intuitionism arose to account for and resolve the crisis in the foundations …   Universalium

  • mathematics, foundations of — Scientific inquiry into the nature of mathematical theories and the scope of mathematical methods. It began with Euclid s Elements as an inquiry into the logical and philosophical basis of mathematics in essence, whether the axioms of any system… …   Universalium

  • philosophy of mathematics — mathematics, philosophy of …   Philosophy dictionary

  • Philosophy of science — is the study of assumptions, foundations, and implications of science. The field is defined by an interest in one of a set of traditional problems or an interest in central or foundational concerns in science. In addition to these central… …   Wikipedia

  • Philosophy of music — Philosophy ( …   Wikipedia

  • Philosophy of language — is the reasoned inquiry into the nature, origins, and usage of language. As a topic, the philosophy of language for Analytic Philosophers is concerned with four central problems: the nature of meaning, language use, language cognition, and the… …   Wikipedia

  • Philosophy of Immanuel Kant —     Philosophy of Immanuel Kant     † Catholic Encyclopedia ► Philosophy of Immanuel Kant     Kant s philosophy is generally designated as a system of transcendental criticism tending towards Agnosticism in theology, and favouring the view that… …   Catholic encyclopedia

  • Philosophy of mathematics — The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of …   Wikipedia

  • Philosophy of mathematics education — The Philosophy of mathematics education is an interdisciplinary area of study and research based on the intersection of the fields of mathematics education and the philosophy of mathematics, the latter being understood in an inclusive sense to… …   Wikipedia

  • Philosophy of Arithmetic (book) — infobox Book | name = Philosophy of Arithmetic title orig = Philosophie der Arithmetik translator = Dallas Willard author = Edmund Husserl illustrator = cover artist = country = Cities: Dordrecht; Boston language = English series = subject =… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”