Platonism and anti-platonism in mathematics - PDF Free DownloadDuring the first half of the 20th century, the philosophy of mathematics was dominated by three views: logicism , intuitionism , and formalism. Given this, it might seem odd that none of these views has been mentioned yet. The reason is that with the exception of certain varieties of formalism these views are not views of the kind discussed above. The views discussed above concern what the sentences of mathematics are really saying and what they are really about. But logicism and intuitionism are not views of this kind at all, and insofar as certain versions of formalism are views of this kind, they are versions of the views described above. How then should logicism, intuitionism, and formalism be characterized? In order to understand these views, it is important to understand the intellectual climate in which they were developed.
Truth and proof: The Platonism of mathematics
Access to the full content is only available to members of institutions that have purchased access. If you belong to such an institution, please log in or find out more about how to order. Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities. These form the subject matter of mathematical discourse: a mathematical statement is true just in case it accurately describes the mathematical facts. An important form of mathematical realism is mathematical Platonism, the view that mathematics is about a collection of independently existing mathematical objects.
The event began with Frenkel giving a presentation about math, kind of an introduction to his wonderful new book Love and Math.
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Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. Mathematical objects are exhaustively defined by their place in such structures. Consequently, structuralism maintains that mathematical objects do not possess any intrinsic properties but are defined by their external relations in a system. For instance, structuralism holds that the integer 1 is exhaustively defined by being the successor of 0 in the structure of the theory of natural numbers. By generalization of this example, any integer is defined by their respective place in this structure of the number line. Other examples of mathematical objects might include lines and planes in geometry , or elements and operations in abstract algebra. Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value.