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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 mathematics and to understand the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts.
Recurrent themes include:
 What are the sources of mathematical subject matter?
 What is the ontological status of mathematical entities?
 What does it mean to refer to a mathematical object?
 What is the character of a mathematical proposition?
 What is the relation between logic and mathematics?
 What is the role of hermeneutics in mathematics?
 What kinds of inquiry play a role in mathematics?
 What are the objectives of mathematical inquiry?
 What gives mathematics its hold on experience?
 What are the human traits behind mathematics?
 What is mathematical beauty?
 What is the source and nature of mathematical truth?
 What is the relationship between the abstract world of mathematics and the material universe?
The terms philosophy of mathematics and mathematical philosophy are frequently used as synonyms.^{[1]} The latter, however, may be used to refer to several other areas of study. One refers to a project of formalising a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labours of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another refers to the working philosophy of an individual practitioner or a likeminded community of practicing mathematicians. Additionally, some understand the term "mathematical philosophy" to be an allusion to the approach taken by Bertrand Russell in his books The Principles of Mathematics and Introduction to Mathematical Philosophy.
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