Platonism
Platonism, rendered as a proper noun, is the philosophy of Plato or the name of other philosophical systems considered closely derived from it. In narrower usage, platonism, rendered as a common noun (with a lower case 'p', subject to sentence case), refers to the philosophy that affirms the existence of abstract objects, which are asserted to "exist" in a "third realm" distinct both from the sensible external world and from the internal world of consciousness, and is the opposite of nominalism (with a lower case "n"). Lower case "platonists" need not accept any of the doctrines of Plato.
In a narrower sense, the term might indicate the doctrine of Platonic realism. The central concept of Platonism, a distinction essential to the Theory of Forms, is the distinction between the reality which is perceptible but unintelligible, and the reality which is imperceptible but intelligible. The forms are typically described in dialogues such as the Phaedo, Symposium and Republic as transcendent, perfect archetypes, of which objects in the everyday world are imperfect copies.
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 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.
The terms philosophy of mathematics and mathematical philosophy are frequently used as synonyms. The latter, however, may be used to refer to several other areas of study. One refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term "mathematical philosophy" to be an allusion to the approach to the foundations of mathematics taken by Bertrand Russell in his books The Principles of Mathematics and Introduction to Mathematical Philosophy.
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