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Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship ...
The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can assert the truth of a statement only by verifying the validity of that ...
The role of mathematics in Western philosophy has grown and expanded from Pythagoras onwards. It is clear that numbers held a particular importance for the Pythagorean school , although it was the later work of Plato that attracts the label of mathematicism from modern philosophers.
The shift in meaning for mathema is likely a result of the rapid categorization during the time of Plato and Aristotle of their mathemata in terms of education: arithmetic, geometry, astronomy, and music (the quadrivium), which the Greeks found to create a "natural grouping" of mathematical (in the modern usage; "doctrina mathematica" in the ancient usage) precepts.
An Introduction to the Philosophy of Mathematics is a 2012 textbook on the philosophy of mathematics by Mark Colyvan. It has a focus on issues in contemporary philosophy , such as the mathematical realism – anti-realism debate and the philosophical significance of mathematical practice, and largely skips over historical debates.
Aristotelian views of (cardinal or counting) numbers begin with Aristotle's observation that the number of a heap or collection is relative to the unit or measure chosen: "'number' means a measured plurality and a plurality of measures ... the measure must always be some identical thing predicable of all the things it measures, e.g. if the things are horses, the measure is 'horse'."
There is no general consensus about the definition of mathematics or its epistemological status—that is, its place inside knowledge. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science.
Logicism is a school of thought, and research programme, in the philosophy of mathematics, based on the thesis that mathematics is an extension of logic or that some or all mathematics may be derived in a suitable formal system whose axioms and rules of inference are 'logical' in nature.