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Three main leading types of definition of mathematics are called as logicist, intuitionist and formalist; each signifying a different philosophical school of thought. All have stern problems, none has extensive acceptance and no settlement looks possible.
An early description or definition of mathematics in terms of logic was Benjamin Peirce's "It is a science that draws required conclusions". In Principia Mathematica, Alfred North Whitehead and Bertrand Russell advanced the philosophical program which is called logicism and attempted to present that all mathematical perceptions, statements and principles can be shown and proven totally in terms of symbolic logic. A definition of logicist of mathematics is Russell's "All Mathematics is Symbolic Logic".
Intuitionist definitions, developing from the philosophy of mathematician L. E. J. Brouwer, recognize mathematics with certain mental phenomena. Specially, whereas other philosophies of mathematics permit objects that can be proven to exist even though they cannot be constructed, intuitivism allows only mathematical objects that one can really construct.
Formalist definitions spot mathematics with its symbols and the set of laws or rules for working on them. Haskell Curry defined mathematics simply as "the science of formal systems."
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