Definitions of mathematics. Different schools of thought, particularly in philosophy, have put forth radically different definitions of mathematics. All are controversial and there is no consensus. Survey of leading definitions[edit] Early definitions[edit] Aristotle defined mathematics as: The science of quantity. In Aristotle's classification of the sciences, discrete quantities were studied by arithmetic, continuous quantities by geometry.[1] Auguste Comte's definition tried to explain the role of mathematics in coordinating phenomena in all other fields:[2] The science of indirect measurement.[3] Auguste Comte 1851 The "indirectness" in Comte's definition refers to determining quantities that cannot be measured directly, such as the distance to planets or the size of atoms, by means of their relations to quantities that can be measured directly.[4] Greater abstraction and competing philosophical schools[edit] Mathematics is the science that draws necessary conclusions.[6] Benjamin Peirce 1870 See also[edit]
Foundations of mathematics. Foundations of mathematics is the study of the basic mathematical concepts (number, geometrical figure, set, function...) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms...) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges. The foundations of mathematics as a whole does not aim to contain the foundations of every mathematical topic.
Historical context[edit] See also: History of logic and History of mathematics. Ancient Greek mathematics[edit] Platonism as a traditional philosophy of mathematics[edit] Middle Ages and Renaissance[edit] Set theory. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
History[edit] Georg Cantor Mathematical topics typically emerge and evolve through interactions among many researchers. Cantor's work initially polarized the mathematicians of his day. Basic concepts and notation[edit] Some ontology[edit] Logic. Logic (from the Ancient Greek: λογική, logike)[1] is the branch of philosophy concerned with the use and study of valid reasoning.[2][3] The study of logic also features prominently in mathematics and computer science. Logic is often divided into three parts: inductive reasoning, abductive reasoning, and deductive reasoning. The study of logic[edit] The concept of logical form is central to logic, it being held that the validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logics. Informal logic is the study of natural language arguments.
The study of fallacies is an especially important branch of informal logic. Logical form[edit] Main article: Logical form Logic is generally considered formal when it analyzes and represents the form of any valid argument type. This is called showing the logical form of the argument. From an observed surprising circumstance is to surmise that. Philosophy of mathematics. 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 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. Recurrent themes[edit] Recurrent themes include: What is the role of Mankind in developing mathematics? History[edit] The origin of mathematics is subject to argument. Some[who?] History of mathematics. A proof from Euclid'sElements, widely considered the most influential textbook of all time.[1] The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC),[2] the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC)[3] and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Prehistoric mathematics[edit] Babylonian mathematics[edit] Egyptian mathematics[edit] Greek mathematics[edit]