The old standard dictionary definition of mathematics was something like, “the study of the properties of numbers and geometrical figures.” This was good enough up to some time in the 19th century. ö — Copernicus. Ã The other sciences study some concrete objects, which are visible, ponderable or detectable by physical apparatus. Mathematics is often erroneously referred to as the science of common sense. Just as physical or chemical knowledge can be independently verified by any competent experimenter, an algebraic or geometric proof can be checked and recognized as a proof by any competent algebraist or geometer. No, as long as it just sits on the shelf, it’s just ink on paper. hƒk aJ( h¯PV h
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— Bertrand Russell17. Mathematics is a game played according to certain rules with meaningless marks on paper. It is just the ideal handling of the problems of life.– Cassius Jackson Keyser 39. heY; aJ( A definition of mathematics should accept the yet-to-be-created new mathematical subjects that are sure to arise in coming decades, not to say centuries. Mathematics is the study of numbers, shapes and patterns.The word comes from the Greek word "μάθημα" (máthema), meaning "science, knowledge, or learning", and is sometimes shortened to maths (in England, Australia, Ireland, and New Zealand) or math (in the United States and Canada). Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as a sensible intelligible world. Below is a list of names and accomplishments of the greatest mathematicians of all time. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.10. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.– John William Navin Sullivan11. Infinite sets were not part of mathematics before Georg Cantor explicitly based them on the notion of one-to-one correspondence.
Mathematics is the science which draws necessary conclusions. – Julian Havel50. George Lakoff and Rafael Nunez, in Where Mathematics Comes From, showed that mathematical proof often can be understood as based on “embodied metaphors.” That explanation of proof cannot be formalized. Mathematics is on the artistic side a creation of new rhythms, orders, designs, harmonies, and on the knowledge side, is a systematic study of various rhythms, orders.– William L. Schaaf 43. (This remark of Imre Lakatos [Proofs and Refutations, introduction], is now a commonplace). Reasoning about mental objects (concepts, ideas) that compels assent (on the part of everyone who understands the concepts involved) is what we call “mathematical”. (If it isn’t, then we’d call it bad teaching.) It becomes mathematics—it comes alive—when somebody starts to read it. Mathematics is written for mathematicians. How has it been decided in the past, that some new branch of study is not just “mathematical” (containing some mathematical features), but really mathematics—requiring to be included within mathematics itself? — Carl Boyer7. Mathematics is concerned with “all possible worlds. Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze.
The things mathematics studies are neither visible nor ponderable nor detectable by physical apparatus. Ã Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof. — Morris Kline35. Mathematical activity or behavior includes: thinking, wondering, dreaming, learning about mathematics; solving math problems, at all levels, from pre-kindergarten up through postdocs and Fields Prize winners; and teaching mathematics, at all levels.
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