“The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument.” StatesGoalStepsObjectsArgumentMathematicsPropertyDefinitionsVariousProofStatementsMathematicalReasoningIntroducingJustifiedConvincingPresentingTheoremsSmall Steps Author:IU?. I. Manin
“Outside observers often assume that the more complicted a piece of mathematics is, the more mathematicians admire it. Nothing could be further from the truth. Mathematicians admire elegance and simplicity above all else, and the ultimate goal in solving a problem is to find the method that does the job in the most efficient manner. Though the major accolades are given to the individual who solves a particular problem first, credit (and gratitude) always goes to those who subsequently find a simpler solution.” FirstsDoeProblemJobsIndividualGivenGoalPiecesParticularGratitudeMajorsSolutionsUltimateMathematicsMethodAssumingSimplicityCreditSolveAdmireEfficientMathematicianObserversEleganceUltimate GoalAccolades Author:Keith Devlin
“The research reported on in our book "A=B", has moved a whole active field of mathematics from the province of human thought to the realm of computer-fodder. It is quite exciting to think about what other fields of pure mathematics, hitherto thought to be reserved to human intelligence, might be moved to that realm next. The goal is to put ourselves out of business completely, and the work is well underway.” ThinkingHumansWellsBookWholeMightScienceNextGoalFieldsPureComputerResearchExcitingMathematicsMovedActiveRealmsReservedProvincesHuman ThoughtHuman IntelligenceFodder Author:Herbert Wilf
“. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories -- specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms (like functions) form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.” ShouldWellsKindPhilosophyFormUnderstandingGoalDealsObjectsFitApproachFunctionRelationMathematicsFoundationObjectivesAlternativesOperationsCategoriesCompositionReplacedOften IsMembership Author:Saunders Mac Lane