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h means in mathematics that if a result or a theory is wrong this can be proved by providing a counterexample similarly as in science theories and results theorems are often obtained from experimentation 36 in mathematics the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects often mind representations without physical support for example when asked how he came about his theorems gauss one of the greatest mathematicians of the 19th century once replied durch planmässiges tattonieren through systematic experimentation f however some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence 37 38 39 40 what precedes is only one aspect of the relationship between mathematics and other sciences other aspects are considered in the next subsections pure and applied mathematics main articles applied mathematics and pure mathematics isaac newton left and gottfried wilhelm leibniz developed infinitesimal calculus until the end of the 19th century the development of mathematics was mainly motivated by the needs of technology and science and there was no clear distinction between pure and applied mathematics for example the natural numbers and arithmetic were introduced for the need of counting and geometry was motivated by surveying architecture and astronomy later isaac newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation moreover most mathematicians were also scientists and many scientists were also mathematicians however a notable exception occurred in ancient greece see pure mathematics ancient greece in the second half ot the 19th century new mathematical theories were introduced which were not related with the physical world at least at that time in particular non euclidean geometries and cantor s theory of transfinite numbers this was one of the starting points of the foundational crisis of mathematics which was eventually solved by the systematization of the axiomatic method for defining mathematical structures so many mathematicians focused their research on internal problems that is pure mathematics and this led to split mathematics into pure mathematics and applied mathematics the latter being often considered as having a lower value during the second half of the 20th century it appeared that many theories issued from applications are also interesting from the point of view of pure mathematics and that many results of pure mathematics have applications outside mathematics see next section in turn the study of these applications may give new insights on the pure theory an example of the first case is the theory of distributions introduced by laurent schwartz for validating computations done in quantum mechanics which became immediately an important tool of pure mathematical analysis an example of the second case is the decidability of the first order theory of the real numbers a problem of pure mathematics that was proved true by alfred tarski with an algorithm that is definitely impossible to implement because of a computational complexity that is much too high for getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities george collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry so the distinction between pure and applied mathematics is presently more a question of personal research aim of mathematicians than a division of mathematics into broad areas the mathematics subject classification does not mention pure mathematics nor applied mathematics however these terms are still used in names of some university departments such as at the faculty of mathematics at the university of cambridge unreasonable effectiveness the unreasonable effectiveness of mathematics 8 is a phenomenon that was named and first made explicit by physicist eugene wigner it is the fact that many mathematical theories even the purest have applications outside their initial object these applications may be completely outside their initial area of mathematics and may concern physical phenomena that were completely unknown when the mathematical theory was introduced a famous example is the prime factorization of natural numbers that was discovered more than 2 000 years before its common use for secure internet communications through the rsa cryptosystem another historical example is the theory of ellipses they were studied by the ancient greek mathematicians as conic sections that is intersections of cones with planes it is almost 2 000 years later that johannes kepler discovered that the trajectories of the planets are ellipses in the 19th century the internal development of geometry pure mathematics lead to define and study non euclidean geometries spaces of dimension higher than three and manifolds at this time these concepts seemed totally disconnected from the physical reality but at the beginning of the 20th century albert einstein developed the theory of relativity that uses fundamentally these concepts in particular spacetime of the special relativity is a non euclidean space of dimension four and spacetime of the general relativity is a curved manifold of dimension four similar examples of unexpected applications of mathematical theories can be found in many areas of mathematics another striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics this is illustrated by the discoveries of the positron and the baryon ω displaystyle omega in both cases the equations of the theories had unexplained solutions which led to conjecture the existence of a unknown particle and to search these particles in both cases these particles were discovered a few years later by specific experiments 41 the connection between abstract mathematics and material reality has led to philosophical debates since at least the time of pythagoras the ancient philosopher plato argued this was possible because material reality reflects abstract objects that exist outside time as a result the view that mathematical objects somehow exist on their own in abstraction is often referred to as platonism although most mathematicians don t typically concern themselves with philosophical questions they may be generally considered as platonists since they think of and talk of their objects of study as real objects 42 nevertheless platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics philosophy proposed definitions main articles definitions of mathematics and philosophy of mathematics there is no general consensus about the exact definition or epistemological status of mathematics 43 44 a great many professional mathematicians take no interest in a definition of mathematics or consider it undefinable 43 there is not even consensus on whether mathematics is an art or a science 44 some just say mathematics is what mathematicians do 43 aristotle defined mathematics as the science of quantity and this definition prevailed until the 18th century however aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics in his view abstraction and studying quantity as a property separable in thought from real instances set mathematics apart 45 in the 19th century when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry which have no clear cut relation to quantity and measurement mathematicians and philosophers began to propose a variety of new definitions 46 to this day philosophers continue to tackle questions in philosophy of mathematics such as the nature of mathematical proof 47 logic and rigor see also logic mathematicians strive to develop their results with systematic reasoning in order to avoid mistaken theorems these false proofs often arise from fallible intuitions and have been common in mathematics history to allow deductive reasoning some basic assumptions need to be admitted explicitly as axioms traditionally these axioms were selected on the grounds of common sense but modern axioms typically express formal guarantees for primitive notions such as simple objects and relations the validity of a mathematical proof is fundamentally a matter of rigour and misunderstanding rigor is a notable cause for some common misconceptions about mathematics mathematical language may give more precision than in everyday speech to ordinary words like or and only other words such as open and field are given new meanings for specific mathematical concepts sometimes mathematicians even coin entirely new words e g homeomorphism this technical vocabulary is both precise and compact making it possible to mentally process complex ideas mathematicians refer to this precision of language and logic as rigor the rigor expected in mathematics has varied over time the ancient greeks expected detailed arguments but in isaac newton s time the methods employed were less rigorous not because of a different conception of mathematics but because of the lack of the mathematical methods that are required for reaching rigor problems inherent in newton s approach were solved only in the second half of the 19th century with the formal definitions of real numbers limits and integrals later in the early 20th century bertrand russell and alfred north whitehead would publish their principia mathematica an attempt to show that all mathematical concepts and statements could be defined then proven entirely through symbolic logic this was part of a wider philosophical program known as logicism which sees mathematics as primarily an extension of logic despite mathematics concision many proofs require hundreds of pages to express the emergence of computer assisted proofs has allowed proof lengths to further expand assisted proofs may be erroneous if the proving software has flaws g 48 on the other hand proof assistants allow for the verification of details that cannot be given in a hand written proof and provide certainty of the correctness of long proofs such as that of the 255 page feit thompson theorem h psychology aesthetic creativity and intuition the validity of a mathematical theorem relies only on the rigor of its proof which could theoretically be done automatically by a computer program this does not mean that there is no place for creativity in a mathematical work on the contrary many important mathematical results theorems are solutions of problems that other mathematicians failed to solve and the invention of a way for solving them may be a fundamental way of the solving process an extreme example is apery s theorem roger apery provided only the ideas for a proof and the formal proof was given only several months later by three other mathematicians creativity and rigor are not the only psychological aspects of the activity of mathematicians many mathematicians see their activity as a game more specifically as solving puzzles this aspect of mathematical activity is emphasized in recreational mathematics many mathematicians give also an aesthetic value to mathematics like beauty it is hard to define it is commonly related to elegance which involves qualities like simplicity symmetry completeness and generality g h hardy in a mathematician s apology expressed the belief that the aesthetic considerations are in themselves sufficient to justify the study of pure mathematics he also identified other criteria such as significance unexpectedness and inevitability which contribute to mathematical aesthetic 49 paul erdős expressed this sentiment more ironically by speaking of the book a supposed divine collection of the most beautiful proofs the 1998 book proofs from the book inspired by erdős is a collection of particularly succinct and revelatory mathematical arguments some examples of particularly elegant results included are euclid s proof that there are infinitely many prime numbers and the fast fourier transform for harmonic analysis some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts 50 one way this difference of viewpoint plays out is in the philosophical debate as to whether mathematical results are created as in art or discovered as in science 41 the popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions in the 20th century the mathematician l e j brouwer even initiated a philosophical perspective known as intuitionism which primarily identifies mathematics with certain creative processes in the mind 51 intuitionism is in turn one flavor of a stance known as constructivism which only considers a mathematical object valid if it can be directly constructed not merely guaranteed by logic indirectly this leads committed constructivists to reject certain results particularly arguments like existential proofs based on the law of excluded middle 52 in the end neither constructivism nor intuitionism displaced classical mathematics or achieved mainstream acceptance however these programs have motivated specific developments such as intuitionistic logic and other foundational insights which are appreciated in their own right 52 education main article mathematics education this section needs expansion with more aspects of mathematics in society such as education math as a career popular culture etc you can help by adding to it june 2022 mathematics has a remarkable ability to cross cultural boundaries and time periods as a human activity the practice of mathematics has a social side which includes education careers recognition popularization and so on in education mathematics is a core part of the curriculum while the content of courses varies many countries in the world teach mathematics to students for significant amounts of time 53 awards and prize problems main category mathematics awards the front side of the fields medal the most prestigious award in mathematics is the fields medal 54 55 established in 1936 and awarded every four years except around world war ii to up to four individuals 56 57 it is considered the mathematical equivalent of the nobel prize 57 other prestigious mathematics awards include the abel prize instituted in 2002 58 and first awarded in 2003 59 the chern medal for lifetime achievement introduced in 2009 60 and first awarded in 2010 61 the wolf prize in mathematics also for lifetime achievement 62 instituted in 1978 63 a famous list of 23 open problems called hilbert s problems was compiled in 1900 by german mathematician david hilbert 64 this list has achieved great celebrity among mathematicians 65 better source needed and as of 2022 at least thirteen of the problems depending how some are interpreted have been solved 66 a ne...