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a population etc uses mathematical knowledge however the choice of counting criteria particularly for unemployment or of models can be subject to controversy 175 176 philosophy main article philosophy of mathematics reality the connection between mathematics and material reality has led to philosophical debates since at least the time of pythagoras the ancient philosopher plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time as a result the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as platonism independently of their possible philosophical opinions modern mathematicians may be generally considered as platonists since they think of and talk of their objects of study as real objects 177 armand borel summarized this view of mathematics reality as follows and provided quotations of g h hardy charles hermite henri poincaré and albert einstein that support his views 141 something becomes objective as opposed to subjective as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together 178 because the language of mathematics is so precise it is ideally suited to defining concepts for which such a consensus exists in my opinion that is sufficient to provide us with a feeling of an objective existence of a reality of mathematics nevertheless platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics as platonism assumes mathematics exists independently but does not explain why it matches reality 179 proposed definitions main article definitions of mathematics there is no general consensus about the definition of mathematics or its epistemological status that is its place inside knowledge a great many professional mathematicians take no interest in a definition of mathematics or consider it undefinable there is not even consensus on whether mathematics is an art or a science some just say mathematics is what mathematicians do 180 181 a common approach is to define mathematics by its object of study 182 183 184 185 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 186 in the 19th century when mathematicians began to address topics such as infinite sets which have no clear cut relation to physical reality a variety of new definitions were given 187 with the large number of new areas of mathematics that have appeared since the beginning of the 20th century defining mathematics by its object of study has become increasingly difficult 188 for example in lieu of a definition saunders mac lane in mathematics form and function summarizes the basics of several areas of mathematics emphasizing their inter connectedness and observes 189 the development of mathematics provides a tightly connected network of formal rules concepts and systems nodes of this network are closely bound to procedures useful in human activities and to questions arising in science the transition from activities to the formal mathematical systems is guided by a variety of general insights and ideas another approach for defining mathematics is to use its methods for example an area of study is often qualified as mathematics as soon as one can prove theorems assertions whose validity relies on a proof that is a purely logical deduction d 190 failed verification rigor see also logic mathematical reasoning requires rigor this means that the definitions must be absolutely unambiguous and the proofs must be reducible to a succession of applications of inference rules e without any use of empirical evidence and intuition f 191 rigorous reasoning is not specific to mathematics but in mathematics the standard of rigor is much higher than elsewhere despite mathematics concision rigorous proofs can require hundreds of pages to express such as the 255 page feit thompson theorem g the emergence of computer assisted proofs has allowed proof lengths to further expand h 192 the result of this trend is a philosophy of the quasi empiricist proof that can not be considered infallible but has a probability attached to it 8 the concept of rigor in mathematics dates back to ancient greece where their society encouraged logical deductive reasoning however this rigorous approach would tend to discourage exploration of new approaches such as irrational numbers and concepts of infinity the method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation in the 18th century social transition led to mathematicians earning their keep through teaching which led to more careful thinking about the underlying concepts of mathematics this produced more rigorous approaches while transitioning from geometric methods to algebraic and then arithmetic proofs 8 at the end of the 19th century it appeared that the definitions of the basic concepts of mathematics were not accurate enough for avoiding paradoxes non euclidean geometries and weierstrass function and contradictions russell s paradox this was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories the re introduction of axiomatic method pioneered by the ancient greeks 8 it results that rigor is no more a relevant concept in mathematics as a proof is either correct or erroneous and a rigorous proof is simply a pleonasm where a special concept of rigor comes into play is in the socialized aspects of a proof wherein it may be demonstrably refuted by other mathematicians after a proof has been accepted for many years or even decades it can then be considered as reliable 193 nevertheless the concept of rigor may remain useful for teaching to beginners what is a mathematical proof 194 training and practice see also computer based mathematics education education main article mathematics education 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 and forms an important element of the stem academic disciplines prominent careers for professional mathematicians include mathematics teacher or professor statistician actuary financial analyst economist accountant commodity trader or computer consultant 195 archaeological evidence shows that instruction in mathematics occurred as early as the second millennium bce in ancient babylonia 196 comparable evidence has been unearthed for scribal mathematics training in the ancient near east and then for the greco roman world starting around 300 bce 197 the oldest known mathematics textbook is the rhind papyrus dated from c 1650 bce in egypt 198 due to a scarcity of books mathematical teachings in ancient india were communicated using memorized oral tradition since the vedic period c 1500 c 500 bce 199 in imperial china during the tang dynasty 618 907 ce a mathematics curriculum was adopted for the civil service exam to join the state bureaucracy 200 following the dark ages mathematics education in europe was provided by religious schools as part of the quadrivium the oldest journal addressing instruction in mathematics was l enseignement mathématique which began publication in 1899 201 the western advancements in science and technology led to the establishment of centralized education systems in many nation states with mathematics as a core component initially for its military applications 202 while the content of courses varies in the present day nearly all countries teach mathematics to students for significant amounts of time 203 during school mathematical capabilities and positive expectations have a strong association with career interest in the field extrinsic factors such as feedback motivation by teachers parents and peer groups can influence the level of interest in mathematics 204 some students studying mathematics may develop an apprehension or fear about their performance in the subject this is known as mathematical anxiety and is considered the most prominent of the disorders impacting academic performance mathematical anxiety can develop due to various factors such as parental and teacher attitudes social stereotypes and personal traits help to counteract the anxiety can come from changes in instructional approaches by interactions with parents and teachers and by tailored treatments for the individual 205 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 206 207 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 208 creativity and rigor are not the only psychological aspects of the activity of mathematicians some mathematicians can see their activity as a game more specifically as solving puzzles 209 this aspect of mathematical activity is emphasized in recreational mathematics mathematicians can find 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 aesthetics 210 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 211 some feel that to consider mathematics a science is to downplay its artistry and history in the seven traditional liberal arts 212 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 141 the popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions cultural impact artistic expression main article mathematics and art notes that sound well together to a western ear are sounds whose fundamental frequencies of vibration are in simple ratios for example an octave doubles the frequency and a perfect fifth multiplies it by 3 2 textstyle frac 3 2 213 214 fractal with a scaling symmetry and a central symmetry humans as well as some other animals find symmetric patterns to be more beautiful 215 mathematically the symmetries of an object form a group known as the symmetry group 216 for example the group underlying mirror symmetry is the cyclic group of two elements z 2 z displaystyle mathbb z 2 mathbb z a rorschach test is a figure invariant by this symmetry 217 as are butterfly and animal bodies more generally at least on the surface 218 waves on the sea surface possess translation symmetry moving one s viewpoint by the distance between wave crests does not change one s view of the sea 219 fractals possess self similarity 220 221 popularization main article popular mathematics popular mathematics is the act of presenting mathematics without technical terms 222 presenting mathematics may be hard since the general public suffers from mathematical anxiety and mathematical objects are highly abstract 223 however popular mathematics writing can overcome this by using applications or cultural links 224 despite this mathematics is rarely the topic of popularization in printed or televised media awards and prize problems main category mathematics awards the front side of the fields medal with an illustration of the greek polymath archimedes the most prestigious award in mathematics is the fields medal 225 226 established by canadian john charles fields in 1936 and awarded every four years except around world war ii to up to four individuals 227 228 it is considered the mathematical equivalent of the nobel prize 228 other prestigious mathematics awards include 229 the abel prize instituted in 2002 230 and first awarded in 2003 231 the chern medal for lifetime achievement introduced in 2009 232 and first awarded in 2010 233 the ams leroy p steele prize awarded since 1970 234 the wolf prize in mathematics also for lifetime achievement 235 instituted in 1978 236 a famous list of 23 open problems called hilbert s problems was compiled in 1900 by german mathematician david hilbert 237 this list has achieved great celebrity among mathematicians 238 and at least thirteen of the problems depending how some are interpreted have been solved 237 a new list of seven important problems titled the millennium prize problems was published in 2000 only one of them the riemann hypothesis duplicates one of hilbert s problems a solution to any of these problems carries a 1 million dollar reward 239 to date only one of these problems the poincaré conjecture has been solved by the russian mathematician grigori perelman 240 see also mathematics portal law mathematics list of mathematical jargon lists of mathematicians list of mathematics books list of mathematics journals lists of mathematics topics mathematical constant mathematical sciences mathematics and art mathematics education philosophy of mathematics relationship between mathematics and physics science technology engineering and mathematics notes here algebra is taken in its modern sense which is roughly speaking the art of manipulating formulas this includes conic sections which are intersections of circular cylinders and planes however some advanced methods of analysis are sometimes used for example methods of complex analysis applied to generating series for example logic belongs to philosophy since aristotle circa the end of the 19th century the foundational crisis of mathematics implied developments of logic that are specific to mathematics this allowed eventually the proof of th...
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