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ematics such as the introduction of variables and symbolic notation attributed to françois viète 1540 1603 97 the introduction of logarithms by john napier in 1614 98 which greatly simplified numerical calculations especially for astronomy and marine navigation the introduction of cartesian coordinates by rené descartes 1596 1650 for reducing geometry to algebra 99 and the development of calculus by isaac newton 1643 1727 and gottfried leibniz 1646 1716 100 leonhard euler 1707 1783 unified and built upon these innovations with a standardized terminology and complemented them with the discovery and proofs of numerous theorems 101 carl friedrich gauss perhaps the foremost mathematician of the 19th century was the german mathematician carl gauss who made numerous contributions to fields such as algebra analysis differential geometry matrix theory number theory and statistics 102 in the early 20th century kurt gödel transformed mathematics by publishing his incompleteness theorems which show in part that any consistent axiomatic system if powerful enough to describe arithmetic will contain true propositions that cannot be proved 57 mathematics has since been greatly extended and there has been a fruitful interaction between mathematics and science to the benefit of both mathematical discoveries continue to be made to this very day according to mikhail b sevryuk in the january 2006 issue of the bulletin of the american mathematical society the number of papers and books included in the mathematical reviews mr database since 1940 the first year of operation of mr is now more than 1 9 million and more than 75 thousand items are added to the database each year the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs 103 symbolic notation and terminology main articles mathematical notation language of mathematics and glossary of mathematics an explanation of the sigma σ summation notation mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise unambiguous and accurate way this notation consists of symbols used for representing operations unspecified numbers relations and any other mathematical objects and then assembling them into expressions and formulas 104 more precisely numbers and other mathematical objects are represented by symbols called variables which are generally latin or greek letters and often include subscripts operation and relations are generally represented by specific symbols or glyphs 105 such as plus multiplication textstyle int integral equal and less than 106 all these symbols are generally grouped according to specific rules to form expressions and formulas 107 normally expressions and formulas do not appear alone but are included in sentences of the current language where expressions play the role of noun phrases and formulas play the role of clauses mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract idealized objects and how they interact it is based on rigorous definitions that provide a standard foundation for communication an axiom or postulate is a mathematical statement that is taken to be true without need of proof if a mathematical statement has yet to be proven or disproven it is termed a conjecture through a series of rigorous arguments employing deductive reasoning a statement that is proven to be true becomes a theorem a specialized theorem that is mainly used to prove another theorem is called a lemma a proven instance that forms part of a more general finding is termed a corollary 108 numerous technical terms used in mathematics are neologisms such as polynomial and homeomorphism 109 other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning for example in mathematics or means one the other or both while in common language it is either ambiguous or means one or the other but not both in mathematics the latter is called exclusive or finally many mathematical terms are common words that are used with a completely different meaning 110 this may lead to sentences that are correct and true mathematical assertions but appear to be nonsense to people who do not have the required background for example every free module is flat and a field is always a ring relationship with sciences mathematics is used in most sciences for modeling phenomena which then allows predictions to be made from experimental laws 111 the independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model 112 inaccurate predictions rather than being caused by invalid mathematical concepts imply the need to change the mathematical model used 113 for example the perihelion precession of mercury could only be explained after the emergence of einstein s general relativity which replaced newton s law of gravitation as a better mathematical model 114 there is still a philosophical debate whether mathematics is a science however in practice mathematicians are typically grouped with scientists and mathematics shares much in common with the physical sciences like them it is falsifiable which 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 115 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 once replied durch planmässiges tattonieren through systematic experimentation 116 however some authors emphasize that mathematics differs from the modern notion of science by not relying on empirical evidence 117 118 119 120 pure and applied mathematics main articles applied mathematics and pure mathematics isaac newton left and gottfried wilhelm leibniz developed infinitesimal calculus until the 19th century there was no clear distinction between pure and applied mathematics as understood today 121 the distinction between developing mathematics for its own sake or for its applications was rather fluid natural numbers and arithmetic were introduced for the need of counting and geometry was motivated by surveying architecture and astronomy but both subjects quickly stood on their own later isaac newton used infinitesimal calculus in part to help explain the movement of the planets and his law of gravitation moreover since antiquity most mathematicians were also scientists and many scientists were also mathematicians 122 nonetheless the western tradition of pure mathematics traces its roots back to ancient greece 123 the problem of integer factorization for example which goes back to euclid in 300 bc had no practical application before its use in the rsa cryptosystem now widely used for the security of computer networks 124 in the 19th century mathematicians such as karl weierstrass and richard dedekind increasingly focused their research on internal problems that is pure mathematics 121 125 this led to split mathematics into pure mathematics and applied mathematics the latter being often considered as having a lower value among mathematical purists however the lines between the two are frequently blurred 126 the aftermath of world war ii led to a surge in the development of applied mathematics in the us and elsewhere 127 128 many of the theories developed for applications were found interesting from the point of view of pure mathematics and many results of pure mathematics were shown to have applications outside mathematics in turn the study of these applications may give new insights on the pure theory 129 130 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 131 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 impossible to implement because of a computational complexity that is much too high 132 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 133 in the present day the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas 134 135 the mathematics subject classification has a section for general applied mathematics but does not mention pure mathematics 9 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 is a phenomenon that was named and first made explicit by physicist eugene wigner 5 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 136 examples of unexpected applications of mathematical theories can be found in many areas of mathematics a notable 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 137 a second 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 was almost 2 000 years later that johannes kepler discovered that the trajectories of the planets are ellipses 138 in the 19th century the internal development of geometry pure mathematics led to definition and study of 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 special relativity is a non euclidean space of dimension four and spacetime of general relativity is a curved manifold of dimension four 139 140 a 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 of the existence of an unknown particle and the search for these particles in both cases these particles were discovered a few years later by specific experiments 141 142 143 specific sciences physics main article relationship between mathematics and physics diagram of a pendulum mathematics and physics have influenced each other over their modern history modern physics uses mathematics abundantly 144 and is also considered to be the motivation of major mathematical developments 145 computing further information theoretical computer science and computational mathematics computing is closely related to mathematics in several ways 146 theoretical computer science is considered to be mathematical in nature 147 communication technologies apply branches of mathematics that may be very old e g arithmetic especially with respect to transmission security in cryptography and coding theory discrete mathematics is useful in many areas of computer science such as complexity theory information theory and graph theory 148 in 1998 the kepler conjecture on sphere packing seemed to also be partially proven by computer 149 statistics and other decision sciences main articles statistics and probability theory whatever the form of a random population distribution μ the sampling mean x̄ tends to a gaussian distribution and its variance σ is given by the central limit theorem of probability theory 150 the field of statistics is a mathematical application that is employed for the collection and processing of data samples using procedures based on mathematical methods such as and especially probability theory statisticians generate data with random sampling or randomized experiments 151 statistical theory studies decision problems such as minimizing the risk expected loss of a statistical action such as using a procedure in for example parameter estimation hypothesis testing and selecting the best in these traditional areas of mathematical statistics a statistical decision problem is formulated by minimizing an objective function like expected loss or cost under specific constraints for example designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence 152 because of its use of optimization the mathematical theory of statistics overlaps with other decision sciences such as operations research control theory and mathematical economics 153 biology and chemistry main articles mathematical and theoretical biology and mathematical chemistry the skin of this giant pufferfish exhibits a turing pattern which can be modeled by reaction diffusion systems biology uses probability extensively in fields such as ecology or neurobiology 154 most discussion of probability centers on the concept of evolutionary fitness 154 ecology heavily uses modeling to simulate population dynamics 154 155 study ecosystems such as the predator prey model measure pollution diffusion 156 or to assess climate change 157 the dynamics of a population can be modeled by coupled differential equations such as the lotka volterra equations 158 statistical hypothesis testing is run on data from clinical trials to determine whether a new treatment works 159 since the start of the 20th century chemistry has used computing to model molecules in three dimensions 160 earth sciences main article geomathematics structural geology and climatology use probabilistic models to predict the risk of natural catastrophes 161 similarly meteorology oceanography and planetology also use mathematics due to their heavy use of models 162 163 164 social sciences further information mathematical economics and historical dynamics areas of mathematics used in the social sciences include probability statistics and differential equations these are used in linguistics economics sociology 165 and psychology 166 supply and demand curves like this one are a staple of mathematical economics often the fundame...
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