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tation 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 fundamental postulate of mathematical economics is that of the rational individual actor homo economicus lit economic man 167 in this model the individual seeks to maximize their self interest 167 and always makes optimal choices using perfect information 168 this atomistic view of economics allows it to relatively easily mathematize its thinking because individual calculations are transposed into mathematical calculations such mathematical modeling allows one to probe economic mechanisms some reject or criticize the concept of homo economicus economists note that real people have limited information make poor choices and care about fairness and altruism not just personal gain 169 without mathematical modeling it is hard to go beyond statistical observations or untestable speculation mathematical modeling allows economists to create structured frameworks to test hypotheses and analyze complex interactions models provide clarity and precision enabling the translation of theoretical concepts into quantifiable predictions that can be tested against real world data 170 at the start of the 20th century there was a development to express historical movements in formulas in 1922 nikolai kondratiev discerned the 50 year long kondratiev cycle which explains phases of economic growth or crisis 171 towards the end of the 19th century mathematicians extended their analysis into geopolitics 172 peter turchin developed cliodynamics in the 1990s 173 mathematization of the social sciences is not without risk in the controversial book fashionable nonsense 1997 sokal and bricmont denounced the unfounded or abusive use of scientific terminology particularly from mathematics or physics in the social sciences 174 the study of complex systems evolution of unemployment business capital demographic evolution of 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 mathemati...
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