In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object. For example, the equations x = cos t y = sin t form a parametric representation of the unit circle, where t is the parameter: A point is on the unit circle if and only if there is a value of t such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: =. Parametric representations are nonunique, so the same quantities may be expressed by a number of different parameterizations. In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, the number of equations being equal to the dimension of the space in which the manifold or variety is considered.
Parametric equations are used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is labeled t. Parameterizations are non-unique. In kinematics, objects' paths through space are described as parametric curves, with each spatial coordinate depending explicitly on an independent parameter. Used in this way, the set of parametric equations for the object's coordinates collectively constitute a vector-valued function for position; such parametric curves can be integrated and differentiated termwise. Thus, if a particle's position is described parametrically as r = its velocity can be found as v = r ′ = and its acceleration as a = r ″ =. Another important use of parametric equations is in the field of computer-aided design. For example, consider the following three representations, all of which are used to describe planar curves; the first two types are known as non-parametric, representations of curves.
In particular, the non-parametric representation depends on the choice of the coordinate system and does not lend itself well to geometric transformations, such as rotations and scaling. These problems can be addressed by rewriting the non-parametric equations in parametric form. Numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclid's parametrization of right triangles such that the lengths of their sides a, b and their hypotenuse c are coprime integers; as a and b are not both one may exchange them to have a and the parameterization is a = 2 m n, b = m 2 − n 2, c = m 2 + n 2, where the parameters m and n are positive coprime integers that are not both odd. By multiplying a, b and c by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths. Converting a set of parametric equations to a single implicit equation involves eliminating the variable t from the simultaneous equations x = f, y = g
Apollonius of Perga
Apollonius of Perga was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry, his definitions of the terms ellipse and hyperbola are the ones in use today. Apollonius worked including astronomy. Most of the work has not survived except in fragmentary references in other authors, his hypothesis of eccentric orbits to explain the aberrant motion of the planets believed until the Middle Ages, was superseded during the Renaissance. For such an important contributor to the field of mathematics, scant biographical information remains; the 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, states: “Apollonius, the geometrician... came from Perga in Pamphylia in the times of Ptolemy Euergetes, so records Herakleios the biographer of Archimedes....” Perga at the time was a Hellenized city of Pamphylia in Anatolia.
The ruins of the city yet stand. It was a center of Hellenistic culture. Euergetes, “benefactor,” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession, his “times” are his regnum, 246-222/221 BC. Times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father; the identity of Herakleios is uncertain. The approximate times of Apollonius are thus certain; the figure Specific birth and death years stated by the various scholars are only speculative. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt. Never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. During the last half of the 3rd century BC, Perga changed hands a number of times, being alternatively under the Seleucids and under the Kingdom of Pergamon to the north, ruled by the Attalid dynasty. Someone designated "of Perga" might well be expected to have worked there.
To the contrary, if Apollonius was identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived and wrote in Alexandria. A letter by the Greek mathematician and astronomer Hypsicles was part of the supplement taken from Euclid's Book XIV, part of the thirteen books of Euclid's Elements. "Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of the bond between them due to their common interest in mathematics. And on one occasion, when looking into the tract written by Apollonius about the comparison of the dodecahedron and icosahedron inscribed in one and the same sphere, to say, on the question what ratio they bear to one another, they came to the conclusion that Apollonius' treatment of it in this book was not correct, but I myself afterwards came across another book published by Apollonius, containing a demonstration of the matter in question, I was attracted by his investigation of the problem.
Now the book published by Apollonius is accessible to all. "For my part, I determined to dedicate to you what I deem to be necessary by way of commentary because you will be able, by reason of your proficiency in all mathematics and in geometry, to pass an expert judgment upon what I am about to write, because, on account of your intimacy with my father and your friendly feeling towards myself, you will lend a kindly ear to my disquisition. But it is time to have done with the preamble and to begin my treatise itself." Apollonius lived toward the end of a historical period now termed the Hellenistic Period, characterized by the superposition of Hellenic culture over extensive non-Hellenic regions to various depths, radical in some places, hardly at all in others. The change was initiated by Philip II of Macedon and his son, Alexander the Great, subjecting all of Greece is a series of stunning victories, went on to conquer the Persian Empire, which ruled territories from Egypt to Pakistan. Philip was assassinated in 336 BC.
Alexander went on to fulfill his plan by conquering the vast Persian empire. The material is located in the surviving false “Prefaces” of the books of his Conics; these are letters delivered to influential friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was requested by a house guest at Alexandria, the geometer, otherwise unknown to history. Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation”, he intended releasing each one as it was completed. Hearing of this plan from Apollonius himself on a subsequent visit of the latter to Pergamon, Eudemus had insisted Apollonius send him each book before release; the circumstances imply that at this stage Apollonius was a young geometer seeking the company and advice of established professionals. Pappus states. Euclid was long gone; this stay had been the final stage of Apollonius’ education.
Eudemus was a senior figure in his earlier education at Pergamon.
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Fibonacci was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is called, "Fibonacci", was made up in 1838 by the Franco-Italian historian Guillaume Libri and is short for filius Bonacci, he is known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano. Fibonacci popularized the Hindu–Arabic numeral system in the Western World through his composition in 1202 of Liber Abaci, he introduced Europe to the sequence of Fibonacci numbers, which he used as an example in Liber Abaci. Fibonacci was born around 1170 to an Italian merchant and customs official. Guglielmo directed a trading post in Algeria. Fibonacci travelled with him as a young boy, it was in Bugia that he learned about the Hindu–Arabic numeral system. Fibonacci travelled around the Mediterranean coast, meeting with many merchants and learning about their systems of doing arithmetic, he soon realised the many advantages of the Hindu-Arabic system which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system.
In 1202, he completed the Liber Abaci. Fibonacci became a guest of Emperor Frederick II. In 1240, the Republic of Pisa honored Fibonacci by granting him a salary in a decree that recognized him for the services that he had given to the city as an advisor on matters of accounting and instruction to citizens; the date of Fibonacci's death is not known, but it has been estimated to be between 1240 and 1250, most in Pisa. In the Liber Abaci, Fibonacci introduced the so-called modus Indorum, today known as the Hindu–Arabic numeral system; the book advocated numeration with the digits place value. The book showed the practical use and value of the new Hindu-Arabic numeral system by applying the numerals to commercial bookkeeping, converting weights and measures, calculation of interest, money-changing, other applications; the book had a profound impact on European thought. No copies of the 1202 edition are known to exist; the 1228 edition, first section introduces the Hindu-Arabic numeral system and compares the system with other systems, such as Roman numerals, methods to convert the other numeral systems into Hindu-Arabic numerals.
Replacing the Roman numeral system, its ancient Egyptian multiplication method, using an abacus for calculations, with a Hindu-Arabic numeral system was an advance in making business calculations easier and faster, which led to the growth of banking and accounting in Europe. The second section explains the uses of Hindu-Arabic numerals in business, for example converting different currencies, calculating profit and interest, which were important to the growing banking industry; the book discusses irrational numbers and prime numbers. Liber Abaci posed and solved a problem involving the growth of a population of rabbits based on idealized assumptions; the solution, generation by generation, was a sequence of numbers known as Fibonacci numbers. Although Fibonacci's Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been described by Indian mathematicians as early as the sixth century. In the Fibonacci sequence, each number is the sum of the previous two numbers.
Fibonacci omitted the "0" included today and began the sequence with 1, 1, 2.... He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377. Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence. In the 19th century, a statue of Fibonacci was raised in Pisa. Today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli. There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, the Pisano period. Beyond mathematics, namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis. Liber Abaci, a book on calculations Practica Geometriae, a compendium of techniques in surveying, the measurement and partition of areas and volumes, other topics in practical geometry.
Flos, solutions to problems posed by Johannes of Palermo Liber quadratorum on Diophantine equations, dedicated to Emperor Frederick II. See in particular congruum and the Brahmagupta–Fibonacci identity. Di minor guisa Commentary on Book X of Euclid's Elements Fibonacci numbers in popular culture Republic of Pisa Adelard of Bath Footnotes Citations Devlin, Keith; the Man of Numbers: Fibonacci's Arithmetic Revolution. Walker Books. ISBN 978-0802779083. Goetzmann, William N. and Rouwenhorst, K. Geert, The Origins of Value: The Financial Innovations That Created Modern Capital Markets, ISBN 0-19-517571-9. Goetzmann, William N. Fibonacci and the Financial Revolution, Yale School of Management International Center for Finance Working Paper No. 03–28 Grimm, R. E. "The Autobiography of Leonardo Pisano", Fibonacci Quarterly, Vol. 11, No. 1, February 1973, pp. 99–104. Horadam, A. F. "Eight hundred years young," The Australian Mathematics Teacher 31 (1975
Moritz Benedikt Cantor was a German historian of mathematics. Cantor was born at Mannheim, he came from a Sephardi Jewish family that had emigrated to the Netherlands from Portugal, another branch of which had established itself in Russia. In his early youth, Moritz Cantor was not strong enough to go to school, his parents decided to educate him at home. However, he was admitted to an advanced class of the Gymnasium in Mannheim. From there he went to the University of Heidelberg in 1848, soon after to the University of Göttingen, where he studied under Gauss and Weber, where Stern awakened in him a strong interest in historical research. After obtaining his Ph. D. at the University of Heidelberg in 1851, he went to Berlin, where he eagerly followed the lectures of Peter Gustav Lejeune Dirichlet. In 1863, he was promoted to the position of assistant professor, in 1877 he became honorary professor. Cantor was one of the founders of the Kritische Zeitschrift für Chemie, Physik und Mathematik. In 1859 he became associated with Schlömilch as editor of the Zeitschrift für Mathematik und Physik, taking charge of the historical and literary section.
Since 1877, through his efforts, a supplement to the Zeitschrift was published under the separate title of Abhandlungen zur Geschichte der Mathematik. Cantor's inaugural dissertation, "Über ein weniger gebräuchliches Coordinaten-System", gave no indication that the history of exact sciences would soon be enriched by a master work by him, his first important work was "Über die Einführung unserer gegenwärtigen Ziffern in Europa", which he wrote for the Zeitschrift für Mathematik und Physik, 1856, vol. i. His greatest work was Vorlesungen über Geschichte der Mathematik; this comprehensive history of mathematics appeared as follows: Volume 1 - From the earliest times until 1200 Volume 2 - From 1200 to 1668 Volume 3 - From 1668 to 1758 Volume 4 - From 1759 to 1799Many historians credit him for founding a new discipline in a field that had hitherto lacked the sound and critical methods of other fields of history. In 1900 Moritz Cantor received the honor of giving a plenary address at the International Congress of Mathematicians in Paris.
Jewish Encyclopedia, 1906 Florian Cajori, Moritz Cantor, The historian of mathematics, Bull. Amer. Math. Soc. 26, pp. 21-28. O'Connor, John J.. Moritz Cantor at the Mathematics Genealogy Project Literature by and about Moritz Cantor in the German National Library catalogue
History of mathematics
The area of study known as the history of mathematics is an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer and Assyria, together with Ancient Egypt and Ebla began using arithmetic and geometry for purposes of taxation, trade and in the field of astronomy and to formulate calendars and record time; the most ancient mathematical texts available are from Mesopotamia and Egypt - Plimpton 322, the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus. All of these texts mention the so-called Pythagorean triples and so, by inference, the Pythagorean theorem, seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry; the study of mathematics as a "demonstrative discipline" begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα, meaning "subject of instruction".
Greek mathematics refined the methods and expanded the subject matter of mathematics. Although they made no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, creation of lunar and solar calendars, arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers; the Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī. Islamic mathematics, in turn and expanded the mathematics known to these civilizations. Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.
Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were followed by centuries of stagnation. Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day; this includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century. At the end of the 19th century the International Congress of Mathematicians was founded and continues to spearhead advances in the field; the origins of mathematical thought lie in the concepts of number and form. Modern studies of animal cognition have shown; such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the "number" concept evolving over time is supported by the existence of languages which preserve the distinction between "one", "two", "many", but not of numbers larger than two.
Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers not being understood until about 500 BC, he writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, some numbers that are multiples of 10." The Ishango bone, according to scholar Alexander Marshack, may have influenced the development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic made use of multiplication by 2.
Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles and Pythagorean triples in their design. All of the above are disputed however, the oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period to the dawn of Christianity; the majority of Babylonian mathematical work comes from two separated periods: The first few hundred years of the second millennium BC, the last few centuries of the first millennium BC. It is named Babylonian mathematics due to the central role of Babylon as a place of study. Under the Arab Empire, Mesopotamia Baghdad, once again became an important center of
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences and engineering disciplines, as well as in the social sciences. A model may help to explain a system and to study the effects of different components, to make predictions about behaviour. Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models; these and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements leads to important advances as better theories are developed.
In the physical sciences, a traditional mathematical model contains most of the following elements: Governing equations Supplementary sub-models Defining equations Constitutive equations Assumptions and constraints Initial and boundary conditions Classical constraints and kinematic equations Mathematical models are composed of relationships and variables. Relationships can be described by operators, such as algebraic operators, differential operators, etc. Variables are abstractions of system parameters of interest. Several classification criteria can be used for mathematical models according to their structure: Linear vs. nonlinear: If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise; the definition of linearity and nonlinearity is dependent on context, linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables.
A differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented by linear equations the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation the model is known as a nonlinear model. Nonlinearity in simple systems, is associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are tied to nonlinearity. Static vs. dynamic: A dynamic model accounts for time-dependent changes in the state of the system, while a static model calculates the system in equilibrium, thus is time-invariant.
Dynamic models are represented by differential equations or difference equations. Explicit vs. implicit: If all of the input parameters of the overall model are known, the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, the corresponding inputs must be solved for by an iterative procedure, such as Newton's method or Broyden's method. In such a case the model is said to be implicit. For example, a jet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties. Discrete vs. continuous: A discrete model treats objects as discrete, such as the particles in a molecular model or the states in a statistical model.
Deterministic vs. probabilistic: A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, variable states are not described by unique values, but rather by probability distributions. Deductive, inductive, or floating: A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them; the floating model rests on neither theory nor observation, but is the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models. Application of catastrophe theory in science has been characterized as a floating model. Mathematical models are of great importance in the natural sciences in physics. Physical theories are invariably expressed using mathematic