This article needs attention from an expert on the subject. Please add a reason or a talk parameter to this template to explain the issue with the article. WikiProject Philosophy may be able to help recruit an expert. (March 2011)

This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (April 2010)

In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules.

For example, Euclidean geometry can be seen as a game whose play consists in moving around certain strings of symbols called axioms according to a set of rules called "rules of inference" to generate new strings. In playing this game one can "prove" that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules.

According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics).

Formalism is associated with rigorous method. In common use, a formalism means the out-turn of the effort towards formalisation of a given limited area. In other words, matters can be formally discussed once captured in a formal system, or commonly enough within something formalisable with claims to be one. Complete formalisation is in the domain of computer science.

Formalism stresses axiomatic proofs using theorems, specifically associated with David Hilbert. A formalist is a person who belongs to the school of formalism, which is a certain mathematical-philosophical doctrine descending from Hilbert.

Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are usually not arbitrary.

Recently, some formalist mathematicians have proposed that all of our formal mathematical knowledge should be systematically encoded in computer-readable formats, so as to facilitate automated proof checking of mathematical proofs and the use of interactive theorem proving in the development of mathematical theories and computer software. Because of their close connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition (see below).

Deductivism[link]

Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one.

This is to say, that if you interpret the strings in such a way that the rules of the game become true then you have to accept that the theorem, or, rather, the interpretation of the theorem you have given it must be a true statement. (The rules of such a game would have to include, for instance, that true statements are assigned to the axioms, and that the rules of inference are truth-preserving, etcetera.)

Under deductivism, the same view is held to be true for all other statements of formal logic and mathematics. Thus, formalism need not mean that these deductive sciences are nothing more than meaningless symbolic games. It is usually hoped that there exists some interpretation in which the rules of the game hold. Compare this position to structuralism.

Taking the deductivist view allows the working mathematician to suspend judgement on the deep philosophical questions and proceed as if solid epistemological foundations were available. Many formalists would say that in practice, the axiom systems to be studied are suggested by the demands of the particular science.

Hilbert's formalism[link]

David Hilbert

A major early proponent of formalism was David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics. Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent (i.e. no contradictions can be derived from the system).

The way that Hilbert tried to show that an axiomatic system was consistent was by formalizing it using a particular language (Snapper, 1979). In order to formalize an axiomatic system, you must first choose a language in which you can express and perform operations within that system. This language must include five components:

• It must include variables such as x, which can stand for some number.
• It must have quantifiers such as the symbol for the existence of an object.
• It must include equality.
• It must include connectives such as ↔ for "if and only if."
• It must include certain undefined terms called parameters. For geometry, these undefined terms might be something like a point or a line, which we still choose symbols for.

Once we choose this language, Hilbert thought that we could prove all theorems within any axiomatic system using nothing more than the axioms themselves and the chosen formal language.

Gödel's conclusion in his incompleteness theorems was that you cannot prove consistency within any axiomatic system rich enough to include classical arithmetic. On the one hand, you must use only the formal language chosen to formalize this axiomatic system; on the other hand, it is impossible to prove the consistency of this language in itself (Snapper, 1979). Hilbert was originally frustrated by Gödel's work because it shattered his life's goal to completely formalize everything in number theory (Reid and Weyl, 1970). However, Gödel did not feel that he contradicted everything about Hilbert's formalist point of view. After Gödel published his work, it became apparent that proof theory still had some use, the only difference is that it could not be used to prove the consistency of all of number theory as Hilbert had hoped (Reid and Weyl, 1970). Present-day formalists use proof theory to further our understanding in mathematics, but perhaps because of Gödel's work, they make no claims about semantic meaning in the work that they do with mathematics. Proofs are simply the manipulation of symbols in our formal language starting from certain rules that we call axioms.

It is important to note that Hilbert is not considered a strict formalist as formalism is defined today. He thought there was some meaning and truth in mathematics, which is precisely why he was trying to prove the consistency of number theory. If number theory turned out to be consistent, then there had to be some sort of truth in it (Goodman, 1979). Strict formalists consider mathematics apart from its semantic meaning. They view mathematics as pure syntax: the manipulation of symbols according to certain rules. They then attempt to show that this set of rules is consistent, much like Hilbert attempted to do (Goodman, 1979). Formalists currently believe that computerized algorithms will eventually take over the task of constructing proofs. Computers will replace humans in all mathematical activities, such as checking to see if a proof is correct or not (Goodman, 1979).

Hilbert was initially a deductivist, but, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.

Axiomatic systems[link]

Other formalists, such as Rudolf Carnap, Alfred Tarski and Haskell Curry, considered mathematics to be the investigation of formal axiom systems. Mathematical logicians study formal systems but are just as often realists as they are formalists.

Principia Mathematica[link]

Perhaps the most serious attempt to prove the consistency of number theory was by the two mathematicians Bertrand Russell and Alfred North Whitehead. They created a work, Principia Mathematica, which derived number theory by the manipulation of symbols using formal logic. This work was very detailed, and Russell and Whitehead spent the better part of a decade in writing it. It wasn't until page 379 of the first volume that they were even able to prove that 1+1=2. The work of Russell and Whitehead came to an end when Gödel published his incompleteness theorems, which stated that the goal of the Principia Mathematica was actually impossible.

Criticisms of formalism[link]

Gödel indicated one of the weak points of formalism by addressing the question of consistency in axiomatic systems. More recent criticisms lie in the assertion of formalists that it is possible to computerize all of mathematics. These criticisms bring up the philosophical question of whether or not computers are able to think. Turing tests, named after Alan Turing, who created the test, are an attempt to provide criteria for judging when a computer is capable of thought. The existence of a computer which in principle could pass a Turing test would prove to formalists that computers will be able to do all of mathematics. However, there are opponents of this claim, such as John Searle, who came up with the "Chinese room" thought experiment. He presented the argument that while a computer may be able to manipulate the symbols that we give it, the machine could attach no meaning to these symbols. Since computers will not be able to deal with semantic content in mathematics (Penrose, 1989), they could not be said to "think."

Further, humans can create several ways to prove the same result, even if they might find it challenging to articulate such methods. Since creativity requires thought having a semantic foundation, a computer would not be able to create different methods of solving the same problem. Indeed, a formalist would not be able to say that these other ways of solving problems exist simply because they have not been formalized (Goodman, 1979).

Another critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above. Formalism is thus silent to the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.

See also[link]

• QED project

References[link]

Goodman, Nicholas D. "Mathematics as an Objective Science." The American Mathematical Monthly 86.7 (1979): 540-551. Print.

Penrose, Roger. The Emperor's New Mind: concerning Computers, Minds, and the Laws of Physics. Oxford: Oxford UP, 1989. Print.

Reid, Constance, and Hermann Weyl. Hilbert. Berlin: Springer-Verlag, 1970. Print.

Snapper, Ernst. "The Three Crises in Mathematics: Logicism, Intuitionism and Formalism." Mathematics Magazine 52.4 (1979): 207-16. Print.

Look up formalism or formalist in Wiktionary, the free dictionary.

Formalism may refer to:

• Form (disambiguation)
• Formal (disambiguation)
• Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary
• Formalism (philosophy), that there is no transcendent meaning to a discipline other than the literal content created by a practitioner
  • Religious formalism, an emphasis on the ritual and observance of religion, rather than its meaning.
  • Formalism (mathematics), that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules.
  • Formalism (art), that a work's artistic value is entirely determined by its form
    • Formalism (music)
    • Formalist film theory, focused on the formal, or technical, elements of a film
    • Formalism (literature)
      • Russian formalism, school of literary criticism in Russia from the 1910s to the 1930s

See also[link]

• Formality

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.^[1]

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of quantity, structure, space, and change.^[2] Mathematicians seek out patterns^[3][4] and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and 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. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics developed at a relatively 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.^[5]

Galileo Galilei (1564–1642) said, 'The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth'.^[6] Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences".^[7] Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions".^[8] David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise."^[9] Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality".^[10]

Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.^[11]

Etymology[link]

The word "mathematics" comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means what one learns, what one gets to know, hence also study and science, and in modern Greek just lesson. The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean to learn. In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study", even in Classical times.^[12] Its adjective is μαθηματικός (mathēmatikós), meaning related to learning or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant the mathematical art.

In Latin, and in English until around 1700, the term "mathematics" more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of "mathematici" meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384-322BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.^[13] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.

History[link]

Greek mathematician Pythagoras (c.570-c.495 BC), commonly credited with discovering the Pythagorean theorem.

The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,^[14] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years.^[15] Elementary arithmetic (addition, subtraction, multiplication and division) naturally followed.

Since numeracy pre-dated writing, further steps were needed for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.^{[citation needed]} Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.^{[citation needed]}

Mayan numerals

The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.^[16] The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.^[17]

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 today. 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 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."^[18]

Inspiration, pure and applied mathematics, and aesthetics[link]

Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.^[19] Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics".^[20] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.^[21] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.^[22] Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to Paul Erdős.^[23][24] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor[link]

Leonhard Euler, who created and popularized much of the mathematical notation used today

Most of the mathematical notation in use today was not invented until the 16th century.^[25] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.^[26] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.

Mathematical language can be difficult to understand for beginners. Words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open and field have been given specialized mathematical meanings. Technical terms such as homeomorphism and integrable have precise meanings in mathematics. Additionally, shorthand phrases such as "iff" for "if and only if" belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.^[27] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.^[28]

Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.^[29]

Fields of mathematics[link]

An abacus, a simple calculating tool used since ancient times.

Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.

Foundations and philosophy[link]

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.^[30] Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer-Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science^{[citation needed]}, as well as to Category Theory.

Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware. A famous problem is the "P=NP?" problem, one of the Millennium Prize Problems.^[31] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.

Failed to parse (Missing texvc executable; please see math/README to configure.): p \Rightarrow q \,
Mathematical logic	Set theory	Category theory	Theory of computation

Pure mathematics[link]

Quantity[link]

The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory.

As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

Failed to parse (Missing texvc executable; please see math/README to configure.): 1, 2, 3\,...\!	Failed to parse (Missing texvc executable; please see math/README to configure.): ...-2, -1, 0, 1, 2\,...\!	Failed to parse (Missing texvc executable; please see math/README to configure.): -2, \frac{2}{3}, 1.21\,\!	Failed to parse (Missing texvc executable; please see math/README to configure.): -e, \sqrt{2}, 3, \pi\,\!	Failed to parse (Missing texvc executable; please see math/README to configure.): 2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!
Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers

Structure[link]

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix}
Combinatorics	Number theory	Group theory	Graph theory	Order theory

Space[link]

The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Convex and discrete geometry was developed to solve problems in number theory and functional analysis but now is pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with the help of computers.

Geometry	Trigonometry	Differential geometry	Topology	Fractal geometry	Measure theory

Change[link]

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Calculus	Vector calculus	Differential equations	Dynamical systems	Chaos theory	Complex analysis

Applied mathematics[link]

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation, study, and use of mathematical models in science, engineering, and other areas of mathematical practice.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Statistics and other decision sciences[link]

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments;^[32] the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data.^[33]

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.^[34] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.^[35]

Computational mathematics[link]

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic matrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

Mathematical physics	Fluid dynamics	Numerical analysis	Optimization	Probability theory	Statistics	Cryptography
Mathematical finance	Game theory	Mathematical biology	Mathematical chemistry	Mathematical economics	Control theory

Mathematics as profession[link]

Arguably the most prestigious award in mathematics is the Fields Medal,^[36][37] established in 1936 and now awarded every 4 years. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize.

The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field.

A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.

Mathematics as science[link]

Carl Friedrich Gauss, known as the "prince of mathematicians".^[38]

Gauss referred to mathematics as "the Queen of the Sciences".^[7] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means a "field of knowledge", and this was the original meaning of "science" in English, also. Of course, mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to natural science follows the rise of Baconian science, which contrasted "natural science" to scholasticism, the Aristotelean method of inquiring from first principles. Of course, the role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as psychology, biology, or physics. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."^[10] More recently, Marcus du Sautoy has called mathematics 'the Queen of Science...the main driving force behind scientific discovery'.^[39]

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.^[40] However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians^[who?] that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."^[41] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.

An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics.^[42] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method.^{[citation needed]}

The opinions of mathematicians on this matter are varied. Many mathematicians^[who?] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others^[who?] feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.^{[citation needed]}

See also[link]

Mathematics portal

	Book: Mathematics
Wikipedia books are collections of articles that can be downloaded or ordered in print.

• Definitions of mathematics
• Mathematics and art
• Mathematics education

Notes[link]

References[link]

Further reading[link]

External links[link]

Find more about Mathematics on Wikipedia's sister projects:
	Definitions and translations from Wiktionary
	Images and media from Commons
	Learning resources from Wikiversity
	News stories from Wikinews
	Quotations from Wikiquote
	Source texts from Wikisource
	Textbooks from Wikibooks

At Wikiversity you can learn more and teach others about Mathematics at:

The School of Mathematics

Rebecca Newberger Goldstein
Born	Rebecca Newberger (1950-02-23) February 23, 1950 (age 62)
Nationality	American
Ethnicity	Jewish
Occupation	Novelist Philosopher

Rebecca Goldstein (born February 23, 1950) is an American novelist and professor of philosophy. She has written five novels, a number of short stories and essays, and biographical studies of mathematician Kurt Gödel and philosopher Baruch Spinoza.

Life and career[link]

Goldstein, born Rebecca Newberger, grew up in White Plains, New York, and did her undergraduate work at Barnard College. She was born into an Orthodox Jewish family. She has one older brother who is an Orthodox Rabbi, and she also has a younger sister. After earning her Ph.D. from Princeton University, she returned to Barnard to teach courses in various philosophical studies. There she published her first novel, The Mind-Body Problem (1983), a serio-comic tale of the conflict between emotion and intelligence, combined with an examination of Jewish tradition and identity. Goldstein said she wrote the book to "...insert 'real life' intimately into the intellectual struggle. In short I wanted to write a philosophically motivated novel."^[1]

Her second novel, The Late-Summer Passion of a Woman of Mind (1989), was also set in academia, though with a far darker tone. Her third novel, The Dark Sister (1993), was something of a departure: a postmodern fictionalization of family and professional issues in the life of William James. Mazel followed in 1995. Properties of Light (2000) is a ghost story about love, betrayal, and quantum physics. Her latest novel is 36 Arguments for the Existence of God: A Work of Fiction (2010). Goldstein has published a collection of short stories, Strange Attractors (1993), that also treated "interactions of thought and feeling," to quote the cover jacket.

Recently Goldstein has turned to biography with her books Incompleteness: The Proof and Paradox of Kurt Gödel (2005) and Betraying Spinoza: The Renegade Jew Who Gave Us Modernity (2006). The books reflect her continuing interests in the relationship between the life of the mind and the demands of everyday existence, and in Jewish perspectives and history.

In addition to Barnard, Goldstein has taught at Columbia and Rutgers. She has been a visiting scholar at Brandeis University, and taught for five years as a visiting professor in the Department of Philosophy at Trinity College in Hartford, Connecticut.

Goldstein lives in Boston and Truro. She divorced her first husband, physicist Sheldon Goldstein, and married^[2] Harvard cognitive psychologist Steven Pinker. She is the mother of the novelist Yael Goldstein Love and the poet Danielle Blau.

Awards and fellowships[link]

• 2011 Humanist of the Year awarded April 2011 by the American Humanist Association
• Fellow, Radcliffe Institute for Advanced Study at Harvard University, 2006–2007
• Guggenheim Fellow, 2006–2007
• Koret Jewish Book Award in Jewish Thought,[1] 2006, for Betraying Spinoza: The Renegade Jew who Gave Us Modernity
• Fellow of the American Academy of Arts and Sciences, 2005^[3]
• MacArthur Fellow, 1996
• National Jewish Book Award, 1995, for Mazel
• Edward Lewis Wallant Award, 1995, for Mazel
• National Jewish Book Award for her book of short stories, Strange Attractors
• Graduated summa cum laude from Barnard College, receiving the Montague Prize for Excellence in Philosophy,
• While at Princeton University, she was awarded a National Science Foundation Fellowship
• Whiting Foundation Fellowship, 1991^[4]

See also[link]

• American philosophy
• List of American philosophers

Notes[link]

External links[link]

This article's use of external links may not follow Wikipedia's policies or guidelines. Please improve this article by removing excessive or inappropriate external links, and converting useful links where appropriate into footnote references. (October 2011)

• Official website
• Rebecca Goldstein, the atheist with a soul in an interview on ThoughtCast, which was also broadcast on WGBH (FM)
• Interview, April 11, 2006
• Rebecca Goldstein discusses her book, "36 Arguments for the Existence of God" on the 7th Avenue Project Radio Show
• An interview with Rebecca Goldstein, about her book "Betraying Spinoza: The Renegade Jew Who Gave Us Modernity" California Literary Review, June 4, 2006
• "Reasonable Doubt": an essay on Baruch Spinoza on Edge.org
• "A Kibitz on Pure Reason": A three-day dialogue on "Betraying Spinoza" between Rebecca Goldstein and Michael Weiss. Jewcy.com, March 17–19, 2007.
• Radio interview on Philosophy Talk
• On Rebecca Goldstein’s New York Times Book Review essay “Theory, Literature, Hoax”
• “Disenchanting Determinism,” commentary on Rebecca Goldstein’s 36 Arguments for the Existence of God

Sir James Alexander Mirrlees FRSE FBA (born 5 July 1936) is a Scottish economist and winner of the 1996 Nobel Memorial Prize in Economic Sciences. He was knighted in 1998.

Born in Minnigaff, Kirkcudbrightshire, Mirrlees was educated at the University of Edinburgh and Trinity College, Cambridge, where he was a very active student debater. One contemporary, Quentin Skinner has suggested that Mirrlees was a member of the Cambridge Apostles along with fellow Nobel Laureate Amartya Sen during this period. Between 1968 and 1976, Mirrlees was a visiting professor at MIT three times.^[1] He taught at both Oxford University (1969–1995) and University of Cambridge (1963- and 1995-).

During his time at Oxford he published papers on economic models for which he would eventually be awarded his Nobel Prize. They centred on situations in which economic information is asymmetrical or incomplete, determining the extent to which they should affect the optimal rate of saving in an economy. Among other results, they demonstrated the principles of "moral hazard" and "optimal income taxation" discussed in the books of William Vickrey. The methodology has since become the standard in the field.

Mirrlees and Vickrey shared the 1996 Nobel Prize for Economics "for their fundamental contributions to the economic theory of incentives under asymmetric information".

Mirrlees is also co-creator, with MIT Professor Peter A. Diamond of the Diamond-Mirrlees Efficiency Theorem, developed in 1971.^[2]

Mirrlees is emeritus Professor of Political Economy at the University of Cambridge, and Fellow of Trinity College, Cambridge. He spends several months a year at the University of Melbourne, Australia. He is currently the Distinguished Professor-at-Large of The Chinese University of Hong Kong as well as University of Macau.^[3] In 2009, he was appointed Master of the Morningside College of The Chinese University of Hong Kong, along with the biologist Samuel Sun Sai-ming.^[4]

Mirrlees is a member of Scotland's Council of Economic Advisers. He also led the The Mirrlees Review, a review of the UK tax system by the Institute for Fiscal Studies.

His students have included eminent academics: Lord Nicholas Stern and Sir Partha Dasgupta.

Publications[link]