Friday, October 12, 2012

Thoughts, and Visions: Mathematics

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.'?[12]?Carl Friedrich Gauss?(1777?1855) referred to mathematics as?'the Queen of the Sciences.'?[13]?Benjamin Peirce?(1809?1880) called mathematics?'the science that draws necessary conclusions'?.[14]?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.'?[15]?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.'?[16] Mathematics is used throughout the world as an essential tool in many fields, including?natural science,?engineering,?medicine, and thesocial 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.[17]

Etymology

The word "mathematics" comes from the?Greek??????? (m?th?ma), 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?m?th?ma?is derived from ??????? (manthano), while the modern Greek equivalent is ??????? (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.[18]?Its adjective is?????????????(math?matik?s), meaning?related to learning?or?studious, which likewise further came to mean?mathematical. In particular,??????????? ??????(math?matik? t?khn?),?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 math?matiques?(and the less commonly used singular derivative?la math?matique), goes back to the Latin neuter plural?mathematica?(Cicero), based on the Greek plural??? ?????????? (ta math?matik?), 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.[19]?In English, the noun?mathematics?takes singular verb forms. It is often shortened to?maths?or, in English-speaking North America,?math.[20]

Definitions of mathematics

Aristotle?defined mathematics as "the science of quantity," and this definition prevailed until the 18th century.[21]?Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as?group theory?and?projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[22]?Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals.[7]?There is not even consensus on whether mathematics is an art or a science.[8]?A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[7]?Some just say, "Mathematics is what mathematicians do."[7] Three leading types of definition of mathematics are called?logicist, intuitionist,?and?formalist,?each reflecting a different philosophical school of thought.[23]?All have severe problems, none has widespread acceptance, and no reconciliation seems possible.[23] Intuitionist?definitions, developing from the philosophy of mathematician?L. E. J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other."[23]?A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct. Formalist?definitions identify mathematics with its symbols and the rules for operating on them.?Haskell Curry?defined mathematics simply as "the science of formal systems."[26]?A?formal system?is a set of symbols, or?tokens,?and some?rules?telling how the tokens may be combined into?formulas. In formal systems, the word?axiom?has a special meaning, different from the ordinary meaning of "a self-evident truth". In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

History

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,[27]?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. 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 forastronomy.[29]?The systematic study of mathematics in its own right began with the?Ancient Greeks?between 600 and 300 BC.[30] 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 theMathematical 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 theirproofs."[31]

Inspiration, pure and applied mathematics, and aesthetics

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?physicistRichard 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.[32]?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 betweenpure 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".[33]?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.[34]?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 methodthat 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.[35]?Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to?Paul Erd?s.[36][37]?The popularity ofrecreational mathematics?is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor

Most of the mathematical notation in use today was not invented until the 16th century.[38]?Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.[39]?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?integrablehave 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.[40]?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.[41] 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?G?del'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.[42]

Fields of mathematics

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

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.[43]?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?G?del'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, G?del 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.

Pure mathematics

Quantity

Structure

Many mathematical objects, such as?sets?of numbers and?functions, exhibit internal structure as a consequence of?operations?orrelations?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?algebrahave very strong interactions in modern mathematics.?Combinatorics?studies ways of enumerating the number of objects that fit a given structure.

Space

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?Poincar? conjecture. Other results in geometry and topology, including the?four color theorem?and?Kepler conjecture, have been proved only with the help of computers.

Change

Applied mathematics

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

Applied mathematics has significant overlap with the discipline of?statistics, whose theory is formulated mathematically, especially withprobability theory. Statisticians (working as part of a research project) "create data that makes sense" with?random sampling?and with randomized?experiments;[45]?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.[46] Statistical theory?studies?decision problems?such as minimizing the?risk?(expected loss) of a statistical action, such as using aprocedure?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.[47]?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.[48]

Computational mathematics

Mathematics as profession

Arguably the most prestigious award in mathematics is the?Fields Medal,[49][50]?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

Gauss referred to mathematics as "the Queen of the Sciences".[13]?In the original Latin?Regina Scientiarum, as well as in?German?K?nigin der Wissenschaften, the word corresponding to?sciencemeans 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."[16]?More recently,?Marcus du Sautoy?has called mathematics 'the Queen of Science...the main driving force behind scientific discovery'.[52] Many philosophers believe that mathematics is not experimentally?falsifiable, and thus not a science according to the definition of?Karl Popper.[53]?However, in the 1930s?G?del'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."[54]?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.[55]?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 andengineering?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

Notes

  1. ^?No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see?Euclid).
  2. ^?a?b?"mathematics,?n.".?Oxford English Dictionary. Oxford University Press. 2012. Retrieved June 16, 2012. "The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis."
  3. ^?Kneebone, G.T. (1963).?Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. pp.?4.?ISBN?0486417123. "Mathematics?is simply the study of abstract structures, or formal patterns of connectedness."
  4. ^?LaTorre, Donald R., John W. Kenelly, Iris B. Reed, Laurel R. Carpenter, and Cynthia R Harris (2011).?Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. pp.?2.?ISBN?1439049572. "Calculus is the study of change?how things change, and how quickly they change."
  5. ^?Ramana (2007).?Applied Mathematics. Tata McGraw-Hill Education. p.?2.10.?ISBN?0070667535. "The mathematical study of change, motion, growth or decay is calculus."
  6. ^?Ziegler, G?nter M.?(2011). "What Is Mathematics?".?An Invitation to Mathematics: From Competitions to Research. Springer. pp.?7.?ISBN?3642195326.
  7. ^?a?b?c?d?Mura, Robert (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences".?Educational Studies in Mathematics?25?(4):?375?385.
  8. ^?a?b?Tobies, Renate and Helmut Neunzert (2012).?Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. pp.?9.?ISBN?3034802293. "It is first necessary to ask what is meant by?mathematics?in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form."
  9. ^?Steen,?L.A.?(April?29,?1988).?The Science of Patterns?Science, 240: 611?616. And summarized at?Association for Supervision and Curriculum Development, www.ascd.org.
  10. ^?Devlin, Keith,?Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe?(Scientific American Paperback Library) 1996,?ISBN 978-0-7167-5047-5
  11. ^?Eves
  12. ^?Marcus du Sautoy,?A Brief History of Mathematics: 1. Newton and Leibniz,?BBC Radio 4, 27/09/2010.
  13. ^?a?b?Waltershausen
  14. ^?Peirce, p.?97.
  15. ^?Hilbert,?D. (1919-20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919-1920 in G?ttingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an English introduction by David?E. Rowe), Basel, Birkh?user (1992).
  16. ^?a?b?Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" He, too, is concerned with?The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
  17. ^?Peterson
  18. ^?Both senses can be found in Plato. Liddell and Scott,s.voce???????????
  19. ^?The Oxford Dictionary of English Etymology,?Oxford English Dictionary,?sub?"mathematics", "mathematic", "mathematics"
  20. ^?"maths,?n."?and?"math,?n.3".?Oxford English Dictionary,?on-line version (2012).
  21. ^?James Franklin, "Aristotelian Realism" in?Philosophy of Mathematics", ed. A.D. Irvine,?p. 104. Elsevier (2009).
  22. ^?Cajori, Florian?(1893).?A History of Mathematics. American Mathematical Society (1991 reprint). pp.?285?6.ISBN?0821821024.
  23. ^?a?b?c?Snapper, Ernst (September 1979), "The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism",Mathematics Magazine?52?(4): 207?16,?doi:10.2307/2689412,JSTOR?2689412.
  24. ^?Peirce, Benjamin?(1882).?Linear Associative Algebra. p.?1.
  25. ^?Bertrand Russell,?The Principles of Mathematics,?p. 5. University Press, Cambridge (1903)
  26. ^?Curry, Haskell?(1951).?Outlines of a Formalist Philosophy of Mathematics. Elsevier. pp.?56.?ISBN?0444533680.
  27. ^?S. Dehaene; G. Dehaene-Lambertz; L. Cohen (Aug 1998). "Abstract representations of numbers in the animal and human brain".?Trends in Neuroscience?21?(8): 355?361.doi:10.1016/S0166-2236(98)01263-6.?PMID?9720604.
  28. ^?See, for example, Raymond L. Wilder,?Evolution of Mathematical Concepts; an Elementary Study,?passim
  29. ^?Kline 1990, Chapter 1.
  30. ^?"A History of Greek Mathematics: From Thales to Euclid". Thomas Little Heath (1981).?ISBN 0-486-24073-8
  31. ^?Sevryuk
  32. ^?Johnson, Gerald W.; Lapidus, Michel L. (2002).?The Feynman Integral and Feynman's Operational Calculus.?Oxford University Press.?ISBN?0-8218-2413-9.
  33. ^?Wigner, Eugene (1960).?"The Unreasonable Effectiveness of Mathematics in the Natural Sciences".?Communications on Pure and Applied Mathematics?13?(1): 1?14.doi:10.1002/cpa.3160130102.
  34. ^?"Mathematics Subject Classification 2010"?(PDF). Retrieved 2010-11-09.
  35. ^?Hardy, G. H. (1940).?A Mathematician's Apology. Cambridge University Press.?ISBN?0-521-42706-1.
  36. ^?Gold, Bonnie; Simons, Rogers A. (2008).?Proof and Other Dilemmas: Mathematics and Philosophy. MAA.
  37. ^?Aigner, Martin;?Ziegler, G?nter?M.?(2001).?Proofs from?The Book. Springer.?ISBN?3-540-40460-0.
  38. ^?Earliest Uses of Various Mathematical Symbols?(Contains many further references).
  39. ^?Kline, p. 140, on?Diophantus; p.261, on?Vieta.
  40. ^?See?false proof?for simple examples of what can go wrong in a formal proof.
  41. ^?Ivars Peterson,?The Mathematical Tourist, Freeman, 1988,ISBN 0-7167-1953-3. p. 4 "A few complain that the computer program can't be verified properly", (in reference to the Haken-Apple proof of the Four Color Theorem).
  42. ^?Patrick Suppes,?Axiomatic Set Theory, Dover, 1972,?ISBN 0-486-61630-4. p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."
  43. ^?Luke Howard Hodgkin & Luke Hodgkin,?A History of Mathematics, Oxford University Press, 2005.
  44. ^?Clay Mathematics Institute, P=NP, claymath.org
  45. ^?Rao, C.R.?(1997)?Statistics and Truth: Putting Chance to Work, World Scientific.?ISBN 981-02-3111-3
  46. ^?Like other mathematical sciences such as?physics?andcomputer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.
  47. ^?Rao, C.?R.?(1981). "Foreword". In Arthanari, T. S.;?Dodge, Yadolah. Wiley Series in Probability and Mathematical Statistics. Wiley. pp.?vii?viii.?ISBN?0-471-08073-X.?MR?607328.
  48. ^?Whittle (1994, pp.?10?11 and 14?18):?Whittle, Peter?(1994)."Almost home". In?Kelly, F.?P..?Probability, statistics and optimisation: A Tribute to Peter Whittle?(previously "A realised path: The Cambridge Statistical Laboratory upto 1993 (revised 2002)" ed.). Chichester: John Wiley. pp.?1?28.?ISBN?0-471-94829-2.
  49. ^?"The Fields Medal is now indisputably the best known and most influential award in mathematics." Monastyrsky
  50. ^?Riehm
  51. ^?Zeidler, Eberhard (2004).?Oxford User's Guide to Mathematics. Oxford, UK: Oxford University Press. p.?1188.?ISBN?0-19-850763-1.
  52. ^?Marcus du Sautoy,?A Brief History of Mathematics: 10. Nicolas Bourbaki,?BBC Radio 4, 01/10/2010.
  53. ^?Shasha, Dennis Elliot; Lazere, Cathy A. (1998).?Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p.?228.
  54. ^?Popper 1995, p. 56
  55. ^?Ziman

References

  • Courant, Richard?and?H. Robbins,?What Is Mathematics??: An Elementary Approach to Ideas and Methods, Oxford University Press, USA; 2 edition (July 18, 1996).?ISBN 0-19-510519-2.
  • Einstein, Albert?(1923).?Sidelights on Relativity: I. Ether and relativity. II. Geometry and experience (translated by G.B. Jeffery, D.Sc., and W. Perrett, Ph.D).. E.P. Dutton & Co., New York.
  • du Sautoy, Marcus,?A Brief History of Mathematics,?BBC Radio 4(2010).
  • Eves, Howard,?An Introduction to the History of Mathematics, Sixth Edition, Saunders, 1990,?ISBN 0-03-029558-0.
  • Kline, Morris,?Mathematical Thought from Ancient to Modern Times, Oxford University Press, USA; Paperback edition (March 1, 1990).ISBN 0-19-506135-7.
  • Monastyrsky, Michael (2001) (PDF).?Some Trends in Modern Mathematics and the Fields Medal. Canadian Mathematical Society. Retrieved 2006-07-28.
  • Oxford English Dictionary, second edition, ed. John Simpson and Edmund Weiner,?Clarendon Press, 1989,?ISBN 0-19-861186-2.
  • The Oxford Dictionary of English Etymology, 1983 reprint.?ISBN 0-19-861112-9.
  • Pappas, Theoni,?The Joy Of Mathematics, Wide World Publishing; Revised edition (June 1989).?ISBN 0-933174-65-9.
  • Peirce, Benjamin?(1881).?Peirce, Charles?Sanders. ed.?"Linear associative algebra".?American Journal of Mathematics?(Johns Hopkins University)?4?(1?4): 97?229.?doi:10.2307/2369153. Corrected, expanded, and annotated revision with an 1875 paper by B.?Peirce and annotations by his son, C.?S.?Peirce, of the 1872 lithograph ed.?Google?Eprint?and as an extract, D.?Van Nostrand, 1882,?Google?Eprint..
  • Peterson, Ivars,?Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001,?ISBN 0-8050-7159-8.
  • Popper, Karl R.?(1995). "On knowledge".?In Search of a Better World: Lectures and Essays from Thirty Years. Routledge.?ISBN?0-415-13548-6.
  • Riehm, Carl (August 2002).?"The Early History of the Fields Medal"?(PDF).?Notices of the AMS?(AMS)?49?(7): 778?782.
  • Sevryuk, Mikhail B.?(January 2006).?"Book Reviews"?(PDF).Bulletin of the American Mathematical Society?43?(1): 101?109.doi:10.1090/S0273-0979-05-01069-4. Retrieved 2006-06-24.
  • Waltershausen, Wolfgang Sartorius von?(1856, repr. 1965).?Gauss zum Ged?chtniss. S?ndig Reprint Verlag H. R. Wohlwend.ISBN?3-253-01702-8.?ISSN?B0000BN5SQ ASIN: B0000BN5SQ.

Further reading

  • Benson, Donald C.,?The Moment of Proof: Mathematical Epiphanies,?Oxford University Press, USA; New Ed edition (December 14, 2000).?ISBN 0-19-513919-4.
  • Boyer, Carl B.,?A History of Mathematics, Wiley; 2nd edition, revised by Uta C. Merzbach, (March 6, 1991).?ISBN 0-471-54397-7.?A concise history of mathematics from the Concept of Number to contemporary Mathematics.
  • Davis, Philip J.?and?Hersh, Reuben,?The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999).?ISBN 0-395-92968-7.
  • Gullberg, Jan,?Mathematics ? From the Birth of Numbers.?W. W. Norton & Company; 1st edition (October 1997).?ISBN 0-393-04002-X.
  • Hazewinkel, Michiel (ed.),?Encyclopaedia of Mathematics.?Kluwer Academic Publishers?2000. ? A translated and expanded version of a Soviet mathematics encyclopedia, in ten

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