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There is debate over whether mathematical objects such as numbers
and points exist naturally or are human creations. The mathematician Benjamin Peirce
called mathematics "the science that draws necessary conclusions". Albert Einstein
, on the other hand, 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."
Through the use of abstraction
and logical reasoning
, mathematics evolved from counting
, and the systematic study of the shapes
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
. Mathematics continued to develop, for example in China in 300 BCE, in India in 100 CE, and in Arabia in 800 CE, until the Renaissance
, when mathematical innovations interacting with new scientific discoveries
led to a rapid increase in the rate of mathematical discovery that continues to the present day.
Mathematics is used throughout the world as an essential tool in many fields, including natural science
, 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 and sometimes leads 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, although practical applications for what began as pure mathematics are often discovered.
The word "mathematics" comes from the Greek
), which means learning
, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.
Its adjective is μαθηματικός
), related to learning
, or studious
, which likewise further came to mean mathematical
. In particular, μαθηματικὴ τέχνη
: ars mathematica
, meant the mathematical art
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
), based on the Greek plural τα μαθηματικά (ta mathēmatiká)
, used by Aristotle
, 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
, which were inherited from the Greek.
In English, the noun mathematics
takes singular verb forms. It is often shortened to maths
or, in English-speaking North America, math
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,
was probably that of numbers
: the realization that a collection of two apples and a collection two oranges (for example) have something in common, namely quantity of their members.
The earliest uses of mathematics were in trading
, land measurement
patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians
began using arithmetic, algebra and geometry for taxation
and other financial calculations, for building and construction, and for astronomy
The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.
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
Inspiration, pure and applied mathematics, and aesthetics
Mathematics arises from many different kinds of problems. At first these were found in commerce
, land measurement
and later astronomy
; nowadays, 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.
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
. 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
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.
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
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.
Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős
often referred to as finding proofs from "The Book" in which God had written down his favorite proofs.
The popularity of recreational mathematics
is another sign of the pleasure many find in solving mathematical questions.
Notation, language, and rigor
, 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.
Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. 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.
can also be hard for beginners. Words such as or
have more precise meanings than in everyday speech. Moreover, words such as open
have been given specialized mathematical meanings. Mathematical jargon
includes technical terms such as homeomorphism
. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
in several typefaces.
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.
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.
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.
Mathematics as science
Carl Friedrich Gauss
, himself known as the "prince of mathematicians",
referred to mathematics as "the Queen of the Sciences".
Carl Friedrich Gauss
referred to mathematics as "the Queen of the Sciences".
In the original Latin Regina Scientiarum
, as well as in German Königin der Wissenschaften
, the word corresponding to science
means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural
science is of later date. If one considers science
to be strictly about the physical world, then mathematics, or at least pure mathematics
, is not a science. 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.
Many philosophers believe that mathematics is not experimentally falsifiable
, and thus not a science according to the definition of Karl Popper
However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of physics
: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."
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.
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
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
In his 2002 book A New Kind of Science
, Stephen Wolfram
argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.
The opinions of mathematicians on this matter are varied. Many mathematicians 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 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
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal
established in 1936 and now awarded every 4 years. It is often considered the equivalent of science's Nobel Prizes
. The Wolf Prize in Mathematics
, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize
, was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such 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.
Fields of mathematics
, 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
, 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
), to the empirical mathematics of the various sciences (applied mathematics
), and more recently to the rigorous study of uncertainty
Many mathematical objects, such as sets
of numbers and functions
, exhibit internal structure as a consequence of operations
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
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
have very strong interactions in modern mathematics.
The study of space originates with geometry
– in particular, Euclidean geometry
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
. 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
and the controversial four color theorem
, whose only proof, by computer, has never been verified by a human.
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.
Some disagreement about the foundations of mathematics continues to 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 on a rigorous axiomatic
framework, and studying the results of such a framework. As such, it is home to Gödel's second incompleteness theorem
, perhaps the most widely celebrated result in logic, which (informally) implies that any 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
). Gödel showed how to construct, whatever the given collection of number-theoretical axioms, a formal statement in the logic that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a true 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
is the common name for the fields of mathematics most generally useful in theoretical computer science
. This includes, on the computer science side, computability theory
, computational complexity theory
, and information theory
. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful 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 rapid advance of computer hardware. 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
As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "P=NP?
" problem, one of the Millennium Prize Problems
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
; 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
; the estimated models and consequential predictions
should be tested
on new data
- ^ 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).
- ^ Steen, L.A. (April 29, 1988). The Science of Patterns. Science, 240: 611–616. and summarized at Association for Supervision and Curriculum Development., ascd.org
- ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 9780716750475
- ^ Jourdain.
- ^ Peirce, p. 97.
- ^ 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.
- ^ Eves
- ^ Peterson
- ^ Both senses can be found in Plato. Liddell and Scott, s.voceμαθηματικός
- ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics"
- ^ S. Dehaene; G. Dehaene-Lambertz; L. Cohen (Aug 1998). "Abstract representations of numbers in the animal and human brain". Trends in Neuroscience 21 (8): pp. 355–361. doi:10.1016/S0166-2236(98)01263-6.
- ^ See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim
- ^ Kline 1990, Chapter 1.
- ^ Sevryuk
- ^ Johnson, Gerald W.; Lapidus, Michel L. (2002). The Feynman Integral and Feynman's Operational Calculus. Oxford University Press.
- ^ Eugene Wigner, 1960, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications on Pure and Applied Mathematics 13(1): 1–14.
- ^ Mathematics Subject Classification 2010
- ^ Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press.
- ^ Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas: Mathematics and Philosophy. MAA.
- ^ Aigner, Martin; Ziegler, Gunter M. (2001). Proofs from the Book. Springer.
- ^ Earliest Uses of Various Mathematical Symbols (Contains many further references).
- ^ Kline, p. 140, on Diophantus; p.261, on Vieta.
- ^ See false proof for simple examples of what can go wrong in a formal proof. The history of the Four Color Theorem contains examples of false proofs accidentally accepted by other mathematicians at the time.
- ^ 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).
- ^ 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."
- ^ Zeidler, Eberhard (2004). Oxford User's Guide to Mathematics. Oxford, UK: Oxford University Press. p. 1188. ISBN 0198507631.
- ^ Waltershausen
- ^ Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228.
- ^ Popper 1995, p. 56
- ^ Ziman
- ^ "The Fields Medal is now indisputably the best known and most influential award in mathematics." Monastyrsky
- ^ Riehm
- ^ Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford University Press, 2005.
- ^ Clay Mathematics Institute, P=NP, claymath.org
- ^ Like other mathematical sciences such as physics and computer 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.
- 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; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Courant, R. 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.
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7. — A gentle introduction to the world of mathematics.
- Einstein, Albert (1923). Sidelights on Relativity (Geometry and Experience). P. Dutton., Co.
- Eves, Howard, An Introduction to the History of Mathematics, Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
- Gullberg, Jan, Mathematics — From the Birth of Numbers. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X. — An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. — A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online.
- Jourdain, Philip E. B., The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover Publications, 2003, ISBN 0-486-43268-8.
- 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. http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf. 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. "Linear Associative Algebra". American Journal of Mathematics (Vol. 4, No. 1/4. (1881). http://books.google.com/books?hl=en&lr=&id=De0GAAAAYAAJ&oi=fnd&pg=PA1&dq=Peirce+Benjamin+Linear+Associative+Algebra+&ots=DauJpcwta-&sig=GfnhyzxFQCUDrB0s-X5HOKqVGAk#v=onepage&q=&f=false. .
- Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
- Paulos, John Allen (1996). A Mathematician Reads the Newspaper. Anchor. ISBN 0-385-48254-X.
- 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. http://www.ams.org/notices/200207/comm-riehm.pdf.
- 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. http://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf. Retrieved 2006-06-24.
- Waltershausen, Wolfgang Sartorius von (1856, repr. 1965). Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 3-253-01702-8. http://www.amazon.de/Gauss-Ged%e4chtnis-Wolfgang-Sartorius-Waltershausen/dp/3253017028.
- Ziman, J.M., F.R.S. (1968). Public Knowledge:An essay concerning the social dimension of science. http://info.med.yale.edu/therarad/summers/ziman.htm.
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