From Wikipedia, the free encyclopedia
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".
[5] 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."
[6]Through the use of
abstraction and
logical reasoning, mathematics evolved 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 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.
[7]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 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.
[8]Etymology
The word "mathematics" comes from the
Greek μάθημα (
máthēma), which means
learning,
study,
science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.
[9] Its adjective is
μαθηματικός (
mathēmatikós),
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.
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, 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.
[10] In English, the noun
mathematics takes singular verb forms. It is often shortened to
maths or, in English-speaking North America,
math.
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,
[11] 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,
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.
[13] 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."
[14]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; 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.
[15] 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."
[16] 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.
[17] 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.
[18] 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.
[19][20] The popularity of
recreational mathematics is another sign of the pleasure many find in solving mathematical questions.
Notation, language, and rigor
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.
[21] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.
[22] 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 also be hard 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.
Mathematical jargon includes technical terms such as
homeomorphism and
integrable. 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".
The
infinity symbol
∞ in several typefaces.
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.
[23] 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.
[24]
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.
[25]
Mathematics as science
Carl Friedrich Gauss, himself known as the "prince of mathematicians",
[26] referred to mathematics as "the Queen of the Sciences".
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".
[27] 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."
[6]
Many philosophers believe that mathematics is not experimentally
falsifiable, and thus not a science according to the definition of
Karl Popper.
[28] 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 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."
[29] 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.
[30] 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] 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.
[citation needed]Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the
Fields Medal,
[31][32] 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
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.
Quantity
-
Structure
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.
-
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. 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.
-
Change
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.
[33] 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.
[citation needed] Modern logic is divided into
recursion theory,
model theory, and
proof theory, and is closely linked to
theoretical computer science.
-
Discrete mathematics
Discrete mathematics 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 and
entropy.
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.
[34]
-
Applied mathematics
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 and
estimation; the estimated models and consequential
predictions should be
tested on
new data.
[35]See also
Notes
- ^ 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.
References
- 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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