Future of mathematics
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The progression of both the nature of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and individual mathematical problems into the future is a widely debated topic; many past predictions about modern mathematics have been misplaced or completely false, so there is reason to believe that many predictions today will follow a similar path. However, the subject still carries an important weight and has been written about by many notable mathematicians. Typically, they are motivated by a desire to set a research agenda to direct efforts to specific problems, or a wish to clarify, update and extrapolate the way that subdisciplines relate to the general discipline of mathematics and its possibilities. Examples of agendas pushing for progress in specific areas in the future, historical and recent, include
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
's
Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
,
Hilbert's problems Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pro ...
,
Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
, and the
Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. According ...
. In the
Mathematics Subject Classification The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. ...
section 01Axx History of mathematics and mathematicians, subsection 01A67 is titled Future prospectives. The accuracy of predictions about mathematics has varied widely and has proceeded very closely to that of technology. Borwein, Jonathan M. (2013)
"The Future of Mathematics: 1965 to 2065."
MAA Centenary Volume. Retrieved 7 February 2019.
As such, it is important to keep in mind that many of the predictions by researchers below may be misguided or turn out to be untrue.


Motivations and methodology for speculation

According to
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
writing in 1908 (English translation), "The true method of forecasting the future of mathematics lies in the study of its history and its present state". The historical approach can consist of the study of earlier predictions, and comparing them to the present state of the art to see how the predictions have fared, e.g. monitoring the progress of Hilbert's problems. A subject survey of mathematics itself however is now problematic: the sheer expansion of the subject gives rise to issues of
mathematical knowledge management Mathematical knowledge management (MKM) is the study of how society can effectively make use of the vast and growing literature on mathematics. It studies approaches such as databases of mathematical knowledge, automated processing of formulae an ...
. The development of technology has also significantly impacted the outcomes of many predictions; because of the uncertain nature of the future of technology, this leads to quite a bit of uncertainty in the future of mathematics. Also entailed by this is that successful predictions about future technology may also result in successful mathematical predictions. Given the support of research by governments and other funding bodies, concerns about the future form part of the rationale of the distribution of funding.
Mathematical education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although rese ...
must also consider changes that are happening in the mathematical requirements of the workplace; course design will be influenced both by current and by possible future areas of application of mathematics.
László Lovász László Lovász (; born March 9, 1948) is a Hungarian mathematician and professor emeritus at Eötvös Loránd University, best known for his work in combinatorics, for which he was awarded the 2021 Abel Prize jointly with Avi Wigderson. He ...
, in ''Trends in Mathematics: How they could Change Education?'' describes how the mathematics community and mathematical research activity is growing and states that this will mean changes in the way things are done: larger organisations mean more resources are spent on overheads (coordination and communication); in mathematics this would equate to more time engaged in survey and expository writing.


Mathematics in general


Subject divisions

Steven G. Krantz Steven George Krantz (born February 3, 1951) is an American scholar, mathematician, and writer. He has authored more than 350 research papers and published more than 150 books. Additionally, Krantz has edited journals such as the ''Notices of th ...
writes in "The Proof is in the Pudding. A Look at the Changing Nature of Mathematical Proof": "It is becoming increasingly evident that the delineations among “engineer” and “mathematician” and “physicist” are becoming ever more vague. It seems plausible that in 100 years we will no longer speak of mathematicians as such but rather of mathematical scientists. It would not be at all surprising if the notion of “the Department of Mathematics” at the college and university level gives way to “the Division of Mathematical Sciences”."


Experimental mathematics

Experimental mathematics Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with th ...
is the use of computers to generate large data sets within which to automate the discovery of patterns which can then form the basis of conjectures and eventually new theories. The paper "Experimental Mathematics: Recent Developments and Future Outlook" describes expected increases in computer capabilities: better hardware in terms of speed and memory capacity; better software in terms of increasing sophistication of
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s; more advanced
visualization Visualization or visualisation may refer to: *Visualization (graphics), the physical or imagining creation of images, diagrams, or animations to communicate a message * Data visualization, the graphic representation of data * Information visualiz ...
facilities; the mixing of numerical and symbolic methods.


Semi-rigorous mathematics

Doron Zeilberger Doron Zeilberger (דורון ציילברגר, born 2 July 1950 in Haifa, Israel) is an Israeli mathematician, known for his work in combinatorics. Education and career He received his doctorate from the Weizmann Institute of Science in 1976, ...
considers a time when computers become so powerful that the predominant questions in mathematics change from proving things to determining how much it would cost: "As wider classes of identities, and perhaps even other kinds of classes of theorems, become routinely provable, we might witness many results for which we would know how to find a proof (or refutation), but we would be unable, or unwilling, to pay for finding such proofs, since “almost certainty” can be bought so much cheaper. I can envision an abstract of a paper, c. 2100, that reads : “We show, in a certain precise sense, that the Goldbach conjecture is true with probability larger than 0.99999, and that its complete truth could be determined with a budget of $10B.”" Some people strongly disagree with Zeilberger's prediction; for example, it has been described as provocative and quite wrongheaded, whereas it has also been stated that choosing which theorems are interesting enough to pay for already happens as a result of funding bodies making decisions as to which areas of research to invest in.


Automated mathematics

In "Rough structure and classification",
Timothy Gowers Sir William Timothy Gowers, (; born 20 November 1963) is a British mathematician. He is Professeur titulaire of the Combinatorics chair at the Collège de France, and director of research at the University of Cambridge and Fellow of Trinity Col ...
writes about three stages: 1) at the moment computers are just slaves doing boring calculations, 2) soon databases of mathematical concepts and proof methods will lead to an intermediate stage where computers are very helpful with theorem proving but unthreatening, and 3) within a century computers will be better than humans at theorem proving.
Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
and
Alessio Figalli Alessio Figalli (; born 2 April 1984) is an Italian mathematician working primarily on calculus of variations and partial differential equations. He was awarded the Prix and in 2012, the EMS Prize in 2012, the Stampacchia Medal in 2015, the ...
(both recipients of Fields Medal) don't agree with Gowers statements, especially those concerning "a threat".


Mathematics by subject

Different subjects of mathematics have very different predictions; for example, while every subject of mathematics is seen to be altered by the computer, some branches are seen to benefit from the use of technology to aid human achievement, while in others computers are predicted to completely replace humans.


Pure mathematics


Combinatorics

In 2001,
Peter Cameron Peter Cameron is the name of: * Peter Cameron (entomologist) (1847–1912), English entomologist who specialised in Hymenoptera * Peter Cameron (minister) (born 1945), Scottish-born Church of Scotland minister convicted of heresy by the Presbyteria ...
in "Combinatorics entering the third millennium" organizes predictions for the future of
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
:
throw some light on present trends and future directions. I have divided the causes into four groups: the influence of the computer; the growing sophistication of combinatorics; its strengthening links with the rest of mathematics; and wider changes in society. What is clear, though, is that combinatorics will continue to elude attempts at formal specification.
Béla Bollobás Béla Bollobás FRS (born 3 August 1943) is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul E ...
writes: "Hilbert, I think, said that a subject is alive only if it has an abundance of problems. It is exactly this that makes combinatorics very much alive. I have no doubt that combinatorics will be around in a hundred years from now. It will be a completely different subject but it will still flourish simply because it still has many, many problems".


Mathematical logic

In the year 2000,
mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
was discussed in "The Prospects For Mathematical Logic In The Twenty-First Century", including
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, mathematical logic in
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, and
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four correspo ...
.


Applied mathematics


Numerical analysis and scientific computing

In 2000,
Lloyd N. Trefethen Lloyd Nicholas Trefethen (born 30 August 1955) is an American mathematician, professor of numerical analysis and head of the Numerical Analysis Group at the Mathematical Institute, University of Oxford. Education Trefethen was born 30 August 19 ...
wrote "Predictions for scientific computing 50 years from now", which concluded with the theme that "Human beings will be removed from the loop" and writing in 2008 in ''
The Princeton Companion to Mathematics ''The Princeton Companion to Mathematics'' is a book providing an extensive overview of mathematics that was published in 2008 by Princeton University Press. Edited by Timothy Gowers with associate editors June Barrow-Green and Imre Leader, it ha ...
'' predicted that by 2050 most numerical programs will be 99% intelligent wrapper and only 1% algorithm, and that the distinction between linear and non-linear problems, and between forward problems (one step) and inverse problems (iteration), and between algebraic and analytic problems, will fade as everything becomes solved by iterative methods inside adaptive intelligent systems that mix and match and combine algorithms as required.


Data analysis

In 1998, Mikhail Gromov in "Possible Trends in Mathematics in the Coming Decades", says that traditional probability theory applies where global structure such as the Gauss Law emerges when there is a lack of structure between individual data points, but that one of today's problems is to develop methods for analyzing
structured data A data model is an abstract model that organizes elements of data and standardizes how they relate to one another and to the properties of real-world entities. For instance, a data model may specify that the data element representing a car be co ...
where classical probability does not apply. Such methods might include advances in
wavelet analysis A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
, higher-dimensional methods and
inverse scattering In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. It is the inverse problem to the direct scattering problem, wh ...
.


Control theory

A list of grand challenges for
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
is outlined in "Future Directions in Control, Dynamics, and Systems: Overview, Grand Challenges, and New Courses".


Mathematical biology

Mathematical biology Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development a ...
is one of the fastest expanding areas of mathematics at the beginning of the 21st century. "Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better" is an essay by
Joel E. Cohen Joel Ephraim Cohen (born February 10, 1944) is a Mathematical and theoretical biology, mathematical biologist. He is currently Abby Rockefeller Mauzé Professor of Populations at the Rockefeller University in New York City and at the Earth Institut ...
.


Mathematical physics

Mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
is an enormous and diverse subject. Some indications of future research directions are given in "New Trends in Mathematical Physics: Selected Contributions of the XVth International Congress on Mathematical Physics".New Trends in Mathematical Physics: Selected Contributions of the XVth International Congress on Mathematical Physics
Editor Vladas Sidoravicius, Springer, 2009, .


See also

*
List of unsolved problems in mathematics Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Eucli ...


References


Further reading


The Future Of Mathematics
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
, 1950
Mathematics: frontiers and perspectives
V. I. Arnold, M. Atiyah, B. Mazur, AMS Bookstore, 2000,
Visions in Mathematics
Editors N. Alon, J. Bourgain, A. Connes, M Gromov, V. Milman, Springer, 2010,
Reflections on the Future of Mathematics
Felix Browder Felix Earl Browder (; July 31, 1927 – December 10, 2016) was an American mathematician known for his work in nonlinear functional analysis. He received the National Medal of Science in 1999 and was President of the American Mathematical Society ...
, JUNE/JULY 2002, NOTICES OF THE AMS
Henri’s Crystal Ball
Philip J. Davis and
David Mumford David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
, April 2008, Notices of the AMS
The nature and growth of modern mathematics
Edna Ernestine Kramer, Princeton University Press, 1982,
Current and future directions in applied mathematics
Editors Mark Alber, Bei Hu, Joachim Rosenthal, Birkhäuser, 1997,
Mathematics unlimited: 2001 and beyond
Editors Björn Engquist, Wilfried Schmid, Springer, 2001,


External links


Math 2.0
forum for all topics related to the future of mathematical publishing.

Felix Breuer, 27 Feb 2012 {{Authority control Futures studies