Future of mathematics

The future of mathematics is a topic that 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 historical and recent include Felix Klein's Erlangen program, Hilbert's problems, Langlands program, and the Millennium Prize Problems. In the Mathematics Subject Classification section 01Axx History of mathematics and mathematicians, the subsection 01A67 is titled Future prospectives.

Motivations and methodology for speculation

According to Henri Poincaré 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".[1] 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.[2] A subject survey of mathematics itself however is now problematic: the sheer expansion of the subject gives rise to issues of mathematical knowledge management.

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.[3] Mathematical education 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.[4] László Lovász, in Trends in Mathematics: How they could Change Education?[5] 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 writes in "The Proof is in the Pudding. A Look at the Changing Nature of Mathematical Proof":[6] "It is becoming increasingly evident that the delinations 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 “Department of Mathematics” at the college and university level gives way to “Division of Mathematical Sciences”."

Experimental mathematics

Experimental mathematics 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 theory. The paper "Experimental Mathematics: Recent Developments and Future Outlook"[7] describes expected increases in computer capabilities: better hardware in terms of speed and memory capacity; better software in terms of increasing sophistication of algorithms; more advanced visualization facilities; the mixing of numerical and symbolic methods.

Semi-rigorous mathematics

Doron Zeilberger 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.”"[8] Some people strongly disagree with Zeilberger's prediction, for example it has been described as provocative and quite wrongheaded,[9] 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",[10] Timothy Gowers 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.

Mathematics by topic area

Combinatorics

On combinatorics: In 2001, Peter Cameron in "Combinatorics entering the third millennium"[11] attempts to "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."" and makes the prediction that What is clear, though, is that combinatorics will continue to elude attempts at formal specification. Béla Bollobás 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".[12]

Numerical analysis and scientific computing

On numerical analysis and scientific computing: In 2000, Lloyd N. Trefethen wrote "Predictions for scientific computing 50 years from now",[13] which concluded with the theme that "Human beings will be removed from the loop" and writing in 2008 in The Princeton Companion to Mathematics 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.[14]

Data analysis

On data analysis: In 1998, Mikhail Gromov in "Possible Trends in Mathematics in the Coming Decades",[15] 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 where classical probability does not apply. Such methods might include advances in wavelet analysis, higher-dimensional methods and inverse scattering.

Control theory

A list of grand challenges for control theory is outlined in "Future Directions in Control, Dynamics, and Systems: Overview, Grand Challenges, and New Courses".[16]

Mathematical logic

Mathematical logic is discussed in "The Prospects For Mathematical Logic In The Twenty-First Century".[17]

Mathematical biology

Mathematical biology 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"[18] is an essay by Joel E. Cohen.

Mathematical physics

Mathematical physics 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".[19]

See also

References

  1. ↑ Henri Poincaré (1908). "The Future of Mathematics". Translation of the French original: "L'avenir des mathématiques". in Revue générale des sciences pures et appliquées 19 (1908), pages 930–939. Also appeared in: Circolo Matematico di Palermo; Bulletin des Sciences mathématiques; Scientia; and Atti del IV° Congresse internazionale dei Matematici. Lecture held at the Eighth International Congress of Mathematicians, Rome, Italy, 1908.
  2. ↑ The honors class: Hilbert's problems and their solvers, Ben Yandell, A K Peters Ltd., 2002, ISBN 978-1-56881-216-8
  3. ↑ Keynote – Mathematics Everywhere, Marja Makarow, ERCIM NEWS 73 April 2008
  4. ↑ Foundations for the future in mathematics education, Editors Richard A. Lesh, Eric Hamilton, James J. Kaput Routledge, 2007, ISBN 978-0-8058-6056-6
  5. ↑ Trends in Mathematics: How they could Change Education?
  6. ↑ The Proof is in the Pudding. A Look at the Changing Nature of Mathematical Proof, Steven G. Krantz, 2008
  7. ↑ Bailey, David H.; Borwein, Jonathan M. (2001). "Experimental Mathematics: Recent Developments and Future Outlook". Mathematics unlimited: 2001 and beyond. Springer. CiteSeerX 10.1.1.138.1705 Freely accessible.
  8. ↑ Doron Zeilberger (1994). "Theorems for a Price: Tomorrow’s Semi-Rigorous Mathematical Culture". The Mathematical Intelligencer 16:4, pages 11–18, December 1994.
  9. ↑ Proof and other dilemmas: mathematics and philosophy, Bonnie Gold, Roger A. Simons, MAA, 2008, ISBN 978-0-88385-567-6
  10. ↑ Rough structure and classification, Timothy Gowers, 1999, https://www.dpmms.cam.ac.uk/~wtg10/gafavisions.ps
  11. ↑ Combinatorics entering the third millennium, Peter J. Cameron, Third draft, July 2001
  12. ↑ "Creative Minds, Charmed Lives", Yu Kiang Leong, World Scientific, 2010
  13. ↑ Predictions for scientific computing 50 years from now, Lloyd N. Trefethen (Mathematics Today, 2000)
  14. ↑ The Princeton Companion to Mathematics, Princeton University Press, 2008, Page 614
  15. ↑ Possible Trends in Mathematics in the Coming Decades, Mikhael Gromov, Notices of the AMS, 1998.
  16. ↑ Future Directions in Control, Dynamics, and Systems: Overview, Grand Challenges, and New Courses, Richard M. Murray, European journal of control, 2003.
  17. ↑ The Prospects For Mathematical Logic In The Twenty-First Century, Samuel R. Buss, Alexander S. Kechris, Anand Pillay, and Richard A. Shore, Bulletin of Symbolic Logic, 2001.
  18. ↑ "Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better", Joel E. Cohen, PLoS Biol, 2004 – biology.plosjournals.org
  19. ↑ New Trends in Mathematical Physics: Selected Contributions of the XVth International Congress on Mathematical Physics, Editor Vladas Sidoravicius, Springer, 2009, ISBN 978-90-481-2809-9.

Further reading

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