The progression of both the nature of 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 may 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'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.
The accuracy of predictions about mathematics' have varied widely and has proceeded very closely to that of technology. 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
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.
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
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, 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 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 “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"
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.”"
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
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 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 in "Combinatorics entering the third millennium" organizes predictions for the future of combinatorics:
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
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 was discussed in "The Prospects For Mathematical Logic In The Twenty-First Century", including set theory, mathematical logic in computer science, and proof theory.
Applied mathematics
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", 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.
Data analysis
On 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 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".
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" 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".
