Mathematics has been being more and more computer dependent. What shape would it possibly take in the coming years?
If we see trends that are emerging from last 50 years it can easily be said that some of the good mathematics is done by those who are not mathematicians and this boundary between mathematician and non-mathematician is doing away. Because of computers and nature of the problems available the mathematics is now devoted to real life problems coming in daily life. One area where mathematical modelling now extensively used is in dynamical systems, bio-science problems and economics including finances.
If we see trends that are emerging from last 50 years it can easily be said that some of the good mathematics is done by those who are not mathematicians and this boundary between mathematician and non-mathematician is doing away. Because of computers and nature of the problems available the mathematics is now devoted to real life problems coming in daily life. One area where mathematical modelling now extensively used is in dynamical systems, bio-science problems and economics including finances.
I am sure, some heuristic methods may come up. Each and every physical situation may be mathematically modeled (like ANN from neurons/brain, cloud computing, bee network, ant network, etc).
Verification of mathematical proofs by computer takes a lot of work out of reviewing mathematical papers. It also makes unwieldy proofs 'wieldy' in a way. This could change the way we do mathematics over the next few decades.
I think that the field of symbolic computation has a potential to grow up further, since untill now it is used only for computing indefinite integrals or solutions of equations of every kind. But, starting from the automated theorem proving in Geometry it could be possible to make an extension to other fields of Mathematics.
I mostly agree with Prof. Mittal. Besides, the the range of knowledge growth depends greatly on life challenges. The more human life involves challenges (e.g. Healthcare or finance as majors), the more practical mathematics, specially statistics, will grow to cope with them. The less human life involves challenges, the more exploratory, theoretical, and pure mathematics will grow.
My guess as a non-mathematic is that new data, new practical questions and the acces to computer applications will bring great future surprises. For example, I imagine that it will be possible to find at other regions of cosmos that light may travel at different non-constant velocities, or perhaps that the exponents of several physical laws -that are normally natural numbers today- may take other real values. The combined power of mathematics, researchers creativeness, and computers would help to make this kind of surprises possible. Thanks, emilio
This is an excellent question. It usually helps to reflect on the future growth familiar areas of mathematics.
It is fairly obvious that we can expect continued growth (significant advances) in varioius forms of geometry and algebra with many possible extensions of currently known results.
In an interview with Felix Browder, American Mathematical Society President, in 1999, who comments on the future growth of mathematics:
http://www.ams.org/notices/199903/comm-browder.pdf
Browder observes that there has been an enormous growth in the practical importance of mathematics in the world, as represented by the computer revolution and the necessary analysis of complex systems, which are everywhere around us. We see this in the genome project , in the mathematics of finance, in the mathematical sophistication and interaction in all realms of theoretical physics, or even practical physics for that matter. What we have to emphasize is that these are not just applications of known mathematics. They are enormous growing points for mathematics [itself] (p. 346].
Here is a summary of Henri Poincare's crystal ball, how Poincare envisioned the future of mathematics:
http://www.ams.org/notices/200804/tx080400458p.pdf
Poinare's specific predictions, pinpointing likely growth areas in mathematics:
Number Theory, foreshadowing the development of Andre Weils introduction of p algebraic geometry and theory of divisors on varieties.
Algebra, considering work on rings of polynomial with integer or other coefficients.
Linear partial differential equations, foreshadowing work on integral equations, infinite-dimensional spaces and extension of linear algebra to such spaces (this, in fact, has happened, thanks to work on Banach spaces (e.g., R.R. Phelps) and Hilbert spaces (e.g., P. Halmos).
Abelian functions, foreshadowing work on Lie groups.
Complex variables, foreshadowing work on pseudo-convexity, Stein manifolds and sheaf theory.
Theory of groups, e.g., Galois groups.
Geometry, foreshadowing work on geometry in higher dimensional spaces.
Set theory, foreshadow work on cardinality of sets, nonstandard analysis and set theory paradoxes.
Axiomatic analysis, foreshadowing work on axiomatic theories such as proximity spaces introduced by Efremovich in the early 1930s.
There is a score card at the end of the Notices article, listing developments foreseen by Poincare and those Poincare missed (see p. 466).
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