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.
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".
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 theories.
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.”"[9] Some people strongly disagree with Zeilberger's prediction; for example, it has been described as provocative and quite wrongheaded,[10] 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.
Terence Tao and Alessio Figalli (both recipients of Fields Medal) don't agree with Gowers statements, especially those concerning "a threat".
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.