Exploratory Experimentation and Computation

We believe the mathematical research community is facing a great challenge to re-evaluate the role of proof in light of recent developments. On one hand, the growing power of current computer systems, of modern mathematical computing packages, and of the growing capacity to data-mine on the Internet, has provided marvelous resources to the research mathematician. On the other hand, the enormous complexity of many modern capstone results such as the Poincar´e conjecture, Fermat’s last theorem, and the classification of finite simple groups has raised questions as to how we can better ensure the integrity of modern mathematics. Yet as the need and prospects for inductive mathematics blossom, the requirement to ensure the role of proof is properly founded remains undiminished.

[1]  Jesus Guillera.  Hypergeometric identities for 10 extended Ramanujan-type series , 2008 .

[2]  L. Franklin,et al.  Exploratory Experiments , 2005, Philosophy of Science.

[3]  S. Pinker.  The Stuff of Thought: Language as a Window into Human Nature , 2007 .

[4]  Jonathan M. Borwein,et al.  Mathematics by experiment - plausible reasoning in the 21st century , 2003 .

[5]  Jonathan M. Borwein,et al.  Surprising Sinc Sums and Integrals , 2008, Am. Math. Mon..

[6]  Marko Petkovšek,et al.  A=B : 等式証明とコンピュータ , 1997 .

[7]  David H. Bailey,et al.  Analysis of PSLQ, an integer relation finding algorithm , 1999, Math. Comput..

[8]  Jonathan M. Borwein,et al.  Higher-Dimensional Box Integrals , 2010, Exp. Math..

[9]  Wadim Zudilin,et al.  Ramanujan-type formulae for 1/pi: a second wind? , 2007, 0712.1332.

[10]  David Bailey,et al.  On the rapid computation of various polylogarithmic constants , 1997, Math. Comput..

[11]  Jonathan M. Borwein,et al.  Ten Problems in Experimental Mathematics , 2006, Am. Math. Mon..

[12]  Jonathan M. Borwein,et al.  The evaluation of character Euler double sums , 2008 .

[13]  D. Smail.  On Deep History and the Brain , 2007 .

[14]  Doron Zeilberger,et al.  Rational functions certify combinatorial identities , 1990 .

[15]  R. Crandall.  Theory of ROOF walks , 2008 .

[16]  P. Corfield.  On Deep History and the Brain , 2008 .

[17]  Jonathan M. Borwein,et al.  Experimentation in Mathematics: Computational Paths to Discovery , 2004 .

[18]  Marcus Giaquinto,et al.  Visual thinking in mathematics , 2007 .

[19]  Wadim Zudilin,et al.  Ramanujan-type formulae for 1/π: a second wind? , 2006 .

[20]  George Polya,et al.  Mathematical discovery : on understanding, learning, and teaching problem solving , 1962 .

[21]  Mario Livio,et al.  Is God a Mathematician , 2009 .

[22]  H. K. Sørensen.  Exploratory experimentation in experimental mathematics : A glimpse at the PSLQ algorithm , 2010 .

[23]  Marcus Giaquinto,et al.  Visual thinking in mathematics : an epistemological study , 2007 .

[24]  Jonathan M. Borwein,et al.  The Computer as Crucible: An Introduction to Experimental Mathematics , 2008 .

[25]  A. Jaffe,et al.  “Theoretical mathematics”: toward a cultural synthesis of mathematics and theoretical physics , 1993, math/9307227.

[26]  Jonathan M. Borwein,et al.  Computer-Assisted Discovery and Proof , 2007 .

[27]  T. Hales.  Formal Proof , 2008 .

[28]  Jeremy Avigad,et al.  Computers in Mathematical Inquiry , 2008 .

[29]  Jonathan M. Borwein,et al.  The computation of previously inaccessible digits of π² and Catalan's constant , 2013 .

[30]  David H. Bailey,et al.  Experimental Mathematics in Action , 2007 .

[31]  Michael Atiyah,et al.  Responses to `Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics', by A. Jaffe and F. Quinn , 1994, math/9404229.