Mathematical Proofs at a Crossroad?

For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomatic-deductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomatic-deductive proofs are not a posteriori work, a luxury we can marginalize nor are computer-assisted proofs bad mathematics. There is hope for integration!

[1]  Cristian S. Calude,et al.  Passages of Proof , 2003, Bull. EATCS.

[2]  Paul J. Campbell Mathematics by Experiment: Plausible Reasoning in the 21st Century (Book) , 2004 .

[3]  E. R. Swart The Philosophical Implications of the Four-Color Problem , 1980 .

[4]  Saunders Mac Lane,et al.  DESPITE PYHSICISTS, PROOF IS ESSENTIAL IN MATHEMATICS , 1997, Synthese.

[5]  C. Perelman,et al.  Traité de l'argumentation , 1958 .

[6]  Andreea S. Calude The Journey of the Four Colour Theorem Through Time , 2001 .

[7]  Gregory J. Chaitin,et al.  The Unknowable , 1999, Nabokov and Indeterminacy.

[8]  G. Pólya,et al.  Mathematics and Plausible Reasoning , 1956 .

[9]  Gregory J. Chaitin,et al.  Exploring RANDOMNESS , 2001, Discrete Mathematics and Theoretical Computer Science.

[10]  Peter W Shor,et al.  Computers and mathematics. Quantum channel capacities. , 2004, Science.

[11]  Reuben Hersh,et al.  What is Mathematics , 2000 .

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

[13]  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.

[14]  Thomas Tymoczko The Four-color Problem and Its Philosophical Significance , 1979 .

[15]  I. Lakatos PROOFS AND REFUTATIONS (I)*† , 1963, The British Journal for the Philosophy of Science.

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

[17]  E. Reed Ways of thinking , 1990, Nature.

[18]  J. Barrow,et al.  IMPOSSIBILITY : THE LIMITS OF SCIENCE AND THE SCIENCE OF LIMITS , 2009 .

[19]  K. Appel,et al.  Every Planar Map Is Four Colorable , 2019, Mathematical Solitaires & Games.

[20]  P. Seymour,et al.  A new proof of the four-colour theorem , 1996 .

[21]  Michael Atiyah,et al.  Responses to: A. Jaffe and F. Quinn, “Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics” [Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 1–13; MR1202292 (94h:00007)] , 1994 .

[22]  Karl Sigmund,et al.  Kepler’s conjecture: How some of the greatest minds in history helped solve one of the oldest math problems in the world , 2004 .

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

[24]  Eugene H. Ehrlich Oxford American Dictionary , 1980 .

[25]  Donald E. Knuth,et al.  Literate Programming , 1984, Comput. J..

[26]  Grzegorz Rozenberg,et al.  Cornerstones of undecidability , 1994, Prentice Hall International Series in Computer Science.

[27]  Silvio Micali,et al.  The knowledge complexity of interactive proof-systems , 1985, STOC '85.

[28]  Steve Marcus No system can be improved in all respects , 1998 .

[29]  M. Kline Mathematics: The Loss of Certainty , 1982 .

[30]  Ch Perelman,et al.  Traité de l'argumentation. La nouvelle rhétorique, Collection de sociologie générale et de philosophie sociale. Éditions de l'Institut de Sociologie de Bruxelles, 2e éd , 1971 .

[31]  A. Jaffe,et al.  Response to comments on ``Theoretical Mathematics'' , 1994, math/9404231.

[32]  Michael Stoeltzner,et al.  What Lakatos Could Teach the Mathematical Physicist , 2002 .

[33]  R. Hersh What is Mathematics, Really? , 1998 .

[34]  Martin Raussen,et al.  Interview with Jean-Pierre Serre , 2003 .