If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt Gödel, 1951) Paper Revised 09–09–04 This paper is an extended version of a presentation made at ICME10, related work is elaborated in references [1–7]. 1 I shall generally explore experimental and heuristic mathematics and give (mostly) accessible, primarily visual and symbolic, examples. The emergence of powerful mathematical computing environments like Maple and Matlab, the growing

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the non-alternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3-loop Feynman diagrams of hep-th/9803091 and subsequently in hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010.

Christopher Koch [34] accurately captures a great scientific distaste for philosophizing: “Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden influence is an acute embarrassment to many researchers, and it is therefore not often acknowledged. ” (Christopher Koch, 2004) That acknowledged, I am of the opinion that mathematical philosophy matters more now than it has in nearly a century. The power of modern computers matched with that of modern mathematical software and the sophistication of current mathematics is changing the way we do mathematics. In my view it is now both necessary and possible to admit quasi-empirical inductive methods fully into mathematical argument. In doing so carefully we will enrich mathematics and yet preserve the mathematical literature’s deserved reputation for reliability—even as the methods and criteria change. What do I mean by reliability? Well, research mathematicians still consult Euler or Riemann to be informed, anatomists only consult Harvey 3 for historical reasons. Mathematicians happily quote old papers as core steps of arguments, physical scientists expect to have to confirm results with another experiment. 1 Mathematical Knowledge as I View It Somewhat unusually, I can exactly place the day at registration that I became a mathematician and I recall the reason why. I was about to deposit my punch cards in the ‘honours history bin’. I remember thinking “If I do study history, in ten years I shall have forgotten how to use the calculus properly. If I take mathematics, I shall still be able to read competently about the War of 1812 or the Papal schism. ” (Jonathan Borwein, 1968) The inescapable reality of objective mathematical knowledge is still with me. Nonetheless, my view then of the edifice I was entering is not that close to my view of the one I inhabit forty years later. 1 The companion web site is at www.experimentalmath.info

Abstract. Is it possible to distinguish algebraic from transcendental real numbers by considering the b-ary expansion in some base b � 2? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number x and for any base g � 2, the g-ary expansion of x should satisfy some of the laws that are shared by almost all numbers. For instance, the frequency where a given finite sequence of digits occurs should depend only on the base and on the length of the sequence. We are very far from such a goal: there is no explicitly known example of a triple (g, a, x), where g � 3 is an integer, a a digit in {0,...,g − 1} and x a real irrational algebraic number, for which one can claim that the digit a occurs infinitely often in the g-ary expansion of x. Hence there is a huge gap between the established theory and the expected state of the art. However, some progress has been made recently, thanks mainly to clever use of Schmidt’s subspace theorem. We review some of these results. 1. Normal Numbers and Expansion of Fundamental Constants 1.1. Borel and Normal Numbers. In two papers, the first [28] published in 1909 and the second [29] in 1950, Borel studied the g-ary expansion of real numbers, where g � 2 is a positive integer. In his second paper, he suggested that this expansion for a real irrational algebraic number should satisfy some of the laws shared by almost all numbers, in the sense of Lebesgue measure. Let g � 2 be an integer. Any real number x has a unique expansion x = a−kg k + ···+ a−1g + a0 + a1g −1 + a2g −2 + ·· ·, where k � 0 is an integer and the ai for i � −k, namely the digits of x in the expansion in base g of x, belong to the set {0, 1,...,g − 1}. Uniqueness is subject to the condition that the sequence (ai)i�−k is not ultimately constant and equal to g − 1. We write this expansion x = a−k ···a−1a0.a1a2 ·· ·. in base 10 (decimal expansion), whereas

Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computer-assisted proofs will be give a special attention. Finally, in a highly speculative part, we will anticipate the evolution of proofs under the assumption that the quantum computer will materialize. We will argue that there is little ‘intrinsic ’ difference between traditional and ‘unconventional ’ types of proofs. 2 Mathematical Proofs: An Evolution in Eight Stages Theory is to practice as rigour is to vigour. D. E. Knuth Reason and experiment are two ways to acquire knowledge. For a long time mathematical

If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. (Kurt

If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics.

(Kantorovich and K.: joint winners Nobel Prize Economics 1975, "for their contributions to the theory of optimum allocation of resources")

Hardy & Wright records elegant forms for the generating functions of the divisor functions k (n) = P djn d k and 2 k (n): 1 X n=1 k (n) n s = (s)(s k) (1) 1 X n=1 2 k (n) n s = (s)(s k) 2 (s 2k) (2s 2k) : (2) We have extended this elegant pair to: Theorem 1 For completely multiplicative f 1 ; f 2 and g 1 ; g 2 , 1 X n=1 (f 1 g 1 )(n) (f 2 g 2 )(n) n s = (3) L f 1 f 2 (s)L g 1 g 2 (s)L f 1 g 2 (s)L g 1 f 2 (s) L f 1 f 2 g 1 g 2 (2s) where f g(n) := P djn f(d)g(n=d) and L f (s) := P 1 n=1 f(n)n s is a Dirichlet series. 2 Let r N (n) be the number of solutions of x 2 1 + +x 2 N = n and let r 2;P (n) be the number of solutions of x 2 + Py 2 = n. One application of Theorem 1 is to obtain closed forms, in terms of (s) and Dirichlet L- functions, for the generating functions of functions such as r N (n); r 2 N (n); r 2;P (n) and r 2;P (n) 2 for certain P and (even) N = 2; 4; 6; 8. We also use these generating functions to obtain asymptotics for the average values of each function for which we obtain a Dirichlet series. We nish by discussing the more vexing case N = 3, and related matters. This is joint work with Stephen Choi (SFU). These transparencies are at www.cecm.sfu.ca/personal/jborwein/talks.html The CECM preprint 01:167 is at: www.cecm.sfu.ca/preprints/2001pp.html 3 OUTLINE 1. Motivation and Background. 2. Theorem on L-series of Squares of Arithmetic Functions. 3. Why Squares not Cubes? 4. Applications to r 4 ; r 6 and r 8 . 5. Applications to Quadratic Forms. 6. Three, Twelve and Twenty Four Squares. 4 1. MOTIVATION and BACKGROUND Evaluating residues of the corresponding g.f.s at their largest real poles and Cauchy's integral theorem leads to X nx r 2...

‘... where almost one quarter hour was spent, each beholding the other with admiration before one word was spoken: at last Mr. Briggs began ”My Lord, I have undertaken this long journey purposely to see your person, and to know by what wit or ingenuity you first came to think of this most excellent help unto Astronomy, viz. the Logarithms: but my Lord, being by you found out, I wonder nobody else found it out before, when now being known it appears so easy.” ’ 1 The emergence of powerful mathematical computing environments, the growing availability of correspondingly powerful (multi-processor) computers and the pervasive presence of the internet allow for mathematicians, students and teachers, to proceed heuristically and ‘quasiinductively’. We may increasingly use symbolic and numeric computation, visualization tools, simulation and data mining. The unique features of our discipline make this both more problematic and more challenging. For example, there is still no truly satisfactory way of displaying mathematical notation on the web; and we care more about the reliability of our literature than does any other science. The traditional role of proof in mathematics is arguably under siege—for reasons both good and bad. 1 Henry Briggs is describing his first meeting in 1617 with Napier whom he had travelled