## Mathematics by Experiment: Plausible Reasoning in the 21st Century, extended second edition, A K (2008)

Venue: | 2008. EXPERIMENTATION AND COMPUTATION 19 |

Citations: | 39 - 15 self |

### BibTeX

@INPROCEEDINGS{Borwein08mathematicsby,

author = {Jonathan M. Borwein},

title = {Mathematics by Experiment: Plausible Reasoning in the 21st Century, extended second edition, A K},

booktitle = {2008. EXPERIMENTATION AND COMPUTATION 19},

year = {2008}

}

### OpenURL

### Abstract

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

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(Show Context)
Citation Context ...s valid to just 12 places. Both are actually transcendental numbers. Here [a] denotes the integer part. Correspondingly the simple continued fractions for tanh(π) and tanh( π 2 ) are respectively and =-=[0, 1, 267, 4, 14, 1, 2, 1, 2, 2, 1, 2, 3, 8, 3, 1]-=- [0, 1, 11, 14, 4, 1, 1, 1, 3, 1, 295, 4, 4, 1, 5, 17, 7] Bill Gosper describes how continued fractions let you “see” what a number is. “[I]t’s completely astounding ... it looks like you are cheating... |

17 | 1999."Emerging Tools for Experimental Mathematics
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(Show Context)
Citation Context ...s valid to just 12 places. Both are actually transcendental numbers. Here [a] denotes the integer part. Correspondingly the simple continued fractions for tanh(π) and tanh( π 2 ) are respectively and =-=[0, 1, 267, 4, 14, 1, 2, 1, 2, 2, 1, 2, 3, 8, 3, 1]-=- [0, 1, 11, 14, 4, 1, 1, 1, 3, 1, 295, 4, 4, 1, 5, 17, 7] Bill Gosper describes how continued fractions let you “see” what a number is. “[I]t’s completely astounding ... it looks like you are cheating... |

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Citation Context ...ing. Paradoxes, incompleteness, undecidability, a gulf between proof and truth. Modern sensibilities and ’chaos’. 5.1 The Main Philosophies of Rigour Such events have spawned four main responses (see =-=[7]-=-). • Everyman: Platonism—stuff exists ( named by Bernais, 1934) • Hilbert: Formalism—math is invented; formal symbolic games without meaning • Brouwer: Intuitionism-—many variants; (embodied cognition... |

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The top 10 algorithms,” Computing
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(Show Context)
Citation Context ...s valid to just 12 places. Both are actually transcendental numbers. Here [a] denotes the integer part. Correspondingly the simple continued fractions for tanh(π) and tanh( π 2 ) are respectively and =-=[0, 1, 267, 4, 14, 1, 2, 1, 2, 2, 1, 2, 3, 8, 3, 1]-=- [0, 1, 11, 14, 4, 1, 1, 1, 3, 1, 295, 4, 4, 1, 5, 17, 7] Bill Gosper describes how continued fractions let you “see” what a number is. “[I]t’s completely astounding ... it looks like you are cheating... |

5 | Algorithms for Bernoulli numbers and Euler numbers - Chen - 2001 |

4 | Making Sense of Experimental Mathematics,” Mathematical Intelligencer, 18, (Fall 1996), 12–18. ∗ [CECM 95:032] 2. Jonathan M. Borwein and Robert Corless, “Emerging Tools for Experimental Mathematics - Borwein, Borwein, et al. - 1999 |

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1 | shedding light on the irrationality of ζ(7)? Recall that ζ(2N + 1) is not proven irrational for N > 1. One of ζ(2n + 3) for n - Perhaps |

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1 | EXP(-1)*sum(k=>0,k^n/k!) - Benoit Cloitre (abcloitre(AT)wanadoo.fr), May 19 2002 a(n) is asymptotic to n!*(2 Pi r^2 exp(r))^(-1/2) exp(exp (r)-1) / r^n, where r is the positive root of r exp(r) = n. - see e - unknown authors |

1 | the Odlyzko reference. a(n) is asymptotic to b^n*exp(b-n-1/2)/sqrt(ln(n)) where b satisfies b*ln(b) = n - 1/2 (see Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed., p. 493) - Benoit Cloitre (abcloitre(AT)wanadoo.fr), Oct 23 2002 Maple: A000110:= - unknown authors |

1 |
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(Show Context)
Citation Context ...ave recently found interesting interpretations in high energy physics, knot theory, combinatorics . . . . Euler found and partially proved theorems on reducibility of depth 2 to depth 1 ζ’s. Indeed, ζ=-=(6, 2)-=- is the lowest weight ‘irreducible’. High precision fast ζ-convolution (see EZFace/Java) allows use of integer relation methods and leads to important dimensional (reducibility) conjectures and amazin... |