## Passages of proof (2008)

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Citations: | 1 - 1 self |

### BibTeX

@MISC{Calude08passagesof,

author = {Cristian S. Calude and Elena Calude and Solomon Marcus},

title = {Passages of proof},

year = {2008}

}

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### Citations

1066 | The Knowledge Complexity of Interactive Proof-systems
- Goldwasser, Micali, et al.
- 1985
(Show Context)
Citation Context ...later by Brent [11]. New types of proofs motivated by the experimental “ideology” have appeared. For example, rather than being a static object, the interactive proof (see Goldwasser, Micali, Rackoff =-=[36]-=-, Blum [10]) is a two–party protocol in which the prover tries to prove a certain fact to the verifier. During the interactive proof the prover and the verifier exchange messages and at the end the ve... |

463 |
Syntactic Structures
- Chomsky
- 1957
(Show Context)
Citation Context ... entity A has simultaneously the tendency to increase its similarity with A and stress its difference from A. To give only one example in this respect, we recall the famous result obtained by Chomsky =-=[29]-=-, in the late 1950s, stating that context–free grammars are not able to generate the English language. This result was accepted by the linguistic and computer science communities until the eighties, w... |

155 |
The Fabric of Reality: The
- Deutsch
- 1997
(Show Context)
Citation Context ...ich is through the physical world. Indeed, our glimpses of mathematics are revealed only through physical objects, human brains, silicon computers, quantum automata, etc., hence, according to Deutsch =-=[32]-=-, they have to obey not only the axioms and the inference rules of the theory, but the laws of physics as well. To complete the picture we need to take into account also the biological dimension. No m... |

135 |
Mathematical thought from Ancient to Modern Times
- Kline
- 1972
(Show Context)
Citation Context ...claim that the Euclidean geometry is a model of deductive thinking, is a big lie. As a matter of fact, the shortcomings to which Dijkstra refers were well-known, as can be seen in Morris Kline’s book =-=[43]-=-, pp. 86–88. In contemporary mathematics we are facing a change of perspective, a change of scenario, replacing the old itinerary definition–theorem–proof by another one (see, for instance, W. Thursto... |

124 | How to prove a theorem so no one else can claim it
- Blum
- 1986
(Show Context)
Citation Context ...ent [11]. New types of proofs motivated by the experimental “ideology” have appeared. For example, rather than being a static object, the interactive proof (see Goldwasser, Micali, Rackoff [36], Blum =-=[10]-=-) is a two–party protocol in which the prover tries to prove a certain fact to the verifier. During the interactive proof the prover and the verifier exchange messages and at the end the verifier prod... |

103 |
Social processes and proofs of theorems and programs
- MILLO, LIPTON, et al.
- 1979
(Show Context)
Citation Context ...A “proof is only one step in the direction of confidence" 4 Landau’s son-in-law. 174The Bulletin of the EATCS argued De Millo, Lipton and Perlis in a classical paper on proofs, theorems and programs =-=[33]-=-. Written in the same spirit is Don Knuth’s warning: “Beware of bugs in the above code: I have only proved it correct, not tried it." 5 Mathematical Proofs: The Semantic Dimension If one must choose b... |

78 |
Stability of groups of odd order
- Feit, Thompson
- 1964
(Show Context)
Citation Context ...of accuracy. Things don’t simply have to make sense to another human being, they must make sense to a computer." Knuth compares his TEX compiler (a document of about 500 pages) with Feit and Thompson =-=[35]-=- theorem that all simple groups of odd order are cyclic. He lucidly argues that the program might not incorporate as much creativity and “daring" as the proof of the theorem, but they come even when c... |

71 |
Randomness and mathematical proof
- Chaitin
- 1975
(Show Context)
Citation Context ...ll go on forever (see Barrow [6], pp. 218–221). In modern times a penetrating insight into the incompleteness phenomenon has been obtained by an information–theoretic analysis pioneered by Chaitin in =-=[24]-=-. Striking results have been obtained by studying the Chaitin’s Omega Number, Ω, the halting probability of a self-delimiting universal Turing machine. This number is not only uncomputable, but also (... |

54 |
Information and Randomness: An Algorithmic Perspective, 2nd Edition, Revised and Extended
- Calude
- 2002
(Show Context)
Citation Context ...proved by ZFC is true, then, ZFC can determine the value of only finitely many bits of Ω, and one can give a bound on the number of bits of Ω which ZFC can determine. Robert Solovay [56] (see more in =-=[17, 15, 16, 14]-=-) has constructed a self-delimiting universal Turing machine such that ZFC, if arithmetically sound, cannot determine any single bit of its halting probability (Ω). Re–phrased, the most powerful forma... |

46 | Exploring Randomness
- Chaitin
- 2001
(Show Context)
Citation Context ...to be self-contained and formal) into a holographic one requires super-linear time. The blend of logical and empirical–experimental arguments (“quasi–empirical mathematics" for Tymoczko [57], Chaitin =-=[25, 26, 28]-=- or “experimental mathematics" for Bailey, Borwein [5], Borwein, Bailey [7], Borwein, Bailey, Girgensohn [8]) may lead to a new way to understand (and practice) mathematics. For example, Chaitin argue... |

43 | The Unknowable - Chaitin - 1999 |

26 | How to believe a machine-checked proof
- Pollack
- 1998
(Show Context)
Citation Context ...thematics fallibility was argued by Lakatos), iii) non–deterministic and probabilistic proofs do not allow mistakes in the applications of rules, they are just indirect forms of checking (see Pollack =-=[50]-=-, p. 210) which correspond to various degrees of rigour, iv) the explanatory component, the understanding ‘emerging’ from proofs, while extremely important from a cognitive point of view, is subjectiv... |

26 | A new proof of the four colour theorem
- Robertson, Sanders, et al.
- 1996
(Show Context)
Citation Context ...ire proof to check its correctness. For Ron Graham, “The real question is this: If no human being can ever hope to check a proof, is it really a proof?" In 1996 Robertson, Sanders, Seymour and Thomas =-=[52]-=- offered a simpler proof involving only 633 configurations. The paper [52] concludes with the following interesting comment (p. 24): “We should mention that both our programs use only integer arithmet... |

24 | Impossibility: The Limits of Science and The Science of Limits - Barrow - 1999 |

22 | Experimental mathematics: recent developments and future outlook
- Bailey, Borwein
- 2001
(Show Context)
Citation Context ...s super-linear time. The blend of logical and empirical–experimental arguments (“quasi–empirical mathematics" for Tymoczko [57], Chaitin [25, 26, 28] or “experimental mathematics" for Bailey, Borwein =-=[5]-=-, Borwein, Bailey [7], Borwein, Bailey, Girgensohn [8]) may lead to a new way to understand (and practice) mathematics. For example, Chaitin argued that we should introduce the Riemann hypothesis as a... |

18 | Computing a glimpse of randomness
- Calude, Dinneen
(Show Context)
Citation Context ...aticians. The “theorem provers" have been very successful as “helpers" in proving many results, from simple theorems of Euclidean geometry to the computation of a few digits of a Chaitin Omega Number =-=[18]-=-. “Artificial" mathematicians are far less ingenious and subtle than human mathematicians, but they surpass their human counterparts by being infinitely more patient and diligent. If a conventional pr... |

18 | Is independence an exception
- Calude, Jürgensen, et al.
- 1994
(Show Context)
Citation Context ...itions in GIT, then the set of true and unprovable statements is topologically “large" (constructively, a set of second Baire category, and in some cases even “larger"), cf. Calude, Jürgensen, Zimand =-=[20]-=-; because theorems proven in such a system have bounded complexity, the probability that an n-bit statement is provable tends to zero when n tends to infinity (see Calude and Jürgensen [19]). There is... |

18 |
Computing with Cells and Atoms
- Calude, Păun
- 2001
(Show Context)
Citation Context ...es may be impossible, e.g. due to the size of the computation. An extreme, and for the time being hypothetical example, is the proof obtained as a result of a quantum computation (see Calude and Păun =-=[22]-=-). The quantum automaton would say “your conjecture is true", but (due to quantum interference) there will be no way to exhibit all trajectories followed by the quantum automaton in reaching that conc... |

18 | The status of the classification of the finite simple groups
- Aschbacher
- 2004
(Show Context)
Citation Context ...ar as the propositions of mathematics 5 Still, there is a controversy in the mathematical community on whether these articles provide a complete and correct proof. For a recent account see Aschbacher =-=[1]-=-. 176The Bulletin of the EATCS are certain, they do not refer to reality, and in so far as they refer to reality, they are not certain", hence (certainty, reality) is a conjugate pair. Obviously, rea... |

18 |
A version of Ω for which ZFC can not predict a single bit
- Solovay
- 2000
(Show Context)
Citation Context ...rem of arithmetic proved by ZFC is true, then, ZFC can determine the value of only finitely many bits of Ω, and one can give a bound on the number of bits of Ω which ZFC can determine. Robert Solovay =-=[56]-=- (see more in [17, 15, 16, 14]) has constructed a self-delimiting universal Turing machine such that ZFC, if arithmetically sound, cannot determine any single bit of its halting probability (Ω). Re–ph... |

14 | Chaitin Ω numbers, Solovay machines and incompleteness, Theoret
- Calude
(Show Context)
Citation Context ...proved by ZFC is true, then, ZFC can determine the value of only finitely many bits of Ω, and one can give a bound on the number of bits of Ω which ZFC can determine. Robert Solovay [56] (see more in =-=[17, 15, 16, 14]-=-) has constructed a self-delimiting universal Turing machine such that ZFC, if arithmetically sound, cannot determine any single bit of its halting probability (Ω). Re–phrased, the most powerful forma... |

13 |
Randomness everywhere
- Calude, Chaitin
- 1999
(Show Context)
Citation Context ...proved by ZFC is true, then, ZFC can determine the value of only finitely many bits of Ω, and one can give a bound on the number of bits of Ω which ZFC can determine. Robert Solovay [56] (see more in =-=[17, 15, 16, 14]-=-) has constructed a self-delimiting universal Turing machine such that ZFC, if arithmetically sound, cannot determine any single bit of its halting probability (Ω). Re–phrased, the most powerful forma... |

13 |
The Four-Color Problem and Its Philosophical Significance”, The
- Tymoczko
- 1979
(Show Context)
Citation Context ...eorem is validated if it has been accepted by a general agreement 6 of the mathematical community (see [54, 55]). The problems raised by the 4CT were discussed by many authors, starting with Tymoczko =-=[57]-=- and Swart [59] (more recent publications are D. MacKenzie [45], J. Casti [23], A.S. Calude [13]). Swart proposed the introduction of a new entity called agnogram, which is “a theorem–like statement t... |

12 |
What is Mathematics
- Hersh
- 1998
(Show Context)
Citation Context ...rd Riemann whose lasting fame does not come (in the first instance) from his theorems or proofs, but from his conjectures, definitions, concepts and examples (see for example, the discussion in Hersh =-=[42]-=-, pp. 50–51). Srinivasa Ramanujan is another famous example of a mathematician who produced more results than proofs. What the mathematical community seems to value most are “ideas". “The most respect... |

11 |
Theory and practice
- Knuth
- 1985
(Show Context)
Citation Context ...t reasoning to ancient Greeks can be illustrated by citing Aristophanes’s comedy The Birds which includes a cameo appearance of Meton, the astronomer, who claims that he had squared the circle. Knuth =-=[44]-=- rhetorically asked: “Where else on earth would a playwright think of including such a scene?" Examples would have been difficult to produce in 1985, but today the situation has changed. Take for exam... |

9 | J.,(2005),The experimental mathematician: the pleasure of discovery and the role of proof - Borwein |

9 | A version of Ω for which ZF C can not predict a single bit - Solovay - 2000 |

8 | Mathematics by
- Borwein, Bailey
- 2008
(Show Context)
Citation Context ...s", “proof animation" or “proof engineering" are just a few examples (see [40]). In some cases an experiment conveys an aesthetic appreciation of mathematics appealing to a much broader audience (cf. =-=[7, 8, 21]-=-). A significant, but simple example of the role an experiment may play in a proof is given by Beyer [9]. He refers to J. North who asked for a computer demonstration that the harmonic series diverges... |

7 | The journey of the four colour theorem through time
- Calude
(Show Context)
Citation Context ...y (see [54, 55]). The problems raised by the 4CT were discussed by many authors, starting with Tymoczko [57] and Swart [59] (more recent publications are D. MacKenzie [45], J. Casti [23], A.S. Calude =-=[13]-=-). Swart proposed the introduction of a new entity called agnogram, which is “a theorem–like statement that we have verified as best we could, but whose truth is not known with the kind of assurance w... |

7 |
Slaying the kraken. The sociohistory of a mathematical proof
- MacKenzie
- 1999
(Show Context)
Citation Context ...nt 6 of the mathematical community (see [54, 55]). The problems raised by the 4CT were discussed by many authors, starting with Tymoczko [57] and Swart [59] (more recent publications are D. MacKenzie =-=[45]-=-, J. Casti [23], A.S. Calude [13]). Swart proposed the introduction of a new entity called agnogram, which is “a theorem–like statement that we have verified as best we could, but whose truth is not k... |

7 | Mathematical proofs at a crossroad
- Calude, Marcus
- 2004
(Show Context)
Citation Context ...s", “proof animation" or “proof engineering" are just a few examples (see [40]). In some cases an experiment conveys an aesthetic appreciation of mathematics appealing to a much broader audience (cf. =-=[7, 8, 21]-=-). A significant, but simple example of the role an experiment may play in a proof is given by Beyer [9]. He refers to J. North who asked for a computer demonstration that the harmonic series diverges... |

6 | Every Planar Graph is Four Colorable - Appel, Haken - 1989 |

6 |
Computation of the regular continued fraction for Euler’s constant
- Brent
- 1977
(Show Context)
Citation Context ...." It is interesting to mention that in 1973 Beyer made together with Mike Waterman a similar attempt to compute Euler’s constant; their experiment failed, but the error was discovered later by Brent =-=[11]-=-. New types of proofs motivated by the experimental “ideology” have appeared. For example, rather than being a static object, the interactive proof (see Goldwasser, Micali, Rackoff [36], Blum [10]) is... |

6 | On the intelligibility of the universe and the notions of simplicity, complexity and irreducibility, http://www.cs.auckland.ac.nz/CDMTCS/ chaitin/bonn.html
- Chaitin
- 2002
(Show Context)
Citation Context ...to be self-contained and formal) into a holographic one requires super-linear time. The blend of logical and empirical–experimental arguments (“quasi–empirical mathematics" for Tymoczko [57], Chaitin =-=[25, 26, 28]-=- or “experimental mathematics" for Bailey, Borwein [5], Borwein, Bailey [7], Borwein, Bailey, Girgensohn [8]) may lead to a new way to understand (and practice) mathematics. For example, Chaitin argue... |

6 |
Modern mathematics: Does it exist
- Thom
- 1973
(Show Context)
Citation Context ...em remains open. A convenient, but fragile, solution is to accept Thom’s pragmatic proposal: a theorem is validated if it has been accepted by a general agreement 6 of the mathematical community (see =-=[54, 55]-=-). The problems raised by the 4CT were discussed by many authors, starting with Tymoczko [57] and Swart [59] (more recent publications are D. MacKenzie [45], J. Casti [23], A.S. Calude [13]). Swart pr... |

5 |
Constructive truth in practice
- Bridges
- 1998
(Show Context)
Citation Context ...al models (of proof) can play an important role in the practice of mathematics. For example, the key feature of constructive mathematics is the identification “existence = computability" (cf. Bridges =-=[12]-=-) and a whole variety of constructive mathematics, the so–called Bishop constructive mathematics, is mathematics with intuitionistic rather than classical underlying logic. 180The Bulletin of the EAT... |

5 |
complexity, randomness and beyond, Minds and Machines
- Incompleteness
(Show Context)
Citation Context |

5 | Computers, paradoxes and the foundations of mathematics
- Chaitin
- 2002
(Show Context)
Citation Context ... Calude and Jürgensen [19]. Hence, incompleteness is natural and inevitable rather then mys169BEATCS no 84 TECHNICAL CONTRIBUTIONS terious and esoteric. This raises the natural question (see Chaitin =-=[26]-=-): How come that in spite of incompleteness, mathematicians are making so much progress? The seventh period belongs to the second half of the 20th century, when algorithmic proofs become acceptable on... |

5 |
Topologie et linguistique
- Thom
- 1970
(Show Context)
Citation Context ...em remains open. A convenient, but fragile, solution is to accept Thom’s pragmatic proposal: a theorem is validated if it has been accepted by a general agreement 6 of the mathematical community (see =-=[54, 55]-=-). The problems raised by the 4CT were discussed by many authors, starting with Tymoczko [57] and Swart [59] (more recent publications are D. MacKenzie [45], J. Casti [23], A.S. Calude [13]). Swart pr... |

4 |
Personal communication to
- Chaitin
- 2002
(Show Context)
Citation Context ...e between rigour and meaning, I shall unhesitatingly choose the latter. R. Thom The above quotation turned slogan as “more rigour, less meaning", or better still, “less rigour, more meaning" (Chaitin =-=[27]-=-) points out the necessity to distinguish between the syntactic and the semantic aspects of proofs. Should proofs belong exclusively to logic, according to the tradition started by Greeks such as Pyth... |

4 |
Pour L’honneur de l’Esprit Humain
- Dieudonné
- 1986
(Show Context)
Citation Context ...icks are for the construction of a building. A building is an articulation of bricks and, analogically, a mathematical work is an articulation of theorems. Motivated by a similar view, Jean Dieudonné =-=[34]-=- defines a mathematician as a person who has proved at least one theorem. In contrast, Arnold’s analogies point out the fact that mathematics is much more than a chain of theorems and proofs, so impli... |

4 |
Towards the animation of proofs–testing proofs by examples, Theoret
- Hayashi, Sumitomo, et al.
(Show Context)
Citation Context ...one can visit, for example, the Mathematica website http://www.wolfram.com. New other systems are produced; “proofs as programs", “proof animation" or “proof engineering" are just a few examples (see =-=[40]-=-). In some cases an experiment conveys an aesthetic appreciation of mathematics appealing to a much broader audience (cf. [7, 8, 21]). A significant, but simple example of the role an experiment may p... |

3 |
A Play, Faber and
- Proof
- 2001
(Show Context)
Citation Context ...ould have been difficult to produce in 1985, but today the situation has changed. Take for example, the movie Pi written and directed by Darren Aronofsky Starring Sean Gullette or Auburn’s play Proof =-=[2]-=- originally produced by the Manhattan Theatre Club on 23rd May 2000. In a more careful description, we observe that deductive mathematics starts with Thales and Pythagoras, while the axiomatic approac... |

3 |
Triangle of Thoughts
- Connes, Linchnerowicz, et al.
- 2001
(Show Context)
Citation Context ...metical statement"? It is a statement about non-negative integers which cannot be invalidated by finding any combination of non-negative integers that contradicts it. In Alain Connes terminology (see =-=[30]-=-, p. 6), a true arithmetical statement is a “primordial mathematical reality". Condition (2) says that the formal system has all the symbols and axioms used in arithmetic, the symbols for 0 (zero), S ... |

3 |
No system can be improved in all respects
- Marcus
- 1998
(Show Context)
Citation Context ...hematical proof to have the status of what was called in physics a “conjugate (complimentary) pair", i.e., a couple of requirements, each of them being satisfied only at the expense of the other (see =-=[47]-=-). Famous prototypes of conjugate pairs are (position, momentum) discovered by W. Heisenberg in quantum mechanics and (consistency, completeness) discovered by K. Gödel in logic. But similar warnings ... |

3 |
Dialogues on Mathematics
- Rényi
- 1967
(Show Context)
Citation Context ...jugate pair: rigour and meaning. In last two situations there is room for manipulation and compromise. Near to the above conjugate pairs is a third one: (rigour, reality), attributed to Socrates (see =-=[51]-=-). A price we have to pay in order to reach rigour is the replacement of the real world by a fictional one. There is no point and no line in the real world, if we take them according to their definiti... |

3 |
The philosophical implications of the four–colour problem
- Swart
- 1980
(Show Context)
Citation Context ...ted if it has been accepted by a general agreement 6 of the mathematical community (see [54, 55]). The problems raised by the 4CT were discussed by many authors, starting with Tymoczko [57] and Swart =-=[59]-=- (more recent publications are D. MacKenzie [45], J. Casti [23], A.S. Calude [13]). Swart proposed the introduction of a new entity called agnogram, which is “a theorem–like statement that we have ver... |

3 |
Probably true theorems, cry wolf
- Babai
- 1994
(Show Context)
Citation Context ...ng the interactive proof the prover and the verifier exchange messages and at the end the verifier produces a verdict “accept" or “reject". A holographic (or probabilistic checkable) proof (see Babai =-=[4]-=-) is still a static object but it is verified probabilistically. Errors become almost instantly apparent after a small part of the proof was checked. 10 The transformation of a classical proof (which ... |

3 |
The Limits of Science and the Science of Limits
- Impossibility
- 1998
(Show Context)
Citation Context ...comes closer to the picture described by GIT. 173BEATCS no 84 TECHNICAL CONTRIBUTIONS ematics, but interpret this in an optimistic way, as a guarantee that mathematics will go on forever (see Barrow =-=[6]-=-, pp. 218–221). In modern times a penetrating insight into the incompleteness phenomenon has been obtained by an information–theoretic analysis pioneered by Chaitin in [24]. Striking results have been... |

2 |
The computer and mathematics
- Beyer
(Show Context)
Citation Context ...nveys an aesthetic appreciation of mathematics appealing to a much broader audience (cf. [7, 8, 21]). A significant, but simple example of the role an experiment may play in a proof is given by Beyer =-=[9]-=-. He refers to J. North who asked for a computer demonstration that the harmonic series diverges. We quote Beyer: “His example illustrates the following principle: Suppose that one has a computer algo... |