## WHITHER MATHEMATICS? (2004)

### BibTeX

@MISC{Davies04whithermathematics?,

author = {E. B. Davies},

title = {WHITHER MATHEMATICS?},

year = {2004}

}

### OpenURL

### Abstract

whither10.tex We describe three successive crises faced by mathematicians during the twentieth century, and their implications for the nature of mathematics. 1

### Citations

468 |
The Emperor's New Mind
- Penrose
- 1990
(Show Context)
Citation Context ...ns as amateur philosophers are no more agreed about the status of their subject than are philosophers. As representatives of many others we cite Roger Penrose as a committed realist (i.e. Platonist), =-=[20, 21]-=-, and Paul Cohen as an anti-realist, [12, 13]. Einstein was clear that mathematics was a product of human thought and that, as far as the propositions of mathematics are certain, they do not refer to ... |

407 | Foundations of constructive analysis
- Bishop
- 1967
(Show Context)
Citation Context ... not total: constructivists adopt a strict, algorithmic notion of existence that is more acceptable to applied mathematicians, numerical analysts and logicians than it is to most pure mathematicians, =-=[7, 8, 9, 15]-=-. Kurt Gödel’s astonishing insights in the 1930s created the first of the three crises to which we refer. He demonstrated that within any sufficiently rich axiomatic system there must exist certain st... |

321 | Shadows of the Mind
- Penrose
- 1994
(Show Context)
Citation Context ...ns as amateur philosophers are no more agreed about the status of their subject than are philosophers. As representatives of many others we cite Roger Penrose as a committed realist (i.e. Platonist), =-=[20, 21]-=-, and Paul Cohen as an anti-realist, [12, 13]. Einstein was clear that mathematics was a product of human thought and that, as far as the propositions of mathematics are certain, they do not refer to ... |

276 |
Every planar map is four colorable
- Appel, Haken, et al.
(Show Context)
Citation Context ...dulging in fantasies. 2 Computer-Assisted Proofs The first example of a major mathematical theorem that depended on computer assistance was the four colour theorem, proved by Appel and Haken in 1976, =-=[1, 2]-=-. It caused great uneasiness among some mathematicians for two reasons. One was that it was considered that one could not be certain that a machine had performed a calculation correctly if one could n... |

137 |
Proofs and refutations : The logic of mathematical discovery
- Lakatos
- 1976
(Show Context)
Citation Context ... pointed out, have been rectified. A famous book of Imre Lakatos is a celebration of the ability of mathematicians to respond to counterexamples to a sequence of flawed statements of Euler’s theorem, =-=[17]-=-. The most famous inconsistency was in Frege’s foundations of mathematics, to which Bertrand Russell found a paradox. Within twenty years the ZFC set theory removed these particular problems, although... |

81 | Spectra and pseudospectra
- Trefethen, Embree
- 2005
(Show Context)
Citation Context ...lf-adjoint matrices came to light as a result of numerical experiments, and has spawned the new field of pseudospectra, which is now being studied as an area of rigorous mathematics in its own right, =-=[28]-=-. Controlled numerical calculations are also playing an essential role as intrinsic parts of papers in various areas of pure mathematics. In some areas of nonlinear PDE rigorous computer-assisted proo... |

56 |
Constructive Analysis, Grundlehren der math. Wissenschaften 279
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ... not total: constructivists adopt a strict, algorithmic notion of existence that is more acceptable to applied mathematicians, numerical analysts and logicians than it is to most pure mathematicians, =-=[7, 8, 9, 15]-=-. Kurt Gödel’s astonishing insights in the 1930s created the first of the three crises to which we refer. He demonstrated that within any sufficiently rich axiomatic system there must exist certain st... |

35 | Naturalism in mathematics - Maddy - 1997 |

28 |
Mathematics as a science of patterns
- Resnik
- 1997
(Show Context)
Citation Context ...t exist. Whole books have been devoted to the discussion of the relationship between ontology and epistemology in mathematics, but it is fair to say that agreement about its solution is not imminent, =-=[5, 6, 25, 26]-=-. Mathematicians as amateur philosophers are no more agreed about the status of their subject than are philosophers. As representatives of many others we cite Roger Penrose as a committed realist (i.e... |

22 |
Platonism and Anti-Platonism in Mathematics
- Balaguer
- 1998
(Show Context)
Citation Context ...t exist. Whole books have been devoted to the discussion of the relationship between ontology and epistemology in mathematics, but it is fair to say that agreement about its solution is not imminent, =-=[5, 6, 25, 26]-=-. Mathematicians as amateur philosophers are no more agreed about the status of their subject than are philosophers. As representatives of many others we cite Roger Penrose as a committed realist (i.e... |

16 | The status of the classification of the finite simple groups
- Aschbacher
- 2004
(Show Context)
Citation Context ..., and in 1990 claims that the classification was complete had to be reconsidered. Eventually this gap was also filled by Aschbacher and Smith and, once again, it seems likely that the proof is sound, =-=[3]-=-. However only about five out of the twelve volumes of the final proof have been published, almost 25 years after the theorem was ‘proved’; see [3, 27] for details. Michael Aschbacher, one of the peop... |

10 |
On finite simple groups and their classification
- Solomon
- 1995
(Show Context)
Citation Context ...once again, it seems likely that the proof is sound, [3]. However only about five out of the twelve volumes of the final proof have been published, almost 25 years after the theorem was ‘proved’; see =-=[3, 27]-=- for details. Michael Aschbacher, one of the people most heavily involved in the project, admits the possibility that a new finite simple group might one day be discovered. If that group has character... |

9 |
Computer-assisted enclosure methods for elliptic differential equations
- Plum
- 2001
(Show Context)
Citation Context ...an essential role as intrinsic parts of papers in various areas of pure mathematics. In some areas of nonlinear PDE rigorous computer-assisted proofs of the existence of solutions have been provided; =-=[22]-=- and [23] provide typical examples. These use interval arithmetic to control the rounding errors in calculations that are conceptually quite straightforward. The key is to provide a rigorous proof of ... |

8 |
Platonism and Anti-Platonism
- Balaguer
- 1998
(Show Context)
Citation Context ...matics, but it is fair to say Brian Davies is professor of mathematics at King’s College London. His email address is E.Brian.Davies@ kcl.ac.uk. that agreement about its solution is not imminent [5], =-=[6]-=-, [25], [26]. Mathematicians as amateur philosophers are no more agreed about the status of their subject than are philosophers. As representatives of many others we cite Roger Penrose as a committed ... |

4 |
Comments on the foundations of set theory
- Cohen
- 1971
(Show Context)
Citation Context ... about the status of their subject than are philosophers. As representatives of many others we cite Roger Penrose as a committed realist (i.e., Platonist) [20], [21] and Paul Cohen as an anti-realist =-=[12]-=-, [13]. Einstein was clear that mathematics was a product of human thought and that, as far as the propositions of mathematics are certain, they do not refer to reality [16]. The author of the present... |

3 |
Highly complex proofs and implications of such proofs
- Aschbacher
(Show Context)
Citation Context ...ly similar to the others, this might not be too disturbing, but he accepts that the discovery of a new finite simple group quite different from the others would throw the problem wide open again; see =-=[4]-=-. Note that Jean-Pierre Serre is also very cautious about accepting the proof, [24]. Aschbacher has noted that the proof seems to be robust. By this he means that every gap so far discovered can be pl... |

3 |
Wieners C: New solutions of the Gelfand problem
- Plum
(Show Context)
Citation Context ...ial role as intrinsic parts of papers in various areas of pure mathematics. In some areas of nonlinear PDE rigorous computer-assisted proofs of the existence of solutions have been provided; [22] and =-=[23]-=- provide typical examples. These use interval arithmetic to control the rounding errors in calculations that are conceptually quite straightforward. The key is to provide a rigorous proof of an inequa... |

3 |
eds) The nature of mathematical proof
- Bundy, Atiyah, et al.
- 2005
(Show Context)
Citation Context ...ect than Gödel’s work ever has. In October 2004 the Royal Society held a two-day discussion meeting in London on “The Nature of Mathematical Proof” to discuss possible ways of responding to them; see =-=[10]-=-. The meeting provided a variety of insights into the issues involved but no solutions. There was evidence of a serious communication problem between the mathematicians and computer scientists present... |

2 |
Deflating Existential Consequence
- Azzouni
- 2004
(Show Context)
Citation Context ...t exist. Whole books have been devoted to the discussion of the relationship between ontology and epistemology in mathematics, but it is fair to say that agreement about its solution is not imminent, =-=[5, 6, 25, 26]-=-. Mathematicians as amateur philosophers are no more agreed about the status of their subject than are philosophers. As representatives of many others we cite Roger Penrose as a committed realist (i.e... |

2 |
Comments on the foundations of set theory. p 9-15
- Cohen
- 1971
(Show Context)
Citation Context ... about the status of their subject than are philosophers. As representatives of many others we cite Roger Penrose as a committed realist (i.e. Platonist), [20, 21], and Paul Cohen as an anti-realist, =-=[12, 13]-=-. Einstein was clear that mathematics was a product of human thought and that, as far as the propositions of mathematics are certain, they do not refer to reality, [16]. The author of the present arti... |

2 | B: A defence of pluralism in mathematics
- Davies
- 2004
(Show Context)
Citation Context ...ot refer to reality, [16]. The author of the present article has always been critical of Platonism, [14]; he now fully accepts the existence of mathematical entities, but only in the Carnapian sense, =-=[15]-=-. This allows mathematical theories to be products of the human imagination, but nevertheless to have definite properties just as chess and Roman law do; it also allows numbers to exist in the same se... |

2 |
Interview with Jean-Pierre Serre
- Raussen, Skau
- 2004
(Show Context)
Citation Context ...e discovery of a new finite simple group quite different from the others would throw the problem wide open again; see [4]. Note that Jean-Pierre Serre is also very cautious about accepting the proof, =-=[24]-=-. Aschbacher has noted that the proof seems to be robust. By this he means that every gap so far discovered can be plugged with only a moderate amount of extra work, leaving the main lines of the proo... |

1 |
Atiyah M and MacIntyre A (eds.): ‘The Nature of
- Bundy, MacKenzie
(Show Context)
Citation Context ...ect than Gödel’s work ever has. In October 2004 the Royal Society held a two day discussion meeting in London on ‘The Nature of Mathematical Proof’ to discuss possible ways of responding to them; see =-=[10]-=-. The meeting provided a variety of insights into the issues involved but no solutions. There was evidence of a serious communication problem between the mathematicians and computer scientists present... |

1 |
Skolem and pessimism about proofs
- Cohen
(Show Context)
Citation Context ... about the status of their subject than are philosophers. As representatives of many others we cite Roger Penrose as a committed realist (i.e. Platonist), [20, 21], and Paul Cohen as an anti-realist, =-=[12, 13]-=-. Einstein was clear that mathematics was a product of human thought and that, as far as the propositions of mathematics are certain, they do not refer to reality, [16]. The author of the present arti... |

1 |
B: ‘Science in the Looking Glass
- Davies
- 2003
(Show Context)
Citation Context ... product of human thought and that, as far as the propositions of mathematics are certain, they do not refer to reality, [16]. The author of the present article has always been critical of Platonism, =-=[14]-=-; he now fully accepts the existence of mathematical entities, but only in the Carnapian sense, [15]. This allows mathematical theories to be products of the human imagination, but nevertheless to hav... |

1 |
Lecture delivered to the Prusian Academy of Sciences
- Einstein
- 1921
(Show Context)
Citation Context ...l Cohen as an anti-realist, [12, 13]. Einstein was clear that mathematics was a product of human thought and that, as far as the propositions of mathematics are certain, they do not refer to reality, =-=[16]-=-. The author of the present article has always been critical of Platonism, [14]; he now fully accepts the existence of mathematical entities, but only in the Carnapian sense, [15]. This allows mathema... |

1 |
Machine computation and proof
- MacPherson
(Show Context)
Citation Context ...er-based part to a more appropriate journal for publication. One of the Annals editors, Robert MacPherson, admitted that the (unpublished) policy of the Annals editors for such papers had failed; see =-=[18]-=-. At the Royal Society meeting there were lively discussions about whether formal proofs of the correctness of programs could have made a contribution to the refereeing process. According to Macpherso... |

1 |
D: Structuralism and the independence of mathematics
- Resnik
- 2004
(Show Context)
Citation Context |

1 |
Lecture delivered to the Prussian Academy of Sciences
- EINSTEIN
- 1921
(Show Context)
Citation Context ...l Cohen as an anti-realist [12], [13]. Einstein was clear that mathematics was a product of human thought and that, as far as the propositions of mathematics are certain, they do not refer to reality =-=[16]-=-. The author of the present article has always been critical of Platonism [14]; he now fully accepts the existence of mathematical entities, but only in the Carnapian sense [15]. This allows mathemati... |