## The Mathematician as a Formalist (1998)

Venue: | in Truth in Mathematics (H.G. Dales and |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Dales98themathematician,

author = {H. G. Dales},

title = {The Mathematician as a Formalist},

booktitle = {in Truth in Mathematics (H.G. Dales and},

year = {1998},

publisher = {Press}

}

### OpenURL

### Abstract

Introduction The existence of this meeting bears testimony to the anodyne remark that there is a continuing debate about what it means to say of a statement in mathematics that it is `true'. This debate began at least 2500 years ago, and will presumably continue at least well into the next millennium; it would be implausible and perhaps presumptuous to suppose that even the union of the talented and distinguished speakers that have been assembled here in Mussomeli will approach any solution to the problem, or even arrive at a consensus of what a solution would amount to. In the end, it falls to the philosophers, with their professional expertise and training, to carry forward the debate and to move us to a fuller understanding of this subtle and elusive matter. Indeed, we are hearing at this meeting a variety of contributions to the debate from different philosophical points of view; also, there is a good number of recent published contributions to the debate (see (Maddy 1990)

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Citation Context ...pect to two norms k \Delta k 1 and k \Delta k 2 . Then the identity map from (A; k \Delta k 1 ) onto (A; k \Delta k 2 ) is automatically continuous. This is Johnson's uniqueness-of-norm theorem; see (=-=Bonsall and Duncan 1973-=-, 25.9). 9. Mathematicians are confident that, if an inconsistency were to emerge in, say, the axioms of ZFC, then a modest modification of the axioms would lead to a similar system without the incons... |

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Citation Context ...we are hearing at this meeting a variety of contributions to the debate from different philosophical points of view; also, there is a good number of recent published contributions to the debate (see (=-=Maddy 1990-=-), for example). What then is the role of the mathematician in this debate? Some mathematicians take the view that, since they are doing mathematics, they certainly know what they are about---that `tr... |

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Citation Context ...sible'. 11. The seminal role of group theory in the great physical theories of the XX th century, including relativity theory and quantum theory, is wellknown; see, for example, the massive treatise (=-=Cornwell 1984-=-). 12. A `quantum group' is not a type of group, but an analogue of a group, and so it is probably misnamed. The group algebra of a finite group and the enveloping algebra of a finite-dimensional Lie ... |

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Citation Context ...ent about real sets. Maybe I will have been enlightened by the end of this conference! The extreme case is to convince me why various (very) large cardinals do or do not exist. It has been suggested (=-=Godel 1947-=-) that we shall resolve the size of the continuum because in time our understanding of sets will evolve to such an extent that eventually an `obviously true' axiom about sets that resolves CH will be ... |

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Citation Context ...however there can be no such general formula for the roots of quintic polynomials. `Galois theory' makes this statement precise, and explains exactly why this is the case. For a popular account, see (=-=Dieudonn'e 1992-=-). 7. Formal definitions that generalize this idea arise in the branch of mathematics called category theory ; see (Mac Lane 1971), for example. 8. Let A be an algebra which is semisimple, and suppose... |

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Citation Context ...ematicians; in its present form, it could take 1000 pages for a full account, proving all the necessary intermediate results. 17. This example is taken from (Dales and Woodin 1996, p. viii). 18. See (=-=Dales 1979-=-) and (Esterle 1978). 19. The argument is the following. Consider the statement (NDH): for each compact space\Omega\Gamma each norm k \Delta k such that (C(\Omega ; k \Delta k) is a normed algebra is ... |

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Citation Context ...41). 15. For an exposition of the theory of `super-real fields', which are the natural generalization of the above concept of the real line R as a complete ordered field, to `bigger' real lines, see (=-=Dales and Woodin 1996-=-). 16. See (Solomon 1995) for a non-technical account. The proof of the classification theory was the work of very many mathematicians; in its present form, it could take 1000 pages for a full account... |

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Citation Context ...s present form, it could take 1000 pages for a full account, proving all the necessary intermediate results. 17. This example is taken from (Dales and Woodin 1996, p. viii). 18. See (Dales 1979) and (=-=Esterle 1978-=-). 19. The argument is the following. Consider the statement (NDH): for each compact space\Omega\Gamma each norm k \Delta k such that (C(\Omega ; k \Delta k) is a normed algebra is equivalent to the u... |

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Citation Context ...ir tenets. For example, finitism has a certain appeal, but this point of view seems to discard much of modern mathematics. The case for constructivism is cogently presented by Bridges in this volume (=-=Bridges 1998-=-); clearly there is much beautiful mathematics here that has a wide appeal---for example, the constructive version of Picard's theorem, described by Bridges, has been much appreciated---but it seems t... |

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Citation Context ...is totally indifferent, or even antagonistic, to the existence of such philosophical musings. The extreme case is that of applied mathematicians and physicists, who, as Effros remarks in his lecture (=-=Effros 1998-=-), whilst valuing our language, often have little patience even for our insistence on rigour in proofs, and so these people are scarcely going to concern themselves with the difference between formali... |

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Citation Context ...at must eventually be produced if the insight is to find its place in the corpus of accepted mathematics, and not just be a private revelation. (This view contrasts with that of Jones in this volume (=-=Jones 1998-=-); it may very well be more applicable in areas of abstract analysis and algebra, which are my natural home, than in such geometric subjects as knot theory.) I think that the success of the major math... |

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Citation Context ...h the working assumptions of modern mathematicians, elucidating the `best practices ', even if only to criticize the inadequacies of this Weltanschauung. (The role of `practitioners' is discussed in (=-=Maddy 1998-=-c).) There are two different aspects of modern mathematics that philosophers must take particular account of. The first of these is the collection of specific theorems that mathematicians have proved ... |