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Burgess’s “scientific” arguments for the existence of mathematical objects| old_uid | 573 |
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| title | Burgess’s “scientific” arguments for the existence of mathematical objects |
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| start_date | 2006/01/30 |
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| schedule | 17h30-19h30 |
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| online | no |
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| location_info | Grande salle |
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| summary | This paper addresses John Burgess’s answer to the “Benacerraf Problem” : How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers ? Burgess’s response is summed up in the motto : “Don’t think, look !” In particular, look at how mathematicians come to accept :
(1) There are prime numbers greater than 10 to the tenth.
That, according to Burgess, is how we come justifiably to believe something implying that there are numbers.
What underlies this argument can be seen more clearly by studying another recent work of Burgess’s (co-written with Gideon Rosen), viz. “Nominalism Reconsidered”. That work presents an argument with three premises : (1) Mathematics abound in theorems that assert the existence of mathematical objets, (2) Mathematicians accept these existence theorems and rely on them in both theoretical and practical contexts, (3) These theorems are proved in an acceptable way. From these premises, the authors conclude that we are justified in believing (to some high degree) in prime numbers greater than a thousand… which is to say that we are justified in disbelieving (to the same high degree) nominalism.
My paper is a rebuttal of the Burgess arguments. It notes that the implications of the above arguments for the concept of evidence and proof are extremely counter-intuitive. It also brings out crucial ambiguities in the premises of the argument that, when seen, expose flaws in reasoning. I also draw upon material in my book A Structural Account of Mathematics to counter specific premises of the argument. |
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| responsibles | Pataut, Dubucs, van Atten |
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