display all the ideas for this combination of texts
5 ideas
| 12211 | It is plausible that x^2 = -1 had no solutions before complex numbers were 'introduced' [Fine,K] |
| Full Idea: It is not implausible that before the 'introduction' of complex numbers, it would have been incorrect for mathematicians to claim that there was a solution to the equation 'x^2 = -1' under a completely unrestricted understanding of 'there are'. | |
| From: Kit Fine (The Question of Ontology [2009]) | |
| A reaction: I have adopted this as the crucial test question for anyone's attitude to platonism in mathematics. I take it as obvious that complex numbers were simply invented so that such equations could be dealt with. They weren't 'discovered'! |
| 9910 | Number-as-objects works wholesale, but fails utterly object by object [Benacerraf] |
| Full Idea: The identification of numbers with objects works wholesale but fails utterly object by object. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], IIIC) | |
| A reaction: This seems to be a glaring problem for platonists. You can stare at 1728 till you are blue in the face, but it only begins to have any properties at all once you examine its place in the system. This is unusual behaviour for an object. |
| 12209 | The indispensability argument shows that nature is non-numerical, not the denial of numbers [Fine,K] |
| Full Idea: Arguments such as the dispensability argument are attempting to show something about the essentially non-numerical character of physical reality, rather than something about the nature or non-existence of the numbers themselves. | |
| From: Kit Fine (The Question of Ontology [2009], p.160) | |
| A reaction: This is aimed at Hartry Field. If Quine was right, and we only believe in numbers because of our science, and then Field shows our science doesn't need it, then Fine would be wrong. Quine must be wrong, as well as Field. |
| 9903 | Number words are not predicates, as they function very differently from adjectives [Benacerraf] |
| Full Idea: The unpredicative nature of number words can be seen by noting how different they are from, say, ordinary adjectives, which do function as predicates. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: He points out that 'x is seventeen' is a rare construction in English, unlike 'x is happy/green/interesting', and that numbers outrank all other adjectives (having to appear first in any string of them). |
| 9904 | The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf] |
| Full Idea: In no consistent theory is there a class of all classes with seventeen members. The existence of the paradoxes is a good reason to deny to 'seventeen' this univocal role of designating the class of all classes with seventeen members. | |
| From: Paul Benacerraf (What Numbers Could Not Be [1965], II) | |
| A reaction: This was Frege's disaster, and seems to block any attempt to achieve logicism by translating numbers into sets. It now seems unclear whether set theory is logic, or mathematics, or sui generis. |