Let L be the following property of a positive integer:
- being large or being greater than the number of carbon atoms in a water molecule.
Necessarily, every positive integer has L. But notice that 1 has L a posteriori, while 10100 has L a priori. For 1 is not large, and it has L because it is greater than the number of carbon atoms in a water molecule, but the latter is an a posteriori fact. On the other hand, it’s a priori that 10100 is large.
If n is a positive number such that it is vague whether it is large (e.g., maybe n = 50), it will be vague whether the fact that n has L is a priori or a posteriori. For the largeness of a number is, I assume, an a priori matter, and so it will be vaguely true that it is a priori true that n is large, and hence that n has L.
4 comments:
I am not sure that 10^100 is large, a priori. There needs to first be a definition of large, doesn't there? For any positive integers, there are infinitely many larger positive integers, so we might, in fact, say that all positive integers are small. As well, 10^100 may seem large to you because you rarely deal with integers of this size, while it may seem not large to me since I deal with much larger numbers all the time (I am a mathematician); so, again, what "large" means is not clear in your statements, so it does not seem to me that the largeness of a number is an a priori matter (unless a definition of large is provided?). On the other hand, I'm not a philosopher...Cheers!
I think it's a question of usage. In ordinary English usage, 10^100 is a large number. In fact, I was being very conservative. Wikipedia says "Some names of large numbers, such as million, billion, and trillion, have real referents in human experience, and are encountered in many contexts."
Ah, I see. I was thinking that a philosopher would be more precise than to rely on mere usage when making these kinds of points. Cheers!
Well, if one were more precise, the argument wouldn't work, since the argument depends precisely on the lack of precision in "large".
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