> On 2/13/2013 6:43 AM, Jesse F. Hughes wrote: >> fom <fomJUNK@nyms.net> writes: >> >>> From "Topics" >>> ------------ >>> >>> " We must say, then, what a definition, a distinctive >>> property, a genus, and a coincident are. >>> >>> "A definition is an account that signifies the essence. >>> One provides either an account to replace a name or an >>> account to replace an account -- for it is also possible >>> to define some of the things signified by an account. >>> Those who merely provide a name, whatever it is, clearly >>> do not provide the definition of the thing, since every >>> definition is an account. Still, this sort of thing -- >>> for example, 'the fine is the fitting' [editor note: Aristotle's >>> reservation here is that replacing one word with another >>> is not really an account] -- should also be counted as >>> definitory. In the same way one should count as definitory >>> a question such as 'Are perception and knowledge the same >>> or different?'; for most of the discussion about definition >>> is occupied with whether things are the same or different. >>> Speaking without qualification, we may count as definitory >>> everything that falls under the same line of inquiry that >>> includes definition. >>> >>> "It is clear immediately that all things just mentioned >>> meet this condition. For if we are able to argue dialectically >>> that things are the same and that they are different, we >>> will in the same way be well supplied to take on >>> definitions; for once we have shown that two things >>> are not the same, we will have undermined the attempted >>> definition. The converse of this point, however, does >>> not hold; for showing that two things are the same is >>> not enough to establish a definition, whereas showing >>> that two things are not the same is enough to destroy a >>> definition." >> >> Thanks for the reference, but I don't think it supports what you said, >> namely >> >> In Aristotle, one finds the discussion that one can never define >> x=y. To be precise, he says that one can never prove a >> definition, but one can destroy a definition. >> >> He does not say "one can never define x=y", since definition of >> equality is not discussed here, > > How is this > > "for most of the discussion about definition > is occupied with whether things are the same or different." > > talking about identity?
Yes, he is talking about identity there, but he is not talking about whether one can define identity.
To determine whether a definition is "correct", it is necessary, but not sufficient, to show that the extension of the two terms is equal.
>> nor do I think this follows from >> anything he does explicitly say. And he also does not say "one can >> never prove a definition", but rather simply that showing two things >> are the same does not establish a definition. > > How do you interpret > > "showing that two things are the same" > > in the context of a mathematical discussion?
I don't understand your question. I suppose that you mean "showing that two things are the same" means showing that they are equal, but I can't guess how this is intended to be relevant to my comments.
>> This means that one >> approach of establishing a definition does not work, but as far as I >> can tell, Aristotle *does* think one can establish definitions as >> correct, or else there would be little point (from his perspective) in >> discussing them. > > He does. It is a relationship between "essence" and > "substance". It has profoundly influenced the nature of > modern mathematics.
Then I'm sure I've no idea why you said, "he says that one can never prove a definition." (I'll let your comment about the nature of modern mathematics go, since I don't know what you mean nor do I think it's relevant for our narrow discussion.)
>> >> Since pretty much every mathematical definition is stipulative, in any >> case (and at least from my perspective), this discussion of >> "establishing" a definition seems a bit off the mark, especially when >> applied to mathematics. >> > > Take a look at "On Constrained Denotation" by Abraham Robinson.
-- Jesse F. Hughes
"This post marks the end of an era in the world of mathematics." -- James S. Harris and the demise of Galois theory