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?
> 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?
> 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.
> > 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.