On 2/13/2013 11:35 AM, Jesse F. Hughes wrote: > fom <fomJUNK@nyms.net> writes: > >> 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.
I see your point here. My language was somewhat inexact -- certainly enough to deserve correction.
> >>> 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. >
Well, it seems that you are focusing on my use of "prove". I am merely observing that showing and proving are very similar in mathematical contexts.
Technically, I am certainly wrong. The more formally one wishes to pursue the meaning of "to prove" the more distant the similarity.
>>> 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."
Aristotle also says that one cannot know first principles. So, his remarks above are not substantiated by any example. It might more accurately portray his position as thinking "one can establish definitions as correct, in principle."