On 2/12/2013 3:46 PM, Jesse F. Hughes wrote: > fom <fomJUNK@nyms.net> writes: > >> 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. But, definitions rely on the notion >> that some word is "the same" as the object toward >> which its language act of referring is directed. > > What part of Aristotle do you have in mind? > > I'm not challenging you. I know only a smidgen of his writings. But > I'd appreciate a pointer to the chapter where he discusses this. > > Thanks. >
Just keep in mind, there are three books in Aristotle that can significantly inform on the modern paradigms. But, Aristotle addresses things differently.
For Aristotle, demonstrative science and dialectical argumentation are an immediate demarcation point. A mathematician should keep to "Posterior Analytics" as much as possible. It is in "Posterior Analytics" that Aristotle justifies a deductive calculus on epistemic grounds. This is the book on demonstrative science.
Unfortunately, there are some things in "Posterior Analytics" that require some more elaboration. Those remarks above concerning definition loosely paraphrase remarks from "Topics". That book is the one which justifies a deductive calculus on dialectical grounds.
The intersection of those analyses is the exposition of a deductive calculus as a matter of rules. This is in "Prior Analytics".
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.
"A distinctive property is one that does not reveal what the subject is, though it belongs only to that subject and is reciprocally predicated of it. [...]"
I threw that last part in for fun. Which is the "essence" and which is the "distinctive property"?
1. A set is determined by its elements.
2. A set is a collection taken as an object.
You can find a nice discussion of identity that actually respects the distinctions between ontological, semantical, and epistemological roles (the role involving definitions is epistemic) in "Understanding Identity Statements" by Thomas V. Morris, ISBN:0-08-030389-7. Frege also has a nice discussion that untangles issues in his "Comments on Sense and Reference".
But, the received paradigm for mathematics grounded on set theory is ontological rather than semantical (forget about epistemic roles). Although Zermelo's 1908 paper began by treating identity as a relation between denotations (semantical) any modern presentation speaks of identity between objects and the general arguments surrounding these matters speak of objects as "self-identical."