Date: Feb 13, 2013 12:35 PM Author: Jesse F. Hughes Subject: Re: infinity can't exist 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.

>> 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.

For what?

--

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