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

fom <> writes:

> On 2/13/2013 6:43 AM, Jesse F. Hughes wrote:
>> fom <> 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