Date: Feb 13, 2013 10:18 AM
Author: fom
Subject: Re: infinity can't exist

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?

> 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.