Date: Feb 13, 2013 12:56 AM
Author: fom
Subject: Re: infinity can't exist

On 2/12/2013 3:46 PM, Jesse F. Hughes wrote:
> fom <> 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

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

"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