fom
Posts:
1,675
Registered:
12/4/12
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Re: infinity can't exist
Posted:
Feb 14, 2013 2:46 PM
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On 2/13/2013 3:19 PM, Jesse F. Hughes wrote:
> fom <fomJUNK@nyms.net> writes:
>
>> On 2/13/2013 11:35 AM, Jesse F. Hughes wrote:
>>> 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.
>>
>> I see your point here. My language was somewhat
>> inexact -- certainly enough to deserve correction.
>>
>>
>>>
>>>>> 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.
>>>
>>
>> Well, it seems that you are focusing on my use
>> of "prove". I am merely observing that showing and
>> proving are very similar in mathematical contexts.
>
> No, that's fine. I read "show" as more or less synonymous with
> "prove", but my comment still stands. I think that since you agreed
> with my comment above ("To determine whether a definition is
> "correct", it is necessary, but not sufficient, to show that the
> extension of the two terms is equal.") that we presumably have no
> disagreement here. That's all I was saying.
>
>> Technically, I am certainly wrong. The more formally
>> one wishes to pursue the meaning of "to prove" the
>> more distant the similarity.
>>
>>
>>>>> 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."
>>
>> Aristotle also says that one cannot know first
>> principles. So, his remarks above are not
>> substantiated by any example. It might more
>> accurately portray his position as thinking
>> "one can establish definitions as correct, in
>> principle."
>>
>> But, I admit to my inaccuracies.
>>
> Eh, it's Usenet. Maybe I shouldn't be so pedantic, but I thought that
> your main comments were pretty far from my reading of A.
>
> Not that I had read that passage before yesterday.
>
I am grateful. Until your remarks, I had not
really understood the sense of "stipulative"
definition in the way that I needed.
Here is where my complexity when interpreting
Aristotle's remarks arise:
Something like
x=y
is called informative identity.
But, within a formal proof, one can never
write that as an assumption. In a formal
proof, one begins with true sentences and
ends with true sentences. So all the
metalogical analysis of meaning outside
of a deduction is "armchair quarterbacking"
whereas what happens in a proof is
"regulation time".
To use an informative identity at the
beginning of a formal proof entails
a quantified formula. That makes it a
description.
Strict transitive relation
AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))
Definition of a bottom
Ax(x=null() <-> Ay(-(xcy <-> x=y)))
So, one has informative identity
being asserted using another transitive,
symmetric, reflexive relation we may
call logical equivalence (LEQ). It is
a very complex set of relationships
that are taken for granted.
Now, if one is formulating the parameters
of an exercise for the purpose of
analysis, one may "stipulate" an
informative identity.
But, if one is considering the use of
identity in relation to foundational
analysis, it is questionable as to
whether or not one is justified in
treating informative identity as
stipulated or not.
It is true that the formulas are quantified
as they should be, but the problem has a
different character from an exercise or
an application.
My prior reference to Abraham Robinson
had to do with how denotation through
descriptions *defines* the diagonal
of a model -- that is, how descriptions
*define* the truth conditions for the
sign of equality.
While there is nothing fundamentally
wrong with the usual notion of identity
through corresponding extensions, the
paradigm in modern set theory seems
to disregard the aspect of naming that
was required in Frege's original
characterization of incomplete symbols
being completed with a name.
For example,
x+2=5
is incomplete because it has no
truth value, whereas the symbol
"3" completes the expression,
3+2=5
It has a truth value because the
symbol "3" purports a reference while
the symbol "x" does not.
Now, in foundations there is another
level of complexity for an assertion
of informative identity
x=y
Because of the use/mention distinction,
each of the symbols has a name,
'x' is the name of x
'y' is the name of y
And, of course, this form of name
goes on ad infinitum.
''x'' is the name of 'x'
''y'' is the name of 'y'
etc.
So, naming, itself, is not indecomposable.
It has an aspect of separation not unlike
the Zermelo sequences
{x}
{{x}}
etc.
Because of your remarks I have a better
criterion for distinguishing between stipulative
informative identity and epistemic informative
identity. Thank you.
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