fom
Posts:
1,037
Registered:
12/4/12
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Re: Matheology � 203
Posted:
Feb 6, 2013 5:10 AM
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On 2/5/2013 9:47 PM, Ralf Bader wrote:
> fom wrote:
>
>> On 2/3/2013 10:50 PM, Ralf Bader wrote:
>>> Virgil wrote:
>>>
>>>> In article
>>>> <bc3c4c0e-d017-49b3-a4f3-22aba84aa3c7@5g2000yqz.googlegroups.com>,
>>>> WM <mueckenh@rz.fh-augsburg.de> wrote:
>>>>
>>>>> On 3 Feb., 22:29, William Hughes <wpihug...@gmail.com> wrote:
>>>>>>>> We can say ?"every line has the property that it
>>>>>>>> does not contain every initial segment of s"
>>>>>>>> There is no need to use the concept "all".
>>>>>>
>>>>>>> Yes, and this is the only sensible way to treat infinity.
>>>>>>
>>>>>> So now we have a way of saying
>>>>>>
>>>>>> s is not a line of L
>>>>>>
>>>>>> e.g. ?0.111... ?is not a line of
>>>>>>
>>>>>> 0.1000...
>>>>>> 0.11000...
>>>>>> 0.111000....
>>>>>> ...
>>>>>>
>>>>>> because every line, l(n), ?has the property that
>>>>>> l(n) does not ?contain every ?initial
>>>>>> segment of 0.111...
>>>>>
>>>>> But that does not exclude s from being in the list. What finite
>>>>> initial segment (FIS) of 0.111... is missing? Up to every line there
>>>>> is some FIS missing, but every FIS is with certainty in some trailing
>>>>> line. And with FIS(n) all smaller FISs are present.
>>>> But with no FIS are all present.
>>>>>
>>>>>> Is there a sensible way of saying
>>>>>> s is a line of L ?
>>>>>
>>>>> There is no sensible way of saying that 0.111... is more than every
>>>>> FIS.
>>>>
>>>> How about "For all f, (f is a FIS) -> (length(0.111...) > length(f))" .
>>>>
>>>> It makes perfect sense to those not permanently encapsulated in
>>>> WMytheology.
>>>
>>> By the way, Mückenheim's crap is as idiotic from an intuitionistic point
>>> of view as it is classically. Intuitionists do not have any problems
>>> distinguishing the numbers 0,1...1 with finitely many digits and the
>>> sequence formed by these numbers resp. the infinite decimal fraction
>>> 0,11....
>>>
>>
>> No. His finitism seems to be more of a mix of Wittgenstein and
>> Abraham Robinson. Although it is not apparent without reading the
>> original sources, it has a certain legitimacy. Names complete
>> Fregean incomplete symbols. So names are the key to model theory.
>> Robinson explains this exact relationship in "On Constrained
>> Denotation". It is, for the most part ignored by the model
>> theory one obtains from textbooks. The model theory that one
>> learns in a textbook parametrizes the quantifier with sets.
>> Thus, the question of definiteness associated with names is
>> directed to the model theory of set theory. In turn, this is
>> questionable by virtue of the Russellian and Quinean arguments
>> for eliminating names by description theory. So, the model
>> theory of sets consists of a somewhat unconvincing discussion
>> of how parameters are constants that vary (see Cohen). If one
>> does not know the history of the subject, then one is simply
>> reading Cohen to learn some wonderful insights and does not
>> question his statements (after all, it is Paul Cohen, right?)
>>
>> In Jech, there is an observation that forcing seems to
>> depend on the definiteness of "objects" in the ground
>> model such as the definiteness of the objects in the
>> constructible universe.
>>
>> If you read Goedel, there is a wonderful footnote explaining
>> the assumption that every object can be given a name in
>> his model of the constructible universe.
>>
>> If you read Tarski, there is an explicit statement that
>> his notion of a formal language is not a purely formal
>> language, but rather one that has formalized a meaningful
>> language--by which one can assume that objects have
>> meaningful names. As for a "scientific" language generated
>> by definition, Tarski has an explicit footnote stating
>> that that is not the kind of language that he is
>> considering.
>>
>> So, we have names being eliminated by Russell and Quine
>> and descriptive names being specifically excluded by the
>> correspondence theory intended to convey truth while the
>> notion of truth in the foundational theory that everyone
>> is using only presumes definiteness through parameters
>> that vary.
>>
>> But, the completion of an incomplete symbol requires
>> a name.
>>
>> Who wouldn't be a little confused?
>
> I am indeed slightly confused about what you wrote and what it has to do
> with the previous discussion.
WM's contentions involve historical issues concerning
mathematical truth. Some of these involve the epistemic
concerns typically associated with finitism or intuitionism.
Others involve naming.
In among everything else, WM keeps directing attention
to the need for names as the only meaningful way to
evaluate the truth of statements about number.
What follows is rather complicated and any knowledgeable
readers are invited to make corrections.
Early on, I asked WM certain questions which directed
attention to Abraham Robinson. For my part, I am
fully aware of certain parts of Robinson's work, and,
in particular, his criticism of Russellian description
theory and the use of names obtained from descriptions
for the express purpose of *defining* the model diagonal.
This is completely different from how "sets" are used
in model theory and Robinson clearly delineates the
difference in his exposition, "On Constrained Denotation."
If one simply begins to look at the historical
record with Frege's discussion of the completion
of incomplete symbols with a name, it is clear that
the importance of this has been lost to the
methods of modern mathematics.
It is simple. The formula
x+3=5
has no truth value because of the presence
of a variable term and is classified as an
incomplete symbol by Frege. The formula
2+3=5
has a truth value and is classified as
a complete symbol because it has no
variable terms.
In his analyses, Frege offered a theory of
names based on descriptions. This differed
from the historical theory of simple
denotation which is referred to as Millian
in the literature. Russell had been dissatisfied
with Frege's solution and offered a different
theory of description in his classic paper "On
Denotation."
For Russell, the important problem was what
has become known as presuppostion failure.
The classic example given by him is
"The present king of France is bald"
Since there is no present king of France,
how does one attribute a truth value?
Russell treated the definite article as
a quantifier in such a manner that the
sentence still has a legitimate truth
value despite the presupposition failure
of its subject.
Given the abstract nature of mathematical
objects, Russell applied his description
theory in "Principia Mathematica". This
is what is referred to as the "no classes"
theory in the literature--an attempt to
write a foundations for mathematics that
would always have a legitimate truth value,
although it might not be determinable.
In part, this had been the reason for
the axiom of reducibility that is often
discounted. Russell attempted to argue
that it was actually a weaker assumption
than set existence, but those few who
had been aware of Russell's work (Skolem
observed that mathematicians had basically
ignored it) did not advocate for it.
Now, in his attempt at full generality,
Russell excluded names from consideration
as an extra-logical affair. And, his
work had certainly been far more influential
than Frege's at that time. Naturally,
the Russellian position on names and
descriptions had been continued among those
who did follow Russell. In "Word and
Object" Quine goes through a convoluted
discussion of how to use description theory to
eliminate names only to be followed
by a discussion of how to use description
theory to re-introduce names. And,
while the distinctions between syntax,
semantics, and pragmatics are generally
attributed to Carnap, the literature
suggests that he was motivated
to rethink his positions by Tarski
who was investigating semantics at
that time. And, even though
Carnap made these distinctions, he
seems to have been strongly motivated
to investigate linguistic form in
such a strictly syntactic manner
that even his fellow researchers
questioned the sensibility of the
work.
What is fundamental to Russellian
description theory are certain arguments
concerning the sign of equality and
its relation to the notion of identity.
Devised at a time prior to model theory,
there is no "diagonal." Russell had
rejected the Fregean notions of sense
and reference (or significance). He
argued that one either had immediate
acquaintance with an object of
reference or that the quantificational
interpretation of the definite article
applied.
For the most part, Russellian description
theory went unchallenged until Strawson
wrote "On Referring" in the 1950's.
That does not, however mean that mathematics
simply adopted Russell's ideas. Zermelo's
first paper on set theory speaks of a
domain of objects for which the sign of
equality is a relation between denoting
symbols. This is more like the Fregean notion
and, if you read Zermelo's paper, you will
find that identity of denotation is enforced
by relation to singletons rather than the
general statement of extensionality.
Returning to the issue of names more directly,
one finds in Tarski's "The Definability of
Concept" an analysis of the kind of definition
for a symbol one would expect in any modern
first-order language. It uses a sign of equality
in the way that logicians might say
x=Socrates
for
x is Socrates
He is examining a method proposed by Padoa
and concludes that "each problem of definability
of a term is reducible without remainder to
a problem of derivability of a specific
sentence."
Yet, in a footnote in the "Concept of
Truth in Formalized Languages" he introduces
a notion of definability of properties
rather than definability of terms and this
is the definability that one finds in books
on set theory. This form of definability
is not the definition of an object which is
a set with the given property that can be
given a name using the syntax for a
description. It is a notion of subtantiality
for a property expressed by a formula
(that is, that a formula with a free variable
is not a vacuous statement).
Having already excluded languages built
upon definitions from his construction,
he makes the observation that this
particular form of definability is
compatible with his procedure.
Now perhaps these are equivalent and
perhaps they are not. I am not certain
for myself, but that is not the real
problem. In the mathematical literature
one speaks of terms, of objects, of
constants, and of parameters. I am
generally savvy enough to make sense
of these references within the contexts
in which I read them. But, in the
area of the truth of mathematical
statements I feel reasonably concerned
that the inconsistent and incompatible
ways in which these phrases are used
makes these investigations questionable.
When Tarski gets away from the language
of the theory of classes, he says
exactly that to which we have all
become accustomed:
"As we saw in section 3, the method
of construction available to us
presupposes first a definition of
another concept of fundamental
importance for the investigations
in the semantics of language. I
mean the notion of the satisfaction
of a sentential function by a
sequence of objects."
Notice, without even an acknowledgement
of how those objects are individuated--
even through some infinitary naming
process--we merely make definite one
set of variables with another set of
variables. That is a long way from
Frege's observations on incomplete
and complete symbols.
Obviously, I do not care about WM's
finitism, although I find the argumentation
interesting in so far as I never actually
met a finitist before. I enjoy all
mathematics and have no problems with any
of it except the model theory for set
theory. The rest of mathematics has come
to depend upon it for a notion of "definite"
object. I sympathize with WM on this
matter because I fully side with Abraham
Robinson's criticisms of Russellian
description theory and Strawson's
criticism of Russellian description
theory and know that the logicism
has misrepresented Leibniz' identity of
indiscernibles which has bearing
on these matters.
For what it is worth, it was a
breath of fresh air when I finally
bit the bullet and purchased the
collected works of Lesniewski.
Finally, I read an author who saw
that Russell's paradox should be
called Russell's mistake.
And, do not take that incorrectly.
Principia Mathematica is an
amazing read.
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