fom
Posts:
1,968
Registered:
12/4/12


Re: Matheology � 203
Posted:
Feb 6, 2013 5:10 AM


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 >>>> <bc3c4c0ed01749b3a4f322aba84aa3c7@5g2000yqz.googlegroups.com>, >>>> WM <mueckenh@rz.fhaugsburg.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 >> languageby 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 literaturean 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 extralogical 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 reintroduce 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 firstorder 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 processwe 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.

