On 3/27/2013 1:12 PM, david petry wrote: > On Wednesday, March 27, 2013 4:41:07 AM UTC-7, Dan wrote: > > >> So falsifiability is either meaningless or trivial for all but the >> simplest arithmetic statements . (I like having statements with >> multiple alternating quantifiers, thank you very much) . > > > I'm not in the mood to argue with you, >
You may expect to "agree to disagree" with most of your discussions of falsifiability.
Because of the nonsense with WM's "crayon marks" I spent some minor effort to investigate one branch of constructive mathematics that I knew to be related. I had some translations of Russian papers and a copy of Sanin's book on constructive real numbers. But I did not have a collection of definitions from which to start.
In seeking those definitions, I ended up purchasing Markov's "Theory of Algorithms" (not what you are thinking). Markov begins precisely by defining his constructive objects in terms of "marks" corresponding to symbols from alphabets. The natural numbers correspond to a particular alphabet and are the very concatenations of symbols you would expect,
1=| 2=|| 3=|||
and so on.
Anyway, this sort of constructive mathematics entails a particular redefinition of the quantifiers.
What follows are the excerpts from Markov's construction that I thought would suffice to convey the basic ideas behind his constructions. Quite possibly, I accidentally excluded some essential elements. Otherwise, it may give you a sense of how the meaning of the quantifiers need to change in order to express something closer to the falsifiability you would take as a criterion.
One thing that should be noted when considering the following excerpts is that Frege originally interpreted universal quantification in terms of arbitrary selections. Markov's explanations seem to be returning to that interpretation by virtue of his use of "given". Historically, this is one aspect of Frege's work that had been objected to by Russell. Thus in so far as "classical" mathematics follows Russell's program, it treats universal quantification along the lines of a course of values rather than as an arbitrary selection.
A second thing to remember is that the construction of syntactic forms from finite alphabets using concatenations serves as a ground for objectively accepted knowledge as it pertains to the kind of objects that may be generally given in his formulation of a constructive mathematics.
From "Theory of Algorithms" by Markov:
"Abstraction of Potential Feasibility
"1. Carrying out constructive processes, we often come up against obstacles connected with a lack of time, space, and material. [...]
"Acting in this manner, we shall disregard the restrictedness of our possibilities concerning time, space, and material. This disregarding is customarily called the abstraction of potential feasibility. [...]
"4. Every application of potential feasibility is an imaginative act. This is equally true for any of the abstractions inherent in mathematics or some other abstract science. Classical mathematics invokes abstractions going much farther than those of constructive mathematics. In particular, it makes use of the abstraction of actual infinity, i.e., allows itself to reason about 'infinite sets' as about given non-constructive 'objects'. The difference between 'classicists' and 'constructivists' consists in their being willing to accept abstractions of different sorts. [...]
"1. An important role in mathematics is played by universal statements beginning with the words <<every>>, <<for every>>, <<whatever be>>,etc. How can they be understood constructively?
"2. The problem is simply solved when there is a list of all objects of the kind to which the universal statement refers. In this case, the statement that every object of this kind satisfies a given requirement can be understood as a many-term conjunction, each of whose terms asserts this about one object in the list, where all objects in the list appear in the conjunction in the sense indicated.
"[...] But if the list contains only one name, we shall then, of course, understand the universal statement as the statement,
<<the unique object named in the list satisfies the given requirement>>
"[...] in the case when the list is empty, i.e., when there are no objects of this kind. We shall then understand our statement as the trivial truth,
nullstring *=* nullstring
[*=* is "graphic equality"]
(independently of what the requirement which an object should satisfy consists of).
"With such an understanding of universal statements in those cases when there is such a list of objects of the kind under consideration, the following holds: When adding a single new object to this kind, the statement that every object of the extended kind satisfies the formulated requirement turns out equivalent to the conjunction of the previous universal statement (pertaining to the initial kind) and the statement that the added object satisfies this requirement.
"3. Everything said in part 2 pertained to the case when there is a list of objects of the kind under consideration. There may, however, be no such list, or it may even be unfeasible because there are 'too many' of these objects.
"Fortunately, there is another possible way of understanding universal statements constructively. For certain objects of the kind under consideration, it may be established by means of suitable arguments that they satisfy the formulated requirments. These arguments may be of such a nature that the possibility of developing analogous arguments for any other object of the kind under consideration will be clear. A conviction will then grow up in us that whatever object of this kind given to us, we shall be able to prove (make obvious through arguments) that this object satisfies the requirement presented. We are claiming that precisely this situation prevails when we say that every object of the given kind satisfies the given requirement.
"Thus, we have arrived at the following understanding of statements to the effect that every constructive object of a given kind satisfies a given requirement. Such a statement claims our ability to prove for any given object of the kind under consideration, that it satisfies the requirement presented.
"The source of our confidence in being able to carry out the required proof, regardless of which object we are given, may be our limited experience with such proofs, provided that this experience has made it clear how we should act in an arbitrary case. This ability of ours to become convinced of the truth of universal statements on the basis of limited experience is what we call our intuition of the universal. [...]
"Direct Negation. Decidable Statements
"1. Studying our ability to carry out constructive processes, we formulate the results of this study in the form of certain statements asserting that at the present time we are able to construct certain objects, we have mastered certain general methods, etc. It is natural to ask what the negations of statements of this kind can look like, negations also saying something about our constructive abilities.
"A naive answer says that the negation of the statement (1)
<<at the present time we are able to construct an object satisfying the requirement...>>
is the statement (2)
<<at the present time we are unable to construct an object satisfying the requirement...>>
that the negation of the statement (3)
<<at the present time we possess a general method for...>>
is the statement (4)
<<at the present time we do not possess a general method for...>>
"This answer must, however, be rejected because statements of the form (2) and (4) may well turn out to be true today and false tomorrow. [...]
"We see, therefore, that a naive treatment of negation can take us beyond the bounds of [constructive] mathematics. Desiring to stay within these bounds, we should be concerned about acquiring a suitable positive understanding of negation. [...] In this connection, we shall naturally require that the negation of a statement always be incompatible with it, i.e. that the conjunction of a statement with its negation be false. [...]
"In general, in the case when statements A and B are such that their conjunction is false, but their disjunction is true, we shall say that B is the direct negation of A. We shall say that a statement A is decidable when we have succeeded in selecting its direct negation.
"In case a statement A is decidable, we have a method for determining whether A is true. We determine this by establishing the truth of the disjunction
<<A or B>>
where B is the direct negation of A. [...]
"Semidecidable Statements. Strengthened Negation
"1. Let us now consider a statement about the existence of a word in a given alphabet satisfying a given requirement expressed by a decidable predicate, i.e., a statement (1)
<<there exists an X such that F>>
Where X is a free verbal variable and F is a decidable, one-place predicate with this variable.
"We shall call statements of this kind semidecidable. [...]
"Investigations analogous to those just carried out can obviously be performed for any semidecidable statement. They suggest that we regard the statement (5)
<<given any X, G holds>>
where G is a direct negation of F, to be a direct negation of the semidecidable statement (1).
"The statement (5) will be called a strengthened negation of the statement (1). [...]
"[...] Thus we call the implication
<<if A, then B>>
understood as the disjunction
<<C or B>>
where C is a direct negation of A, the material implication with premise A and conclusion B. [...]
"1. Let us now consider implications with a semidecidable premise, i.e., statements of the form (1)
<<if there exists an X such that F, then A>>
"[...] Let us try to interpret the implication (1) in the spirit of material implications as the disjunction (2)
<<at the present time we do not possess a method for constructing a word in the given alphabet, satisfying the predicate F, or A>>
"One finds that this disjunction can be true at a given moment, but is not insured against refutation in the future. This will be the case when A is false and we are unable to construct a desired word now, but shall be able to do so later. [...]
"Note, however, that the statement (2) would be insured against refutation if the statement (3)
<<given any X, if F, then A>>
were proven, where the implication obtained from the phrase (4)
<<if F, then A>>
after replacement of the variable X [in F] by any word in our alphabet should be understood as a material implication. [...]
"All this suggests that precisely the statement (3) be considered as the interpretation of the implication (1). We shall call the implication (1), understood in this way, the strengthened implication with the premise
<<there exists an X such that F>>
and the conclusion A.
"According to the definition, a strengthened implication always has a semidecidable premise."