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Topic: Mathematics and the Roots of Postmodern Thought
Replies: 2   Last Post: Apr 5, 2013 12:54 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Mathematics and the Roots of Postmodern Thought
Posted: Mar 27, 2013 9:54 PM
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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.
[...]



"Universal Statements

"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).
[...]



"Material Implication

"[...] 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. [...]



"Strengthened Implication

"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."





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