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An example of Philosophical Inspiration and Philosophical Content in formal Achievements

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A Perception of philosophical contents depend on the goal ... a true scholar is curious [or: very, very curious] of something over and above his/her career! ... – PowerPoint PPT presentation

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Title: An example of Philosophical Inspiration and Philosophical Content in formal Achievements


1
An example of Philosophical Inspiration and
Philosophical Content in formal Achievements
Andrzej Grzegorczyk (Warszawa)?
  • The first general psychological remark
  • A Perception of philosophical contents depend on
    the goal (taste, predilection) of a scholar,
    i.e. on
  • What a scholar is looking for ?
  • (We, of course, presume that a true scholar is
    curious or very, very curious of something
    over and above his/her career!)

2
Let me begin by the distinction of Two main
categories (of science)
  • Formal Science
  • about our constructions which are tools for
    speaking on reality
  • Curiosity for the constructions which are
    invented by ourselves
  • (by mathematicians themselves)
  • Real Science about reality
  • Curiosity for the reality which is independent on
    ourselves
  • Curiosity for the phenomenon's of
  • life, history, structure of the matter, etc.
    peculiarities of reality
  • Imaginary Science

The attitude of Scholars
Let me think about this distinction a little more
.
3
We see easily that The values in sciences
  • When we describe reality we are interested in
    good description. The value of a good
    description is called
  • truth
  • When we invent formal tools to describe a
    reality, then the value of our construction is
  • possible real existence (i.e. consistency) and
    something else what may be called
  • beauty and/or intellectual joke (of the
    construction)?

4
A value means this what we appreciatein real
science in formal sciencewe appreciate
  • A good description of objects (events)?
  • The value (of a good description) is called
  • Truth
  • (reality may be also beautyifull )?
  • Invention of some constructions
  • The value are
  • consistency applicability and
  • Joke or Beauty
  • (construction sometimes may occur as something
    existing)?

We get to know the reality by means of invented
constructions
5
In formal science one may discover a distinction
(also relevant to the taste of a scholar). I
mean an alleged distinction between preferences
of Mathematicians and Logicians
  • Mathematicians like procedures of counting
  • Mathematicians were always proud of calculating
    functions. e.g. using procedure of recursion
    f(0)a, f(n1) F(f,n)
  • Calculation is the fundamental joke (a result of
    an active procedure) obtained by a math.
    construction
  • Logicians like vision of the inside
  • Logicians are more philosophers, and were always
    proud of seeing and defining properties or
    relations. They define using primitives, logical
    connectives and quantifiers
  • Definition exhibits a fundamental beauty the
    contemplative vision of a logical construction

6
An illustration for the last distinction Let
compare the approach to meta-mathematics of two
men about 80 years ago (in ? 1929-1931)?
  • Logician
  • Alfred Tarski
  • begins by philosophical analysis of texts and
    finds concatenation as basic operation on texts.
  • Hence Tarski opens a philosophical natural way to
    consider the decidability of T as
  • Empirical Discernibility.
  • Discernibility is a kind of vision of texts which
    belong to T.
  • Mathematician
  • Kurt Gödel
  • translates texts of theory T into numbers N ( T
    )?
  • He also introduces the
  • General recursiveness and defines decidability
    of T as calculability of characteristic function
  • of N( T )
  • ( a result of activity of counting-procedure)?

7
Alas, Tarski missed the possibility to build
decidability purely on concatenation
(discovered by himself or together with
Lesniewski) But we may develop Tarskis
initial idea and now draw out the
followingConsequences of Tarskis original
logical intuition
  • 1. The property a (concatenation of b with
    c) is directly Empirical Discernible (shortly
    EmD)?
  • 2. Definition using propositional connectives
    does not lead out of EmD.
  • 3. Definition by quantification relativised to
    subtexts of a given text does not lead out of EmD
  • 4. Definition by dual quantification does not
    lead out of EmD. and at the end we draw a
    conclusion
  • 5. The class of EmD decidables may be defined
    as
  • the smallest which satisfies 1.- 4. above
    conditions.
  • (Let me look at the above items more closely)?

8
A first consequence of Tarskis idea1.
Empirical discernibility of concatenation
  • Let take as the symbol of concatenation.
  • Hence if x and y are some texts then xy is
    defined as the text composed of the texts x and y
    in such a manner that
  • the text y follows immediately the text x
  • then e.g. we have follow follow but
  • it is not true that follow foll llow
  • The relation of concatenation is evidently
    empirical discernible. (it is a psychological
    evidence)?

9
the second evidence in following Tarskis
idea2. Definition using propositional
connectives does not lead out of the class of
Empirical Discernibles.
  • Discernibility of the occurrence involves the
    discernibility of nonoccurence and vice-versa.
    Hence discernibility of P discernibility of P
  • The discernibility of a conjunction P ? Q may
    be comprehended as
  • discernibility of P and discernibility of Q
  • Proof Hence all logical connectives evidently
    do not lead out of EmD

10
the third consequence (following Tarskis
idea)3. Quantification relativised to subtexts
of a given text does not lead out of EmD
  • Suppose R?EmD and a new property S is defined
    by Quantification relativised to subtexts of a
    given text t . This means that
  • where subtexts are defined as follows
  • y subtext of t (yt or ?w,z (ywt or
    zyt or zywt )) (where
    concatenation)
  • then also S ?EmD .
  • Proof the set of subtexts of a finite text is
    effectively finite.
  • S (x..) ?y (y subtext of t R (y..) )?

11
the fourth consequence (following Tarskis idea)
4. Definition by dual quantification does not
lead out of Empirical Discernibles.
  • Defining by dual quantification means that a new
    property S is definable in two ways
  • S (x..) ?y R (y,x,..) and S (x..) ?y R
    (y,x..)
  • where the properties R and R are already
    EmD.
  • Proof from the both equivalences we get that the
    following holds
  • ?x ?y (R (y,x,..) or R (y,x..) ) from excluded
    middle
  • Hence for every x we can find the first y such
    that
  • R (y,x,..) or R (y,x..), because the texts
    may be lexicographically ordered. Just we can
    discern S (x) or S (x).
  • It is of course a translation of the Emil Post
    proof of the Complement Theorem.

12
The fifth consequence (following Tarskis idea)
5. A possibility of the construction of
set-theoretical definition of Empirical
Discernibles
  • The class EmD the smallest class of definable
    properties of Texts
  • which contains and is closed under the
    definitions which use the operations
  • Propositional connectives
  • Quantification relativized to subtexts
  • Dual Quantification.

Compare with Computability A different range of
imagination Computability Procedural Algorithms
of Calculation Discernibility Vision of logical
order . These are Different ranges of Cognitive
tools. A different philosophy.
13
Do we get in practice the same result? in the
both approaches?Of course we do! As an example
it is easy to show multiplication as an
discernible property
  • Definition of numbers let select a sign 1
    hence numbers may be conceived as the texts
  • 1,11, 111, .... composed only of 1
  • x?N ?y ( y subtext x ? 1 subtext y )?
  • Definition of addition (no trouble)
  • Addition identifies with concatenation
  • xy x y
  • Definition of multiplication is more complicated.

14
The definition of multiplication as an example of
the possibility of uttering inductive
constructions using concatenation
  • We add some 3 new signs , then we can
    define a set of finite inductive sequence of
    triples (shortly fist(x) )
  • the first triple is x , 1 , x ,
  • the second is x , 11 , xx etc.
  • The first part of each triple contains always
    only one x.
  • The second part of each triple is composed of
    several copies of 1 ( It is a Number according
    to the preceding slide 13)
  • The thrid part of each triple is composed of
    the element x repeated as many times as 1 is
    repeated in the second part of the triple .
  • (The last property will be assured by inductive
    condition in the following definition of
    fist(x) )?

15
  • A general Definition of fist may be writen in
    the following way
  • A text s is fist (x) iff
  • 1. The initial condition the first triple is
  • x , 1 , x
  • 2. The inductive condition
  • If((x,k,z and x,k,z are the
    neighbor triples in the sequence s ) then
  • ( k k1 and z zx )).
  • Where two triples are called neighbor when there
    is nothing between them in s and the second is
    just the next one.
  • Then we define the relation of multiplication
    (not a function!)?

16
Multiplication proves to be discernible
  • Using finite inductive sequences of triples
    (fist) of the preceding slide, we can define the
    relation of multiplication as follows (
    M(x,y,z) means x.yz )?
  • M(x, y, z) ? s ( s is fist(x) x,1,x
    is the first triple of s x,y,z is the
    last triple of s )?.
  • Multiplication may be also defined as follows
  • M(x, y, z) ? s,u (( s is fist(x)
    x,1,x is the first triple of s
    x,y,u is the last triple of s ) ? uz)?
  • Proof Multiplication is definable by dual
    quantification. Hence is discernible.

17
What are texts?There are two conceptions of
text1. Materialistic and 2.
IdealisticThere are many separate material
things which are issues of One paper
  • In Maths we speak on Texts as Idealistic
    entities
  • Texts are abstract entities which are obtained by
    two (simultaneous) identifications
  • We identify two materialistic texts which have
  • 1) the same order of letters, and
  • 2) corresponding letters (atomic texts) have the
    same shape .
  • This intuition may be generalized! (as
    follows)?

18
Formal (set-theoretical) definition of TEXTS
  • We consider a (set-theoretical) Universe U
    and
  • Family F of ordered pairs ?X, RX? where X?U and
    RX is a relation which orders the set X.
  • Family L of Letters L is a family of disjoint
    sets which cover U
  • a?U ? W (W?L and a?W) L is a classification of
    characters
  • W,Z?L ? ( W?Z Ø or WZ )?
  • if a,b?W and W?L then a and b are treated as
    the same letter (of the same character).
  • We may say that two pairs ?X, RX? and ?Y,
    RY? are similar iff there is a 1-1
    function f mapping X on Y in such a way that
  • 1. The function f preserves the type of ordering
  • If ?a,b ( a,b?X ? ( a RXb f(a) RY f(b) )
    and
  • 2. The function f preserves the characters
  • ?a, W ( W?L ? ( a?W f(a)?W )?
  • The classes of abstraction of this similarity
    may be called TEXTS,
  • This may be formally written as follows (in the
    next slaid).

19
Formal (set-theoretical) definition of TEXTS
(continued)?
  • First we define ?X, RX? as the TEXT
    (idealistic) determined by a (materialistic)
    set-theoretical pair ?X, RX?
  • ?X, RX? ?Y, RY? ?Y, RY? similar to ?X, RX?
  • Now the Definition of TEXTS is simply as follows
  • P ?TEXTS for some X, RX P ?X, RX?
  • and the Definition of concatenation
  • ?Z,Rz ? ?X, RX? ?Y, RY? iff for
    some X, RX, Y, RY ?X,RX?? ?X, RX?
    ?Y, RY?? ?Y, RY? X?YØ ZX?Y
    ?a,b?Z (aRzb ((a?X b?Y) or
  • (a,b?X a RXb) or (a,b? Y a RYb) )?

20
The axioms of the elementary theory of
TEXTS(Following Tarski 1931)?
  • A1 x(yz)(xy)z (connectivity )?
  • A2 xyzu ? ((xz ? yu) V
  • ?w((wuy ? xwz) V (zwx ?
    wyu)))? (the axiom of 'editor')?
  • A3 ? ? xy A4 ? ? xy
  • A5 ? ? ?
  • some results on A1-A5 are in the paper
    Grzegorczyk Zdanowski Undecidability and
    Concatenation , in the book dedicated to Andrzej
    Mostowski (to appear)

21
For the end an interesting open problem ! A
conjecture
  • For every model ?M, M? of the theory A1-A5 there
    exist a set-theoretical universe U and the
    related families F,L such that
  • ?TEXTSFL, ? is isomorphic with ?M, M? .
  • ?
  • It may be a theorem of representation similar to
    the theorem of representation of Boolean
    algebras. ?
  • The end.
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