An example of Philosophical Inspiration and Philosophical Content in formal Achievements

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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!)

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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
.
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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)?

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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
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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

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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)?

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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)?

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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)?

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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

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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..) )?

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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.

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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.
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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.

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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) )?

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• 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!)?

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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.

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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)?

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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).

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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) )?

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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)

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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.