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1 An example of Philosophical Inspiration and Philosophical Content in formal Achievements The first general psychological remark : 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!) Andrzej Grzegorczyk (Warszawa)

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

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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: When we describe reality we are interested in good description. The value of a good description is called: truth truth When we invent formal tools to describe a ‘reality’, then the value of our construction is: 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: possible real existence (i.e. consistency) and something else what may be called: beauty and/or i ntellectual joke (of the construction) beauty and/or i ntellectual joke (of the construction)

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4 A value means this what we appreciate in real science in formal science we appreciate A good description of objects (events) A good description of objects (events) The value (of a good description) is called: Truth Truth (reality may be also beautyifull ) (reality may be also beautyifull ) Invention of some constructions Invention of some constructions The value are: consistency = applicability and consistency = applicability and Joke or Beauty Joke or Beauty (construction sometimes may occur as something existing) (construction sometimes may occur as something existing) We get to know the reality by means of invented constructions

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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 like procedures of counting Mathematicians were always proud of calculating functions. e.g. using procedure of recursion: f(0)=a, f(n+1)= F(f,n)… Mathematicians were always proud of calculating functions. e.g. using procedure of recursion: f(0)=a, f(n+1)= F(f,n)… Calculation is the fundamental joke (a result of an active procedure) obtained by a math. construction Calculation is the fundamental joke (a result of an active procedure) obtained by a math. construction Logicians like vision of the inside 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 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 Definition exhibits a fundamental beauty: the contemplative vision of a logical construction

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6 An illustration for the last distinction: L et compare the approach to meta-mathematics of two men: about 80 years ago: (in 1929-1931) Mathematician Mathematician Kurt Gödel : 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 ) of N( T ) ( a result of activity of counting-procedure) Logician: Logician: Alfred Tarski : Alfred Tarski : begins by philosophical analysis of texts and finds concatenation as basic operation on texts. 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. Empirical Discernibility. Discernibility is a kind of vision of texts which belong to T.

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7 Alas, Tarski missed the possibility to build decidability purely on concatenation (discovered by himself [or together with Lesniewski]) But we may develop Tarski’s initial idea and now draw out the following Consequences of Tarski’s original logical intuition: 1. The property : „a = (concatenation of b with c)” is directly Empirical Discernible ( shortly: EmD) 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. 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 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 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 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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8 A first consequence of Tarski’s idea: 1. Empirical discernibility of concatenation Let take „ ^” as the symbol of concatenation. Let take „ ^” as the symbol of concatenation. Hence: if x and y are some texts then: x^y is defined as the text composed of the texts x and y in such a manner that Hence: if x and y are some texts then: x^y 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” = „fol”^”low” but: then e.g. we have: „follow” = „fol”^”low” but: it is not true that:„follow” =„foll” ^„llow” The relation of concatenation is evidently empirical discernible. (it is a psychological evidence) The relation of concatenation is evidently empirical discernible. (it is a psychological evidence)

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9 the second evidence in following Tarski’s idea: 2. 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 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 ٨ Q may be comprehended as: The discernibility of a conjunction: P ٨ Q may be comprehended as: discernibility of P and discernibility of Q discernibility of P and discernibility of Q Proof: Hence all logical connectives evidently do not lead out of EmD

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10 the third consequence (following Tarski’s idea): 3. Quantification relativised to subtexts of a given text does not lead out of EmD the third consequence (following Tarski’s 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: 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 ≡ (y=t or w,z (y^w=t or z^y=t or z^y^w=t )) (where „ ^ “= concatenation) y subtext of t ≡ (y=t or w,z (y^w=t or z^y=t or z^y^w=t )) (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..) ) S (x..) ≡ y (y subtext of t & R (y..) )

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11 the fourth consequence (following Tarski’s idea) : 4. Definition by dual quantification does not lead out of Empirical Discernibles. the fourth consequence (following Tarski’s 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: 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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12 The fifth consequence (following Tarski’s 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: The class EmD = the smallest class of definable properties of Texts: which contains ^ and is closed under the definitions which use the operations: which contains ^ and is closed under the definitions which use the operations: Propositional connectives Propositional connectives Quantification relativized to subtexts Quantification relativized to subtexts Dual Quantification. 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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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: Definition of numbers: let select a sign : „1” hence numbers may be conceived as the texts: 1,1^1, 1^1^1,.... composed only of „ 1”: x N ≡ y ( y subtext x → „ 1 ” subtext y ) Definition of addition (no trouble) : Definition of addition (no trouble) : Addition identifies with concatenation: x+y = x ^ y Definition of multiplication is more complicated. Definition of multiplication is more complicated.

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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) ): 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 first triple is «^ x^, ^ 1^, ^ x^ », the second is «^x^, ^1^ 1^, x^ x^ » etc. the second is «^x^, ^1^ 1^, x^ x^ » etc. The first part of each triple contains always only one x. 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 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 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) ) (The last property will be assured by inductive condition in the following definition of fist(x) )

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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: 1. The initial condition: the first triple is: « ^ x ^, ^ 1 ^, ^ x ^ » « ^ x ^, ^ 1 ^, ^ x ^ » 2. The inductive condition: 2. The inductive condition: If((«^x^,^k^,^z^ » and «^x^,^k`^,^z`^ » are the neighbor triples in the sequence s ) then: If((«^x^,^k^,^z^ » and «^x^,^k`^,^z`^ » are the neighbor triples in the sequence s ) then: ( k`= k^1 and z`= z^x )). ( k`= k^1 and z`= z^x )). 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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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.y=z ) 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.y=z ) 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 ) . 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 ) → u=z) 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 ) → u=z) Proof: Multiplication is definable by dual quantification. Hence is discernible.

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

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18 Formal (set-theoretical) definition of TEXTS We consider: a (set-theoretical) Universe: U and: We consider: a (set-theoretical) Universe: U and: Family F of ordered pairs: X, R X where X U and R X is a relation which orders the set X. Family F of ordered pairs: X, R X where X U and R X is a relation which orders the set X. Family L of „Letters”: L is a family of disjoint sets which cover U: 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 a U ≡ W (W L and a W)L is a classification of characters W,Z L → ( W Z= Ø or W=Z ) W,Z L → ( W Z= Ø or W=Z ) if a,b W and W L then a and b are treated as the same letter (of the same character). 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, R X and Y, R Y 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 R X b ≡ f(a) R Y f(b) ) and 2.The function f preserves the characters: a, W ( W L → ( a W ≡ f(a) W ) a, W ( W L → ( a W ≡ f(a) W ) The classes of abstraction of this similarity may be called TEXTS, 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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19 Formal (set-theoretical) definition of TEXTS (continued) First we define [ X, R X ] as the TEXT (idealistic) determined by a (‘materialistic’) set-theoretical pair X, R X : First we define [ X, R X ] as the TEXT (idealistic) determined by a (‘materialistic’) set-theoretical pair X, R X : [ X, R X ] = { Y, R Y : Y, R Y similar to X, R X } [ X, R X ] = { Y, R Y : Y, R Y similar to X, R X } Now the Definition of TEXTS is simply as follows: Now the Definition of TEXTS is simply as follows: P TEXTS ≡ for some X, R X : P = [ X, R X ] and the Definition of concatenation ^ : [ Z,R z ] = [ X, R X ] ^ [ Y, R Y ] iff for some X`, R` X, Y`, R` Y : X`,R` X` [ X, R X ] & Y`, R` Y` [ Y, R Y ]& X` Y`=Ø & Z=X` Y` & a,b Z (aR z b ≡ ((a X`& b Y`) or (a,b X`& a R` X` b) or (a,b Y`& a R` Y` b) )

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20 The axioms of the elementary theory of TEXTS (Following Tarski [1931]) A1 x^(y^z)=(x^y)^z (connectivity ) A2 x^y=z^u → ( (x=z Λ y=u) V w ( (w^u=y Λ x^w=z) V (z^w=x Λ w^y=u) ))( the axiom of: 'editor' ) w ( (w^u=y Λ x^w=z) V (z^w=x Λ w^y=u) ))( the axiom of: 'editor' ) A3 ≠ x^y A4 ≠ x^y 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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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: 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: TEXTS FL, ^ is isomorphic with M, ^ M . TEXTS FL, ^ 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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EE1J2 – Discrete Maths Lecture 7

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