11

An example of An example of Philosophical Inspiration and Philosophical Philosophical Inspiration and Philosophical

Content Content in formal Achievementsin formal Achievements

The The firstfirst general psychological general psychological remarkremark::

A PerceptionA Perception of philosophical contents depend of philosophical contents depend on the on the goal goal (taste, predilection) of a scholar, (taste, predilection) of a scholar,

i.e. on:i.e. on:What a scholar is looking for ? What a scholar is looking for ?

(We, of course, presume that: a true scholar is curious [or: very, very curious] of (We, of course, presume that: a true scholar is curious [or: very, very curious] of

something over and above his/her career!)something over and above his/her career!)

Andrzej Grzegorczyk (Warszawa)

22

Let me begin by the distinction ofLet me begin by the distinction of

Two main Two main categoriescategories (of (of science)science)::

Curiosity Curiosity for the for the phenomenon's of:phenomenon's of:

life, history, structure life, history, structure of the matter, etc. of the matter, etc. peculiarities of realitypeculiarities 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 .

33

We see easily that:We see easily that: The values in sciences :The values in sciences :

When we When we describe describe realityreality we are interested in we are interested in good description. The value of a good good description. The value of a good description is called:description is called:

truthtruth When we When we invent formalinvent formal tools to describe a tools to describe a

‘reality’, then the ‘reality’, then the valuevalue of our of our constructionconstruction is: is:

possible possible realreal existenceexistence (i.e. (i.e. consistencyconsistency) ) and and something else what may be called: something else what may be called:

beauty beauty and/or and/or iintellectualntellectual jokejoke (of the construction)(of the construction)

44

A value A value meansmeans this what we appreciate this what we appreciatein real science in formal in real science in formal

sciencesciencewe appreciatewe appreciate

A good A good descriptiondescription of objects (events)of objects (events)

The valueThe value (of a good (of a good description) description) is called:is called:

TruthTruth (reality may be also (reality may be also

beautyifullbeautyifull ) )

InventionInvention of some of some constructionsconstructions

The value are:The value are: consistency = consistency =

applicability applicability and and JokeJoke or or BeautyBeauty

(construction (construction sometimes may occur sometimes may occur as something existing)as something existing)

We get to know the reality by means of invented constructions

55

In formal science one may discover a In formal science one may discover a distinction: distinction: ((also relevant to the also relevant to the taste taste of a of a scholar). I mean an alleged distinction between scholar). I mean an alleged distinction between

preferences of preferences of MathematiciansMathematicians and and Logicians :Logicians :

MathematicianMathematicianss like like procedures of countingprocedures of counting Mathematicians were Mathematicians were

always proud of always proud of calculatingcalculating functions. e.g. usingfunctions. e.g. using procedure of recursionprocedure of recursion: : f(0)=a, f(n+1)= F(f,n)…f(0)=a, f(n+1)= F(f,n)…

CalculationCalculation is the is the fundamental fundamental jokejoke (a result of an (a result of an activeactive procedureprocedure) ) obtained by a math. obtained by a math. constructionconstruction

LogicianLogicianss like vision like vision of the of the insideinside Logicians are more Logicians are more

philosophers, and were always philosophers, and were always proud of seeing and proud of seeing and definingdefining propertiesproperties or or relationsrelations. They . They definedefine using: primitives, using: primitives, logical logical connectivesconnectives and and quantifiers quantifiers

DefinitionDefinition exhibits a exhibits a fundamental fundamental beauty:beauty: the the contemplativecontemplative visionvision of a logical of a logical constructionconstruction

66

An illustration for the last distinction: LAn illustration for the last distinction: Let compare et compare the the approachapproach to meta-mathematics of to meta-mathematics of two two

menmen: : aboutabout 80 years ago: 80 years ago: (in (in 1929-1931) 1929-1931)

MathematicianMathematician Kurt Kurt GödelGödel:: translatestranslates textstexts of of

theory theory T T into numbersinto numbers N ( N ( TT ) )

He also He also introducesintroduces the the General recursiveness General recursiveness

and defines and defines decidabilitydecidability of of TT as as calculabilitycalculability of of characteristiccharacteristic function function

of of N(N( T T ) ) ( a result of activity of ( a result of activity of

counting-counting-procedureprocedure))

LogicianLogician: : AlfredAlfred TarskiTarski: :

beginsbegins by philosophical by philosophical analysisanalysis of of texts and findstexts and finds concatenationconcatenation as basic operation as basic operation on texts.on texts.

Hence Tarski opens a Hence Tarski opens a philosophical natural way philosophical natural way to consider the to consider the decidabilitydecidability of of TT as as

Empirical Discernibility.Empirical Discernibility.Discernibility is a kind of Discernibility is a kind of visionvision

of texts which belong to of texts which belong to T.T.

77

Alas, Tarski missed the possibility to build Alas, Tarski missed the possibility to build decidabilitydecidability purely purely on on concatenationconcatenation (discovered by himself [or together with Lesniewski]) (discovered by himself [or together with Lesniewski])

But we may develop Tarski’s initial idea and now draw out the But we may develop Tarski’s initial idea and now draw out the followingfollowing

Consequences of Tarski’s original logical intuition:Consequences of Tarski’s original logical intuition: 1. The property : „1. The property : „aa = (concatenation of = (concatenation of bb with with cc))” ”

is directly is directly EmEmpirical pirical DDiscernibleiscernible ( (shortly: shortly: EmDEmD)) 2. Definition using 2. Definition using propositional connectivespropositional connectives

does not lead out of does not lead out of EmDEmD. . 3. Definition by 3. Definition by quantification relativisedquantification relativised to to

subtexts of a given text does not lead out of subtexts of a given text does not lead out of EmDEmD 4. Definition by 4. Definition by dual quantificationdual quantification does not lead does not lead

out of out of EmDEmD. . and at the end we draw a conclusionand at the end we draw a conclusion 5. The class of 5. The class of EmDEmD = = decidablesdecidables may be defined may be defined

asasthe smallest which satisfies 1.- 4. above conditions.the smallest which satisfies 1.- 4. above conditions.

(Let me look at the above items more closely)(Let me look at the above items more closely)

88

A first consequence of Tarski’s idea:A first consequence of Tarski’s idea:

11. . Empirical discernibility of Empirical discernibility of concatenationconcatenation

Let take „ Let take „ ^̂”” as the symbol of concatenation. as the symbol of concatenation. Hence: if Hence: if xx and and yy are some texts then: are some texts then: x^y x^y is is

defineddefined as the text as the text composedcomposed of the texts of the texts x x and and yy in such a manner that in such a manner that

the text the text yy follows immediately the text follows immediately the text xx

then e.g. then e.g. we havewe have: : „follow”„follow” = = „fol”^”low”„fol”^”low” but: but: it is it is not true that:not true that: „follow” =„foll” ^„llow”„follow” =„foll” ^„llow”

The relation of concatenation is evidently The relation of concatenation is evidently empirical discernible. empirical discernible. (it is a psychological evidence)(it is a psychological evidence)

99

the second evidence in following Tarski’s idea:the second evidence in following Tarski’s idea:2. Definition using 2. Definition using propositional propositional

connectivesconnectives does not lead out of the does not lead out of the class ofclass of Empirical DiscerniblesEmpirical Discernibles..

Discernibility of theDiscernibility of the occurrence occurrence involves the involves the discernibility of discernibility of nonoccurencenonoccurence and vice- and vice-versa. Hence: versa. Hence: discernibility of P discernibility of P ≡≡ discernibility discernibility of of ¬¬P P

The The discernibility ofdiscernibility of a a conjunctionconjunction: : P P ٨٨ QQ may may be comprehended as: be comprehended as:

discernibility of P discernibility of P andand discernibility of Q discernibility of Q Proof:Proof: Hence all logical connectives evidently Hence all logical connectives evidently

do not lead out of do not lead out of EmDEmD

1010

the third consequence (following Tarski’s idea):the third consequence (following Tarski’s idea):3. Quantification relativised to subtexts of a 3. Quantification relativised to subtexts of a

given text does not lead out of given text does not lead out of EmDEmD Suppose Suppose RREmDEmD and a new property and a new property SS

is defined by is defined by Quantification relativised to Quantification relativised to subtexts of a given text subtexts of a given text t t . This means that:. This means that:

where where subtextssubtexts are defined as follows: are defined as follows: y subtext of y subtext of tt ≡ (y= ≡ (y=tt or or w,z (yw,z (y^̂w=w=tt or z or z^̂y=y=tt

or z^y^w=or z^y^w=tt )) )) ((where „ ^where „ ^ “= concatenation) “= concatenation) then then also also S S EmDEmD . . Proof: the set of subtexts of a finite text is effectively Proof: the set of subtexts of a finite text is effectively

finite. finite.

S S (x..) ≡ (x..) ≡ y (y subtext of y (y subtext of tt && R R (y..) )(y..) )

1111

the fourth consequence (following Tarski’s idea) :the fourth consequence (following Tarski’s idea) :4. Definition by 4. Definition by dual quantificationdual quantification

does not lead out of does not lead out of Empirical Empirical DiscerniblesDiscernibles..

Defining by Defining by dual quantification dual quantification means that means that a new property a new property SS is definable in two ways: 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.

1212

The fifth consequence (following Tarski’s idea):The fifth consequence (following Tarski’s idea): 5. A possibility of the construction of set-5. A possibility of the construction of set-

theoretical theoretical definitiondefinition of of Empirical Empirical DiscerniblesDiscernibles

The class The class EmDEmD = the smallest class of = the smallest class of definable properties of Texts:definable properties of Texts:

which contains which contains ^̂ and is closed under and is closed under the definitions which use the operations:the definitions which use the operations: Propositional connectives Propositional connectives Quantification relativized to subtextsQuantification relativized to subtexts Dual Quantification.Dual Quantification.

Compare with Computability: A different range of imaginationComputability = Procedural Algorithms of CalculationDiscernibility = Vision of logical order . These are Different ranges of Cognitive tools. A different philosophy.

1313

Do we get Do we get in practice in practice the same result? the same result? in the both approaches?in the both approaches?Of course we do!Of course we do! As As an example it is easy toan example it is easy to show multiplication as an discernible show multiplication as an discernible

property property Definition of Definition of numbersnumbers: : let select a signlet select a sign: „: „11” ”

hence numbers may be conceived as the texts:hence numbers may be conceived as the texts:11,,1^11^1,, 1^1^1 1^1^1, ...., .... composed only of „composed only of „11”:”:

xxNN ≡ ≡ yy (( yy subtext subtext xx → → „„11” ” subtextsubtext y y ))

Definition of Definition of additionaddition (no trouble) :(no trouble) :Addition identifies with concatenation:Addition identifies with concatenation:

x+y = xx+y = x ^̂ yy Definition of Definition of multiplicationmultiplication is more is more

complicatedcomplicated..

1414

The definition of multiplication as an example of The definition of multiplication as an example of the possibility of uttering inductive constructions the possibility of uttering inductive constructions

using concatenationusing concatenation We add some 3 new signs:We add some 3 new signs: « , » « , » then we can define a set then we can define a set

of: of: ffiniteinite i inductive nductive ssequence of equence of triplestriples ( (shortly : shortly : fist(xfist(x) ):) ): the first triple is the first triple is ««^ x^ ^ x^ , , ^̂1^ 1^ , , ^̂x^ x^ »» , , the second isthe second is « «^x^ ^x^ , , ^1^^1^1^ 1^ , , x^x^x^ x^ »» etc. etc. The first part of each triple contains always only one The first part of each triple contains always only one xx. . The second part of each triple is composed of several The second part of each triple is composed of several

copies of copies of 1 1 ( It is a Number ( It is a Number [according to the preceding slide [according to the preceding slide 13]) 13])

The The thrid part of each triple is composed of the elementthrid part of each triple is composed of the element x x repeated as many times as repeated as many times as 1 1 is repeated in the second is repeated in the second part of the triple . part of the triple .

(The last property will be assured by inductive condition in (The last property will be assured by inductive condition in the following definition of the following definition of fist(xfist(x) )) )

1515

A general Definition of A general Definition of fistfist may be writen in the may be writen in the following way:following way:

A textA text s s is is fist fist (x) (x) iff iff 1. The initial condition: 1. The initial condition: the first triple is:the first triple is:

««^̂ x x^̂ , , ^̂11^̂ , , ^̂xx^̂ »» 2.2. The inductive condition:The inductive condition:

If((If((««^x^^x^,,^k^^k^,,^z^ ^z^ »» and and ««^x^^x^,,^k`^^k`^,,^z`^ ^z`^ »» are the are the neighborneighbor triples in the sequence triples in the sequence s s )) then: then:

( ( k`= k^1 k`= k^1 and and z`= zz`= z^̂xx )). )). Where two triples are called neighbor when there is Where two triples are called neighbor when there is

nothing between them in nothing between them in ss and the second is just the and the second is just the next one.next one.

Then we define the Then we define the relationrelation of multiplication: of multiplication: ((not a not a function!function!))

1616

Multiplication proves to be discernibleMultiplication proves to be discernible Using finite inductive sequences of triples (Using finite inductive sequences of triples (fistfist) )

of the preceding slide, we can define the relation of the preceding slide, we can define the relation of multiplication as follows:of multiplication as follows: ( M(x,y,z) means x.y=z )( M(x,y,z) means x.y=z )

M(x, y, z) M(x, y, z) ≡≡ s s ( ( s s isis fist(fist(xx)) & & ««^x^^x^,,^1^^1^,,^x^^x^»» is the first is the first triple of triple of s s && ««^x^^x^,,^y^^y^,,^z^^z^»» is the last triple of is the last triple of s s ))..

Multiplication may be also defined as follows:Multiplication may be also defined as follows: M(x, y, z) M(x, y, z) ≡ ≡ s,s,uu (( (( s s isis fist(fist(xx)) & & ««^̂x^x^,,^1^^1^,,^x^^x^»» is the is the

first triple of first triple of s s && ««^x^^x^,,^y^^y^,,^u^^u^»» is the last triple of is the last triple of s s ) ) → → u=z)u=z)

Proof: Multiplication is definable by dual Proof: Multiplication is definable by dual quantification.quantification. Hence is discernible.Hence is discernible.

1717

What are texts?What are texts?There are two conceptions of There are two conceptions of texttext::1. 1. MaterialisticMaterialistic and 2. and 2.

IdealisticIdealisticThere are many separate material There are many separate material things which things which

are are issues of issues of OneOne paper paper In Maths we speak on Texts as In Maths we speak on Texts as IdealisticIdealistic entities: entities: Texts are Texts are abstract entitiesabstract entities which are obtained which are obtained

by two (simultaneous) identifications: by two (simultaneous) identifications: We We identify two materialistic textsidentify two materialistic texts: : which have: which have: 1) the same 1) the same orderorder of letters, of letters, and and 2) corresponding 2) corresponding lettersletters (atomic texts) have the same (atomic texts) have the same shapeshape . . This intuition may be generalized! This intuition may be generalized! (as follows)(as follows)

1818

Formal (set-theoretical) definition of TEXTSFormal (set-theoretical) definition of TEXTS We consider: a (set-theoretical) We consider: a (set-theoretical) UniverseUniverse: U and: : U and: Family F of ordered pairsFamily F of ordered pairs: : X, RX, RXX where X where XU and RU and RX X is a relation is a relation

which which orderorders thes the set set X. X. Family L of „Family L of „LettersLetters”: L is a family of disjoint sets which cover U:”: L is a family of disjoint sets which cover U:

aaU U ≡ ≡ W (W W (WL and aL and aW)W) L is L is a classification of charactersa classification of characters W,ZW,ZL → ( WL → ( WZ= Ø or W=Z )Z= Ø or W=Z ) if a,bif a,bW and WW and WL then a and b are treated as the same letter (of the L then a and b are treated as the same letter (of the

same character). same character). We may say that : two pairs: We may say that : two pairs: X, RX, RXX and and Y, RY, RYY are are similarsimilar iffiff there is a there is a

1-1 function f mapping X on Y in such a way that: 1-1 function f mapping X on Y in such a way that: 1.1. The function f preserves the type of ordering:The function f preserves the type of ordering:

If If a,b ( a,ba,b ( a,bX → ( a X → ( a RRXXb b ≡ f(a) ≡ f(a) RRYY f(b) ) f(b) ) and and 2.2. The function f preserves theThe function f preserves the characters characters: :

a, W ( Wa, W ( WL L → ( a→ ( aW ≡ f(a)W ≡ f(a)W )W ) The The classes of abstractionclasses of abstraction of this of this similaritysimilarity may be called TEXTS, may be called TEXTS,This may be formally written as follows (in the next slaid). This may be formally written as follows (in the next slaid).

1919

Formal (set-theoretical) definition of TEXTS Formal (set-theoretical) definition of TEXTS (continued)(continued)

First we define First we define [[X, RX, RXX ] ] as the TEXT (idealistic) determined by a as the TEXT (idealistic) determined by a (‘materialistic’) set-theoretical pair (‘materialistic’) set-theoretical pair X, RX, RXX : :

[[X, RX, RXX ] = { ] = {Y, RY, RYY: : Y, RY, RYY similar to similar to X, RX, RXX } } Now the Definition of TEXTS is simply as follows:Now the Definition of TEXTS is simply as follows:P P TEXTS TEXTS ≡ for some ≡ for some X, RX, RXX : P = [ : P = [X, RX, RXX ] ] and the and the Definition of concatenation ^Definition of concatenation ^ : :

[[Z,RZ,Rzz ] = [] = [X, RX, RXX ] ^ [ ] ^ [Y, RY, RYY ] ] iffiff for some X`, R` for some X`, R`XX, Y`, R`, Y`, R`Y Y : : X`,R`X`,R`X`X` [ [X, RX, RXX ] & ] & Y`, R`Y`, R`Y`Y` [ [Y, RY, RYY ]& X` ]& X`Y`=Y`=Ø & Ø & Z=X`Z=X`Y` & Y` & a,ba,bZ (aRZ (aRzzb b ≡≡ ((a ((aX`& bX`& bY`) or Y`) or

(a,b(a,bX`& a R`X`& a R`X`X`b) or (a,bb) or (a,b Y`& a R` Y`& a R`Y`Y`b) )b) )

2020

The axioms of the elementary theory The axioms of the elementary theory of TEXTSof TEXTS

(Following Tarski [1931])(Following Tarski [1931])A1 x^(y^z)=(x^y)^z A1 x^(y^z)=(x^y)^z ( (connectivity )connectivity )A2 x^y=z^u A2 x^y=z^u →→ (((x=z (x=z ΛΛ y=u) y=u) VV ww(((w^u=y (w^u=y ΛΛ x^w=z) x^w=z) VV (z^w=x (z^w=x ΛΛ

w^y=u)w^y=u))))) ((the axiom of: 'the axiom of: 'editor'editor'))A3A3 ≠ ≠ x^y x^y A4 A4 ≠≠ x^y x^y

A5A5 ≠≠ [some results on A1-A5 are in the paper: Grzegorczyk & Zdanowski [some results on A1-A5 are in the paper: Grzegorczyk & Zdanowski

Undecidability and Concatenation , Undecidability and Concatenation , in the book dedicated to in the book dedicated to Andrzej Mostowski (to appear) ]Andrzej Mostowski (to appear) ]

2121

For the endFor the end an interesting open problem !an interesting open problem !

A conjecture: A conjecture: For every model For every model M, ^M, ^MM of the theory of the theory A1-A5A1-A5

there exist a set-theoretical universe U and there exist a set-theoretical universe U and the related families F,L such that: the related families F,L such that: TEXTSTEXTSFLFL, ^, ^ is isomorphic with is isomorphic with M, ^M, ^MM . .

??It may be a theorem of representation similar to the It may be a theorem of representation similar to the

theorem of representation of Boolean algebras. ?theorem of representation of Boolean algebras. ?The end.The end.