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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!)

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

.

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)?

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

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

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)?

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)?

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)?

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

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..) )?

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.

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.

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.

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) )?

- 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!)?

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.

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)?

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).

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) )?

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)

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.