tq- C- P- A- B- @|- Problem Set

1)      Write down the axioms schema and the three shortest axioms in the tq-system

R: xT-Qx is an axiom, whenever x is a hyphen string.

    -t-q-

    --t-q--

    ---t-q---

2)      Write down the sole rule of inference for the tq-system and apply it to the well-formed string -----t-----q-----

R: Rule: Suppose that x, y and z are all hyphen-strings. And suppose that xTyQz is an old theorem. Then, xTy-qzx a new theorem.

(1) -----t-----q----- axiom

(2) -----t------q---------- from (1) using the rule

3)      Reasoning in I-mode, argue that the string you produced in the previous item is not a theorem in the tq-system

R: It cannot be a theorem because 5x6 != 11. 

4)      Working in M-mode, show that -----t ---q --------------- is a theorem in the tq-system.

R: Use the rule to go back and find the axiom, if it is an axiom, than it is a theorem.

(1)    -----t---q---------------   theorem

(2)    -----t--q---------- from (1) using the inverse of the rule

(3)    -----t-q----- from (2) using the inverse of the rule

-----t-q------ = xT-Qx

From this we ca conclude that ------t---q--------------- is a theorem in the tq-system.

5)      What are the two rules of C-system

R: 
1) Suppose x, y and z are hyphens-strings. If x-ty-qz is a theorem then Cz is a theorem.
 2) Suppose that x, y and z are all hyphen-strings. And suppose that xTyQz is an old theorem. Then, xTy-qzx a new theorem.

6)      Working within the C-system, argue that C-------- is a theorem of the system.

R: --t-q-- axiom
(2) --t--q---- from axiom using rule 2
(3) --t---q----- from (2) using rule 2
(4) --t----q-------- from (3) using rule 2
(5) C-------- from (4) using rule 1

7)      Does adding the following rule to the C-system constitute a Post production system for determining primes? Please explaining your response. Suppose x is a hyphen-string. If Cx is not a theorem, then Px is a theorem.

R: No, because the rule does not entail a purely typographical operation.

8)      Hofstadter writes. When a figure or “positive space” (e.g., human form, or a letter, or a still life) is drawn inside a frame,an unavoidable consequence is that its complementary shape – also called the “ground” , or “background”, or “negative space” – has also been drawn.  According to this view, the quiche pan shown below that I computationally rendered would be considered negative space. Explain how this is so.

R: The drawn consist in the black space and the withe space is the ground.

9)      Consider the A-system as defined by the following axiom and theorem. Axiom A-- , Rule: suppose that x is a hyphen-string, If Ax is a theorem , so is Ax--

a)      Show that A -------- is a theorem of the A-system by working within the system

R: (1) A - - axiom

     (2) A - - - - from (1) using the rule I

      (3) A - - - - - - from (2) using the rule I

      (4) A -------- from(3) using the rule I

b)      Specify a decision procedure for determining theorem hood in the A-system

R: Suppose you have Ax, if you can use the rule to go back to A -- (The only axiom) you can say that Ax is a theorem.

c)       Provide an I-mode argument that the string A------------ is not a theorem of the A-system

R: All theorems in the A-system have an even number of hyphens.

d)      What subset of the natural numbers do you think it was my intent to capture with the A-system

R: All even numbers.

      

10)   Consider the as yet to be formally defined B-system which you should imagine is intended to capture precisely all of the natural numbers that the A-system does not capture.

a)      Propose, by analogy with the rule on page 66 of GEB, an invalid rule for producing theorems in the B-system.

R: Suppose x is a hyphen-string. If Ax is not a theorem, then Bx is a theorem.

b)      Define a (valid) Post production system for the B-system in terms of one axiom and one rule

R: Axiom: B-

 Rule I: Suppose that x is a hyphen-string. If Bx is a theorem, so is Bx--

c)       Derive B----------- within the B-system.

R: (1) B- axiom

      (2) B--- from(1) using the rule I

      (3) B----- from (2) using the rule I

      (4) B------- from (3) using the rule I

      (5) B--------- from (4) using the rule I

      (6) B----------- from(5) using the rule I

d)      What subset of the natural numbers does the B-system capture?

R: All odd numbers.

11)   Under interpretation, what does the A-system theorem A -------- say? Under interpretation, what does the B-system theorem B----------- say?

R: Say that 8 is an even number and 11 is a odd number.

12)   What does mean for a set to be “recursively enumerable”? What does it mean for a set to be “recursive”?

R: Recursively Enumerable: Means that it can be generated according to typographical rules.

     Recursive: A set is recursive if you can write a program that tells you whether the given input is in the set or not.

13)   Argue that the set of even number is recursively enumerable.

R: Using the A-system you can produce every even number. The A-system use a typographical rule

14)   Argue that the set of even number is recursive

R: It is recursively enumerable and its complement (odd numbers) is recursive because you can use B-system to get all odd numbers.

15)   Argue that the set of prime numbers is recursively enumerable.

R:You could use rule of the Px-system to get all recursive enumerable.

16)   Argue that the set of prime numbers is recursive.

R: It is recursively enumerable and its complement (odd numbers) is recursive because you can use C- system to the rest of the numbers.

17)   In a sentence or two, explain why you think that I am not asking you in this problem set to derive something lime P----- within the P-system?

R: Because the P system it is not a formal system that have a typographical rule.

18)   Consider the following Post production system, a system that we will call the @?- system:

AXIOM: @ | | - ||| -

Rules: Suppose x is a string made up of zero or more symbols drawm from the set { @ , - , |} then:

1: if you have x-||-|||, you can add –

2: if you have x-|||-||, you can add –

3: if you have x|-||-||, you can add |

4: if you have x|-|||-|, you can add |

5: if you have x||-||-|, you can add |

6: if you have x|-|||-, you can add |

7: if you have x|||-||-, you can add |

8: you can replace @ by C

9: you can replace @ by G

10: you can replace @ by D

11: you can replace C| by CD

12y you can replace C- by CV

13: you can replace V| by VW

14: you can replace V- by VD

15: you can replace D| by DE

16: you can replace D- by DW

17: you can replace W| by WF

18: you can replace W- by WE

19: you can replace E| by EX

20: you can replace E- by EF

21: you can replace F| by FG

22: you can replace F- by FX

23: you can replace X| by XY

24: you can replace X- by XG

25: you can replace G| by GA

26: you can replace G- by GY

27: you can replace Y| by YZ

28: you can replace Y- by YA

29: you can replace A| by AB

30: you can replace A- by AZ

31: you can replace Z| by ZC

32: you can replace Z- by ZB

33: you can replace B| by BV

34: you can replace B- by BC

a)      How many axioms in the system? How many rules in the system? How many theorems in the system?

R:  1 Axiom. 34 Rules. Infinite theorems.

b)      Show that @||-|||-||-|||- is a theorem of the @|- system by performing a derivation within the system.

R: @||-|||-  axiom

     (2) @||-|||-| from(1) by rule 6

     (3) @||-|||-|| from(2) by rule 4

     (4) @||-|||-||- from(3) by rule 2

     (5)@||-|||-||-| from(4) by rule 7

     (6)@||-|||-||-|| from(5) by rule 5

     (7)@||-|||-||-||| from(6) by rule 3

     (8) @||-|||-||-|||- from(7) by rule 1

c)       Show that CDEFGABC  is a theorem of the @|- system by performing a derivation within the system

R: (1) @||-|||-  axiom

     (2) C||-|||- from (1) by rule 8

     (3) CD|-|||- from (2) by rule 11

     (4) CDE-|||- from (3) by rule 15

     (5)CDEF|||- from (4) by rule 20

     (6)CDEFG||-from (5) by rule 21

     (7)CDEFGA|-from (6) by rule 25

     (8)CDEFGAB - from (7) by rule 29

      (9) CDEFGABC from (8) by rule 34

d)      Show that GABCDEXG  is a theorem of the @|-system by performing a derivation within the system

R: (1) @||-|||-  axiom

     (2) G||-|||- from (1) by rule 9

     (3) GA|-|||- from (2) by rule 25

     (4) GAB-|||- from (3) by rule 29

     (5)GABC|||- from (4) by rule 34

     (6)GABCD||-from (5) by rule 11

     (7)GABCDE|-from (6) by rule 15

     (8)GABCDEX - from (7) by rule 19

      (9) GABCDEXG from (8) by rule 24

e)      Show that DEXGABVD is a theorem of the @|-system by performing a derivation within the system

R: (1) @||-|||-  axiom

     (2) D||-|||- from (1) by rule 10

     (3) DE|-|||- from (2) by rule 15

     (4) DEX-|||- from (3) by rule 19

     (5)DEXG|||- from (4) by rule 24

     (6)DEXGA||-from (5) by rule 25

     (7)DEXGAB|-from (6) by rule 29

     (8)DEXGABV - from (7) by rule 33

      (9) DEXGABVD from (8) by rule 14

f)       What do you think I had in mind when I invented this system?

 R:Generate musical scales.

Click here to download the PDF

Markov song

I love you ( Barney Song )
GW EW GW GW EW GW AW GW FW EW DW EW FW EW FW GW CQ CQ CQ CQ DW EW FW GW GW DW DW FW EW DW CQ

31 Notes.
6 Different notes.
2 Times ( Q and W).

	GW	EW	FW	DW	CQ	AW
GW	1	2	1	1	1	1
EW	2	0	3	2	0	0
FW	2	3	0	0	0	0
DW	0	2	1	1	1	0
CQ	0	0	0	1	3	0
AW	0	0	0	0	1	0

	GW	EW	FW	DW	CQ	AW
GW	0.142	0.288	0.142	0.142	0.142	0.142
EW	2	0.288	0.428	0.288	0	0
FW	0.4	0.6	0	0	0	0
DW	0	0.4	0.2	0.2	0.2	0
CQ	0	0	0	0.25	0.75	0
AW	0	0	0	0	1	0

	GW	EW	FW	DW	CQ	AW
GW	0.142	0.427	0.569	0.711	0.853	1
EW	0.288	0.716	1	1	1	1
FW	0.4	1	1	1	1	1
DW	0	0.4	0.6	0.8	1	1
CQ	0	0	0	0.25	1	1
AW	0	0	0	0	1	1

I did a java code to generate the variations:

Variation 1
GW AW CQ DW EW FW EW EW FW EW FW EW FW GW CQ CQ DW DW FW EW FW EW FW GW GW AW CQ CQ DW CQ

Variation 2
GW GW DW DW EW FW GW AW CQ CQ DW EW FW GW FW GW CQ CQ CQ CQ CQ CQ CQ DW CQ CQ DW FW GW GW EW

Improvising : Original + Var1 + Original + Var2 + Original.
Code
Original
Variation 1
Variation 2
Full Song
PDF

The pq-System Problem Set

1)      What is the formal system of Chapter 2 called?

R:  The formal system presented in this chapter is called pq-system.

2)      What are the distinct symbols of this formal system?

R: p q -.

3)      How many axioms in the pq-system?

R: Infinite number of axioms.

4)      Write down the axiom schema for the pq-system.

R: xP – Qx – is an axiom, whenever x is composed of hyphens only.

5)      What is a “schema”? Define the term.

R: A mold to define something. How to do something.

6)      Write down the three shortest axioms in the pq-system.

R:- p - q- -  / - -p -  q - - - / - - - p  - q - - - -

 

7)      Write down the sole rule of production for the pq-system

R:  Suppose x, y and z all stand for particular strings containing only hyphens. And suppose that x p y q z is known to be a theorem. Then x p y – q z – is a theorem

8)      Show that - -p - -q - - - - is a theorem of the pq-system. That is, derive it from an axiom and repeated application of the rule.

R: (1) - - p - q - - -    axiom

     (2) - - p - - q - - - -  from (1)

     

9)      Show that - - - - -p- - - -q- - - - - - - - - is a theorem of the pq-system. That is, derive it from an axiom and repeated application of the rule.

R: (1) - - - - - p - q - - - - - -     axiom

     (2) - - - - -p - - q - - - - - - -   from(1)

     (3) - - - - -p - - - q- - - - - - - from(2)

     (4) - - - - -p - - - -q - - - - - - - - - from(3)

10)   Write down a string of symbols in the pq-system which is not well formed

R: - - p - - p - -p - - q - - - - - - - -

11)   State a decision procedure for the pq-system

R: Take your theorem, iterate backwards, find the beginning and check whether it`s an axiom or not, if it is an axiom, then it is by definition a theorem, and the test is over.

12)   In the longest paragraph on page 48, Hofstadter engages in some “top-down” reasoning. In one sentence, articulate exactly what it is that he demonstrates with his top-down reasoning in this paragraph?

R: How to find the beginning of all theorems, the axiom schemata or maybe you will find something that is not an axiom.

13)   In one sentence, characterize “top-down” reasoning

R: Start in the theorems and find the axioms.

14)   In one sentence, characterize “bottom-up” reasoning

R: Start in the axioms and find the theorems

15)   Consider the procedure for generating theorems of the pq-system given at the top of page 49. What will be in the bucket after executing statements (1a) and (1b) and (2a) and (2b) and(3a) and (3b) – after all six of these statements have been executed!

R: (1a) - p - q - -

     (1b) - p - - q - - -

     Bucket:   - p - q - -; - p - - q - - -

     (2a) - - p - q - - -

     (2b) - p - - q - - - ; - p - - - q - - - -; - - p - - q - - - -

     Bucket: - p - q - -;  - p - - q - - -;- p - - - q - - - -;  - - p - q - - -; -- p - - q - - -- ;

     (3a) - - - p - q - - - -

     (3b) - p - q - -;  - p - - q - - -; - p - - - q - - - -; - p - - - - q - - - - - ;   - - p - q - - -;   - - p - - q - - - -;

; - - p - - - q - - - - -;  - - -p  -  q - - - -; - - -p - -q - - - - -;

16)   16 What role does the procedure introduced on the top of page 49 ply in Hofstander`s presentation of the pq-system and related matters? Answer in just one sentence!

17)   What is an isomorphism?

R: Two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure.

18)   What is an interpretation in the context of a formal system?

R: The symbol-word correspondence that you give to your formal system.

19)   When was Linear B deciphered?

20)   How many meaningful interpretations of the pq-system did Hofstadter present in this chapter?

R: 2

21)   How many meaningless presentations of the pq-system are there?

R: Infinite.

22)   In 50 plus or minus 20 words, summarize what Hofstadter says in the section titled “Formal systems and reality.”

R: Hofstadter says that is possible combine reality and formal systems, and maybe the reality is just a big formal system. Affirm that the reality is deterministic is something hard to accept.

PDF

Pseudocode Assignment

1) What is pseudocode?

                Pseudocode is a detailed description of what a computer program or algorithm must do, expressed in a formally-styled natural language rather than in a programming language.

Pseudocode:

IF today is Monday THEN
        Roll a dice/Randomize 1 to 6.
        IF {1,2,3,4} THEN set A
        IF {5} THEN set B

        IF {6} THEN set C
IF today is Tuesday or Wednesday or Thursday or Friday THEN
        Roll a dice/Randomize 1 to 6.
        IF {3,4}and the last day is different from B & C THEN set B
        ELSE set A

        IF {5,6}and the last day is different from C THEN set C

        IF {1,2} THEN set A


PDF

Biography Day Assignment

                                                                                                                                 

Emil Post

  

“I study Mathematics as a product of the human mind and not as absolute.” (Post – 1920)

Life:

                Emil Leon Post was born in Augustów, a Russian city, in February 11 1897 and died April 21, 1954. His parents, a Jewish family, immigrated to America in May 1904 when he was a kid. In his childhood his first love was astronomy, after Post suffer an accident - a car crashed into other car, and he was in the middle – he lost his left arm, and because of this, he could not follow this profession. He started to show interest to math and attended to Townsend Harris High School, graduated in City College of New York in Math and obtained his Ph.D. in Math at Columbia University.

Work:

                In his first paper Post talked about the Principia Mathematica and proved the consistency as well as the completeness of the propositional calculus as developed in Whitehead and Russell’s Principia Mathematica. He did that using truth-tables (Introduced by C. S Peirce and Schröder). From this paper came general notions of completeness and consistency. For post, a system is complete if every well-formed formula can be proved after introducing a well-formed formula that is not provable. A system is consistent if well-formed formula consisting of only a propositional variable is provable. He also introduced in this paper multivalued systems of propositional logic and introduced multivalued truth tables in analyzing them.

                Post also created the “Post production system” – cited in the MU-puzzle, GEB book – a system that work with strings basically. Post production system is a string-manipulation system that start with a string (A collection of characters) and then applying some rules you can transform/modify this string – Like the MU-puzzle-, this system generates what we call “Formal language”, and this language is composed by the tokens, symbols or letters formed by the rules. From this, we can obtain the recursively enumerable languages – If you have a set S and you have a recursive function whose domain is exactly S then you have a recursively enumerable language.

                He also did some work in 1920 obtaining the same results that Kurt Godel obtained in his theory in 1930. Post theory was fundamental to the progress of the theory of recursive functions.

References:

Web site:

http://www.ualberta.ca/~francisp/Phil428.526/UrquhartPost.pdf ; Accessed: 2/4/2015

http://www.encyclopedia.com/doc/1G2-2830903487.html ; Accessed: 2/4/2015

http://www-history.mcs.st-andrews.ac.uk/Biographies/Post.html ; Accessed: 2/4/2015

Image:

http://www.pucsp.br/~logica/Post.jpg ; Accessed: 2/4/2015

If you want download the pdf, click here.

Slide

MU-system Problem Set

                                                                                                                           

COG 356. MU-system Problem Set.

1) What, does Hofstadter claim, is one of the most central notions running through GEB?

R: In the book, Hofstadter claim that “Formal system” is one of the most central notions running through GEB.

2) Who invented the sort of formal system that Hofstadter features in his book (the sort of system that the MIU-system exemplifies), and when did this invention take place?

R: The formal system that Hofstadter features in his book was invented by Emil Post in 1920’s.

3) In one four-word interrogative question, state the puzzle that is feature in this chapter.

R: “Can you produce MU?”

4) What is the given string in the MIU-system?

R: The given string is MI.

5) What is the goal string of the MU-puzzle?

R: The goal string of the puzzle is MU.

6) How many rules in the MIU-system?

R: The MIU-system has 4 rules.

7) Carefully, precisely, write down the first rule of the MIU-system, and give two example of its use, one directly from the chapter, and one that does not appear explicitly in the chapter.

R: Rule I: If you possess a string whose last letter is I, you can add on a U at the end.

                Example: From MIIII, you may get MIIIIU

                                   From MIUI, you may get MIUIU

8) Carefully, precisely, write down the second rule of the MIU-system, and give two example of its use, one directly from the chapter, and one that does not appear explicitly in the chapter.

R: Rule II: Suppose you have Mx. Then you may add Mxx to your collection.

                Example: From MIU, you may get MIUIU

                                   From MUUI, you may get MUUIUUI

9) Carefully, precisely, write down the third rule of the MIU-system, and give two example of its use, one directly from the chapter, and one that does not appear explicitly in the chapter.

R: Rule III: If III occurs in one of the strings in your collection, you may make a new string with U in place of III.

                Example: From MIII, you may get MU

                                   From MUUIIIUU, you may get MUUUUU

10) Carefully, precisely, write down the fourth rule of the MIU-system, and give two example of its use, one directly from the chapter, and one that does not appear explicitly in the chapter.

R: Rule IV: IF UU occurs inside one of your strings, you can drop it.

                Example: From UUU, you may get U

                                   From MIUUI, you may get MII

11) What is the word used to describe strings that are producible by the rules of a formal system from strings that have already been produced?

R: The word used to describe string that are producible by the rules is “Theorem.”

12) What is the technical term for the string MI in the MIU-system?

R: The technical term for the string MI in the MIU-system is “Axiom.”

13) In a formal system, is it more appropriate to say that theorems are proven or that theorems are produced?

R: In a formal system, it is more appropriate to say that theorems are merely produced according to some rules.

14) How does Hofstadter define the term derivation?

R: Hofstadter defines derivation as “An explicit, line-by-line demonstration of how to produce that theorem according to the rules of the formal system.” (Page 36 – GEB)

15) Reproduce, line by line, character by character (including “reasons” (rule citations)) Hofstadter‘s derivation of the string MUIIU.

R: (1) MI                               axiom

     (2) MII                            from (1) by rule II

     (3) MIIII                         from (2) by rule II

     (4) MIIIIU                      from (3) by rule I

     (5) MUIU                        from (4) by rule III

     (6) MUIUUIU                from (5) by rule II

     (7) MUIIU                       from (6) by rule IV

16) Write down, line by line (Including “reasons” (rule citations)) a derivation of the string MIIUIIU.

R: (1) MI                               axiom

     (1) MII                             from (1) by rule II

     (1) MIIU                          from (2) by rule I

     (1) MIIUIIU                    from (3) by rule I

    

17) On page 37, Hofstadter claims that there is a fundamental difference between a machine and a human? What is that difference?

R: Hofstadter claims that it is possible for a machine to act 100% unobservant and for human this is impossible. Humans can sometimes act unobservant but not 100% like a machine could.

18) With respect to formals systems, what is the difference between working “inside the system” and working “outside the system?”

R: Working “outside the system” means that something is escaping/changing a situation in some way. The book presents the following example: Someone is reading a book and then stop doing this and go to sleep. The person jumped off the system. Working “inside the system” basically means: Do what you are supposed to do (or programmed to do, if it is a machine).

19) Are there any theorems in the MIU-system that do not start with the letter M?

R: No. All the theorems in the MIU-system start with the letter M because all theorems must begin with an M (Page 39 – Decision Procedures).

20) How is the previous question answered, by working within the system or by working outside the system?

R: Outside the system.

21) What does “M-mode” refer to? What does “I-mode” refer to?

R: M-mode: Mechanical model, do the work to solve the puzzle like a machine

    I-mode: Intelligent mode, do the work to solve the puzzle using your intelligence/skills or any other thing that comes in your mind.

22) Do you think that humans can work in M-mode?

R: I think humans can work in M-mode sometimes, if you are really used to a task you may not really pay attention in what you are doing because you already know what you are supposed to do. Example: If you put your hand in the fire (For accident), you will move it out without thinking.

23) Do you think that machines can work in I-mode?

R: I think in the future machines will work in I-mode, but today in my opinion machines cannot work in 100% I-mode. Some machines can think (AI machines), but they cannot be compared to a human mind, not yet.

24) Two of the rules of the MIU-system are lengthening rules. What does this mean? Two of the rules of the MIU-system are shortening rules. What does this mean?

R: Lengthening rules: Rules that can increase the length of your string. For example: Rules I, II.

   Shortening rules: Rules that can decrease the length of your string. For example: Rules III, IV.

25) Define “decision procedure” with respect to a formal system.

R: Decision procedure is a test for theorem hood that always terminate in a finite amount of time. If you have a decision procedure, you have a very precisely/solid description of the nature of all theorems in the system that you are testing.

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First GEB Reading Assignment

Ten things that I found interesting in the first chapter:

   1)  Frederick the Great – King of Prussia - was an intellectual person and really loved music (Compose and Play flute).Moreover, he was a big fan of Bach.

   2)   Johann Sebastian Bach was really good in improvisation playing organ and creating compositions that seems impossible to be composed. He was one of the greatest composers of his time.

  3)      Isomorphism: Describe the same information transforming it in something different. Basically, present the information in a different manner but maintaining the meaning.

  4)      The Musical Offering (Created by Bach in offering to Frederick) is a masterpiece that goes beyond all rules used to compose fugues and canons.

  5)      Strange loops: Start in somewhere, go forward or backward and then you reach the start again (Notice that this result was not expected). Looks like a paradox, an infinite loop that you never can escape.

  6)      Gödel created the Gödel’s Theorem that basically can be summarized in: Transform the numbers in characters or symbols, so you can have a group of symbols, or a sequence. Give to this sequence a unique reference (Gödel number) and then you can understand this sequence in two manners: declarations of number theory or declarations of declarations of number theory.

  7)      Principia Mathematica is the foundation of mathematic. Gödel discovered that it is a half-finished system (Some rules/statements cannot be proved or unproved) based in his Theorem.

  8)      Strange loops are so constant in set theories that Russell saw a need to eliminate them, creating a hierarchy for the groups. It was not a “100% correct” way to solve that, but it solved in some way.

  9)      From the beginning of the computer era people were already aware about the risk/danger of creating an artificial intelligence (I share this feeling.)

 10)   Determine an exactly border - A rule that with total assurance can separate two groups – between intelligent and non-intelligent behavior is hard or probably impossible.

My opinion about the chapter "Three-Part Invention":

                I found the chapter “Three-Part Invention” really mysterious and intriguing. The way that the author presented the environment and the characters is quite interesting. If you stop to think about what Zeno was talking about – his theory “Motion Unexists” – and understand his paradox, like the Tortoise, you may start to think that he was correct about the illusion of motion. In my opinion, I think that this was the Author's point, make you to believe in this, so you would look for an answer to proof if he was or was not wrong, like Achilles. I am already in love with this book.



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