quipus

Principia Mathematica in Quipus

§ 82.The system of the PRINCIPIA MATHEMATICA (Whitehead, Alfred North and Russell, Bertrand Arthur William) translated to our Quipus:(The complete version of § 82.  is in : Miguel Calvo R.  Manual of writing in Quipus.E-mail: guelcal@gmail.com. PDF format.  )

 

 

As we know the precedent work “The Principia...” is the outmost work of the xx century in the field of Mathematical logic. Here we are going to sate a little fragment of this great work, writing in symbolic logic (notation) of Łukasiewicz by A.N.Prior2@) . Then –after quoted- we are going to translate it to our quipu-grams and lastly we will put it into the huasca(cord) at present drawing , but with no problem  for materialized – if the reader so wishes it  in real cords and having a wonderful fragment or piece of our contemporaneous Aristotle( simile  with Russell) and Plato (regarding to Whitehead) .

 

82.1: Structure of the PM. System:

  i :     Rule of detachment: If a formula (alpha: α) is a thesis, and  the implication Cαβ (If alpha then  beta) is also a thesis , then the consequent  β can be detached from the implication as a posterior thesis.

  ii:     Undefined Operators: A, N.

  iii :   Definition: 1.01. C= AN.

  iv :   Axioms

1.2: CAppp.

1.3: CqApq.

1.4: CApqAqp.

1.5: CApAqrAqApr.

1.6: CCqrCApqApr.

 

  v :    Theorems:

 

              1.2. p/Np x 1.01 = 2.01.

 

2.01.      CCpNpNp.

 

              1.3. p/Np x 1.01 = 2.02.


 

 

2.02.      CqCpq.

 

              1.4. p/Np,q/Nq x 1.01 = 2.03.

 

2.03.      CCpNqCqNp.

 

              1.5. p/Np,q/Nq x 1.01 = 2.04.

 

2.04.      CCpCqrCqCpr.

 

              1.6. p/Np x 1.01 = 2.05.

 

2.05.      CCqrCCpqCpr.

 

              2.04. p/Cqr, q/Cpq, r/Cpr = C 2.05____ 2..06.

 

2.06.      CCpq CCqrCpr.

 

              1.3. q/p = 2.07.

 

2.07.      Cp App.

 

              2.05. q/App, r/p = C 1.2 ____ C 2.07 ____ 2.08

 

2.08.      Cpp.

82.2...............

82.3.............

 

Miguel Calvo R.  Manual of writing in Quipus.E-mail: guelcal@gmail.com

 

 

82.4.: Demonstration of the theorem  2.01. by means of  quipus:

  i:      1.2.CAppp (first  axiom of the  P.M.) See: 82.1.: iv;(If  p or p then p).

  ii:     1.01.C=AN ( Or well no-p or  q; It is defined as: if  p then q).

 iii:     r x 1.01.: "Apply ii in "

 iv:     Np. (False proposition).

  v:     Knotting "i": We proceed to form a quipu correspondent  to the axiom 1.2.: CAppp. = vi.

 vi:     Fig. 163. of  i.

   vii:   In i: p/Np = viii.

 viii:    CANpNpNp.

viii.i:   50B1 = σ

viii.ii:   5H1 = τ

viii.iii: 70B0 = ε

1+) ix  : In vi: σ/ ετ σ.= x.          

 

x...:    Fig. 164 of viii:

 

 

 

xi  : iii: r/viii = xii.

 

xii : CCpNpNp (Theorem 2.01.)

xii.i: α = 3Ñ.

xii.ii: π = αα.

(+) xiii: In x: ατ ετ / πτ = xiv.      

xiv : Fig. 165 of xii.

 

 

 

 

 

 

 


     2

@)A.N.PRIOR, FORMAL LOGIC, Pages: 26, 27 y 28; Oxford at the Clarendon Press.

In all the context of the §82 the “C”, represents the “implication”; the “A”, the Inclusive disjunction and the “N” the negation logic.

  1+) for better understanding the characters that appeared in ut supra: 82.4 (ix, xiii) to see and compare the tables of the Fig. 141 and of the Fig. 142, 143, 144.