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					Synthetic concepts a priori

Marie Duží,
VSB-Technical University, Ostrava

Pavel Materna
Czech Academy of Sciences, Prague


                                    1
    Stating the problem
    (From the intuitionistic point of view
     the problem has been formulated
     by Per Martin-Löf.)

   In Kritik der reinen Vernunft, A, 6 - 7,
    Kant defines synthetic judgments a priori.
    Analytical judgments are those in which the
    predicate is contained in the subject.
    The others are called by Kant synthetic.


                                                  2
  Stating the problem
 Kant‟squestion whether there are
 judgments that are a priori, but
 (surprisingly) synthetic, is not trivial:
      it might seem that if a judgment is true
      independently of the state of the world, i.e.,
      a priori, then it is true due to its predicate
      being contained in the subject.
      Kant tries to show that it is not so.

                                                   3
Kant’s example
Kant‟s attempt to prove the existence of synthetic
  a priori judgments by considering
                     7 + 5 = 12
  shows the weakness of his assumption that
  each sentence can be understood as an
  application of a predicate to a subject. In
  mathematics such a reduction is untenable.
This attempt has been analyzed and criticized at
  the beginning of 20th century by the French
  mathematician L. Couturat.


                                                     4
   Kant, a rational core
Modification of Kant‟s problem:
 Concept of the number 12 is not contained in the concept
  7 + 5.
Or, in other words:
 The concept 7 + 5 is not itself sufficient to identify the
  number 12.

  Intuitively it is obvious that in this case such a statement is
  not true.
  Consider, however, some other mathematical notions that
  are not as simple as the notions used in Kant‟s example:

                                                                5
 Synthetic concepts a priori;
 the problem
Question: do the following concepts „sufficiently
  identify (or present)‟ the respective entities?
 The number of prime twins 
                   (natural or transfinite) number
 The number of prime twins is infinite 
                   Truth-value
 Fermat’s last theorem  Truth-value
 Theorems of the 2nd order predicate logic 
             a class of formulas
 The number   irrational number


                                                     6
    Synthetic concepts a priori
                 The answer depends on the way we define
              concept, concept a priori, concept a posteriori,
                         and on the way we explicate
                 ‘concept itself sufficiently identifies …’.

   We are going to define „feasibly executable concepts‟ in terms of the
    structure of concepts (without any reference to psychological content of
    any-being‟s capacities).
   Obviously, any set-theoretical theory of concepts (e.g. Fregean) cannot
    be competent:
      „there is nothing about a set in virtue of which it may be said to present
       something (Zalta);
      each (general) concept is in such a theory identified with the respective set.
      We wish to distinguish between a concept of an entity and the entity itself
   Moreover, a concept cannot be conceived as an expression, but as an
    extra-linguistic, abstract object.


                                                                                  7
    Procedural theory of concepts
   Inspiration by Frege, Church:
     expression (has its) sense = concept = mode of
      the presentation (of the denoted entity D)
Expression  concept  entity D
              expresses          identifies

   Concept = procedure (instruction), the output of
    which (if any) is an entity D
   Concept, being a procedure, is structured;
    it consists of constituents subprocedures,
    never of non-procedural objects

                                                       8
    Procedural theory of concepts
   Pavel Tichý (1968): „Sense and Procedure‟, later
    in „Intensions in terms of Turing machines‟
    formulated the idea of structured meanings:
    meaning of an expression is a procedure
    (structured in an algorithmic way), a way of
    arriving at the denoted entity; TIL construction
   Pavel Materna (1988): „Concepts and Objects‟:
    concept is a closed construction
   Y. Moschovakis (1994, 2003): sense and
    denotation as algorithm and value


                                                       9
    Concepts: a priori, a posteriori
   Each concept, even an empirical one, identifies the
    respective entity a priori: the output of the procedure does
    not depend on the state of the world.
   Empirical concepts are, however, a posteriori with
    respect to the value of the identified intension: they identify
    the denoted entity D a priori, but D is an intension: a
    function, the value (reference of an expression) of which
    depends on the state of the world; this value cannot be
    determined without an experience
   Mathematical concepts are a priori: D is an extension
    (not a function from possible worlds…)



                                                                10
    Concepts: synthetic, analytic
 Empirical concept a posteriori  synthetic:
  identifies an intension.
 Mathematical concept Ca priori analytic:
    C   identifies an extension E without mediation of
      any other concepts but its constituents;
     The procedure C is complete, it is itself sufficient
      to produce its output: 7+5 identifies 12
          Understanding the instruction 7+5, we don‟t need any
           other concepts but the concepts of the function +, and
           of natural numbers 7 and 5 to identify the number 12


                                                                11
Mathematical concepts: analytic ?
   The number of prime twins
   The number of prime twins is infinite
   The number  (= the ratio of …)
   abcn (n  2  (an + bn = cn))
   Theorem of the 2nd order predicate logic
   Goldbach‟s conjecture
       We understand the above expressions; we know the concepts
        (instructions, what to do)
       The respective entity D (truth value, number, set of formulas) is
        exactly determined
       Yet, we do not have to know D,
       the procedure is not complete, we need „a help‟ of some other
        concepts to identify D


                                                                            12
       Synthetic concepts:
     non-executable instructions ?
   Platonic (realist) answer: abstract entities exist; the
    instructions are always executable. If not by a
    human being, then by a hypothetical being whose
    intellectual capacities exceed our limited ones.
   Intuitionist‟s answer (Fletcher): “for me, only those
    abstract entities exist that are well defined ”
   Question: in which sense can the definition be
    insufficient?


                                                         13
     How to logically handle structured
       meanings?
   TIL constructions
   Specification in TIL: Montague-like lambda terms
    (with a fixed intended interpretation) that denote, not
    the function constructed, but the construction itself
   Rich ontology: entities organized in an infinite
    ramified hierarchy of types
     any entity of any type of any order (even a construction)
      can be mentioned within the theory without generating
      paradox.

                                                              14
     Constructions - structured meanings
     A direct contact with an object:
a)   variables x, y, z, w, t … v-construct entities of any type
b)   trivialisation 0X           constructs X
     Composed way to an object:
c)   composition [X        X1 ... Xn ] the value of the function / 
                (1…n) 1       n
d)   closure [ x1...xn X]   constructs a function / ( 1…n)
                1 n 
Examples:      „primes‟: 0prime
               „primes are numbers with exactly two factors‟:
                       0prime = x [[0Card y [0Factor y x]] = 02]

               „the successor function‟: x [0+ x 01]

                                                                     15
    Concepts  definitions
   Concept is a closed construction
   An atomic concept : does not have any other sub-
    concepts (used as constituents to identify an object)
    but itself:
       Trivialisation  05, 07, 0+, 0prime, …, and
       construction of an identity function  x.x
   A composed concept: does have other constituents
    …
       composition  [0+ 05 07], [x [0+ x 07] 05]  number 12
       closure  x [0+ x 07]  adding number 7 to any number


                                                            16
  Concepts  definitions
Definition of an entity E: a non-empty composed concept of E
      [0+ 05 07], [x [0+ x 07] 05] define the number 12
      x [0+ x 07] defines the function adding 7
      [0: 05 00] is empty; it is not a definition, does not
                                      identify anything
      [0Card xy [[0prime x]  [0prime y] 
                   !z [[0prime z]  [x  z  y]] ]
                                      defines the number of prime twins 
                   but we are not able to determine the number
                   in a finite number of steps;

       Is it a good definition?
       In other words, is the last concept analytic ?


                                                                            17
    Analytic concepts  definition
1st attempt:
 An a priori concept C is analytic if it identifies
 the respective object in finitely many steps
 using just its constituents; otherwise C is
 synthetic
But: 0prime  a one-step instruction: grasp the
 actual infinity ! Only God can execute this
 step

                                                18
Analytic concepts  definition
   x [0+ x 01]  a three step instruction:
       Identify the function +
       Identify the number 1
       For any number k apply + to the pair k,1
   Three executable steps ?
   Yes, providing the number k is a rational
    number; in case of an irrational number k the
    third instruction step involves infinite number of
    non-executable steps !

                                                         19
Analytic concepts  definition
 2nd attempt:
  An a priori concept C is analytic if it
  identifies the respective object in an
  effective way using just its constituents;
  otherwise C is synthetic
 „effective way‟ has to be explicated:
 Consider 0prime (ineffective way) vs.
  x [0Card y [0Factor y x] = 02]

                                               20
    Analytic concepts  definition
     x [[0Card y [0Factor y x]] = 02]
    Consists of „finitary‟ instruction steps:
    1.   For any natural number (x) do:
    2.   Compute the finite set F of factors of x
          y [0Factor y x]
    3.   Compute the number N of elements of F
          [0Card y [0Factor y x]]
    4.   If N=2 output True, otherwise False
    The procedure does not involve the actual infinity;
     for any number x it decides whether x is a prime;
     potential infinity is involved

                                                       21
    Analytic concepts  definition
   3rd attempt:
    An a priori concept C is analytic if it identifies the
    respective object in a finitary way using just its
    constituents; otherwise C is synthetic
   Finitary way actual infinity is not involved
   Fletcher: the very simplest type of construction
    allows just a single atom (call it „0‟) and a single
    combination rule (given a construction x we may
    construct Sx) with no associated conditions

                                                         22
    Problem: trivialisation
   Question:
    analytic = -computable = recursive definition ?
   x [0Card y [0Factor y x] = 02]
   [0Factor y x]  Factor(of) / () is an infinite binary
    relation on natural numbers; doesn‟t [0Factor y x] involve
    actual infinity?
    Yet, for any numbers x, y the procedure is executable in a
    finite number of steps;
    providing we “know what to do”
   Shouldn‟t we replace the atomic concept 0Factor with a
    definition of the relation?
   y is a factor of x iff y divides x without a remainder


                                                             23
    Problem: trivialisation
   But then we‟d have to define the relation of
    „dividing without a remainder‟
   Where to stop this refining?
   Fletcher: „0‟, Sx [0Successor x]
   But: 0Successor returns actual infinity !
    Though [0Successor x] is perfectly executable for
    any number;
   Intuitionistic approach:
    “end up” with the construction and “cut off” the
    constructed entity

                                                        24
Problem: trivialisation
   Our proposal:
     using   (de dicto) trivialisation of actual infinity,
      e.g., 0Successor – synthetic
     using (de re) trivialisation of infinity like in
      x [0Successor x] constructs only potential
      infinity – analytic




                                                              25
    Analytic concepts and recursive
    functions
   Analytic a priori concepts are those that identify n-ary
    (n  0) recursive functions (in the finitary way)
   Consequence: there are more synthetic than analytic
    concepts a priori
     Thereare uncountably many functions, but only countably
      many recursive functions
   There are also synthetic concepts a priori that identify
    recursive functions in an ‘non-effective’ way


                                                           26
    Problem: an analytic counterpart of
    a synthetic concept a priori
   If a synthetic concept identifies a recursive function R in a
    non-finitary way, then there is an analytic equivalent concept
    that identifies R in a finitary way.
   A synthetic concept specifies a problem; one feature of the
    development of mathematical theories consists just in
    seeking and finding an analytic concept (solution of the
    problem) equivalent to the respective synthetic one.
   Among many examples we can adduce the discovery of a
    finitary calculation of any member of the infinite expansion of
    the number .


                                                               27
    Problem: an analytic counterpart of
    a synthetic concept a priori
   To understand this creative process we must be aware of
    the following fact:
   The possibility of discovering a new concept is limited by the
    resource of atomic (simple) concepts that are at our disposal.
   A conceptual system S is given by a set of simple
    concepts, from which all other complex concepts belonging
    to S are composed.
   It happens frequently that an analytic counterpart of the
    synthetic concept cannot be defined within the given
    conceptual system S. But later on some extension and/or
    modification of S comes into being; the new system S’
    makes it possible to find the analytic counterpart. A classical
    example:

                                                               28
    Example: Fermat‟s Last Theorem
   The concept given by the original formulation of
    Fermat‟s Last Theorem, i.e., by
                abcn (n > 2  (an + bn = cn))
   is synthetic in that it is impossible to calculate the
    respective truth-value.
   The concept given by the famous proof of FLT
    can be construed as the analytic counterpart of
    the former concept but the conceptual system that
    made it possible to construct the proof is an
    essential expansion of the system used by
    mathematics long after Fermat‟s LT.

                                                        29
Synthetic concepts a priori



    Thank you for your attention !




                                     30

				
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