Synthetic concepts a priori

1
Synthetic concepts a priori

• Marie Duží,
• VSB-Technical University, Ostrava
• Pavel Materna
• Czech Academy of Sciences, Prague

2
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.

3
Stating the problem

• Kants question 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.

4
Kants example

• Kants 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.

5
Kant, a rational core

• Modification of Kants 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 Kants example

6
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
• Fermats last theorem ? Truth-value
• Theorems of the 2nd order predicate logic ? a
  class of formulas
• The number ? ? irrational number

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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-beings 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.

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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

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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

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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)

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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 75 identifies
  12
• Understanding the instruction 75, we dont need
  any other concepts but the concepts of the
  function , and of natural numbers 7 and 5 to
  identify the number 12

12
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
• Goldbachs 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

13
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.
• Intuitionists answer (Fletcher) for me, only
  those abstract entities exist that are well
  defined
• Question in which sense can the definition be
  insufficient?

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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.

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Constructions - structured meanings

• A direct contact with an object
• variables x, y, z, w, t v-construct entities
  of any type
• trivialisation 0X constructs X
• Composed way to an object
• composition X X1 ... Xn the
  value of the function / ?
• (??1?n) ?1 ?n
• 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

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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

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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 ?

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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

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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 !

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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

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Analytic concepts ? definition

• ?x 0Card ?y 0Factor y x 02
• Consists of finitary instruction steps
• For any natural number (?x) do
• Compute the finite set F of factors of x ?y
  0Factor y x
• Compute the number N of elements of F 0Card ?y
  0Factor y x
• If N2 output True, otherwise False
• The procedure does not involve the actual
  infinity for any number x it decides whether x
  is a primepotential infinity is involved

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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

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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 doesnt
  0Factor y x involve actual infinity?Yet, for
  any numbers x, y the procedure is executable in a
  finite number of stepsproviding we know what
  to do
• Shouldnt 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

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Problem trivialisation

• But then wed 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

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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

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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
• There are 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

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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 ?.

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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

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Example Fermats Last Theorem

• The concept given by the original formulation of
  Fermats Last Theorem, i.e., by
• ?abcn (n gt 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 Fermats LT.

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Synthetic concepts a priori

• Thank you for your attention !