Transcript Synthetic concepts

Synthetic concepts a priori
Marie Duží, Pavel Materna
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
question whether there are
judgments that are a priori, but
(surprisingly) synthetic, is not trivial:
 Kant’s

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
c)
d)
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]
15
Concepts  definitions


Concept is a closed construction
An atomic concept : does not have any other subconcepts (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.
2.
3.
4.

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 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
 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
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