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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.) n 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 n Kant’squestion whether there are judgments that are a priori, but (surprisingly) synthetic, is not trivial: Ø 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: n Concept of the number 12 is not contained in the concept 7 + 5. Or, in other words: n 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? n The number of prime twins ® (natural or transfinite) number n The number of prime twins is infinite ® Truth-value n Fermat’s last theorem ® Truth-value n Theorems of the 2nd order predicate logic ® a class of formulas n The number p ® 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 …’. n 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). n 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 n Moreover, a concept cannot be conceived as an expression, but as an extra-linguistic, abstract object. 7 Procedural theory of concepts n Inspiration by Frege, Church: ¨ expression (has its) sense = concept = mode of the presentation (of the denoted entity D) Expression ¾¾® concept ¾¾¾® entity D expresses identifies n Concept = procedure (instruction), the output of which (if any) is an entity D n Concept, being a procedure, is structured; it consists of constituents ¾subprocedures, never of non-procedural objects 8 Procedural theory of concepts n 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 n Pavel Materna (1988): ‘Concepts and Objects’: concept is a closed construction n Y. Moschovakis (1994, 2003): sense and denotation as algorithm and value 9 Concepts: a priori, a posteriori n 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. n 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 n Mathematical concepts are a priori: D is an extension (not a function from possible worlds…) 10 Concepts: synthetic, analytic n Empirical concept ¾a posteriori ¾ synthetic: identifies an intension. n Mathematical concept C¾a priori ¾analytic: ¨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 n 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 ? n The number of prime twins n The number of prime twins is infinite n The number p (= the ratio of …) n "abcn (n > 2 É Ø(an + bn = cn)) n Theorem of the 2nd order predicate logic n Goldbach’s conjecture Ø We understand the above expressions; we know the concepts (instructions, what to do) Ø The respective entity D (truth value, number, set of formulas) is exactly determined Ø Yet, we do not have to know D, Ø the procedure is not complete, we need ‘a help’ of some other concepts to identify D 12 Synthetic concepts: non-executable instructions ? n 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. n Intuitionist’s answer (Fletcher): “for me, only those abstract entities exist that are well defined ” n Question: in which sense can the definition be insufficient? 13 How to logically handle structured meanings? n TIL constructions n Specification in TIL: Montague-like lambda terms (with a fixed intended interpretation) that denote, not the function constructed, but the construction itself n 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 / a (ab1…bn) b1 bn d) closure [l x1...xn X] constructs a function / (a b1…bn) b1 bn a Examples: ‘primes’: 0prime ‘primes are numbers with exactly two factors’: 0 prime = lx [[0Card ly [0Factor y x]] = 02] ‘the successor function’: lx [0+ x 01] 15 Concepts ¾ definitions n Concept is a closed construction n An atomic concept : does not have any other sub- concepts (used as constituents to identify an object) but itself: t Trivialisation ¾ 05, 07, 0+, 0prime, …, and t construction of an identity function ¾ lx.x n A composed concept: does have other constituents … t composition ¾ [0+ 05 07], [lx [0+ x 07] 05] ® number 12 t closure ¾ lx [0+ x 07] ® adding number 7 to any number 16 Concepts ¾ definitions Definition of an entity E: a non-empty composed concept of E t [0+ 05 07], [lx [0+ x 07] 05] define the number 12 t lx [0+ x 07] defines the function adding 7 t [0: 05 00] is empty; it is not a definition, does not identify anything t [0Card lxy [[0prime x] Ù [0prime y] Ù $!z [Ø[0prime z] Ù [x £ z £ y]] ] defines the number of prime twins ¾ 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 ¾ definition n1st 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 ¾ a one-step instruction: grasp the actual infinity ! Only God can execute this step 18 Analytic concepts ¾ definition n lx [0+ x 01] ¾ a three step instruction: t Identify the function + t Identify the number 1 t For any number k apply + to the pair ák,1ñ n Three executable steps ? n 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 ¾ definition n 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 n ‘effective way’ has to be explicated: n Consider 0prime (ineffective way) vs. lx [0Card ly [0Factor y x] = 02] 20 Analytic concepts ¾ definition n lx [[0Card ly [0Factor y x]] = 02] n Consists of ‘finitary’ instruction steps: ¨ For any natural number (lx) do: ¨ Compute the finite set F of factors of x ly [0Factor y x] ¨ Compute the number N of elements of F [0Card ly [0Factor y x]] ¨ If N=2 output True, otherwise False t 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 ¾ definition n 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 t Finitary way ¾actual infinity is not involved t 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 n Question: analytic = l-computable = recursive definition ? n lx [0Card ly [0Factor y x] = 02] n [0Factor y x] ¾ Factor(of) / (onn) 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” n Shouldn’t we replace the atomic concept 0Factor with a definition of the relation? n y is a factor of x iff y divides x without a remainder 23 Problem: trivialisation n But then we’d have to define the relation of ‘dividing without a remainder’ n Where to stop this refining? n Fletcher: ‘0’, Sx ¾[0Successor x] n But: 0Successor returns actual infinity ! Though [0Successor x] is perfectly executable for any number; n Intuitionistic approach: “end up” with the construction and “cut off” the constructed entity 24 Problem: trivialisation n Our proposal: ¨ using (de dicto) trivialisation of actual infinity, e.g., 0Successor – synthetic ¨ using (de re) trivialisation of infinity like in lx [0Successor x] constructs only potential infinity – analytic 25 Analytic concepts and recursive functions n Analytic a priori concepts are those that identify n-ary (n ³ 0) recursive functions (in the finitary way) n Consequence: there are more synthetic than analytic concepts a priori ¨ Thereare uncountably many functions, but only countably many recursive functions n 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 n 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. n 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. n Among many examples we can adduce the discovery of a finitary calculation of any member of the infinite expansion of the number p. 27 Problem: an analytic counterpart of a synthetic concept a priori n To understand this creative process we must be aware of the following fact: n The possibility of discovering a new concept is limited by the resource of atomic (simple) concepts that are at our disposal. n A conceptual system S is given by a set of simple concepts, from which all other complex concepts belonging to S are composed. n 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 n The concept given by the original formulation of Fermat’s Last Theorem, i.e., by "abcn (n > 2 ® Ø(an + bn = cn)) n is synthetic in that it is impossible to calculate the respective truth-value. n 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