Cognitive Systems Research. This is the author's updated version of DOI: 10.1016/j.cogsys.2016.02.004 The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning The Evidence-Based Argument for Lucas' Gödelian Thesis Bhupinder Singh Anand Last updated on April 17, 2016 Abstract. We consider the argument that Tarski's classic definitions permit an intelligence-whether human or mechanistic-to admit finitary evidence-based definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways: (1) in terms of classical algorithmic verifiabilty; and (2) in terms of finitary algorithmic computability. We then show that the two definitions correspond to two distinctly different assignments of satisfaction and truth to the compound formulas of PA over N-IPA(N, SV ) and IPA(N, SC). We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both IPA(N, SV ) and IPA(N, SC). We then show: (a) that if we assume the satisfaction and truth of the compound formulas of PA are always non-finitarily decidable under IPA(N, SV ), then this assignment corresponds to the classical non-finitary putative standard interpretation IPA(N, S) of PA over the domain N; and (b) that the satisfaction and truth of the compound formulas of PA are always finitarily decidable under the assignment IPA(N, SC), from which we may finitarily conclude that PA is consistent. We further conclude that the appropriate inference to be drawn from Gödel's 1931 paper on undecidable arithmetical propositions is that we can define PA formulas which-under interpretation-are algorithmically verifiable as always true over N, but not algorithmically computable as always true over N. We conclude from this that Lucas' Gödelian argument is validated if the assignment IPA(N, SV ) can be treated as circumscribing the ambit of human reasoning about 'true' arithmetical propositions, and the assignment IPA(N, SC) as circumscribing the ambit of mechanistic reasoning about 'true' arithmetical propositions. Keywords. Algorithmic computability, Algorithmic verifiability, Arithmetical satisfaction, Arithmetical truth, Arithmetical provability, Classical arithmetical reasoning, Cauchy sequence, Consistency of Arithmetic, Evidence-based interpretation, Finitary arithmetical reasoning, Gödel's β-function, Gödelian undecidability, Human Reasoning, Lucas' Gödelian argument, Hilbert's First Problem, Mechanistic Reasoning, Peano Arithmetic PA, Poincaré-Hilbert debate, Standard interpretation, Tarski's definitions, Uncomputability. 2010 Mathematics Subject Classification. 03B10 Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Defining algorithmic verifiability and algorithmic computability . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Reviewing Tarski's inductive assignment of truth-values under an interpretation . . . . . . . . . . . . . . . 5 4. The ambiguity in the classical putative standard interpretation of PA over the domain N of the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5. The standard verifiable interpretation IPA(N, SV ) of PA over N . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.A. The PA axioms are algorithmically verifiable as true under IPA(N, SV ) . . . . . . . . . . . . . . . . . . . . . . . . . 7 6. The standard computable interpretation IPA(N, SC) of PA over N . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6.A. The PA axioms are algorithmically computable as true under IPA(N, SC) . . . . . . . . . . . . . . . . . . . . . . . 9 7. Bridging PA Provability and Computability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 8. An evidence-based perspective of Lucas' Gödelian argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 9. Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 1. Introduction 1. Introduction We briefly consider a philosophical challenge that arises when an intelligence-whether human or mechanistic-accepts arithmetical propositions as true under an interpretation-either axiomatically or on the basis of subjective self-evidence-without any specified methodology for evidencing such acceptance1. For instance conventional wisdom, whilst accepting Alfred Tarski's classical definitions of the satisfiability and truth of the formulas of a formal language under an interpretation as adequate to the intended purpose, postulates that under the classical putative standard interpretation IPA(N, Standard, Cl− assical) 2 of the first-order Peano Arithmetic PA3 over the domain N of the natural numbers: (i) The satisfiability/truth of the atomic formulas of PA can be assumed as uniquely decidable under IPA(N, S); (ii) The PA axioms can be assumed to uniquely interpret as satisfied/true under IPA(N, S); (iii) The PA rules of inference-Generalisation and Modus Ponens-can be assumed to uniquely preserve such satisfaction/truth under IPA(N, S); (iv) Aristotle's particularisation4 can be assumed to hold under IPA(N, S). We shall argue that the seemingly innocent and self-evident assumptions of uniqueness in (i) to (iii)-as also the seemingly innocent assumption in (iv) which, despite being obviously non-finitary, is unquestioningly accepted in classical literature5 as equally self-evident under any logically unexceptionable interpretation of the classical first-order logic FOL-conceal an ambiguity with far-reaching consequences. The ambiguity is revealed if we note6 that Tarski's classic definitions permit both human and mechanistic intelligences to admit finitary7 evidence-based definitions of the satisfaction and truth of the atomic formulas of PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways: (1a) In terms of classical algorithmic verifiabilty; and (1b) In terms of finitary algorithmic computability. We shall show8 that: (2a) The two definitions correspond to two distinctly different assignments of satisfaction and truth to the compound formulas of PA over N-say IPA(N, Standard, V erifiable) and IPA(N, Standard, Computable)9; where 1For a brief recent review of such challenges, see [Fe06], [Fe08]; also [An04] and Rodrigo Freire's informal essay on 'Interpretation and Truth in Cantorian Set Theory '. 2See Section 9., Appendix A. We shall refer to this henceforth as IPA(N, S). 3We take this to be the first-order theory S defined in any standard text such as [Me64], p.102. 4See Section 9., Appendix A. Informally, Aristotle's particularisation is the non-finitary assumption that an assertion such as, 'There exists an x such that F (x) holds'-usually denoted symbolically by '(∃x)F (x)'-can always be validly inferred in the classical logic of predicates from the assertion, 'It is not the case that: for any given x, F (x) does not hold'-usually denoted symbolically by '¬(∀x)¬F (x)' ([HA28], pp.58-59). 5See Section 9., Appendix A. 6See [An12] and [An15]. 7We mean 'finitary' in the sense that ". . . there should be an algorithm for deciding the truth or falsity of any mathematical statement" . . . http://en.wikipedia.org/wiki/Hilbert's program. For a brief review of 'finitism' and 'constructivity' in the context of this paper see [Fe08]. 8cf. [An12] and [An15]. 9We shall refer to these henceforth as IPA(N, SV ) and IPA(N, SC) respectively. B. S. Anand, The Evidence-Based Argument for Lucas' Gödelian Thesis 3 (2b) The PA axioms are true over N, and the PA rules of inference preserve truth over N, under both IPA(N, SV ) (Section 5.A.) and IPA(N, SC) (Section 6.A.). We shall then show that10: (3a) If we assume the satisfaction and truth of the compound formulas of PA are always non-finitarily decidable under the assignment IPA(N, SV ), then this assignment defines a non-finitary interpretation of PA in which Aristotle's particularisation always holds over N; and which may be taken to correspond to the intended (putative) classical non-finitary standard interpretation IPA(N, S) of PA over the domain N-from which only a human intelligence may non-finitarily conclude that PA is consistent; whilst (3b) The satisfaction and truth of the compound formulas of PA are always finitarily decidable under the assignment IPA(N, SC), which thus defines a finitary interpretation of PA-from which both intelligences may finitarily conclude that PA is consistent11. We shall show further that both intelligences would logically conclude that: (4a) The assignment IPA(N, SC) defines a subset of PA formulas that are algorithmically computable as true under the putative standard interpretation IPA(N, S) if, and only if, the formulas are PA provable; (4b) PA is not ω-consistent12; and (4c) PA is categorical with respect to algorithmic computability. Both intelligences would also logically conclude that: (5a) Since PA is not ω-consistent, Gödel's argument in [Go31] (p.28(2))-that "Neg(17Gen r) is not κ-PROVABLE"13-does not yield a 'formally undecidable proposition' in PA14; (5b) The appropriate conclusion to be drawn from Gödel's argument in [Go31] (p.27(1))- that "17Gen r is not κ-PROVABLE"-is that his 'undecidable arithmetical proposition' is an instantiation of the argument15 that we can define number-theoretic formulas which are algorithmically verifiable as always true, but not algorithmically computable as always true. We shall finally conclude from this that: 10cf. [An12] and [An15]. 11As sought by David Hilbert for the second of the twenty three problems that he highlighted at the International Congress of Mathematicians in Paris in 1900. 12See Section 9., Appendix A. 13The reason we prefer to consider Gödel's original argument (rather than any of its subsequent avatars) is that, for a purist, Gödel's remarkably self-contained 1931 paper-it neither contained, nor needed, any formal citations- remains unsurpassed in mathematical literature for thoroughness, clarity, transparency and soundness of exposition, from first principles (thus avoiding any implicit mathematical or philosophical assumptions), of his notion of arithmetical 'undecidability' as based on his Theorems VI and XI and their logical consequences. 14We note that if PA is not ω-consistent, then Aristotle's particularisation does not hold in any finitary interpretation of PA over N. Now, J. Barkeley Rosser's 'undecidable' arithmetical proposition in [Ro36] is of the form [(∀y)(Q(h, y)→ (∃z)(z ≤ y∧S(h, z)))]. Thus his 'extension' of Gödel's proof of undecidability too does not yield a 'formally undecidable proposition' in PA, since it assumes that Aristotle's particularisation holds when interpreting [(∀y)(Q(h, y)→ (∃z)(z ≤ y ∧ S(h, z)))] under a finitary interpretation over N ([Ro36], Theorem II, pp.233-234; [Kl52], Theorem 29, pp.208-209; [Me64], Proposition3.32, pp.145-146). 15Corresponding to Cantor's diagonal argument and Turing's halting argument as reflected in Theorem 2.1. 4 2. Defining algorithmic verifiability and algorithmic computability Lucas' Gödelian argument16 is validated if the assignment IPA(N, SV ) can be treated as circumscribing the ambit of human reasoning about 'true' arithmetical propositions, and the assignment IPA(N, SC) can be treated as circumscribing the ambit of mechanistic reasoning about 'true' arithmetical p[ropositions. 2. Defining algorithmic verifiability and algorithmic computability We begin by introducing the following two concepts: Definition 1. Algorithmic verifiability: A number-theoretical relation F (x) is algorithmically verifiable if, and only if, for any given natural number n, there is an algorithm AL(F, n) which can provide objective evidence 17 for deciding the truth/falsity of each proposition in the finite sequence {F (1), F (2), . . . , F (n)}. Definition 2. Algorithmic computability: A number theoretical relation F (x) is algorithmically computable if, and only if, there is an algorithm ALF that can provide objective evidence for deciding the truth/falsity of each proposition in the denumerable sequence {F (1), F (2), . . .}. We note that algorithmic computability implies the existence of an algorithm that can finitarily decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions18, whereas algorithmic verifiability does not imply the existence of an algorithm that can finitarily decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions. The following argument shows that although every algorithmically computable relation is algorithmically verifiable, the converse is not true. Theorem 2.1. There are number theoretic functions that are algorithmically verifiable but not algorithmically computable. Proof : (a) Since any real number R is mathematically definable as the unique limit of a correspondingly unique Cauchy sequence {Σni=0r(i).2−i : n = 0, 1, . . .} of rational numbers:  Let r(n) denote the nth digit in the decimal expression of the real numberR = Ltn→∞Σ n i=0r(i).2 −i in binary notation.  Then, for any given natural number n, Gödel's β-function19 defines an algorithm AL(R, n) that can verify the truth/falsity of each proposition in the finite sequence: 16Which Lucas advanced in [Lu61]. 17cf. [Mu91]: "It is by now folklore . . . that one can view the values of a simple functional language as specifying evidence for propositions in a constructive logic . . . ". 18We note that the concept of 'algorithmic computability' is essentially an expression of the more rigorously defined concept of 'realizability' in [Kl52], p.503. 19In Theorem VII of his 1931 paper Gödel defined ([Go31], p.31, Lemma 1; see also [Me64], p.131, Proposition 3.21) a primitive recursive function-Gödel's β-function-as: β(x1, x2, x3) = rm(1 + (x3 + 1) ? x2, x1) where rm(x1, x2) denotes the remainder obtained on dividing x2 by x1. Gödel then showed that, for any non-terminating sequence of values f(x1, 0), f(x1, 1), . . ., we can construct natural numbers b, c such that: (i) j = max(n, f(x1, 0), f(x1, 1), . . . , f(x1, n)); (ii) c = j!; (iii) β(b, c, i) = f(x1, i) for 0 ≤ i ≤ n. and that β(x1, x2, x3) is strongly represented in PA by [Bt(x1, x2, x3, x4)], which is defined as follows: [(∃w)(x1 = ((1 + (x3 + 1) ? x2) ? w + x4) ∧ (x4 < 1 + (x3 + 1) ? x2))]. B. S. Anand, The Evidence-Based Argument for Lucas' Gödelian Thesis 5 {r(0) = 0, r(1) = 0, . . . , r(n) = 0}.  Hence, for any real number R, the relation r(x) = 0 is algorithmically verifiable trivially. (b) Since it follows from Alan Turing's Halting argument20 that there are algorithmically uncomputable real numbers:  Let r(n) denote the nth digit in the decimal expression of an algorithmically uncomputable real number R in binary notation.  By (a), the relation r(x) = 0 is algorithmically verifiable trivially.  However, by definition there is no algorithm ALR that can decide the truth/falsity of each proposition in the denumerable sequence: {r(0) = 0, r(1) = 0, . . .}.  Hence the relation r(x) = 0 is algorithmically verifiable but not algorithmically computable.  3. Reviewing Tarski's inductive assignment of truth-values under an interpretation We shall essentially follow standard expositions21 of Tarski's inductive definitions on the 'satisfiability' and 'truth' of the formulas of a formal language under an interpretation where: Definition 3. If [A] is an atomic formula [A(x1, x2, . . . , xn)] 22 of a formal language S, then the denumerable sequence (a1, a2, . . .) in the domain D of an interpretation IS(D) of S satisfies [A] if, and only if: (i) [A(x1, x2, . . . , xn)] interprets under IS(D) as a unique relation A∗(x1, x2, . . . , xn) in D for any witness WD of D; (ii) there is a Satisfaction Method that provides objective evidence23 by which any witness WD of D can objectively define for any atomic formula [A(x1, x2, . . . , xn)] of S, and any given denumerable sequence (b1, b2, . . .) of D, whether the proposition A∗(b1, b2, . . . , bn) holds or not in D; (iii) A∗(a1, a2, . . . , an) holds in D for any WD. Witness: From a constructive perspective, the existence of a 'witness' as in (i) above is implicit in the usual expositions of Tarski's definitions. Satisfaction Method: From a constructive perspective, the existence of a Satisfaction Method as in (ii) above is also implicit in the usual expositions of Tarski's definitions. A constructive perspective: We highlight the word 'define' in (ii) above to emphasise the constructive perspective underlying this paper; which is that the concepts of 'satisfaction' and 'truth' under an interpretation are to be explicitly viewed as objective assignments by a convention that is witness-independent. A Platonist perspective would substitute 'decide' for 'define', thus implicitly suggesting that these concepts can 'exist', in the sense of needing to be discovered by some witness-dependent means-eerily akin to a 'revelation'-if the domain D is N. 20[Tu36], p.132, §8. 21See Section 9., Appendix A. 22We shall use square brackets to indicate that the contents represent a symbol or a formula of a formal theory, generally assumed to be well-formed unless otherwise indicated by the context. 23In the sense of [Mu91]. 6 5. The standard verifiable interpretation IPA(N, SV ) of PA over N We further define the truth values of 'satisfaction', 'truth', and 'falsity' for the compound formulas of a first-order theory S under the interpretation IS(D) in terms of only the satisfiability of the atomic formulas of S over D as usual24. We now show how Tarski's definitions yield two distinctly different 'standard' interpretations of the first-order Peano Arithmetic PA. 4. The ambiguity in the classical putative standard interpretation of PA over the domain N of the natural numbers The classical putative standard interpretation IPA(N, S) of PA over the domain N of the natural numbers is obtained if, in IS(D): (a) we define S as PA with standard first-order predicate calculus as the underlying logic25; (b) we define D as the set N of natural numbers; (c) we assume for any atomic formula [A(x1, x2, . . . , xn)] of PA, and any given sequence (b∗1, b ∗ 2, . . . , b ∗ n) of N, that the proposition A∗(b∗1, b∗2, . . . , b∗n) is decidable in N; (d) we define the witness W(N, Standard, Classical) informally as the 'mathematical intuition' of a human intelligence for whom, classically, (c) has been implicitly accepted as objectively 'decidable' in N. (e) we postulate that Aristotle's particularisation holds over N26. Clearly, (e) does not form any part of Tarski's inductive definitions of the satisfaction, and truth, of the formulas of PA under the above interpretation. Moreover, its inclusion makes IPA(N, S) extraneously non-finitary27. We shall show that the implicit acceptance in (d) conceals an ambiguity that needs to be made explicit since: Lemma 4.1. A∗(x1, x2, . . . , xn) is both algorithmically verifiable and algorithmically computable in N by W(N, Standard, Classical). Proof (i) It follows from the argument in Theorem 5.1 (below) that A∗(x1, x2, . . . , xn) is algorithmically verifiable in N by W(N, Standard, Classical). (ii) It follows from the argument in Theorem 6.1 (below) that A∗(x1, x2, . . . , xn) is algorithmically computable in N by W(N, Standard, Classical). The lemma follows.  We note without proof that28 (i) is consistent with, whilst (ii) is inconsistent with, the assumption of Aristotle's particularisation. 24See Section 9., Appendix A. 25Where the string [(∃ . . .)] is defined as-and is to be treated as an abbreviation for-the PA formula [¬(∀ . . .)¬]. We do not consider the case where the underlying logic is Hilbert's formalisation of Aristotle's logic of predicates in terms of his ε-operator ([Hi27], pp.465-466). 26This postulates that a PA formula such as [(∃x)F (x)] can always be taken to interpret under IPA(N, S) as 'There is some natural number n such that F (n) holds in N. 27As argued by Brouwer in [Br08]. 28For a more detailed argument see [An12] and [An15]. B. S. Anand, The Evidence-Based Argument for Lucas' Gödelian Thesis 7 5. The standard verifiable interpretation IPA(N, SV ) of PA over N We now consider a standard verifiable interpretation IPA(N, SV ) of PA, under which we define: Definition 4. An atomic formula [A] of PA is satisfiable under the interpretation IPA(N, SV ) if, and only if, [A] is algorithmically verifiable under IPA(N, SV ). We note that: Theorem 5.1. The atomic formulas of PA are algorithmically verifiable as true or false under the standard verifiable interpretation IPA(N, SV ). Proof It follows from Gödel's definition of the primitive recursive relation xBy29-where x is the Gödel number of a proof sequence in PA whose last term is the PA formula with Gödel-number y- that, if [A] is an atomic formula of PA, we can algorithmically verify which one of the PA formulas [A] and [¬A] is necessarily PA-provable and, ipso facto, true under IPA(N, SV ).  We note that the interpretation IPA(N, SV ) cannot claim to be finitary30. Reason: It follows from Theorem 2.1 that we cannot conclude finitarily from Tarski's Definitions 3 to 10 whether or not a quantified PA formula [(∀xi)R] is algorithmically verifiable as true under IPA(N, SV ) if [R] is algorithmically verifiable but not algorithmically computable under the interpretation31. 5.A. The PA axioms are algorithmically verifiable as true under IPA(N, SV ) The significance of defining satisfaction in terms of algorithmic verifiability under IPA(N, SV ) is that: Lemma 5.2. The PA axioms PA1 to PA8 are algorithmically verifiable as true over N under the interpretation IPA(N, SV ). Proof Since [x+y], [x?y], [x = y], [x′] are defined recursively32, the PA axioms PA1 to PA8 interpret as recursive relations that do not involve any quantification. The lemma follows straightforwardly from Theorem 5.1 and Tarski's Definitions 3 to 10.  Lemma 5.3. For any given PA formula [F (x)], the Induction axiom schema [F (0) → (((∀x)(F (x) → F (x′))) → (∀x)F (x))] interprets as an algorithmically verifiable true formula under IPA(N, SV ). Proof (a) If [F (0)] interprets as an algorithmically verifiable false formula under IPA(N, SV ) the lemma is proved. Reason: Since [F (0) → (((∀x)(F (x) → F (x′))) → (∀x)F (x))] interprets as an algorithmically verifiable true formula under IPA(N, SV ) if, and only if, either [F (0)] interprets as an algorithmically verifiable false formula or [((∀x)(F (x) → F (x′))) → (∀x)F (x)] interprets as an algorithmically verifiable true formula under under IPA(N, SV ). 29[Go31], p. 22(45). 30See [An12] and [An15] for a proof that IPA(N, SV ) is non-finitary, since it defines a model of PA if, and only if, PA is ω-consistent and so we may always non-finitarily conclude from [(∃x)R(x)] the existence of some numeral [n] such that [R(n)]. 31Although a proof that such a PA formula exists is not obvious, we shall show that Gödel's 'undecidable' arithmetical formula [R(x)] is algorithmically verifiable but not algorithmically computable under the interpretation IPA(N, SV ). 32cf. [Go31], p.17. 8 6. The standard computable interpretation IPA(N, SC) of PA over N (b) If [F (0)] interprets as an algorithmically verifiable true formula, and [(∀x)(F (x) → F (x′))] interprets as an algorithmically verifiable false formula, under IPA(N, SV ), the lemma is proved. (c) If [F (0)] and [(∀x)(F (x) → F (x′))] both interpret as algorithmically verifiable true formulas under IPA(N, SV ) then, for any natural number n, there is an algorithm which (by Definition 1) will evidence that [F (n) → F (n′)] is an algorithmically verifiable true formula under IPA(N, SV ). (d) Since [F (0)] interprets as an algorithmically verifiable true formula under IPA(N, SV ), it follows for any natural number n that there is an algorithm which will evidence that each of the formulas in the finite sequence {[F (0), F (1), . . . , F (n)}] is an algorithmically verifiable true formula under the interpretation. (e) Hence [(∀x)F (x)] is an algorithmically verifiable true formula under IPA(N, SV ). Since the above cases are exhaustive, the lemma follows.  We note that if [F (0)] and [(∀x)(F (x)→ F (x′))] both interpret as algorithmically verifiable true formulas under IPA(N, SV ), then we can only conclude that, for any natural number n, there is an algorithm which will give evidence for any m ≤ n that the formula [F (m)] is true under IPA(N, S). We cannot conclude that there is an algorithm which, for any natural number n, will give evidence that the formula [F (n)] is true under IPA(N, S). Lemma 5.4. Generalisation preserves algorithmically verifiable truth under IPA(N, SV ). Proof The two meta-assertions: '[F (x)] interprets as an algorithmically verifiable true formula under IPA(N, SV )33' and '[(∀x)F (x)] interprets as an algorithmically verifiable true formula under IPA(N, SV )' both mean: [F (x)] is algorithmically verifiable as always true under IPA(N, SV ).  It is also straightforward to see that: Lemma 5.5. Modus Ponens preserves algorithmically verifiable truth under IPA(N, SV ).  We thus have that: Theorem 5.6. The axioms of PA are always algorithmically verifiable as true under the interpretation IPA(N, SV ), and the rules of inference of PA preserve the properties of algorithmically verifiable satisfaction/truth under IPA(N, SV ).  By Theorem 5.1 we conclude that: Theorem 5.7. If the PA formulas are algorithmically verifiable as true or false under IPA(N, SV ), then PA is consistent.  We note that, like Gentzen's argument, such a proof of consistency would be debatably 'finitary', since we cannot conclude from Theorem 5.1 that the quantified formulas of PA are 'finitarily' decidable as true or false under the interpretation IPA(N, SV ). 33See Definition 9 B. S. Anand, The Evidence-Based Argument for Lucas' Gödelian Thesis 9 6. The standard computable interpretation IPA(N, SC) of PA over N We next consider a standard computable interpretation IPA(N, SC) of PA, under which we define: Definition 5. An atomic formula [A] of PA is satisfiable under the interpretation IPA(N, SC) if, and only if, [A] is algorithmically computable under IPA(N, SC). We note that: Theorem 6.1. The atomic formulas of PA are algorithmically computable as true or as false under the standard computable interpretation IPA(N, SC). Proof If [A(x1, x2, . . . , xn)] is an atomic formula of PA then, for any given sequence of numerals [b1, b2, . . . , bn], the PA formula [A(b1, b2, . . . , bn)] is an atomic formula of the form [c = d], where [c] and [d] are atomic PA formulas that denote PA numerals. Since [c] and [d] are recursively defined formulas in the language of PA, it follows from a standard result34 that [c = d] is algorithmically computable as either true or false in N since there is an algorithm that, for any given sequence of numerals [b1, b2, . . . , bn], will give evidence whether [A(b1, b2, . . . , bn)] interprets as true or false in N. The lemma follows.  We note that the interpretation IPA(N, SC) is finitary since: Lemma 6.2. The formulas of PA are algorithmically computable finitarily as true or as false under IPA(N, SC). Proof The Lemma follows by finite induction from Definition 2, Tarski's Definitions 3 to 10, and Theorem 6.1.  6.A. The PA axioms are algorithmically computable as true under IPA(N, SC) The significance of defining satisfaction in terms of algorithmic computability under IPA(N, SC) as above is that: Lemma 6.3. The PA axioms PA1 to PA8 are algorithmically computable as true under the interpretation IPA(N, SC). Proof Since [x+y], [x?y], [x = y], [x′] are defined recursively35, the PA axioms PA1 to PA8 interpret as recursive relations that do not involve any quantification. The lemma follows straightforwardly from Definitions 3 to 10 in Section 3. and Theorem 5.1.  Lemma 6.4. For any given PA formula [F (x)], the Induction axiom schema [F (0) → (((∀x)(F (x) → F (x′))) → (∀x)F (x))] interprets as an algorithmically computable true formula under IPA(N, SC). Proof By Definitions 3 to 10: (a) If [F (0)] interprets as an algorithmically computable false formula under IPA(N, SC) the lemma is proved. 34For any natural numbers m, n, if m 6= n, then PA proves [¬(m = n)] ([Me64], p.110, Proposition 3.6). The converse is obviously true. 35cf. [Go31], p.17. 10 6. The standard computable interpretation IPA(N, SC) of PA over N Since [F (0) → (((∀x)(F (x) → F (x′))) → (∀x)F (x))] interprets as an algorithmically computable true formula if, and only if, either [F (0)] interprets as an algorithmically computable false formula, or [((∀x)(F (x) → F (x′))) → (∀x)F (x)] interprets as an algorithmically computable true formula, under IPA(N, SC). (b) If [F (0)] interprets as an algorithmically computable true formula, and [(∀x)(F (x) → F (x′))] interprets as an algorithmically computable false formula, under IPA(N, SC), the lemma is proved. (c) If [F (0)] and [(∀x)(F (x) → F (x′))] both interpret as algorithmically computable true formulas under IPA(N, SC), then by Definition 2 there is an algorithm which, for any natural number n, will give evidence that the formula [F (n) → F (n′)] is an algorithmically computable true formula under IPA(N, SC). (d) Since [F (0)] interprets as an algorithmically computable true formula under IPA(N, SC), it follows that there is an algorithm which, for any natural number n, will give evidence that [F (n)] is an algorithmically computable true formula under the interpretation. (e) Hence [(∀x)F (x)] is an algorithmically computable true formula under IPA(N, SC). Since the above cases are exhaustive, the lemma follows.  The Poincaré-Hilbert debate: We note that Lemma 6.4 appears to dissolve the Poincaré-Hilbert debate36 since: (i) the algorithmically verifiable non-finitary interpretation IPA(N, SV ) of PA validates Poincaré's argument that the PA Axiom Schema of Finite Induction could not be justified finitarily with respect to algorithmic verifiability under the classical putative standard interpretation of arithmetic, as any such argument would necessarily need to appeal to some form of infinite induction37; whilst (ii) the algorithmically computable finitary interpretation IPA(N, SC) of PA validates Hilbert's belief that a finitary justification of the Axiom Schema was possible under some finitary interpretation of an arithmetic such as PA. Lemma 6.5. Generalisation preserves algorithmically computable truth under IPA(N, SC). Proof The two meta-assertions: '[F (x)] interprets as an algorithmically computable true formula under IPA(N, SC)38' and '[(∀x)F (x)] interprets as an algorithmically computable true formula under IPA(N, SC)' both mean: [F (x)] is algorithmically computable as always true under IPA(N, S).  It is also straightforward to see that: Lemma 6.6. Modus Ponens preserves algorithmically computable truth under IPA(N, SC).  We thus have that39: 36See [Hi27], p.472; also [Br13], p.59; [We27], p.482; [Pa71], p.502-503. 37cf. Gerhard Gentzen's non-finitary proof of consistency for PA, which involves a non-finitary Rule of Infinite Induction that admits appeal to the well-ordering property of transfinite ordinals. 38See Definition 9 39Without appeal, moreover, to Aristotle's particularisation. B. S. Anand, The Evidence-Based Argument for Lucas' Gödelian Thesis 11 Theorem 6.7. The axioms of PA are always algorithmically computable as true under the interpretation IPA(N, SC), and the rules of inference of PA preserve the properties of algorithmically computable satisfaction/truth under IPA(N, SC).  We thus have a finitary proof that: Theorem 6.8. PA is consistent.  7. Bridging PA Provability and Computability We now show that PA can have no non-standard model, since it is 'computably' complete in the sense that: Theorem 7.1. (Provability Theorem for PA) A PA formula [F (x)] is PA-provable if, and only if, [F (x)] is algorithmically computable as always true in N. Proof We have by definition that [(∀x)F (x)] interprets as true under the interpretation IPA(N, SC) if, and only if, [F (x)] is algorithmically computable as always true in N. By Theorem 6.7, IPA(N, SC) defines a finitary model of PA over N such that: If [(∀x)F (x)] is PA-provable, then [F (x)] is algorithmically computable as always true in N; If [¬(∀x)F (x)] is PA-provable, then it is not the case that [F (x)] is algorithmically computable as always true in N. Now, we cannot have that both [(∀x)F (x)] and [¬(∀x)F (x)] are PA-unprovable for some PA formula [F (x)], as this would yield the contradiction: (i) There is a finitary model-say I ′PA(N, SC)-of PA+[(∀x)F (x)] in which [F (x)] is algorithmically computable as always true in N. (ii) There is a finitary model-say I ′′PA(N, SC)-of PA+[¬(∀x)F (x)] in which it is not the case that [F (x)] is algorithmically computable as always true in N. The lemma follows.  Corollary 7.2. PA is categorical with respect to algorithmic computability. 8. An evidence-based perspective of Lucas' Gödelian argument We finally note that: Lemma 8.1. If IPA(N, M) defines a model of PA over N, then there is a PA formula [F ] which is algorithmically verifiable as always true over N under IPA(N, M) even though [F ] is not PA-provable. Proof Gödel has shown how to construct an arithmetical formula with a single variable-say [R(x)]40- such that [R(x)] is not PA-provable41, but [R(n)] is instantiationally PA-provable for any given PA 40Gödel refers to the formula [R(x)] only by its Gödel number r ([Go31], p.25(12)). 41Gödel's aim in [Go31] was to show that [(∀x)R(x)] is not P-provable; by Generalisation it follows, however, that [R(x)] is also not P-provable. 12 8. An evidence-based perspective of Lucas' Gödelian argument numeral [n]. Hence, for any given numeral [n], Gödel's primitive recursive relation xBd[R(n)]e42 must hold for some x. The lemma follows.  By the argument in Theorem 7.1 it follows that: Corollary 8.2. The PA formula [¬(∀x)R(x)] defined in Lemma 8.1 is PA-provable.  Corollary 8.3. In any model of PA, Gödel's arithmetical formula [R(x)] interprets as an algorithmically verifiable, but not algorithmically computable, tautology over N. Proof Gödel has shown that [R(x)]43 always interprets as an algorithmically verifiable tautology over N44. By Corollary 8.2 [R(x)] is not algorithmically computable as always true in N.  Corollary 8.4. PA is not ω-consistent.45 Proof Gödel has shown that if PA is consistent, then [R(n)] is PA-provable for any given PA numeral [n]46. By Corollary 8.2 and the definition of ω-consistency, if PA is consistent then it is not ωconsistent.  Corollary 8.5. The putative standard interpretation IPA(N, S) of PA does not define a model of PA47. Proof If PA is consistent but not ω-consistent, then Aristotle's particularisation does not hold over N. Since the putative standard interpretation of PA appeals to Aristotle's particularisation, the lemma follows.  We conclude from this that Lucas' Gödelian argument48 can validly claim that: Thesis 1. There can be no mechanist model of human reasoning if the assignment IPA(N, SV ) can be treated as circumscribing the ambit of human reasoning about 'true' arithmetical propositions49, and the assignment IPA(N, SC) can be treated as circumscribing the ambit of mechanistic reasoning about 'true' arithmetical propositions. 42Where d[R(n)]e denotes the Gödel-number of the PA formula [R(n)]. 43Gödel refers to the formula [R(x)] only by its Gödel number r; [Go31], p.25, eqn.12. 44[Go31], p.26(2): "(n)¬(nBκ(17Gen r)) holds" 45This conclusion is contrary to accepted dogma. See, for instance, Davis' remarks in [Da82], p.129(iii) that ". . . there is no equivocation. Either an adequate arithmetical logic is ω-inconsistent (in which case it is possible to prove false statements within it) or it has an unsolvable decision problem and is subject to the limitations of Gödel's incompleteness theorem". 46[Go31], p.26(2). 47I note that finitists of all hues-ranging from Brouwer [Br08], to Wittgenstein [Wi78], to Alexander Yessenin-Volpin [He04]-have persistently questioned the assumption that the putative standard interpretation IPA(N, S) can be treated as well-defining a model of PA; see also [Brm07]. 48Although Lucas' original thesis ([Lu61] deserves consideration that lies beyond the immediate scope of this investigation, we draw attention to his informal defence of it from a philosophical perspective in The Gödelian Argument: Turn Over the Page, where he concludes with the argument that: "Thus, though the Gödelian formula is not a very interesting formula to enunciate, the Gödelian argument argues strongly for creativity, first in ruling out any reductionist account of the mind that would show us to be, au fond, necessarily unoriginal automata, and secondly by proving that the conceptual space exists in which it intelligible to speak of someone's being creative, without having to hold that he must be either acting at random or else in accordance with an antecedently specifiable rule". 49Such a thesis can be justified by the argument in [An13] and [An15a] that: (i) the assignment IPA(N, SV ) can be viewed as corresponding to the way human intelligence conceptualises, symbolically represents, and logically reasons about, those sensory perceptions that are triggered by physical processes which can be treated as representable-not necessarily finitarily-by algorithmically verifiable formulas, where a physical process is effectively computable if, and only if, it is algorithmically verifiable; whilst: (ii) the assignment IPA(N, SC) can be viewed as corresponding to the way human intelligence conceptualises, symbolically represents, and logically reasons about, only those sensory perceptions that are triggered by physical processes which can be treated as representable-finitarily-by algorithmically computable formulas, where a physical process is effectively computable if, and only if, it is algorithmically computable. We suggest how such a perspective offers a resolution to the EPR paradox. B. S. Anand, The Evidence-Based Argument for Lucas' Gödelian Thesis 13 Argument: Gödel has shown how to construct an arithmetical formula with a single variable- say [R(x)]50-such that [R(x)] is not PA-provable, but [R(n)] is instantiationally PA-provable for any given PA numeral [n]. Hence, for any given numeral [n], Gödel's primitive recursive relation xBd[R(n)]e51 must hold for some natural number m. If we assume that any mechanical witness can only reason finitarily then although, for any given numeral [n], a mechanical witness can give evidence under the assignment IPA(N, SC) that the PA formula [R(n)] holds in N, no mechanical witness can conclude finitarily under the assignment IPA(N, SC) that, for any given numeral [n], the PA formula [R(n)] holds in N. However, if we assume that a human witness can also reason non-finitarily, then a human witness can conclude under the assignment IPA(N, SV ) that, for any given numeral [n], the PA formula [R(n)] holds in N. 9. Appendix A Aristotle's particularisation: Aristotle's particularisation is the implicit non-finitary assumption that the classical first-order logic FOL is ω-consistent, and so we may always interpret the formal expression '[(∃x) F (x)]' of a formal language under an interpretation as 'There exists an object s in the domain of the interpretation such that F (s)'. We note that Aristotle's particularisation is a non-finitary but fundamental tenet of classical logic unrestrictedly adopted as intuitively obvious by standard literature52. However, L. E. J. Brouwer had noted in his seminal 1908 paper on the unreliability of logical principles53 that the commonly accepted interpretation of this formula is ambiguous if interpretation is intended over an infinite domain. Brouwer essentially argued that: (i) Even supposing the formula '[P (x)]' of a formal Arithmetical language interprets as an arithmetical relation denoted by 'P ∗(x)'; and (ii) the formula '[¬(∀x)¬P (x)]' interprets as the arithmetical proposition denoted by '¬(∀x)¬P ∗(x)'; (iii) the formula '[(∃x)P (x)]'-which is formally defined as '[¬(∀x)¬P ∗(x)]'-need not interpret as the arithmetical proposition denoted by the usual abbreviation '(∃x)P ∗(x)'; and (iv) that such postulation is invalid as a general logical principle in the absence of a means for constructing some putative object a for which the proposition P ∗(a) holds in the domain of the interpretation. Hence we shall follow the convention that the assumption that '(∃x)P ∗(x)' is the intended interpretation of the formula '[(∃x)P (x)]'-which is essentially the assumption that Aristotle's particularisation holds over the domain of the interpretation-must always be explicit. 50Gödel refers to this formula only by its Gödel number r ([Go31], p.25(12)). 51Where xBy denotes Gödel's primitive recursive relation 'x is the Gödel-number of a proof sequence in PA whose last term is the PA formula with Gödel-number y' ([Go31], p. 22(45)); and d[R(n)]e denotes the Gödel-number of the PA formula [R(n)]. 52See [Hi25], p.382; [HA28], p.48; [Sk28], p.515; [Go31], p.32.; [Kl52], p.169; [Ro53], p.90; [BF58], p.46; [Be59], pp.178 & 218; [Su60], p.3; [Wa63], p.314-315; [Qu63], pp.12-13; [Kn63], p.60; [Co66], p.4; [Me64], p.52(ii); [Nv64], p.92; [Li64], p.33; [Sh67], p.13; [Da82], p.xxv; [Rg87], p.xvii; [EC89], p.174; [Mu91]; [Sm92], p.18, Ex.3; [BBJ03], p.102; [Cr05], p.6. 53[Br08]. 14 9. Appendix A ω-consistency: A formal system S is ω-consistent if, and only if, there is no S-formula [F (x)] for which, first, [¬(∀x)F (x)] is S-provable and, second, [F (a)] is S-provable for any given S-term [a]. In order to avoid intuitionistic objections to his reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions54, Gödel did not assume that the classical putative standard assignment IPA(N, S) of PA yields a model of PA. Instead, Gödel introduced the syntactic property of ω-consistency as an explicit assumption in his formal reasoning55. Gödel explained at some length56 that his reasons for introducing ω-consistency as an explicit assumption in his formal reasoning was to avoid appealing to the semantic concept of classical arithmetical truth-a concept which is implicitly based on an intuitionistically objectionable logic that assumes Aristotle's particularisation is valid over N. However, we note that if we assume the classical putative standard assignment IPA(N, S) of PA yields a model of PA, then PA is consistent if, and only if, it is ω-consistent. It can thus be argued that Gödel's Platonism was perhaps rooted (justifiably within the context of the implicit non-finitary assumption of Aristotle's particularisation in classical theory) in his implicitly held57 non-finitary belief that any first-order axiomatic theory of arithmetic or set theory is ω-consistent. Standard interpretation of PA: The classical putative standard interpretation IPA(N, S) of PA over the domain N of the natural numbers is the one in which the logical constants have their 'usual' interpretations58 in Aristotle's logic of predications59 (which subsumes Aristotle's particularisation), and60: (a) The set of non-negative integers is the domain; (b) The symbol [0] interprets as the integer 0; (c) The symbol [′] interprets as the successor operation (addition of 1); (d) The symbols [+] and [?] interpret as ordinary addition and multiplication; (e) The symbol [=] interprets as the identity relation. The axioms of first-order Peano Arithmetic (PA) PA1 [(x1 = x2) → ((x1 = x3) → (x2 = x3))]; PA2 [(x1 = x2) → (x′1 = x′2)]; PA3 [0 6= x′1]; PA4 [(x ′ 1 = x ′ 2) → (x1 = x2)]; PA5 [(x1 + 0) = x1]; PA6 [(x1 + x ′ 2) = (x1 + x2) ′]; PA7 [(x1 ? 0) = 0]; PA8 [(x1 ? x ′ 2) = ((x1 ? x2) + x1)]; PA9 For any well-formed formula [F (x)] of PA: [F (0) → (((∀x)(F (x) → F (x′))) → (∀x)F (x))]. Generalisation in PA If [A] is PA-provable, then so is [(∀x)A]. Modus Ponens in PA If [A] and [A→ B] are PA-provable, then so is [B]. 54[Go31]. 55[Go31], p.23 and p.28. 56In his introduction on p.9 of [Go31]. 57[Go31], p.28. 58We essentially follow the definitions in [Me64], p.49. 59See http://plato.stanford.edu/entries/aristotle-logic/, §4.3. 60cf. [Me64], p.107. B. S. Anand, The Evidence-Based Argument for Lucas' Gödelian Thesis 15 Hilbert's Second Problem: "When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. . . . But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results. In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. . . . On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms."61 In this paper, we treat Hilbert's intent62 behind the enunciation of his Second Problem as essentially seeking a finitary proof for the consistency of arithmetic when formalised in a language such as the first order Peano Arithmetic PA. Tarski's inductive definitions: We shall assume that truth values of 'satisfaction', 'truth', and 'falsity' are assignable inductively-whether finitarily or non-finitarily-to the compound formulas of a first-order theory S under the interpretation IS(D) in terms of only the satisfiability of the atomic formulas of S over D as usual63: Definition 6. A denumerable sequence s of D satisfies [¬A] under IS(D) if, and only if, s does not satisfy [A]; Definition 7. A denumerable sequence s of D satisfies [A→ B] under IS(D) if, and only if, either it is not the case that s satisfies [A], or s satisfies [B]; Definition 8. A denumerable sequence s of D satisfies [(∀xi)A] under IS(D) if, and only if, given any denumerable sequence t of D which differs from s in at most the i'th component, t satisfies [A]; Definition 9. A well-formed formula [A] of D is true under IS(D) if, and only if, given any denumerable sequence t of D, t satisfies [A]; Definition 10. A well-formed formula [A] of D is false under IS(D) if, and only if, it is not the case that [A] is true under IS(D). References [BBJ03] George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic (4th ed). Cambridge University Press, Cambridge. [Be59] Evert W. Beth. 1959. The Foundations of Mathematics. Studies in Logic and the Foundations of Mathematics. Edited by L. E. J. Brouwer, E. W. Beth, A. Heyting. 1959. North Holland Publishing Company, Amsterdam. [Bee07] Carlo Beenakker. 2007. Hempels dilemma and the physics of computation. In Knowledge in Ferment: Dilemmas in Science, Scholarship and Society (pp.65-71). eds. Adriaan in t Groen, Henk Jan de Jonge, Eduard Klasen, Hilje Papma, Piet van Slooten. 2007. Leiden University Press, Leiden. [BF58] Paul Bernays and Abraham A. Fraenkel. 1958. Axiomatic Set Theory. Studies in Logic and the Foundations of Mathematics. Edited by L. E. J. Brouwer, E. W. Beth, A. Heyting. 1959. North Holland Publishing Company, Amsterdam. 61Excerpted from Maby Winton Newson's English translation [Nw02] of David Hilbert's address [Hi00] at the International Congress of Mathematicians in Paris in 1900. 62Compare Curtis Franks' thesis in [Fr09] that Hilbert's intent behind the enunciation of his Second Problem was essentially to establish the autonomy of arithmetical truth without appeal to any debatable philosophical considerations. 63See [Me64], p.51; [Mu91]. 16 References [Brm07] Manuel Bremer. 2007. Varieties of Finitism. In Metaphysica. October 2007, Volume 8, Issue 2, pp 131-148. Springer, Netherlands. [Br08] L. E. J. Brouwer. 1908. The Unreliability of the Logical Principles. English translation in A. Heyting, Ed. L. E. J. Brouwer: Collected Works 1: Philosophy and Foundations of Mathematics. Amsterdam: North Holland / New York: American Elsevier (1975): pp. 107-111. [Br13] L. E. J. Brouwer. 1913. Intuitionism and Formalism. Inaugural address at the University of Amsterdam, October 14, 1912. Translated by Professor Arnold Dresden for the Bulletin of the American Mathematical Society, Volume 20 (1913), pp.81-96. 1999. Electronically published in Bulletin (New Series) of the American Mathematical Society, Volume 37, Number 1, pp.55-64. [Co66] Paul J. Cohen. 1966. Set Theory and the Continuum Hypothesis. (Lecture notes given at Harvard University, Spring 1965) W. A. Benjamin, Inc., New York. [Cr05] John N. Crossley. 2005. What is Mathematical Logic? A Survey. Address at the First Indian Conference on Logic and its Relationship with Other Disciplines held at the Indian Institute of Technology, Powai, Mumbai from January 8 to 12. Reprinted in Logic at the Crossroads: An Interdisciplinary View Volume I (pp.3-18). ed. Amitabha Gupta, Rohit Parikh and Johan van Bentham. 2007. Allied Publishers Private Limited, Mumbai. [Da82] Martin Davis. 1958. Computability and Unsolvability. 1982 ed. Dover Publications, Inc., New York. [EC89] Richard L. Epstein, Walter A. Carnielli. 1989. Computability: Computable Functions, Logic, and the Foundations of Mathematics. Wadsworth & Brooks, California. [Fe06] Solomon Feferman. 2006. Are There Absolutely Unsolvable Problems? Gödel's Dichotomy. Philosophia Mathematica (2006) 14 (2): 134-152. [Fe08] Solomon Feferman. 2008. Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert's program. in Special Issue: Gödel's dialectica Interpretation Dialectica, Volume 62, Issue 2, June 2008, pp. 245-290. [Fr09] Curtis Franks. 2009. The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited. Cambridge University Press, New York. [Go31] Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. [Gu11] Gualtiero Piccinini. 2011. The Physical Church-Turing Thesis: Modest or Bold? In The British Journal for the Philosophy of Science (2011) 62 (4): 733-769. Oxford University Press, Inc., New York. [Gu12] Gualtiero Piccinini. 2012. Computationalism. In the The Oxford Handbook of Philosophy of Cognitive Science (Chapter 10, pp.222-250). eds. Eric Margolis, Richard Samuels, and Stephen P. Stitch. 2012. Oxford University Press, Inc., New York. [HA28] David Hilbert & Wilhelm Ackermann. 1928. Principles of Mathematical Logic. Translation of the second edition of the Grundzüge Der Theoretischen Logik. 1928. Springer, Berlin. 1950. Chelsea Publishing Company, New York. [He04] Catherine Christer-Hennix. 2004. Some remarks on Finitistic Model Theory, Ultra-Intuitionism and the main problem of the Foundation of Mathematics. ILLC Seminar, 2nd April 2004, Amsterdam. [Hi00] David Hilbert. 1900. Mathematical Problems. An address delivered before the International Congress of Mathematicians at Paris in 1900. The French translation by M. L. Laugel "Sur les problmes futurs des mathmatiques" appeared in Compte Rendu du Deuxime Congrs International des Mathmaticiens, pp. 58-114, Gauthier-Villars, Paris, 1902. Dr. Maby Winton Newson translated this address into English with the author's permission for Bulletin of the American Mathematical Society 8 (1902), 437-479. A reprint appears in Mathematical Developments Arising from Hilbert Problems, edited by Felix Brouder, American Mathematical Society, 1976. The original address "Mathematische Probleme" appeared in Gttinger Nachrichten, 1900, pp. 253-297, and in Archiv der Mathematik und Physik, (3) 1 (1901), 44-63 and 213-237. [A fuller title of the journal Gttinger Nachrichten is Nachrichten von der Knigl. Gesellschaft der Wiss. zu Gttingen.] An HTML version of Dr. Newson's translation was prepared for the web by D. Joyce with only minor modifications, mainly, more complete references, and is accessible at http://aleph0.clarku.edu/ djoyce/hilbert/problems.html#note1. [Hi25] David Hilbert. 1925. On the Infinite. Text of an address delivered in Münster on 4th June 1925 at a meeting of the Westphalian Mathematical Society. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 1931. Harvard University Press, Cambridge, Massachusetts. [Hi27] David Hilbert. 1927. The Foundations of Mathematics. Text of an address delivered in July 1927 at the Hamburg Mathematical Seminar. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 1931. Harvard University Press, Cambridge, Massachusetts. [Kl52] Stephen Cole Kleene. 1952. Introduction to Metamathematics. North Holland Publishing Company, Amsterdam. [Kn63] G. T. Kneebone. 1963. Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. D. Van Norstrand Company Limited, London. [Li64] A. H. Lightstone. 1964. The Axiomatic Method. Prentice Hall, NJ. [Lu61] J. R. Lucas. 1961. Minds, Machines and Gödel . Philosophy, XXXVI, 1961, pp.112-127; reprinted in The Modeling of Mind. Kenneth M.Sayre and Frederick J.Crosson, eds., Notre Dame Press, 1963, pp.269-270; and Minds and Machines, ed. Alan Ross Anderson, Prentice-Hall, 1954, pp.43-59. B. S. Anand, The Evidence-Based Argument for Lucas' Gödelian Thesis 17 [Me64] Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand, Princeton. [Mu91] Chetan R. Murthy. 1991. An Evaluation Semantics for Classical Proofs. Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, (also Cornell TR 91-1213), 1991. [Nv64] P. S. Novikov. 1964. Elements of Mathematical Logic. Oliver & Boyd, Edinburgh and London. [Nw02] Maby Winton Newson. 1902. Mathematical Problems: Lecture delivered before the International Congress of Mathematicians at Paris in 1900 by Professor David Hilbert. Bulletin of the American Mathematical Society 8 (1902), 437-479. HTML version provided at http://aleph0.clarku.edu/ djoyce/hilbert/problems.html. [Pa71] Rohit Parikh. 1971. Existence and Feasibility in Arithmetic. The Journal of Symbolic Logic, Vol.36, No. 3 (Sep., 1971), pp. 494-508. [Qu63] Willard Van Orman Quine. 1963. Set Theory and its Logic. Harvard University Press, Cambridge, Massachusette. [Rg87] Hartley Rogers Jr. 1987. Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge, Massachusetts. [Ro36] J. Barkley Rosser. 1936. Extensions of some Theorems of Gödel and Church. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from The Journal of Symbolic Logic. Vol.1. pp.87-91. [Ro53] J. Barkley Rosser. 1953. Logic for Mathematicians. McGraw Hill, New York. [Ru53] Walter Rudin. 1953. Principles of Mathematical Analysis. McGraw Hill, New York. [Sh67] Joseph R. Shoenfield. 1967. Mathematical Logic. Reprinted 2001. A. K. Peters Ltd., Massachusetts. [Sk28] Thoralf Skolem. 1928. On Mathematical Logic. Text of a lecture delivered on 22nd October 1928 before the Norwegian Mathematical Association. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 1931. Harvard University Press, Cambridge, Massachusetts. [Sm92] Raymond M. Smullyan. 1992. Gödel's Incompleteness Theorems. Oxford University Press, Inc., New York. [Su60] Patrick Suppes. 1960. Axiomatic Set Theory. Van Norstrand, Princeton. [Ta33] Alfred Tarski. 1933. The concept of truth in the languages of the deductive sciences. In Logic, Semantics, Metamathematics, papers from 1923 to 1938 (p152-278). ed. John Corcoran. 1983. Hackett Publishing Company, Indianapolis. [Tu36] Alan Turing. 1936. On computable numbers, with an application to the Entscheidungsproblem. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546. [Wa63] Hao Wang. 1963. A survey of Mathematical Logic. North Holland Publishing Company, Amsterdam. [We27] Hermann Weyl. 1927. Comments on Hilbert's second lecture on the foundations of mathematics. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 1931. Harvard University Press, Cambridge, Massachusetts. [Wi78] Ludwig Wittgenstein. 1937. Remarks on the Foundations of Mathematics. 1978 ed., MIT Press, Cambridge. [An04] Bhupinder Singh Anand. 2004. Do Gödels incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably? Invited article in the Special Issue on Minds, Brains and Mathematics, NeuroQuantology, [S.l.], v. 2, n. 2, 2004, pp.60-100. ISSN 1303-5150. DOI: 10.14704/nq.2004.2.2.39. [An12] . . . 2012. Evidence-Based Interpretations of PA. Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK. Accessible at: http://events.cs.bham.ac.uk/turing12/proceedings/04.pdf. [An13] . . . 2013. A suggested mathematical perspective for the EPR argument. Presented on 7'th April at the workshop on 'Logical Quantum Structures' at UNILOG'2013, 4th World Congress and School on Universal Logic, 29th March 2013 7th April 2013, Rio de Janeiro, Brazil. Accessible at: http://alixcomsi.com/42 Resolving EPR Update.pdf (paper); http://alixcomsi.com/42 Resolving EPR UNILOG 2013 Presentation.pdf (presentation). [An15] . . . 2015. Why Hilbert's and Brouwer's interpretations of quantification are complementary and not contradictory. Presentation on 10th June at the Epsilon 2015 workshop on 'Hilbert's Epsilon and Tau in Logic, Informatics and Linguistics', 10th June 2015 12th June 2015, University of Montpellier, France. [An15a] . . . 2015. Algorithmically Verifiable Logic vis à vis Algorithmically Computable Logic: Could resolving EPR need two complementary Logics? Presented on 26'th June at the workshop on 'Emergent Computational Logics' at UNILOG'2015, 5th World Congress and School on Universal Logic, 20th June 2015 30th June 2015, Istanbul, Turkey. Bhupinder Singh Anand, #1003 B Wing, Lady Ratan Tower, Dainik Shivner Marg, Gandhinagar, Worli, Mumbai 400 018, Maharashtra, India. e-mail: bhup.anand@gmail.com Personal archive: http://alixcomsi.com/Index01.pdf. On-going work pages: https://foundationalperspectives.wordpress.com/2013/11/11/the-holy-grail-of-arithmetic-bridging-provability-and-computability/