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A foundational argument for defining Effective Computability formally, and weakening the Church and Turing Theses – II

(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

$\S 1$ The Logical Issue

In the previous posts we addressed first the computational issue, and second the philosophical issue—concerning the informal concept of effective computability’—that seemed implicit in Selmer Bringsjord’s narrational case against Church’s Thesis [1].

We now address the logical issue that leads to a formal definability of this concept which—arguably—captures our intuitive notion of the concept more fully.

We note that in this paper on undecidable arithmetical propositions we have shown how it follows from Theorem VII of Gödel’s seminal 1931 paper that every recursive function $f(x_{1}, x_{2})$ is representable in the first-order Peano Arithmetic PA by a formula $[F(x_{1}, x_{2}, x_{3})]$ which is algorithmically verifiable, but not algorithmically computable, if we assume (Aristotle’s particularisation) that the negation of a universally quantified formula of the first-order predicate calculus is always indicative of the existence of a counter-example under the standard interpretation of PA.

In this earlier post on the Birmingham paper, we have also shown that:

We shall argue in this post that the standard postulation of the Church-Turing Thesis—which postulates that the intuitive concept of effective computability’ is completely captured by the formal notion of algorithmic computability’—does not hold if we formally define a number-theoretic formula as effectively computable if, and only if, it is algorithmically verifiable; and it therefore needs to be replaced by a weaker postulation of the Thesis as an instantiational equivalence.

$\S 2$ Weakening the Church and Turing Theses

We begin by noting that the following theses are classically equivalent [1]:

Standard Church’s Thesis: [2] A number-theoretic function (or relation, treated as a Boolean function) is effectively computable if, and only if, it is recursive [3].

Standard Turing’s Thesis: [4] A number-theoretic function (or relation, treated as a Boolean function) is effectively computable if, and only if, it is Turing-computable [5].

In this paper we shall argue that, from a foundational perspective, the principle of Occam’s razor suggests the Theses should be postulated minimally as the following equivalences:

Weak Church’s Thesis: A number-theoretic function (or relation, treated as a Boolean function) is effectively computable if, and only if, it is instantiationally equivalent to a recursive function (or relation, treated as a Boolean function).

Weak Turing’s Thesis: A number-theoretic function (or relation, treated as a Boolean function) is effectively computable if, and only if, it is instantiationally equivalent to a Turing-computable function (or relation, treated as a Boolean function).

$\S 2.1$ The need for explicitly distinguishing between instantiational’ and uniform’ methods

Why Church’s Thesis?

It is significant that both Kurt Gödel (initially) and Alonzo Church (subsequently—possibly under the influence of Gödel’s disquietitude) enunciated Church’s formulation of effective computability’ as a Thesis because Gödel was instinctively uncomfortable with accepting it as a definition that minimally captures the essence of intuitive effective computability’ [6].

Kurt Gödel’s reservations

Gödel’s reservations seem vindicated if we accept that a number-theoretic function can be effectively computable instantiationally (in the sense of being algorithmically verifiable as defined in the Birmingham paper, reproduced in this post), but not by a uniform method (in the sense of being algorithmically computable as defined in the Birmingham paper, reproduced in this post).

The significance of the fact (considered in the Birmingham paper, reproduced in this post) that truth’ too can be effectively decidable both instantiationally and by a uniform method under the standard interpretation of PA is reflected in Gödel’s famous 1951 Gibbs lecture[7], where he remarks:

“I wish to point out that one may conjecture the truth of a universal proposition (for example, that I shall be able to verify a certain property for any integer given to me) and at the same time conjecture that no general proof for this fact exists. It is easy to imagine situations in which both these conjectures would be very well founded. For the first half of it, this would, for example, be the case if the proposition in question were some equation $F(n) = G(n)$ of two number-theoretical functions which could be verified up to very great numbers $n$.” [8]

Alan Turing’s perspective

Such a possibility is also implicit in Turing’s remarks [9]:

“The computable numbers do not include all (in the ordinary sense) definable numbers. Let P be a sequence whose n-th figure is 1 or 0 according as n is or is not satisfactory. It is an immediate consequence of the theorem of $\S8$ that P is not computable. It is (so far as we know at present) possible that any assigned number of figures of P can be calculated, but not by a uniform process. When sufficiently many figures of P have been calculated, an essentially new method is necessary in order to obtain more figures.”

Boolos, Burgess and Jeffrey’s query

The need for placing such a distinction on a formal basis has also been expressed explicitly on occasion [10].

Thus, Boolos, Burgess and Jeffrey [11] define a diagonal halting function, $d$, any value of which can be decided effectively, although there is no single algorithm that can effectively compute $d$.

Now, the straightforward way of expressing this phenomenon should be to say that there are well-defined number-theoretic functions that are effectively computable instantiationally but not uniformly. Yet, following Church and Turing, such functions are labeled as uncomputable [12]!

However, as Boolos, Burgess and Jeffrey note quizically:

“According to Turing’s Thesis, since $d$ is not Turing-computable, $d$ cannot be effectively computable. Why not? After all, although no Turing machine computes the function $d$, we were able to compute at least its first few values, For since, as we have noted, $f_{1} = f_{2} = f_{3} =$ the empty function we have $d(1) = d(2) = d(3) = 1$. And it may seem that we can actually compute $d(n)$ for any positive integer $n$—if we don’t run out of time.” [13]

Why should Chaitin’s constant $\Omega$ be labelled uncomputable’?

The reluctance to treat a function such as $d(n)$—or the function $\Omega(n)$ that computes the $n^{th}$ digit in the decimal expression of a Chaitin constant $\Omega$ [14]—as computable, on the grounds that the time’ needed to compute it increases monotonically with $n$, is curious [15]; the same applies to any total Turing-computable function $f(n)$![16]

Moreover, such a reluctance to treat instantiationally computable functions such as $d(n)$ as not effectively computable’ is difficult to reconcile with a conventional wisdom that holds the standard interpretation of the first order Peano Arithmetic PA as defining an intuitively sound model of PA.

Reason: We have shown in the Birmingham paper (reproduced in this post) that ‘satisfaction’ and ‘truth’ under the standard interpretation of PA is definable constructively in terms of algorithmic verifiability (instantiational computability).

$\S 2.2$ Distinguishing between algorithmic verifiability and algorithmic computability

We now show in Theorem 1 that if Aristotle’s particularisation is presumed valid over the structure $\mathbb{N}$ of the natural numbers—as is the case under the standard interpretation of the first-order Peano Arithmetic PA—then it follows from the instantiational nature of the (constructively defined [17]) Gödel $\beta$-function that a primitive recursive relation can be instantiationally equivalent to an arithmetical relation, where the former is algorithmically computable over $\mathbb{N}$, whilst the latter is algorithmically verifiable (i.e., instantiationally computable) but not algorithmically computable over $\mathbb{N}$.[18]

$\S 2.2.1$ Significance of Gödel’s $\beta$-function

We note first that in Theorem VII of his seminal 1931 paper on formally undecidable arithmetical propositions Gödel showed that, given a total number-theoretic function $f(x)$ and any natural number $n$, we can construct a primitive recursive function $\beta(z, y, x)$ and natural numbers $b_{n}, c_{n}$ such that $\beta(b_{n}, c_{n}, i)$ $= f(i)$ for all $0 \leq i \leq n$.

In this paper we shall essentially answer the following question affirmatively:

Query 3: Does Gödel’s Theorem VII admit construction of an arithmetical function $A(x)$ such that:

(a) for any given natural number $n$, there is an algorithm that can verify $A(i) = f(i)$ for all $0 \leq i \leq n$ (hence $A(x)$ may be said to be algorithmically verifiable if $f(x)$ is recursive);

(b) there is no algorithm that can verify $A(i) = f(i)$ for all $0 \leq i$ (so $A(x)$ may be said to be algorithmically uncomputable)?

$\S 2.2.2$ Defining effective computability

Now, in the Birmingham paper (reproduced in this post), we have formally defined what it means for a formula of an arithmetical language to be:

(i) Algorithmically verifiable;

(ii) Algorithmically computable.

under an interpretation.

We shall thus propose the definition:

Effective computability: A number-theoretic formula is effectively computable if, and only if, it is algorithmically verifiable.

Intuitionistically unobjectionable: We note first that since every finite set of integers is recursive, every well-defined number-theoretical formula is algorithmically verifiable, and so the above definition is intuitionistically unobjectionable; and second that the existence of an arithmetic formula that is algorithmically verifiable but not algorithmically computable (Theorem 1) supports Gödel’s reservations on Alonzo Church’s original intention to label his Thesis as a definition [19].

The concept is well-defined, since we have shown in the Birmingham paper (reproduced in this post) that the algorithmically verifiable and the algorithmically computable PA formulas are well-defined under the standard interpretation of PA and that:

(a) The PA-formulas are decidable as satisfied / unsatisfied or true / false under the standard interpretation of PA if, and only if, they are algorithmically verifiable;

(b) The algorithmically computable PA-formulas are a proper subset of the algorithmically verifiable PA-formulas;

(c) The PA-axioms are algorithmically computable as satisfied / true under the standard interpretation of PA;

(d) Generalisation and Modus Ponens preserve algorithmically computable truth under the standard interpretation of PA;

(e) The provable PA-formulas are precisely the ones that are algorithmically computable as satisfied / true under the standard interpretation of PA.

$\S 3$ Gödel’s Theorem VII and algorithmically verifiable, but not algorithmically computable, arithmetical propositions

In his seminal 1931 paper on formally undecidable arithmetical propositions, Gödel defined a curious primitive recursive function—Gödel’s $\beta$-function—as [20]:

Definition 1: $\beta (x_{1}, x_{2}, x_{3}) = rm(1+(x_{3}+ 1) \star x_{2}, x_{1})$

where $rm(x_{1}, x_{2})$ denotes the remainder obtained on dividing $x_{2}$ by $x_{1}$.

Gödel showed that the above function has the remarkable property that:

Lemma 1: For any given denumerable sequence of natural numbers, say $f(k, 0),\ f(k, 1),\ \ldots$, and any given natural number $n$, we can construct natural numbers $b, c, j$ such that:

(i) $j = max(n, f(k, 0), f(k, 1), \ldots, f(k, n))$;

(ii) $c = j$!;

(iii) $\beta(b, c, i) = f(k, i)$ for $0 \leq i \leq n$.

Proof: This is a standard result [21]. $\Box$

Now we have the standard definition [22]:

Definition 2: A number-theoretic function $f(x_{1}, \ldots, x_{n})$ is said to be representable in PA if, and only if, there is a PA formula $[F(x_{1}, \dots, x_{n+1})]$ with the free variables $[x_{1}, \ldots, x_{n+1}]$, such that, for any given natural numbers $k_{1}, \ldots, k_{n+1}$:

(i) if $f(k_{1}, \ldots, k_{n}) = k_{n+1}$ then PA proves: $[F(k_{1}, \ldots, k_{n}, k_{n+1})]$;

(ii) PA proves: $[(\exists_{1} x_{n+1})F(k_{1}, \ldots, k_{n}, x_{n+1})]$.

The function $f(x_{1}, \ldots, x_{n})$ is said to be strongly representable in PA if we further have that:

(iii) PA proves: $[(\exists_{1} x_{n+1})F(x_{1}, \ldots, x_{n}, x_{n+1})]$

Interpretation of $[\exists_{1}]$‘: The symbol $[\exists_{1}]$‘ denotes uniqueness’ under an interpretation which assumes that Aristotle’s particularisation holds in the domain of the interpretation.

Formally, however, the PA formula:

$[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$

is merely a short-hand notation for the PA formula:

$[\neg(\forall x_{3})\neg F(x_{1}, x_{2}, x_{3}) \wedge (\forall y)(\forall z)(F(x_{1}, x_{2}, y) \wedge F(x_{1}, x_{2}, z) \rightarrow y=z)]$.

We then have:

Lemma 2 $\beta(x_{1}, x_{2}, x_{3})$ is strongly represented in PA by $[Bt(x_{1}, x_{2}, x_{3}, x_{4})]$, which is defined as follows:

$[(\exists w)(x_{1} = ((1 + (x_{3} + 1)\star x_{2}) \star w + x_{4}) \wedge (x_{4} < 1 + (x_{3} + 1) \star x_{2}))]$.

Proof: This is a standard result [23]. $\Box$

Gödel further showed (also under the tacit, but critical, presumption of Aristotle’s particularisation [24] that:

Lemma 3: If $f(x_{1}, x_{2})$ is a recursive function defined by:

(i) $f(x_{1}, 0) = g(x_{1})$

(ii) $f(x_{1}, (x_{2}+1)) = h(x_{1}, x_{2}, f(x_{1}, x_{2}))$

where $g(x_{1})$ and $h(x_{1}, x_{2}, x_{3})$ are recursive functions of lower rank [25] that are represented in PA by well-formed formulas $[G(x_{1}, x_{2})]$ and $[H(x_{1}, x_{2}, x_{3}, x_{4})]$,

then $f(x_{1}, x_{2})$ is represented in PA by the following well-formed formula, denoted by $[F(x_{1}, x_{2}, x_{3})]$:

$[(\exists u)(\exists v)(((\exists w)(Bt(u, v, 0, w) \wedge G(x_{1}, w))) \wedge Bt(u, v, x_{2}, x_{3}) \wedge (\forall w)(w < x_{2} \rightarrow (\exists y)(\exists z)(Bt(u, v, w, y) \wedge Bt(u, v, (w+1), z) \wedge H(x_{1}, w, y, z)))].$

Proof: This is a standard result [26]. $\Box$

$\S 4.1$ What does “$[(\exists_{1} x_{3})F(k, m, x_{3})]$ is provable” assert under the standard interpretation of PA?

Now, if the PA formula $[F(x_{1}, x_{2}, x_{3})]$ represents in PA the recursive function denoted by $f(x_{1}, x_{2})$ then by definition, for any given numerals $[k], [m]$, the formula $[(\exists_{1} x_{3})F(k, m, x_{3})]$ is provable in PA; and true under the standard interpretation of PA.

We thus have that:

Lemma 4:$[(\exists_{1} x_{3})F(k, m, x_{3})]$ is true under the standard interpretation of PA” is the assertion that:

Given any natural numbers $k, m$, we can construct natural numbers $t_{(k, m)}, u_{(k, m)}, v_{(k, m)}$—all functions of $k, m$—such that:

(a) $\beta(u_{(k, m)}, v_{(k, m)}, 0) = g(k)$;

(b) for all $i, $\beta(u_{(k, m)}, v_{(k, m)}, i) = h(k, i, f(k, i))$;

(c) $\beta(u_{(k, m)}, v_{(k, m)}, m) = t_{(k, m)}$;

where $f(x_{1}, x_{2})$, $g(x_{1})$ and $h(x_{1}, x_{2}, x_{3})$ are any recursive functions that are formally represented in PA by $F(x_{1}, x_{2}, x_{3}), G(x_{1}, x_{2})$ and $H(x_{1}, x_{2}, x_{3}, x_{4})$ respectively such that:

(i) $f(k, 0) = g(k)$

(ii) $f(k, (y+1)) = h(k, y, f(k, y))$ for all $y

(iii) $g(x_{1})$ and $h(x_{1}, x_{2}, x_{3})$ are recursive functions that are assumed to be of lower rank than $f(x_{1}, x_{2})$.

Proof: For any given natural numbers $k$ and $m$, if $[F(x_{1}, x_{2}, x_{3})]$ interprets as a well-defined arithmetical relation under the standard interpretation of PA, then we can define a deterministic Turing machine $TM$ that can construct’ the sequences:

$f(k, 0), f(k, 1), \ldots, f(k, m)$

and:

$\beta(u_{(k, m)}, v_{(k, m)}, 0), \beta(u_{(k, m)}, v_{(k, m)}, 1), \ldots, \beta(u_{(k, m)}, v_{(k, m)}, m)$

and give evidence to verify the assertion. $\Box$[27]

We now see that:

Theorem 1: Under the standard interpretation of PA $[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$ is algorithmically verifiable, but not algorithmically computable, as always true over $\mathbb{N}$.

Proof: It follows from Lemma 4 that:

(1) $[(\exists_{1} x_{3})F(k, m, x_{3})]$ is PA-provable for any given numerals $[k, m]$. Hence $[(\exists_{1} x_{3})F(k, m, x_{3})]$ is true under the standard interpretation of PA. It then follows from the definition of $[F(x_{1}, x_{2}, x_{3})]$ in Lemma 3 that, for any given natural numbers $k, m$, we can construct some pair of natural numbers $u_{(k, m)}, v_{(k, m)}$—where $u_{(k, m)}, v_{(k, m)}$ are functions of the given natural numbers $k$ and $m$—such that:

(a) $\beta(u_{(k, m)}, v_{(k, m)}, i) = f(k, i)$ for $0 \leq i \leq m$;

(b) $F^{*}(k, m, f(k, m))$ holds in $\mathbb{N}$.

Since $\beta(x_{1}, x_{2}, x_{3})$ is primitive recursive, $\beta(u_{(k, m)}, v_{(k, m)}, i)$ defines a deterministic Turing machine $TM$ that can construct’ the denumerable sequence $f'(k, 0), f'(k, 1), \ldots$ for any given natural numbers $k$ and $m$ such that:

(c) $f(k, i) = f'(k, i)$ for $0 \leq i \leq m$.

We can thus define a deterministic Turing machine $TM$ that will give evidence that the PA formula $[(\exists_{1} x_{3})F(k, m, x_{3})]$ is true under the standard interpretation of PA.

Hence $[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$ is algorithmically verifiable over $\mathbb{N}$ under the standard interpretation of PA.

(2) Now, the pair of natural numbers $u_{(x_{1}, x_{2})}, v_{(x_{1}, x_{2})}$ are defined such that:

(a) $\beta(u_{(x_{1}, x_{2})}, v_{(x_{1}, x_{2})}, i) = f(x_{1}, i)$ for $0 \leq i \leq x_{2}$;

(b) $F^{*}(x_{1}, x_{2}, f(x_{1}, x_{2}))$ holds in $\mathbb{N}$;

where $v_{(x_{1}, x_{2})}$ is defined in Lemma 3 as $j$!, and:

(c) $j = max(n, f(x_{1}, 0), f(x_{1}, 1), \ldots, f(x_{1}, x_{2}))$;

(d) $n$ is the number’ of terms in the sequence $f(x_{1}, 0), f(x_{1}, 1), \ldots, f(x_{1}, x_{2})$.

Since $j$ is not definable for a denumerable sequence $\beta(u_{(x_{1}, x_{2})}, v_{(x_{1}, x_{2})}, i)$ we cannot define a denumerable sequence $f'(x_{1}, 0), f'(x_{1}, 1), \ldots$ such that:

(e) $f(k, i) = f'(k, i)$ for all $i \geq 0$.

We cannot thus define a deterministic Turing machine $TM$ that will give evidence that the PA formula $[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$ interprets as true under the standard interpretation of PA for any given sequence of numerals $[(a_{1}, a_{2})]$.

Hence $[(\exists_{1} x_{3})F(x_{1}, x_{2}, x_{3})]$ is not algorithmically computable over $\mathbb{N}$ under the standard interpretation of PA.

The theorem follows. $\Box$

Corollary 1: If the standard interpretation of PA is sound, then the classical Church and Turing theses are false.

The above theorem now suggests the following definition:

Definition 2: (Effective computability) A number-theoretic function is effectively computable if, and only if, it is algorithmically verifiable.

Such a definition of effective computability now allows the classical Church and Turing theses to be expressed as the weak equivalences in $\S 2$—rather than as identities—without any apparent loss of generality.

References

BBJ03 George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic (4th ed). Cambridge University Press, Cambridge.

Bri93 Selmer Bringsjord. 1993. The Narrational Case Against Church’s Thesis. Easter APA meetings, Atlanta.

Ch36 Alonzo Church. 1936. An unsolvable problem of elementary number theory. In M. Davis (ed.). 1965. The Undecidable Raven Press, New York. Reprinted from the Am. J. Math., Vol. 58, pp.345-363.

Ct75 Gregory J. Chaitin. 1975. A Theory of Program Size Formally Identical to Information Theory. J. Assoc. Comput. Mach. 22 (1975), pp. 329-340.

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.

Go51 Kurt Gödel. 1951. Some basic theorems on the foundations of mathematics and their implications. Gibbs lecture. In Kurt Gödel, Collected Works III, pp.304-323.\ 1995. Unpublished Essays and Lectures. Solomon Feferman et al (ed.). Oxford University Press, New York.

Ka59 Laszlo Kalmár. 1959. An Argument Against the Plausibility of Church’s Thesis. In Heyting, A. (ed.) Constructivity in Mathematics. North-Holland, Amsterdam.

Kl36 Stephen Cole Kleene. 1936. General Recursive Functions of Natural Numbers. Math. Annalen vol. 112 (1936) pp.727-766.

Me64 Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand, Princeton.

Me90 Elliott Mendelson. 1990. Second Thoughts About Church’s Thesis and Mathematical Proofs. Journal of Philosophy 87.5.

Pa71 Rohit Parikh. 1971. Existence and Feasibility in Arithmetic. The Journal of Symbolic Logic, Vol.36, No. 3 (Sep., 1971), pp. 494-508.

Si97 Wilfried Sieg. 1997. Step by recursive step: Church’s analysis of effective calculability Bulletin of Symbolic Logic, Volume 3, Number 2.

Sm07 Peter Smith. 2007. Church’s Thesis after 70 Years. A commentary and critical review of Church’s Thesis After 70 Years. In Meinong Studies Vol 1 (Ontos Mathematical Logic 1), 2006 (2013), Eds. Adam Olszewski, Jan Wolenski, Robert Janusz. Ontos Verlag (Walter de Gruyter), Frankfurt, Germany.

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.

An12 … 2012. Evidence-Based Interpretations of PA. In 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.

Notes

Return to 2: Church’s (original) Thesis: The effectively computable number-theoretic functions are the algorithmically computable number-theoretic functions Ch36.

Return to 4: After describing what he meant by “computable” numbers in the opening sentence of his 1936 paper on Computable Numbers Tu36, Turing immediately expressed this thesis—albeit informally—as: “… the computable numbers include all numbers which could naturally be regarded as computable”.

Return to 8: Parikh’s paper Pa71 can also be viewed as an attempt to investigate the consequences of expressing the essence of Gödel’s remarks formally.

Return to 9: Tu36, $\S9(II)$, p.139.

Return to 10: Parikh’s distinction between decidability’ and feasibility’ in Pa71 also appears to echo the need for such a distinction.

Return to 12: The issue here seems to be that, when using language to express the abstract objects of our individual, and common, mental concept spaces’, we use the word exists’ loosely in three senses, without making explicit distinctions between them (see An07).

Return to 14: Chaitin’s Halting Probability is given by $0 < \Omega = \sum2^{-|p|} < 1$, where the summation is over all self-delimiting programs $p$ that halt, and $|p|$ is the size in bits of the halting program $p$; see Ct75.

Return to 16: The only difference being that, in the latter case, we know there is a common program’ of constant length that will compute $f(n)$ for any given natural number $n$; in the former, we know we may need distinctly different programs for computing $f(n)$ for different values of $n$, where the length of the program will, sometime, reference $n$.

Return to 18: Analagous distinctions in analysis: The distinction between algorithmically computable, and algorithmically verifiable but not algorithmically computable, number-theoretic functions seeks to reflect in arithmetic the essence of uniform methods (formally detailed in the Birmingham paper (reproduced in this post) and in its main consequence—the Provability Theorem for PA—as detailed in this post), classically characterised by the distinctions in analysis between: (a) uniformly continuous, and point-wise continuous but not uniformly continuous, functions over an interval; (b) uniformly convergent, and point-wise convergent but not uniformly convergent, series.

A limitation of set theory and a possible barrier to computation: We note, further, that the above distinction cannot be reflected within a language—such as the set theory ZF—which identifies equality’ with equivalence’. Since functions are defined extensionally as mappings, such a language cannot recognise that a set which represents a primitive recursive function may be equivalent to, but computationally different from, a set that represents an arithmetical function; where the former function is algorithmically computable over $\mathbb{N}$, whilst the latter is algorithmically verifiable but not algorithmically computable over $\mathbb{N}$.

Return to 19: See the Provability Theorem for PA in this post.

Return to 20: cf. Go31, p.31, Lemma 1; Me64, p.131, Proposition 3.21.

Return to 21: cf. Go31, p.31, Lemma 1; Me64, p.131, Proposition 3.22.

Return to 24: The implicit assumption being that the negation of a universally quantified formula of the first-order predicate calculus is indicative of “the existence of a counter-example”—Go31, p.32.

Return to 27: A critical philosophical issue that we do not address here is whether the PA formula $[F(x_{1}, x_{2}, x_{3}]$ can be considered to interpret under a sound interpretation of PA as a well-defined predicate, since the denumerable sequences $\{f(k, 0), f(k, 1), \ldots, f(k, m), m_{p}: p>0$ and $m_{p}$ is not equal to $m_{q}$ if $p$ is not equal to $q\}$—are represented by denumerable, distinctly different, functions $\beta(u_{p_{1}}, v_{p_{2}}, i)$ respectively. There are thus denumerable pairs $(u_{p_{1}}, v_{p_{2}})$ for which $\beta(u_{p_{1}}, v_{p_{2}}, i)$ yields the sequence $f(k, 0), f(k, 1), \ldots, f(k, m)$.

Author’s working archives & abstracts of investigations

A foundational argument for defining Effective Computability formally, and weakening the Church and Turing Theses – I

(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

$\S 1.1$ The Philosophical Issue

In a previous post we have argued that standard interpretations of classical theory may inadvertently be weakening a desirable perception—of mathematics as the lingua franca of scientific expression—by ignoring the possibility that since mathematics is, indeed, indisputably accepted as the language that most effectively expresses and communicates intuitive truth, the chasm between formal truth and provability must, of necessity, be bridgeable.

We further queried whether the roots of such interpretations may lie in removable ambiguities that currently persist in the classical definitions of foundational elements; ambiguities that allow the introduction of non-constructive—hence non-verifiable, non-computational, ambiguous and essentially Platonic—elements into the standard interpretations of classical mathematics.

Query 1: Are formal classical theories essentially unable to adequately express the extent and range of human cognition, or does the problem lie in the way formal theories are classically interpreted at the moment?

We noted that the former addressed the question of whether there are absolute limits on our capacity to express human cognition unambiguously; the latter, whether there are only temporal limits—not necessarily absolute—to the capacity of classical interpretations to communicate unambiguously that which we intended to capture within our formal expression.

We argued that, prima facie, applied science continues, perforce, to interpret mathematical concepts Platonically, whilst waiting for mathematics to provide suitable, and hopefully reliable, answers as to how best it may faithfully express its observations verifiably.

$\S 1.2$ Are axiomatic computational concepts really unambiguous?

This now raises the corresponding philosophical question that is implicit in Selmer Bringsjord’s narrational case against Church’s Thesis [1]:

Query 2 Is there a duality in the classical acceptance of non-constructive, foundational, concepts as axiomatic?

We now argue that beyond the question raised in an earlier post of whether—as computer scientist Lance Fortnow believes—Turing machines can capture everything we can compute’, or whether—as computer scientists Peter Wegner and Dina Goldin suggest—they are inappropriate as a universal foundation for computational problem solving’, we also need to address the philosophical question—implicit in Bringsjord’s paper—of whether, or not, the concept of effective computability’ is capable of a constructive, and intuitionistically unobjectionable, definition; and the relation of such definition to that of formal provability and to the standard perceptions of the Church and Turing Theses as reviewed here by at heart Trinity mathmo and by profession Philosopher Peter Smith.

We therefore consider the case for introduction of such a definition from a philosophical point of view, and consider some consequences.

$\S 1.2.1$ Mendelson’s thesis

We note that Elliott Mendelson [2] is quoted by Bringsjord in his paper as saying (italicised parenthetical qualifications added):

(i) “Here is the main conclusion I wish to draw:

it is completely unwarranted to say that CT is unprovable just because it states an equivalence between a vague, imprecise notion (effectively computable function) and a precise mathematical notion (partial-recursive function)”.

(ii) “The concepts and assumptions that support the notion of partial-recursive function are, in an essential way, no less vague and imprecise (non-constructive, and intuitionistically objectionable) than the notion of effectively computable function; the former are just more familiar and are part of a respectable theory with connections to other parts of logic and mathematics.

(The notion of effectively computable function could have been incorporated into an axiomatic presentation of classical mathematics, but the acceptance of CT made this unnecessary.) …

Functions are defined in terms of sets, but the concept of set is no clearer (not more non-constructive, and intuitionistically objectionable) than that of function and a foundation of mathematics can be based on a theory using function as primitive notion instead of set.

Tarski’s definition of truth is formulated in set-theoretic terms, but the notion of set is no clearer (not more non-constructive, and intuitionistically objectionable), than that of truth.

The model-theoretic definition of logical validity is based ultimately on set theory, the foundations of which are no clearer (not more non-constructive, and intuitionistically objectionable) than our intuitive (non-constructive, and intuitionistically objectionable) understanding of logical validity”.

(iii) “The notion of Turing-computable function is no clearer (not more non-constructive, and intuitionistically objectionable) than, nor more mathematically useful (foundationally speaking) than, the notion of an effectively computable function …”

where:

(a) The Church-Turing Thesis, CT, is formulated as:

“A function is effectively computable if and only if it is Turing-computable”.

(b) An effectively computable function is defined to be the computing of the function by an algorithm.

(c) The classical notion of an algorithm is expressed by Mendelson as:

“… an effective and completely specified procedure for solving a whole class of problems. …

An algorithm does not require ingenuity; its application is prescribed in advance and does not depend upon any empirical or random factors”.

and, where Bringsjord paraphrases (iii) as:

(iv) “The notion of a formally defined program for guiding the operation of a TM is no clearer than, nor more mathematically useful (foundationally speaking) than, the notion of an algorithm”.

(v) “This proposition, it would then seem, is the very heart of the matter.

If (iv) is true then Mendelson has made his case; if this proposition is false, then his case is doomed, since we can chain back by modus tollens and negate (iii)”.

$\S 1.2.2$ The concept of constructive, and intuitionistically unobjectionable’

Now, prima facie, any formalisation of a vague and imprecise’, intuitive’ concept—say C—would normally be intended to capture the concept C both faithfully and completely within a constructive, and intuitionistically unobjectionable [3], language L.

Clearly, we could disprove the thesis—that C and its formalisation L are interchangeable, hence equivalent—by showing that there is a constructive aspect of C that is formalisable in a constructive language L’, but that such formalisation cannot be assumed expressible in L without introducing inconsistency.

However, equally clearly, there can be no way of proving the equivalence as this would contradict the premise that the concept is vague and imprecise’, hence essentially open-ended in a non-definable way, and so non-formalisable.

Obviously, Mendelson’s assertion that there is no justification for claiming Church’s Thesis as unprovable must, therefore, rely on an interpretation that differs significantly from the above; for instance, his concept of provability may appeal to the axiomatic acceptability of vague and imprecise’ concepts—as suggested by his remarks.

Now, we note that all the examples cited by Mendelson involve the decidability (computability) of an infinitude of meta-mathematical instances, where the distinction between the constructive meta-assertion that any given instance is individually decidable (instantiationally computable), and the non-constructive meta-assertion that all the instances are jointly decidable (uniformly computable), is not addressed explicitly.

However, $\S 1.2.1(a)$, $\S 1.2.1(b)$ and $\S 1.2.1(c)$ appear to suggest that Mendelson’s remarks relate implicitly to non-constructive meta-assertions.

Perhaps the real issue, then, is the one that emerges if we replace Mendelson’s use of implicitly open-ended concepts such as vague and imprecise’ and intuitive’ by the more meta-mathematically meaningful concept of non-constructive, and intuitionistically objectionable’, as italicised and indicated parenthetically.

The essence of Mendelson’s meta-assertion $\S 1.2.1(iii)$ then appears to be that, if the classically accepted definitions of foundational concepts such as partial recursive function’, function’, Tarskian truth’ etc. are also non-constructive, and intuitionistically objectionable, then replacing one non-constructive concept by another may be psychologically unappealing, but it should be meta-mathematically valid and acceptable.

$\S 1.2.3$ The duality

Clearly, meta-assertion $\S 1.2.1(iii)$ would stand refuted by a non-algorithmic’ effective method that is constructive.

However, if it is explicitly—and, as suggested by the nature of the arguments in Bringsjord’s paper, widely—accepted at the outset that any effective method is necessarily algorithmic (i.e. uniform as stated in $\S 1.2.1(c)$ above), then any counter-argument to CT can, prima facie, only offer non-algorithmic methods that may, paradoxically, be effective’ intuitively but in a non-constructive, and intuitionistically objectionable, way only!

Recognition of this dilemma is implicit in the admission that the various arguments, as presented by Bringsjord in the case against Church’s Thesis—including his narrational case—are open to reasonable, but inconclusive, refutations.

Nevertheless, if we accept Mendelson’s thesis that the inter-changeability of non-constructive concepts is valid in the foundations of mathematics, then Bringsjord’s case against Church’s Thesis, since it is based similarly on non-constructive concepts, should also be considered conclusive classically (even though it cannot, prima facie, be considered constructively conclusive in an intuitionistically unobjectionable way).

There is, thus, an apparent duality in the—seemingly extra-logical—decision as to whether an argument based on non-constructive concepts may be accepted as classically conclusive or not.

That this duality may originate in the very issues raised in Mendelson’s remarks—concerning the non-constructive roots of foundational concepts that are classically accepted as mathematically sound—is seen if we note that these issues may be more significant than is, prima facie, apparent.

$\S 1.2.4$ Definition of a formal mathematical object, and consequences

Thus, if we define a formal mathematical object as any symbol for an individual letter, function letter or a predicate letter that can be introduced through definition into a formal theory without inviting inconsistency, then it can be argued that unrestricted, non-constructive, definitions of non-constructive, foundational, set-theoretic concepts—such as mapping’, function’, recursively enumerable set’, etc.—in terms of constructive number-theoretic concepts—such as recursive number-theoretic functions and relations—may not always correspond to formal mathematical objects.

In other words, the assumption that every definition corresponds to a formal mathematical object may introduce a formal inconsistency into standard Peano Arithmetic and, ipso facto, into any Axiomatic Set Theory that models standard PA (an inconsistency which, loosely speaking, may be viewed as a constructive arithmetical parallel to Russell’s non-constructive impredicative set).

Since it can also be argued that the non-constructive element in Tarski’s definitions of satisfiability’ and truth’, and in Church’s Thesis, originate in a common, but removable, ambiguity in the interpretation of an effective method, perhaps it is worth considering whether Bringsjord’s acceptance of the assumption—that every constructive effective method is necessarily algorithmic, in the sense of being a uniform procedure as in $\S 1.2.1(c)$ above—is mathematically necessary, or even whether it is at all intuitively tenable.

(Uniform procedure: A property usually taken to be a necessary condition for a procedure to qualify as effective.)

Thus, we may argue that we can explicitly and constructively define a non-algorithmic’ effective method as one that, in any given instance, is instantiationally computable if, and only if, it terminates finitely with a conclusive result; and an algorithmic’ effective method as one that is uniformly computable if, and only if, it terminates finitely, with a conclusive result, in any given instance. [4]

$\S 1.2.5$ Bringsjord’s case against CT

Apropos the specific arguments against CT it would seem, prima facie, that a non-algorithmic effective—even if not obviously constructive—method could be implicit in the following argument considered by Bringsjord:

“Assume for the sake of argument that all human cognition consists in the execution of effective processes (in brains, perhaps). It would then follow by CT that such processes are Turing-computable, i.e., that computationalism is true. However, if computationalism is false, while there remains incontrovertible evidence that human cognition consists in the execution of effective processes, CT is overthrown”.

Assuming computationalism is false, the issue in this argument would, then, be whether there is a constructive, and adequate, expression of human cognition in terms of individually effective methods.

An appeal to such a non-algorithmic effective method may, in fact, be implicit in Bringsjord’s consideration of the predicate H, defined by:

$H(P, i)$ iff $(\exists n)S(P, i, n)$

where the predicate $S(P, u, n)$ holds if, and only if, TM M, running program P on input $u$, halts in exactly $n$ steps ($= MP : u =>n$ halt).

Bringsjord’s Total Computability

Bringsjord defines S as totally (and, implicitly, uniformly) computable in the sense that, given some triple $(P, u, n)$, there is some (uniform) program P* which, running on some TM M*, can infallibly give us a verdict, Y (yes’) or N (no’), for whether or not S is true of this triple.

He then notes that, since the ability to (uniformly) determine, for a pair $(P, i)$, whether or not H is true of it, is equivalent to solving the full halting problem, H is not totally computable.

Bringsjord’s Partial Computability

However, he also notes that there is a program (implicitly non-uniform, and so, possibly, effective individually) which, when asked whether or not some TM M run by P on $u$ halts, will produce Y iff $MP : u =>n$ halt. For this reason H is declared partially (implicitly, individually) computable.

More explicitly, Bringsjord remarks that Laszlo Kalmár’s refutation of CT [5] is classically inconclusive mainly because it does not admit any uniform effective method, but appeals to the existence of an infinitude of individually effective methods.

Kalmár’s Perspective of CT

Considering that it always seeks to calculate $g(n)$ (defined below) constructively even in the absence of a uniform procedure—not necessarily within a fixed postulate system—we shall show that Kalmár’s finitary argument ([6]as reproduced below from Bringsjord’s paper) makes a pertinent observation:

“First, he draws our attention to a function $g$ that isn’t Turing-computable, given that $f$ is [7]:

$g(x) = \mu y(f(x, y) = 0)$ = {the least $y$ such that $f(x, y) = 0$ if $y$ exists; and $0$ if there is no such $y$}

Kalmár proceeds to point out that for any $n$ in $N$ for which a natural number $y$ with $f(n, y) = 0$ exists,

an obvious method for the calculation of the least such $y$ … can be given,’

namely, calculate in succession the values $f(n, 0), f(n, 1), f(n, 2), \ldots$ (which, by hypothesis, is something a computist or TM can do) until we hit a natural number $m$ such that $f(n, m) = 0$, and set $y = m$.

On the other hand, for any natural number $n$ for which we can prove, not in the frame of some fixed postulate system but by means of arbitrary—of course, correct—arguments that no natural number $y$ with $f(n, y) = 0$ exists, we have also a method to calculate the value $g(n)$ in a finite number of steps.

Kalmár goes on to argue as follows. The definition of $g$ itself implies the tertium non datur, and from it and CT we can infer the existence of a natural number $p$ which is such that

(*) there is no natural number $y$ such that $f(p, y)= 0$; and

(**) this cannot be proved by any correct means.

Kalmár claims that (*) and (**) are very strange, and that therefore CT is at the very least implausible.”

Distinguishing between individually effective computability and uniformly effective computability

Now, the significant point that emerges from Bringsjord’s and Kalmár’s philosophical arguments is the need to distinguish formally between non-algorithmic (i.e., instantiational) computability (or, more precisely, algorithmic verifiability as defined here) and algorithmic (i.e., uniform) computability (or, more precisely, algorithmic computability as defined here) as highlighted in the Birmingham paper.

In the next post we note that this point has also been raised from a more formal, logical, perspective; and consider what is arguably an intuitively-more-adequate formal definability of effective computability’ in terms of ‘algorithmic verifiability’ under which the classical Church and Turing Theses do not hold, but weakened Church and Turing Theses do!

References

BBJ03 George S. Boolos, John P. Burgess, Richard C. Jeffrey. 2003. Computability and Logic (4th ed). Cambridge University Press, Cambridge.

Bri93 Selmer Bringsjord. 1993. The Narrational Case Against Church’s Thesis. Easter APA meetings, Atlanta.

Ch36 Alonzo Church. 1936. An unsolvable problem of elementary number theory. In M. Davis (ed.). 1965. The Undecidable Raven Press, New York. Reprinted from the Am. J. Math., Vol. 58, pp.345-363.

Ct75 Gregory J. Chaitin. 1975. A Theory of Program Size Formally Identical to Information Theory. J. Assoc. Comput. Mach. 22 (1975), pp. 329-340.

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.

Go51 Kurt Gödel. 1951. Some basic theorems on the foundations of mathematics and their implications. Gibbs lecture. In Kurt Gödel, Collected Works III, pp.304-323.\ 1995. Unpublished Essays and Lectures. Solomon Feferman et al (ed.). Oxford University Press, New York.

Ka59 Laszlo Kalmár. 1959. An Argument Against the Plausibility of Church’s Thesis. In Heyting, A. (ed.) Constructivity in Mathematics. North-Holland, Amsterdam.

Kl36 Stephen Cole Kleene. 1936. General Recursive Functions of Natural Numbers. Math. Annalen vol. 112 (1936) pp.727-766.

Me64 Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand, Princeton.

Me90 Elliott Mendelson. 1990. Second Thoughts About Church’s Thesis and Mathematical Proofs. Journal of Philosophy 87.5.

Pa71 Rohit Parikh. 1971. Existence and Feasibility in Arithmetic. The Journal of Symbolic Logic, Vol.36, No. 3 (Sep., 1971), pp. 494-508.

Si97 Wilfried Sieg. 1997. Step by recursive step: Church’s analysis of effective calculability Bulletin of Symbolic Logic, Volume 3, Number 2.

Sm07 Peter Smith. 2007. Church’s Thesis after 70 Years. A commentary and critical review of Church’s Thesis After 70 Years. In Meinong Studies Vol 1 (Ontos Mathematical Logic 1), 2006 (2013), Eds. Adam Olszewski, Jan Wolenski, Robert Janusz. Ontos Verlag (Walter de Gruyter), Frankfurt, Germany.

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.

An12 … 2012. Evidence-Based Interpretations of PA. In 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.

Notes

Return to 3: The terms constructive’ and constructive, and intuitionistically unobjectionable’ are used synonymously both in their familiar linguistic sense, and in a mathematically precise sense. Mathematically, we term a concept as constructive, and intuitionistically unobjectionable’ if, and only if, it can be defined in terms of pre-existing concepts without inviting inconsistency. Otherwise, we understand it to mean unambiguously verifiable, by some effective method’, within some finite, well-defined, language or meta-language. It may also be taken to correspond, broadly, to the concept of constructive, and intuitionistically unobjectionable’ in the sense apparently intended by Gödel in his seminal 1931 paper Go31, p.26.

Return to 4: We note that the possibility of a distinction between the interpreted number-theoretic meta-assertions, For any given natural number $x$, $F(x)$ is true’ and $F(x)$ is true for all natural numbers $x$‘, is not evident unless these are expressed symbolically as, $(\forall x)(F(x)$ is true)’ and $(\forall x)F(x)$ is true’, respectively. The issue, then, is whether the distinction can be given any mathematical significance. For instance, under a constructive formulation of Tarski’s definitions, we may qualify the latter by saying that it can be meaningfully asserted as a totality only if $F$‘ is a well-defined mathematical object.

Return to 7: Bringsjord notes that the original proof can be found on page 741 of Kleene Kl36.

Return to 8: We detail a formal proof of this Thesis in this post.

Author’s working archives & abstracts of investigations

$\S 1$ The Holy Grail of Arithmetic: Bridging Provability and Computability

(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)

Peter Wegner and Dina Goldin

In a short opinion paper, Computation Beyond Turing Machines‘, Computer Scientists Peter Wegner and Dina Goldin (Wg03) advanced the thesis that:

A paradigm shift is necessary in our notion of computational problem solving, so it can provide a complete model for the services of today’s computing systems and software agents.’

We note that Wegner and Goldin’s arguments, in support of their thesis, seem to reflect an extraordinarily eclectic view of mathematics, combining both an implicit acceptance of, and implicit frustration at, the standard interpretations and dogmas of classical mathematical theory:

(i) … Turing machines are inappropriate as a universal foundation for computational problem solving, and … computer science is a fundamentally non-mathematical discipline.’

(ii) (Turing’s) 1936 paper … proved that mathematics could not be completely modeled by computers.’

(iii) … the Church-Turing Thesis … equated logic, lambda calculus, Turing machines, and algorithmic computing as equivalent mechanisms of problem solving.’

(iv) Turing implied in his 1936 paper that Turing machines … could not provide a model for all forms of mathematics.’

(v) … Gödel had shown in 1931 that logic cannot model mathematics … and Turing showed that neither logic nor algorithms can completely model computing and human thought.’

These remarks vividly illustrate the dilemma with which not only Theoretical Computer Sciences, but all applied sciences that depend on mathematics—for providing a verifiable language to express their observations precisely—are faced:

Query: Are formal classical theories essentially unable to adequately express the extent and range of human cognition, or does the problem lie in the way formal theories are classically interpreted at the moment?

The former addresses the question of whether there are absolute limits on our capacity to express human cognition unambiguously; the latter, whether there are only temporal limits—not necessarily absolute—to the capacity of classical interpretations to communicate unambiguously that which we intended to capture within our formal expression.

Prima facie, applied science continues, perforce, to interpret mathematical concepts Platonically, whilst waiting for mathematics to provide suitable, and hopefully reliable, answers as to how best it may faithfully express its observations verifiably.

Lance Fortnow

This dilemma is also reflected in Computer Scientist Lance Fortnow’s on-line rebuttal of Wegner and Goldin’s thesis, and of their reasoning.

Thus Fortnow divides his faith between the standard interpretations of classical mathematics (and, possibly, the standard set-theoretical models of formal systems such as standard Peano Arithmetic), and the classical computational theory of Turing machines.

He relies on the former to provide all the proofs that matter:

Not every mathematical statement has a logical proof, but logic does capture everything we can prove in mathematics, which is really what matters’;

and, on the latter to take care of all essential, non-provable, truth:

… what we can compute is what computer science is all about’.

Can faith alone suffice?

However, as we shall argue in a subsequent post, Fortnow’s faith in a classical Church-Turing Thesis that ensures:

… Turing machines capture everything we can compute’,

may be as misplaced as his faith in the infallibility of standard interpretations of classical mathematics.

The reason: There are, prima facie, reasonably strong arguments for a Kuhnian (Ku62) paradigm shift; not, as Wegner and Goldin believe, in the notion of computational problem solving, but in the standard interpretations of classical mathematical concepts.

However, Wegner and Goldin could be right in arguing that the direction of such a shift must be towards the incorporation of non-algorithmic effective methods into classical mathematical theory (as detailed in the Birmingham paper); presuming, from the following remarks, that this is, indeed, what external interactions’ are assumed to provide beyond classical Turing-computability:

(vi) … that Turing machine models could completely describe all forms of computation … contradicted Turing’s assertion that Turing machines could only formalize algorithmic problem solving … and became a dogmatic principle of the theory of computation’.

(vii) … interaction between the program and the world (environment) that takes place during the computation plays a key role that cannot be replaced by any set of inputs determined prior to the computation’.

(viii) … a theory of concurrency and interaction requires a new conceptual framework, not just a refinement of what we find natural for sequential [algorithmic] computing’.

(ix) … the assumption that all of computation can be algorithmically specified is still widely accepted’.

A widespread notion of particular interest, which seems to be recurrently implicit in Wegner and Goldin’s assertions too, is that mathematics is a dispensable tool of science, rather than its indispensable mother tongue.

Elliott Mendelson

However, the roots of such beliefs may also lie in ambiguities, in the classical definitions of foundational elements, that allow the introduction of non-constructive—hence non-verifiable, non-computational, ambiguous, and essentially Platonic—elements into the standard interpretations of classical mathematics.

For instance, in a 1990 philosophical reflection, Elliott Mendelson’s following remarks (in Me90; reproduced from Selmer Bringsjord (Br93)), implicitly imply that classical definitions of various foundational elements can be argued as being either ambiguous, or non-constructive, or both:

Here is the main conclusion I wish to draw: it is completely unwarranted to say that CT is unprovable just because it states an equivalence between a vague, imprecise notion (effectively computable function) and a precise mathematical notion (partial-recursive function). … The concepts and assumptions that support the notion of partial-recursive function are, in an essential way, no less vague and imprecise than the notion of effectively computable function; the former are just more familiar and are part of a respectable theory with connections to other parts of logic and mathematics. (The notion of effectively computable function could have been incorporated into an axiomatic presentation of classical mathematics, but the acceptance of CT made this unnecessary.) … Functions are defined in terms of sets, but the concept of set is no clearer than that of function and a foundation of mathematics can be based on a theory using function as primitive notion instead of set. Tarski’s definition of truth is formulated in set-theoretic terms, but the notion of set is no clearer than that of truth. The model-theoretic definition of logical validity is based ultimately on set theory, the foundations of which are no clearer than our intuitive understanding of logical validity. … The notion of Turing-computable function is no clearer than, nor more mathematically useful (foundationally speaking) than, the notion of an effectively computable function.’

Consequently, standard interpretations of classical theory may, inadvertently, be weakening a desirable perception—of mathematics as the lingua franca of scientific expression—by ignoring the possibility that, since mathematics is, indeed, indisputably accepted as the language that most effectively expresses and communicates intuitive truth, the chasm between formal truth and provability must, of necessity, be bridgeable.

Cristian Calude, Elena Calude and Solomon Marcus

The belief in the existence of such a bridge is occasionally implicit in interpretations of computational theory.

For instance, in an arXived paper Passages of Proof, Computer Scientists Cristian Calude, Elena Calude and Solomon Marcus remark that:

“Classically, there are two equivalent ways to look at the mathematical notion of proof: logical, as a finite sequence of sentences strictly obeying some axioms and inference rules, and computational, as a specific type of computation. Indeed, from a proof given as a sequence of sentences one can easily construct a Turing machine producing that sequence as the result of some finite computation and, conversely, given a machine computing a proof we can just print all sentences produced during the computation and arrange them into a sequence.”

In other words, the authors seem to hold that Turing-computability of a proof’, in the case of an arithmetical proposition, is equivalent to provability of its representation in PA.

Wilfrid Sieg

We now attempt to build such a bridge formally, which is essentially one between the arithmetical ‘Decidability and Calculability’ described by Philosopher Wilfrid Sieg in his in-depth and wide-ranging survey ‘On Comptability‘, in which he addresses Gödel’s lifelong belief that an iff bridge between the two concepts is ‘impossible’ for ‘the whole calculus of predicates’ (Wi08, p.602).

$\S 2$ Bridging provability and computability: The foundations

In the paper titled “Evidence-Based Interpretations of $PA$” that was presented to the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, held from $2^{nd}$ to $6^{th}$ July 2012 at the University of Birmingham, UK (reproduced in this post) we have defined what it means for a number-theoretic function to be:

We have shown there that:

(i) The standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of the first order Peano Arithmetic PA is finitarily sound if, and only if, Aristotle’s particularisation holds over $N$; and the latter is the case if, and only if, PA is $\omega$-consistent.

(ii) We can define a finitarily sound algorithmic interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA over the domain $N$ where, if $[A]$ is an atomic formula $[A(x_{1}, x_{2}, \ldots, x_{n})]$ of PA, then the sequence of natural numbers $(a_{1}, a_{2}, \ldots, a_{n})$ satisfies $[A]$ if, and only if $[A(a_{1}, a_{2}, \ldots, a_{n})]$ is algorithmically computable under $\mathcal{I}_{PA(N,\ Algorithmic)}$, but we do not presume that Aristotle’s particularisation is valid over $N$.

(iii) The axioms of PA are always true under the finitary interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$, and the rules of inference of PA preserve the properties of satisfaction/truth under $\mathcal{I}_{PA(N,\ Algorithmic)}$.

We concluded that:

Theorem 1: The interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ of PA is finitarily sound.

Theorem 2: PA is consistent.

$\S 3$ Extending Buss’ Bounded Arithmetic

One of the more significant consequences of the Birmingham paper is that we can extend the iff bridge between the domain of provability and that of computability envisaged under Buss’ Bounded Arithmetic by showing that an arithmetical formula $[F]$ is PA-provable if, and only if, $[F]$ interprets as true under an algorithmic interpretation of PA.

$\S 4$ A Provability Theorem for PA

We first show that PA can have no non-standard model (for a distinctly different proof of this convention-challenging thesis see this post and this paper), since it is algorithmically’ complete in the sense that:

Theorem 3: (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 $[(\forall x)F(x)]$ interprets as true under the interpretation $\mathcal{I}_{PA(N,\ Algorithmic)}$ if, and only if, $[F(x)]$ is algorithmically computable as always true in $N$.

Since $\mathcal{I}_{PA(N,\ Algorithmic)}$ is finitarily sound, it defines a finitary model of PA over $N$—say $\mathcal{M}_{PA(\beta)}$—such that:

If $[(\forall x)F(x)]$ is PA-provable, then $[F(x)]$ is algorithmically computable as always true in $N$;

If $[\neg(\forall x)F(x)]$ 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 $[(\forall x)F(x)]$ and $[\neg(\forall x)F(x)]$ are PA-unprovable for some PA formula $[F(x)]$, as this would yield the contradiction:

(i) There is a finitary model—say $M1_{\beta}$—of PA+$[(\forall x)F(x)]$ in which $[F(x)]$ is algorithmically computable as always true in $N$.

(ii) There is a finitary model—say $M2_{\beta}$—of PA+$[\neg(\forall x)F(x)]$ in which it is not the case that $[F(x)]$ is algorithmically computable as always true in $N.$

The lemma follows. $\Box$

$\S 5$ The holy grail of arithmetic

We thus have that:

Corollary 1: PA is categorical finitarily.

Now we note that:

Lemma 2: If PA has a sound interpretation $\mathcal{I}_{PA(N,\ Sound)}$ over $N$, then there is a PA formula $[F]$ which is algorithmically verifiable as always true over $N$ under $\mathcal{I}_{PA(N,\ Sound)}$ even though $[F]$ is not PA-provable.

Proof In his seminal 1931 paper on formally undecidable arithmetical propositions, Kurt Gödel has shown how to construct an arithmetical formula with a single variable—say $[R(x)]$ [1]—such that $[R(x)]$ is not PA-provable [2], but $[R(n)]$ is instantiationally PA-provable for any given PA numeral $[n]$. Hence, for any given numeral $[n]$, the PA formula $xB \lceil [R(n)] \rceil$ must hold for some $x$. The lemma follows. $\Box$

By the argument in Theorem 3 it follows that:

Corollary 2: The PA formula $[\neg(\forall x)R(x)]$ defined in Lemma 2 is PA-provable.

Corollary 3: Under any sound interpretation of PA, Gödel’s $[R(x)]$ interprets as an algorithmically verifiable, but not algorithmically computable, tautology over $N$.

Proof Gödel has shown that $[R(x)]$ [3] interprets as an algorithmically verifiable tautology [4]. By Corollary 2 $[R(x)]$ is not algorithmically computable as always true in $N$. $\Box$

Corollary 4: PA is not $\omega$-consistent. [5]

Proof Gödel has shown that if PA is consistent, then $[R(n)]$ is PA-provable for any given PA numeral $[n]$ [6]. By Corollary 2 and the definition of $\omega$-consistency, if PA is consistent then it is not $\omega$-consistent. $\Box$

Corollary 5: The standard interpretation $\mathcal{I}_{PA(N,\ Standard)}$ of PA is not finitarily sound, and does not yield a finitary model of PA [7].

Proof If PA is consistent but not $\omega$-consistent, then Aristotle’s particularisation does not hold over $N$. Since the standard’, interpretation of PA appeals to Aristotle’s particularisation, the lemma follows. $\Box$

Since formal quantification is currently interpreted in classical logic [8] so as to admit Aristotle’s particularisation over $N$ as axiomatic [9], the above suggests that we may need to review number-theoretic arguments [10] that appeal unrestrictedly to classical Aristotlean logic.

$\S 6$ The Provability Theorem for PA and Bounded Arithmetic

In a 1997 paper [11], Samuel R. Buss considered Bounded Arithmetics obtained by:

(a) limiting the applicability of the Induction Axiom Schema in PA only to functions with quantifiers bounded by an unspecified natural number bound $b$;

(b) weakening’ the statement of the axiom with the aim of differentiating between effective computability over the sequence of natural numbers, and feasible polynomial-time’ computability over a bounded sequence of the natural numbers [12].

Presumably Buss’ intent—as expressed below—is to build an iff bridge between provability in a Bounded Arithmetic and Computability so that a $\Pi_{k}$ formula, say $[(\forall x)f(x)]$, is provable in the Bounded Arithmetic if, and only if, there is an algorithm that, for any given numeral $[n]$, decides the $\Delta_{(k/(k-1))}$ formula $[f(n)]$ as true’:

If $[(\forall x)(\exists y)f(x, y)]$ is provable, then there should be an algorithm to find $y$ as a function of $x$ [13].

Since we have proven such a Provability Theorem for PA in the previous section, the first question arises:

$\S 7$ Does the introduction of bounded quantifiers yield any computational advantage?

Now, one difference [14] between a Bounded Arithmetic and PA is that we can presume in the Bounded Arithmetic that, from a proof of $[(\exists y)f(n, y)]$, we may always conclude that there is some numeral $[m]$ such that $[f(n, m)]$ is provable in the arithmetic; however, this is not a finitarily sound conclusion in PA.

Reason: Since $[(\exists y)f(n, y)]$ is simply a shorthand for $[\neg (\forall y)\neg f(n, y)]$, such a presumption implies that Aristotle’s particularisation holds over the natural numbers under any finitarily sound interpretation of PA.

To see that (as Brouwer steadfastly held) this may not always be the case, interpret $[(\forall x)f(x)]$ as [15]:

There is an algorithm that decides $[f(n)]$ as true’ for any given numeral $[n]$.

In such case, if $[(\forall x)(\exists y)f(x, y)]$ is provable in PA, then we can only conclude that:

There is an algorithm that, for any given numeral $[n]$, decides that it is not the case that there is an algorithm that, for any given numeral $[m]$, decides $[\neg f(n, m)]$ as true’.

We cannot, however, conclude—as we can in a Bounded Arithmetic—that:

There is an algorithm that, for any given numeral $[n]$, decides that there is an algorithm that, for some numeral $[m]$, decides $[f(n, m)]$ as true’.

Reason: $[(\exists y)f(n, y)]$ may be a Halting-type formula for some numeral $[n]$.

This could be the case if $[(\forall x)(\exists y)f(x, y)]$ were PA-unprovable, but $[(\exists y)f(n, y)]$ PA-provable for any given numeral $[n]$.

Presumably it is the belief that any finitarily sound interpretation of PA requires Aristotle’s particularisation to hold in $N$, and the recognition that the latter does not admit linking provability to computability in PA, which has led to considering the effect of bounding quantification in PA.

However, as we have seen in the preceding sections, we are able to link provability to computability through the Provability Theorem for PA by recognising precisely that, to the contrary, any interpretation of PA which requires Aristotle’s particularisation to hold in $N$ cannot be finitarily sound!

The postulation of an unspecified bound in a Bounded Arithmetic in order to arrive at a provability-computability link thus appears dispensible.

The question then arises:

$\S 8$ Does weakening’ the PA Induction Axiom Schema yield any computational advantage?

Now, Buss considers a bounded arithmetic $S_{2}$ which is, essentially, PA with the following weakened’ Induction Axiom Schema, PIND [16]:

$[\{f(0)\ \&\ (\forall x)(f(\lfloor \frac{x}{2} \rfloor) \rightarrow f(x))\} \rightarrow (\forall x)f(x)]$

However, PIND can be expressed in first-order Peano Arithmetic PA as follows:

$[\{f(0)\ \&\ (\forall x)(f(x) \rightarrow (f(2*x)\ \&\ f(2*x+1)))\} \rightarrow (\forall x)f(x)]$.

Moreover, the above is a particular case of PIND($k$):

$[\{f(0)\ \&\ (\forall x)(f(x) \rightarrow (f(k*x)\ \&\ f(k*x+1)\ \&\ \ldots\ \&\ f(k*x+k-1)))\}$ $\rightarrow (\forall x)f(x)]$.

Now we have the PA theorem:

$[(\forall x)f(x) \rightarrow \{f(0)\ \&\ (\forall x)(f(x) \rightarrow f(x+1))\}]$

It follows that the following is also a PA theorem:

$[\{f(0)\ \&\ (\forall x)(f(x) \rightarrow f(x+1))\} \rightarrow$ $\{f(0)\ \&\ (\forall x)(f(x) \rightarrow (f(k*x)\ \&\ f(k*x+1)\ \&\ \ldots\ \&\ f(k*x+k-1)))\}]$

In other words, for any numeral $[k]$, PIND($k$) is equivalent in PA to the standard Induction Axiom of PA!

Thus, the Provability Theorem for PA suggests that all arguments and conclusions of a Bounded Arithmetic can be reflected in PA without any loss of generality.

References

Br93 Selmer Bringsjord 1993. The Narrational Case Against Church’s Thesis. Easter APA meetings, Atlanta.

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.

Bu97 Samuel R. Buss. 1997. Bounded Arithmetic and Propositional Proof Complexity. In Logic of Computation. pp.67-122. Ed. H. Schwichtenberg. Springer-Verlag, Berlin.

CCS01 Cristian S. Calude, Elena Calude and Solomon Marcus. 2001. Passages of Proof. Workshop, Annual Conference of the Australasian Association of Philosophy (New Zealand Division), Auckland. Archived at: http://arxiv.org/pdf/math/0305213.pdf. Also in EATCS Bulletin, Number 84, October 2004, viii+258 pp.

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.

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.

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.

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.

Ku62 Thomas S. Kuhn. 1962. The structure of Scientific Revolutions. 2nd Ed. 1970. University of Chicago Press, Chicago.

Me90 Elliott Mendelson. 1990. Second Thoughts About Church’s Thesis and Mathematical Proofs. In Journal of Philosophy 87.5.

Pa71 Rohit Parikh. 1971. Existence and Feasibility in Arithmetic. In The Journal of Symbolic Logic,>i> Vol.36, No. 3 (Sep., 1971), pp. 494-508.

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.

Si08 Wilfrid Sieg. 2008. On Computability in Handbook of the Philosophy of Science. Philosophy of Mathematics. pp.525-621. Volume Editor: Andrew Irvine. General Editors: Dov M. Gabbay, Paul Thagard and John Woods. Elsevier BV. 2008.

WG03 Peter Wegner and Dina Goldin. 2003. Computation Beyond Turing Machines. Communications of the ACM, 46 (4) 2003.

Notes

Return to 1: Gödel refers to this formula only by its Gödel number $r$ (Go31, p.25(12)).

Return to 2: Gödel’s immediate aim in Go31 was to show that $[(\forall x)R(x)]$ is not P-provable; by Generalisation it follows, however, that $[R(x)]$ is also not P-provable.

Return to 3: Gödel refers to this formula only by its Gödel number $r$ (Go31, p.25, eqn.12).

Return to 4: Go31, p.26(2): “$(n)\neg(nB_{\kappa}(17Gen\ r))$ holds”.

Return to 5: This 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 $\omega$-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”.

Return to 7: I note that finitists of all hues—ranging from Brouwer Br08 to Alexander Yessenin-Volpin He04—have persistently questioned the finitary soundness of the `standard’ interpretation $\mathcal{I}_{PA(N,\ Standard)}$.

Return to 8: See Hi25, p.382; HA28, p.48; Be59, pp.178 \& 218.

Return to 9: In the sense of being intuitively obvious. See, for instance, Da82, p.xxiv; Rg87, p.308 (1)-(4); EC89, p.174 (4); BBJ03, p.102.

Return to 10: For instance Rosser’s construction of an undecidable arithmetical proposition in PA (see Ro36)—which does not explicitly assume that PA is $\omega$-consistent—implicitly presumes that Aristotle’s particularisation holds over $N$.

Return to 15: We have seen in the earlier sections that such an interpretation is finitarily sound.

Return to 16: Where $\lfloor \frac{x}{2} \rfloor$ denotes the largest natural number lower bound of the rational $\frac{x}{2}$.

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