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

In this post I address two critical issues, as raised in private correspondence with researchers, which may illuminate some objections to Gödel’s reasoning and conclusions that have been raised elsewhere by Wittgenstein, Floyd, Putnam et al.:

(i) By Rosser’s reasoning, doesn’t simple consistency suffice for defining an undecidable arithmetical proposition?

(ii) Doesn’t Gödel’s undecidable formula assert its own unprovability?

NOTE: The following correspondence refers copiously to this paper that was presented in June 2015 at the workshop on Hilbert’s Epsilon and Tau in Logic, Informatics and Linguistics, University of Montpellier, France.

Subsequently, most of the cited results were detailed formally in the following paper due to appear in the December 2016 issue of ‘Cognitive Systems Research‘:

The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The evidence-based argument for Lucas’ Gödelian thesis‘.

A: Doesn’t simple consistency suffice for defining Rosser’s undecidable arithmetical proposition?

You claim that the PA system is \omega-inconsistent, and that Gödel’s first theorem holds vacuously. But by Rosser’s result, simple consistency suffices.

Well, it does seem surprising that Rosser’s claim—that his ‘undecidable’ proposition only assumes simple consistency—has not been addressed more extensively in the literature. Number-theoretic expositions of Rosser’s proof have generally remained either implicit or sketchy (see, for instance, this post).

Note that Rosser’s proposition and reasoning involve interpretation of an existential quantifier, whilst Gödel’s proposition and reasoning only involve interpretation of a universal quantifier.

The reason why Rosser’s claim is untenable is that—in order to interpret the existential quantifier as per Hilbert’s \epsilon-calculus—Rosser’s argument needs to assume his Rule C (see Elliott Mendelson, Introduction to Mathematical Logic, 1964 ed., p.73), which implicitly implies that Gödel’s arithmetic P—in which Rosser’s argumentation is grounded—is \omega-consistent .

See, for instance, this analysis of (a) Wang’s outline of Rosser’s argument on p.5, (b) Beth’s outline of Rosser’s argument on p.6, and (c) Mendelson’s exposition of Rosser’s argument in Section 4.2 on p.8.

Moreover, the assumption is foundationally fragile, because Rule C invalidly assumes that we can introduce an ‘unspecified’ formula denoting an ‘unspecified’ numeral into PA even if the formula has not been demonstrated to be algorithmically definable in terms of the alphabet of PA.

See Theorem 8.5 and following remarks in Section 8, pp.7-8 of this paper that was presented in June 2015 at the workshop on Hilbert’s Epsilon and Tau in Logic, Informatics and Linguistics, University of Montpellier, France.

B: As I see it, rule C is only a shortcut.

As I see it, rule C is only a shortcut; it is totally eliminable. Moreover, it is part of predicate logic, not of the Peano’s arithmetic.

Assuming that Rule C is a short cut which can always be eliminated is illusory, and is tantamount to invalidly (see Corollary 8.6, p.17 of the Epsilon 2015 paper) claiming that Hilbert’s \epsilon calculus is a conservative extension of the first-order predicate calculus.

Reason: Application of Rule C invalidly (see Theorem 8.5 and following remarks in Section 8, pp.7-8 of the Epsilon 2015 paper) involves introduction of a new individual constant, say [d], in a first-order theory K (see Mendelson 1964, p.74, I(iv)); ‘invalidly’ since Rule C does not qualify that [d] must be algorithmically computable from the alphabet of K—which is necessary if K is first-order.

Notation: We use square brackets to indicate that the expression within the brackets denotes a well-formed formula of a formal system, say K, that is to be viewed syntactically merely as a first-order string of K—i.e, one which is finitarily constructed from the alphabet of the language of K—without any reference to its meaning under any interpretation of K.

Essentially, Rule C mirrors in K the intuitionistically objectionable postulation that the formula [(\exists x)F(x)] of K can always be interpreted as:

F'(a) holds for some element a

in the domain of the interpretation of K under which the formula [F(x)] interprets as the relation F'(x).

The Epsilon 2015 paper shows that this is not a valid interpretation of the formula [(\exists x)F(x)] under any finitary, evidence-based, interpretation of K.

That, incidentally, is a consequence of the proof that PA is not \omega-consistent; which itself is a consequence of (Theorem 7.1, p.15, of the Epsilon 2015 paper):

Provability Theorem for PA: A PA formula [F(x)] is provable if, and only if, [F(x)] interprets as an arithmetical relation F'(x) that is algorithmically computable as always true (see Definition 3, p.7, of the Epsilon 2015 paper) over the structure \mathbb{N} of the natural numbers.

Compare with what Gödel has essentially shown in his famous 1931 paper on formally undecidable arithmetical propositions, which is that (Lemma 8.1, p.16, of the Epsilon 2015 paper):

Gödel: There is a PA formula [R(x, p)]—which Gödel refers to by its Gödel number r—which is not provable in PA, even though [R(x, p)] interprets as an arithmetical relation that is algorithmically verifiable as always true (see Definition 4, p.7, of the Epsilon 2015 paper) over the structure \mathbb{N} of the natural numbers.

C: If you by-pass the intuitionist objections, would all logicist and post-formalist theories hold?

If I have understood correctly, you claim that the PA system is \omega-inconsistent from an intuitionistic point of view? If you by-pass the intuitionist objections, would all logicist and post-formalist theories hold?

There is nothing to bypass—the first-order Peano Arithmetic PA is a formal axiomatic system which is \omega-inconsistent as much for an intuitionist, as it is for a realist, a finitist, a formalist, a logicist or a nominalist.

Philosophers may differ about beliefs that are essentially unverifiable; but the \omega-incompleteness of PA is a verifiable logical meta-theorem that none of them would dispute.

D: Isn’t Gödel’s undecidable formula [(\forall x)R(x, p)]—which Gödel refers to by its Gödel number 17Gen\ r—self-referential?

Isn’t Gödel’s undecidable formula [(\forall x)R(x, p)]—which Gödel refers to by its Gödel number 17Gen\ r—self-referential and covertly paradoxical?

According to Wittgenstein it interprets in any model as a sentence that is devoid of sense, or even meaning. I think a good reason for this is that the formula is simply syntactically wrongly formed: the provability of provability is not defined and can not be consistently defined.

What you propose may be correct, but for automation systems of deduction wouldn’t \omega-inconsistency be much more problematic than undecidability?

How would you feel if a syntax rule is proposed, that formulas containing numerals are instantiations of open formulas that may not be part of the canonical language? Too daring, may be?

Let me briefly respond to the interesting points that you have raised.

1. The \omega-inconsistency of PA is a meta-theorem; it is a Corollary of the Provability Theorem of PA (Theorem 7.1, p.15, of the Epsilon 2015 paper).

2. Gödel’s PA-formula [(\forall x)R(x, p)] is not an undecidable formula of PA. It is merely unprovable in PA.

3. Moreover, Gödel’s PA-formula [\neg(\forall x)R(x, p)] is provable in PA, which is why the PA formula [(\forall x)R(x, p)] is not an undecidable formula of PA.

4. Gödel’s PA-formula [(\forall x)R(x, p)] is not self-referential.

5. Wittgenstein correctly believed—albeit purely on the basis of philosophical considerations unrelated to whether or not Gödel’s formal reasoning was correct—that Gödel was wrong in stating that the PA formula [(\forall x)R(x, p)] asserts its own unprovability in PA.

Reason: We have for Gödel’s primitive recursive relation Q(x, y) that:

Q(x, p) is true if, and only if, the PA formula [R(x, p)] is provable in PA.

However, in order to conclude that the PA formula [(\forall x)R(x, p)] asserts its own unprovability in PA, Gödel’s argument must further imply—which it does not—that:

(\forall x)Q(x, p) is true (and so, by Gödel’s definition of Q(x, y), the PA formula [(\forall x)R(x, p)] is not provable in PA) if, and only if, the PA formula [(\forall x)R(x, p)] is provable in PA.

In other words, for the PA formula [(\forall x)R(x, p)] to assert its own unprovability in PA, Gödel must show—which his own argument shows is impossible, since the PA formula [(\forall x)R(x, p)] is not provable in PA—that:

The primitive recursive relation Q(x, p) is algorithmically computable as always true if, and only if, the arithmetical relation R'(x, p) is algorithmically computable as always true (where R'(x, p) is the arithmetical interpretation of the PA formula [R(x, p)] over the structure \mathbb{N} of the natural numbers).

6. Hence, Gödel’s PA-formula [(\forall x)R(x, p)] is not covertly paradoxical.

7. IF Wittgenstein believed that the PA formula [(\forall x)R(x, p)] is empty of meaning and has no valid interpretation, then he was wrong, and—as Gödel justifiably believed—he could not have properly grasped Gödel’s formal reasoning that:

(i) ‘17Gen\ r is not \kappa-provable’ is a valid meta-theorem if PA is consistent, which means that:

‘If PA is consistent and we assume that the PA formula [(\forall x)R(x, p)] is provable in PA, then the PA formula [\neg(\forall x)R(x, p)] must also be provable in PA; from which we may conclude that the PA formula [(\forall x)R(x, p)] is not provable in PA’

(ii) ‘Neg(17Gen\ r) is not \kappa-provable’ is a valid meta-theorem ONLY if PA is \omega-consistent, which means that:

‘If PA is \omega-consistent and we assume that the PA formula [\neg(\forall x)R(x, p)] is provable in PA, then the PA formula [(\forall x)R(x, p)] must also be provable in PA; from which we may conclude that the PA formula [\neg(\forall x)R(x, p)] is not provable in PA’.

8. In fact the PA formula [(\forall x)R(x, p)] has the following TWO meaningful interpretations (the first of which is a true arithmetical meta-statement—since the PA formula [R(n)] is provable in PA for any PA-numeral [n]—but the second is not—since the PA formula [(\forall x)R(x, p)] is not provable in PA):

(i) For any given natural number n, there is an algorithm which will verify that each of the arithmetical meta-statements ‘R'(1, p) is true’, ‘R'(2, p) is true’, …, ‘R'(n, p) is true’ holds under the standard, algorithmically verifiable, interpretation \mathbb{M} of PA (see \S 5, p.11 of the Epsilon 2015 paper);

(ii) There is an algorithm which will verify that, for any given natural number n, the arithmetical statement ‘R'(n, p) is true’ holds under the finitary, algorithmically computable, interpretation \mathbb{B} of PA (see \S 6, p.13 of the Epsilon 2015 paper).

9. IF Wittgenstein believed that the PA formula [(\forall x)R(x, p)] is not a well-defined PA formula, then he was wrong.

Gödel’s definition of the PA formula [(\forall x)R(x, p)] yields a well-formed formula in PA, and cannot be treated as ‘syntactically wrongly formed’.

10. The Provability Theorem for PA shows that both ‘proving something in PA’ and ‘proving that something is provable in PA’ are finitarily well-defined meta-mathematical concepts.

11. The Provability Theorem for PA implies that PA is complete with respect to the concepts of satisfaction, truth and provability definable in automated deduction systems, which can only define algorithmically computable truth.

12. The Provability Theorem for PA implies that PA is categorical, so you can introduce your proposed syntax rule ONLY if it leads to a conservative extension of PA.

13. Whether ‘daring’ or not, why would you want to introduce such a rule?

E: Consider these two statements of yours …

Consider these two statements of yours:

“(iv): p is the Gödel-number of the formula [(\forall x)][R(x, y)] of PA” and

“D(4): Gödel’s PA-formula [(\forall x)R(x, p)] is not self-referential.”

If ‘p‘ is the Gödel-number of the open formula in para (iv), and the second argument of the closed formula R in para D(4) is ‘p‘, then the second formula is obtained by instantiating the variable ‘y‘ in the first with its own Gödel-number.

So how would you call, in one word, the relation between the entire formula (in D(4)) and its second argument?

Para D(4) is an attempt to clarify precisely this point.

1. Apropos the first statement ‘(iv)’ cited by you:

From a pedantic perspective, the “relation between the entire formula (in D(4)) and its second argument” cannot be termed self-referential because the “second argument”, i.e., p, is the Gödel-number of the PA formula [(\forall x)R(x, y)], and not that of “the entire formula (in 4)”, i.e., of the formula [(\forall x)R(x, p)] itself (whose Gödel number is 17Gen\ r).

Putting it crudely, 17Gen\ r is neither self-referential—nor circularly defined—because it is not defined in terms of 17Gen\ r, but in terms of p.

2. Apropos the second statement ‘D(4)’ cited by you:

I would interpret:

Gödel’s PA-formula [(\forall x)R(x, p)] is self-referential

to mean, in this particular context, that—as Gödel wrongly claimed:

[(\forall x)R(x, p)] asserts its own unprovability in PA.

Now, if we were to accept the claim that [(\forall x)R(x, p)] is self-referential in the above sense, then (as various critics of Gödel’s reasoning have pointed out) we would have to conclude further that Gödel’s argument leads to the contradiction:

(\forall x)Q(x, p) is true—and so, by Gödel’s definition of Q(x, y)—the PA formula [(\forall x)R(x, p)] is not provable in PA—if, and only if, the PA formula [(\forall x)R(x, p)] is provable in PA.

However, in view of the Provability Theorem of PA (Theorem 7.1, p.15, of the Epsilon 2015 paper), this contradiction would only follow if Gödel’s argument were to establish (which it does not) that:

The primitive recursive relation Q(x, p) is algorithmically computable as always true if, and only if, the arithmetical interpretation R'(x, p) of the PA formula [R(x, p)] is algorithmically computable as always true over the structure \mathbb{N} of the natural numbers.

The reason Gödel cannot claim to have established the above is that his argument only proves the much weaker meta-statement:

The arithmetical interpretation R'(x, p) of the PA formula [R(x, p)] is algorithmically verifiable as always true over the structure \mathbb{N} of the natural numbers.

Ergo—contrary to Gödel’s claim— Gödel’s PA-formula [(\forall x)R(x, p)] is not self-referential (and so, even though Gödel’s claimed interpretation of what his own reasoning proves is wrong, there is no paradox in Gödel’s reasoning per se)!

F: Is the PA system \omega-inconsistent without remedy?

Is the PA system \omega-inconsistent without remedy? Is it possible to introduce a new axiom or new rule which by-passes the problematic unprovable statements of the Gödel-Rosser Theorems?

1. Please note that the first-order Peano Arithmetic PA is:

(i) consistent (Theorem 7.3, p.15, of the Epsilon 2015 paper); which means that for any PA-formula [A], we cannot have that both [A] and [\neg A] are Theorems of PA;

(ii) complete (Theorem 7.1, p.15, of the Epsilon 2015 paper); which means that we cannot add an axiom to PA which is not a Theorem of PA without inviting inconsistency;

(iii) categorical (Theorem 7.2, p.15, of the Epsilon 2015 paper); which means that if \mathbb{M} is an interpretation of PA over a structure \mathbb{S}, and \mathbb{B} is an interpretation of PA over a structure \mathbb{T}, then \mathbb{S} and \mathbb{T} are identical and denote the structure \mathbb{N} of the natural numbers defined by Dedekind’s axioms; and so PA has no model which contains an element that is not a natural number (see Footnote 54, p.16, of the Epsilon 2015 paper).

2. What this means with respect to Gödel’s reasoning is that:

(i) PA has no undecidable propositions, which is why it is not \omega-consistent (Corollary 8.4, p.16, of the Epsilon 2015 paper);

(ii) The Gödel formula [(\forall x)R(x, p)] is not provable in PA; but it is algorithmically verifiable as true (Corollary 8.3, p.16, of the Epsilon 2015 paper) under the algorithmically verifiable standard interpretation \mathbb{M} of PA (see Section 5, p.11, of the Epsilon 2015 paper) over the structure \mathbb{N} of the natural numbers;

(iii) The Gödel formula [(\forall x)R(x, p)] is not provable in PA; and it is algorithmically computable as false (Corollary 8.3, p.16, of the Epsilon 2015 paper) under the algorithmically computable finitary interpretation \mathbb{B} of PA (see Section 6, p.13, of the Epsilon 2015 paper) over the structure \mathbb{N} of the natural numbers;

(iv) The Gödel formula [\neg(\forall x)R(x, p)] is provable in PA; and it is therefore also algorithmically verifiable as true under the algorithmically verifiable standard interpretation \mathbb{M} of PA over the structure \mathbb{N} of the natural numbers—which means that the logic by which the standard interpretation of PA assigns values of ‘satisfaction’ and ‘truth’ to the formulas of PA (under Tarski’s definitions) may be paraconsistent (see http://plato.stanford.edu/entries/logic-paraconsistent) since PA is consistent;

(v) The Gödel formula [\neg(\forall x)R(x, p)] is provable in PA; and it is therefore algorithmically computable as true (Corollary 8.2, p.16, of the Epsilon 2015 paper) under the algorithmically computable finitary interpretation \mathbb{B} of PA over the structure \mathbb{N} of the natural numbers.

3. It also means that:

(a) The “Gödel-Rosser Theorem” is not a Theorem of PA;

(b) The “unprovable Gödel sentence” is not a “problematic statement”;

(c) The “PA system” does not require a “remedy” just because it is “\omega-inconsistent”;

(d) No “new axiom or new rule” can “by-pass the unprovable sentence”.

4. Which raises the question:

Why do you see the “unprovable Gödel sentence” as a “problematic statement” that requires a “remedy” which must “by-pass the unprovable sentence”?

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

Ferguson’s and Priest’s thesis

In a brief, but provocative, review of what they term as “the enduring evolution of logic” over the ages, the authors of Oxford University Press’ recently released ‘A Dictionary of Logic‘, philosophers Thomas Macaulay Ferguson and Graham Priest, take to task what they view as a Kant-influenced manner in which logic is taught as a first course in most places in the world:

“… as usually ahistorical and somewhat dogmatic. This is what logic is; just learn the rules. It is as if Frege had brought down the tablets from Mount Sinai: the result is God-given, fixed, and unquestionable.”

Ferguson and Priest conclude their review by remarking that:

“Logic provides a theory, or set of theories, about what follows from what, and why. And like any theoretical inquiry, it has evolved, and will continue to do so. It will surely produce theories of greater depth, scope, subtlety, refinement—and maybe even truth.”

However, it is not obvious whether that is prescient optimism, or a tongue-in-cheek exit line!

A nineteenth century parody of the struggle to define ‘truth’ objectively

For, if anything, the developments in logic since around 1931 has—seemingly in gross violation of the hallowed principle of Ockham’s razor, and its crude, but highly effective, modern avatar KISS—indeed produced a plethora of theories of great depth, scope, subtlety, and refinement.

These, however, seem to have more in common with the, cynical, twentieth century emphasis on subjective, unverifiable, ‘truth’, rather than with the concept of an objective, evidence-based, ‘truth’ that centuries of philosophers and mathematicians strenuously struggled to differentiate and express.

A struggle reflected so eloquently in this nineteenth century quote:

“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”

“The question is,” said Alice, “whether you can make words mean so many different things.”

“The question is,” said Humpty Dumpty, “which is to be master—that’s all.”

… Lewis Carroll (Charles L. Dodgson), ‘Through the Looking-Glass’, chapter 6, p. 205 (1934 ed.). First published in 1872.

Making sense of mathematical propositions about infinite processes

It was, indeed, an epic struggle which culminated in the nineteenth century standards of rigour successfully imposed—in no small measure by the works of Augustin-Louis Cauchy and Karl Weierstrasse—on verifiable interpretations of mathematical propositions about infinite processes involving real numbers.

A struggle, moreover, which should have culminated equally successfully in similar twentieth century standards—on verifiable interpretations of mathematical propositions containing references to infinite computations involving integers—sought to be imposed in 1936 by Alan Turing upon philosophical and mathematical discourse.

The Liar paradox

For it follows from Turing’s 1936 reasoning that where quantification is not, or cannot be, explicitly defined in formal logical terms—eg. the classical expression of the Liar paradox as ‘This sentence is a lie’—a paradox cannot per se be considered as posing serious linguistic or philosophical concerns (see, for instance, the series of four posts beginning here).

Of course—as reflected implicitly in Kurt Gödel’s seminal 1931 paper on undecidable arithmetical propositions—it would be a matter of serious concern if the word ‘This’ in the English language sentence, ‘This sentence is a lie’, could be validly viewed as implicitly implying that:

(i) there is a constructive infinite enumeration of English language sentences;

(ii) to each of which a truth-value can be constructively assigned by the rules of a two-valued logic; and,

(iii) in which ‘This’ refers uniquely to a particular sentence in the enumeration.

Gödel’s influence on Turing’s reasoning

However, Turing’s constructive perspective had the misfortune of being subverted by a knee-jerk, anti-establishment, culture that was—and apparently remains to this day—overwhelmed by Gödel’s powerful Platonic—and essentially unverifiable—mathematical and philosophical 1931 interpretation of his own construction of an arithmetical proposition that is formally unprovable, but undeniably true under any definition of ‘truth’ in any interpretation of arithmetic over the natural numbers.

Otherwise, I believe that Turing could easily have provided the necessary constructive interpretations of arithmetical truth—sought by David Hilbert for establishing the consistency of number theory finitarily—which is addressed by the following paper due to appear in the December 2016 issue of ‘Cognitive Systems Research‘:

The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The evidence-based argument for Lucas’ Gödelian thesis‘.

What is logic: using Ockham’s razor

Moreover, the paper endorses the implicit orthodoxy of an Ockham’s razor influenced perspective—which Ferguson and Priest seemingly find wanting—that logic is simply a deterministic set of rules that must constructively assign the truth values of ‘truth/falsity’ to the sentences of a language.

It is a view that I expressed earlier as the key to a possible resolution of the EPR paradox in the following paper that I presented on 26’th June at the workshop on Emergent Computational Logics at UNILOG’2015, Istanbul, Turkey:

Algorithmically Verifiable Logic vis à vis Algorithmically Computable Logic: Could resolving EPR need two complementary Logics?

where I introduced the definition:

A finite set \lambda of rules is a Logic of a formal mathematical language \mathcal{L} if, and only if, \lambda constructively assigns unique truth-values:

(a) Of provability/unprovability to the formulas of \mathcal{L}; and

(b) Of truth/falsity to the sentences of the Theory T(\mathcal{U}) which is defined semantically by the \lambda-interpretation of \mathcal{L} over a structure \mathcal{U}.

I showed there that such a definitional rule-based approach to ‘logic’ and ‘truth’ allows us to:

\bullet Equate the provable formulas of the first order Peano Arithmetic PA with the PA formulas that can be evidenced as `true’ under an algorithmically computable interpretation of PA over the structure \mathbb{N} of the natural numbers;

\bullet Adequately represent some of the philosophically troubling abstractions of the physical sciences mathematically;

\bullet Interpret such representations unambiguously; and

\bullet Conclude further:

\bullet First that the concept of infinity is an emergent feature of any mechanical intelligence whose true arithmetical propositions are provable in the first-order Peano Arithmetic; and

\bullet Second that discovery and formulation of the laws of quantum physics lies within the algorithmically computable logic and reasoning of a mechanical intelligence whose logic is circumscribed by the first-order Peano Arithmetic.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

A Economist: The return of the machinery question

In a Special Report on Artificial Intelligence in its issue of 25th June 2016, ‘The return of the machinery question‘, the Economist suggests that both cosmologist Stephen Hawking and enterpreneur Elon Musk share to some degree the:

“… fear that AI poses an existential threat to humanity, because superintelligent computers might not share mankind’s goals and could turn on their creators”.

B Our irrational propensity to fear that which we are drawn to embrace

Surprising, since I suspect both would readily agree that, if anything should scare us, it is our irrational propensity to fear that which we are drawn to embrace!

And therein should lie not only our comfort, but perhaps also our salvation.

For Artificial Intelligence is constrained by rationality; Human Intelligence is not.

An Artificial Intelligence must, whether individually or collectively, create and/or destroy only rationally. Humankind can and does, both individually and collectively, create and destroy irrationally.

C Justifying irrationality

For instance, as the legatees of logicians Kurt Goedel and Alfred Tarski have amply demonstrated, a Human Intelligence can easily be led to believe that some statements of even the simplest of mathematical languages—Arithmetic—must be both ‘formally undecidable’ and ‘true’, even in the absence of any objective yardstick for determining what is ‘true’!

D Differentiating between Human reasoning and Mechanistic reasoning

An Artificial Intelligence, however, can only treat as true that which can be proven—by its rules—to be true by an objective assignment of ‘truth’ and ‘provability’ values to the propositions of the language that formally expresses its mechanical operations—Arithmetic.

The implications of the difference are not obvious; but that the difference could be significant is the thesis of this paper which is due to appear in the December 2016 issue of Cognitive Systems Research:

The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning‘.

E Respect for evidence-based ‘truth’ could be Darwinian

More importantly, the paper demonstrates that both Human Intelligence—whose evolution is accepted as Darwinian—and Artificial Intelligence—whose evolution it is ‘feared’ may be Darwinian—share a common (Darwinian?) respect for an accountable concept of ‘truth’.

A respect that should make both Intelligences fitter to survive by recognising what philosopher Christopher Mole describes in this invitational blogpost as the:

“… importance of the rapport between an organism and its environment”

—an environment that can obviously accommodate the birth, and nurture the evolution, of both intelligences.

So, it may not be too far-fetched to conjecture that the evolution of both intelligences must also, then, share a Darwinian respect for the kind of human values—towards protecting intelligent life forms—that, no matter in how limited or flawed a guise, is visibly emerging as an inherent characteristic of a human evolution which, no matter what the cost could, albeit optimistically, be viewed as struggling to incrementally strengthen, and simultaneously integrate, individualism (fundamental particles) into nationalism (atoms) into multi-nationalism (molecules) and, possibly, into universalism (elements).

F The larger question: Should we fear an extra-terrestrial Intelligence?

From a broader perspective yet, our apprehensions about the evolution of a rampant Artificial Intelligence created by a Frankensteinian Human Intelligence should, perhaps, more rightly be addressed—as some have urged—within the larger uncertainty posed by SETI:

Is there a rational danger to humankind in actively seeking an extra-terrestrial intelligence?

I would argue that any answer would depend on how we articulate the question and that, in order to engage in a constructive and productive debate, we need to question—and reduce to a minimum—some of our most cherished mathematical and scientific beliefs and fears which cannot be communicated objectively.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

We investigate whether the probabilistic distribution of prime numbers can be treated as a heuristic model of quantum behaviour, since it too can be treated as a quantum phenomena, with a well-defined binomial probability function that is algorithmically computable, where the conjectured values of \pi(n) differ from actual values with a binomial standard deviation, and where we define a phenomena as a quantum phenomena if, and only if, it obeys laws that can only be represented mathematically by functions that are algorithmically verifiable, but not algorithmically computable.

1. Thesis: The concept of ‘mathematical truth’ must be accountable

The thesis of this investigation is that a major philosophical challenge—which has so far inhibited a deeper understanding of the quantum behaviour reflected in the mathematical representation of some laws of nature (see, for instance, this paper by Eamonn Healey)—lies in holding to account the uncritical acceptance of propositions of a mathematical language as true under an interpretation—either axiomatically or on the basis of subjective self-evidence—without any specified methodology of accountability for objectively evidencing such acceptance.

2. The concept of ‘set-theoretical truth’ is not accountable

Since current folk lore is that all scientific truths can be expressed adequately, and communicated unambiguously, in the first order Set Theory ZF, and since the Axiom of Infinity of ZF cannot—even in principle—be objectively evidenced as true under any putative interpretation of ZF (as we argue in this post), an undesirable consequence of such an uncritical acceptance is that the distinction between the truths of mathematical propositions under interpretation which can be objectively evidenced, and those which cannot, is not evident.

3. The significance of such accountability for mathematics

The significance of such a distinction for mathematics is highlighted in this paper due to appear in the December 2016 issue of Cognitive Systems Research, where we address this challenge by considering the two finitarily accountable concepts of algorithmic verifiability and algorithmic computability (first introduced in this paper at the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, Birmingham, UK).

(i) 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 for deciding the truth/falsity of each proposition in the finite sequence \{F(1), F(2), \ldots, F(n)\}.

(ii) Algorithmic computability

A number theoretical relation F(x) is algorithmically computable if, and only if, there is an algorithm AL_{F} that can provide objective evidence for deciding the truth/falsity of each proposition in the denumerable sequence \{F(1), F(2), \ldots\}.

(iii) Algorithmic verifiability vis à vis algorithmic computability

We note that algorithmic computability implies the existence of an algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions, whereas algorithmic verifiability does not imply the existence of an algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions.

From the point of view of a finitary mathematical philosophy—which is the constraint within which an applied science ought to ideally operate—the significant difference between the two concepts could be expressed by saying that we may treat the decimal representation of a real number as corresponding to a physically measurable limit—and not only to a mathematically definable limit—if and only if such representation is definable by an algorithmically computable function (Thesis 1 on p.9 of this paper that was presented on 26th June at the workshop on Emergent Computational Logics at UNILOG’2015, 5th World Congress and School on Universal Logic, Istanbul, Turkey).

We note that although every algorithmically computable relation is algorithmically verifiable, the converse is not true.

We show in the CSR paper how such accountability helps define finitary truth assignments that differentiate human reasoning from mechanistic reasoning in arithmetic by identifying two, hitherto unsuspected, Tarskian interpretations of the first order Peano Arithmetic PA, under both of which the PA axioms interpret as finitarily true over the domain N of the natural numbers, and the PA rules of inference preserve such truth finitarily over N.

4. The ambit of human reasoning vis à vis the ambit of mechanistic reasoning

One corresponds to the classical, non-finitary, putative standard interpretation of PA over N, and can be treated as circumscribing the ambit of human reasoning about ‘true’ arithmetical propositions.

The other corresponds to a finitary interpretation of PA over N that circumscibes the ambit of mechanistic reasoning about ‘true’ arithmetical propositions, and establishes the long-sought for consistency of PA (see this post); which establishes PA as a mathematical language of unambiguous communication for the mathematical representation of physical phenomena.

5. The significance of such accountability for the mathematical representation of physical phenomena

The significance of such a distinction for the mathematical representation of physical phenomena is highlighted in this paper that was presented on 26th June at the workshop on Emergent Computational Logics at UNILOG’2015, 5th World Congress and School on Universal Logic, Istanbul, Turkey, where we showed how some of the seemingly paradoxical elements of quantum mechanics may resolve if we define:

Quantum phenomena: A phenomena is a quantum phenomena if, and only if, it obeys laws that can only be represented mathematically by functions that are algorithmically verifiable but not algorithmically computable.

6. The mathematical representation of quantum phenomena that is determinate but not predictable

By considering the properties of Gödel’s \beta function (see \S4.1 on p.8 of this preprint)—which allows us to strongly represent any non-terminating sequence of natural numbers by an arithmetical function—it would follow that, since any projection of the future values of a quantum-phenomena-associated, algorithmically verifiable, function is consistent with an infinity of algorithmically computable functions, all of whose past values are identical to the algorithmically verifiable past values of the function, the phenomena itself would be essentially unpredicatable if it cannot be represented by an algorithmically computable function.

However, since the algorithmic verifiability of any quantum phenomena shows that it is mathematically determinate, it follows that the physical phenomena itself must observe determinate laws.

7. Such representation does not need to admit multiverses

Hence (contrary to any interpretation that admits unverifiable multiverses) only one algorithmically computable extension of the function is consistent with the law determining the behaviour of the phenomena, and each possible extension must therefore be associated with a probability that the next observation of the phenomena is described by that particular extension.

8. Is the probability of the future behaviour of quantum phenomena definable by an algorithmically computable function?

The question arises: Although we cannot represent quantum phenomena explicitly by an algorithmically computable function, does the phenomena lend itself to an algorithmically computable probability of its future behaviour in the above sense?

9. Can primes yield a heuristic model of quantum behaviour?

We now show that the distribution of prime numbers denoted by the arithmetical prime counting function \pi(n) is a quantum phenomena in the above sense, with a well-defined probability function that is algorithmically computable.

10. Two prime probabilities

We consider the two probabilities:

(i) The probability P(a) of selecting a number that has the property of being prime from a given set S of numbers;

Example 1: I have a bag containing 100 numbers in which there are twice as many composites as primes. What is the probability that the first number you blindly pick from it is a prime. This is the basis for setting odds in games such as roulette.

(ii) The probability P(b) of determining a proper factor of a given number n.

Example 2: I give you a 5-digit combination lock along with a 10-digit number n. The lock only opens if you set the combination to a proper factor of n which is greater than 1. What is the probability that the first combination you try will open the lock. This is the basis for RSA encryption, which provides the cryptosystem used by many banks for securing their communications.

11. The probability of a randomly chosen number from the set of natural numbers is not definable

Clearly the probability P(a) of selecting a number that has the property of being prime from a given set S of numbers is definable if the precise proportion of primes to non-primes in S is definable.

However if S is the set N of all integers, and we cannot define a precise ratio of primes to composites in N, but only an order of magnitude such as O(\frac{1}{log_{_{e}}n}), then equally obviously P(a) = P(n\ is\ a\ prime) cannot be defined in N (see Chapter 2, p.9, Theorem 2.1, here).

12. The prime divisors of a natural number are independent

Now, the following paper proves P(b) = \frac{1}{\pi(\sqrt{n})}, since it shows that whether or not a prime p divides a given integer n is independent of whether or not a prime q \neq p divides n:

Why Integer Factorising cannot be polynomial time

We thus have that \pi(n) \approx n.\prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}}), with a binomial standard deviation.

Hence, even though we cannot define the probability P(n\ is\ a\ prime) of selecting a number from the set N of all natural numbers that has the property of being prime, \prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}}) can be treated as the putative non-heuristic probability that a given n is a prime.

13. The distribution of primes is a quantum phenomena

The distribution of primes is thus determinate but unpredictable, since it is representable by the algorithmically verifiable but not algorithmically computable arithmetical number-theoretic function Pr(n) = p_{_{n}}, where p_{_{n}} is the n‘th prime.

The Prime Number Generating Theorem and the Trim and Compact algorithms detailed in this 1964 investigation illustrate why the arithmetical number-theoretic function Pr(n) is algorithmically verifiable but not algorithmically computable (see also this Wikipedia proof that no non-constant polynomial function Pr(n) with integer coefficients exists that evaluates to a prime number for all integers n.).

Moreover, although the distribution of primes is a quantum phenomena with probabilty \prod_{_{i = 1}}^{^{\pi(\sqrt{n})}}(1-\frac{1}{p_{_{i}}}), it is easily seen (see Figs. 7-11 on pp.23-26 of this preprint) that the generation of the primes is algorithmically computable.

14. Why the universe may be algorithmically computable

By analogy, this suggests that although the measurable values of some individual properties of particles in the universe over time may represent a quantum phenomena, the universe itself may be algorithmically computable if the laws governing the generation of all the particles in the universe over time are algorithmically computable.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

A. A mathematical physicist’s conception of thinking about infinity in consistent ways

John Baez is a mathematical physicist, currently working at the math department at U. C. Riverside in California, and also at the Centre for Quantum Technologies in Singapore.

Baez is not only academically active in the areas of network theory and information theory, but also socially active in promoting and supporting the Azimuth Project, which is a platform for scientists, engineers and mathematicians to collaboratively do something about the global ecological crisis.

In a recent post—Large Countable Ordinals (Part 1)—on the Azimuth Blog, Baez confesses to a passionate urge to write a series of blogs—that might even eventually yield a book—about the infinite, reflecting both his fascination with, and frustration at, the challenges involved in formally denoting and talking meaningfully about different sizes of infinity:

“I love the infinite. … It may not exist in the physical world, but we can set up rules to think about it in consistent ways, and then it’s a helpful concept. … Cantor’s realization that there are different sizes of infinity is … part of the everyday bread and butter of mathematics.”

B. Why thinking about infinity in a consistent way must be constrained by an objective, evidence-based, perspective

I would cautiously submit however that (as I briefly argue in this blogpost), before committing to any such venture, whether we can think about the “different sizes of infinity” in “consistent ways“, and to what extent such a concept is “helpful“, are issues that may need to be addressed from an objective, evidence-based, computational perspective in addition to the conventional self-evident, intuition-based, classical perspective towards formal axiomatic theories.

C. Why we cannot conflate the behaviour of Goodstein’s sequence in Arithmetic with its behaviour in Set Theory

Let me suggest why by briefly reviewing—albeit unusually—the usual argument of Goodstein’s Theorem (see here) that every Goodstein sequence over the natural numbers must terminate finitely.

1. The Goodstein sequence over the natural numbers

First, let g(1, m, [2]), g(2, m, [3]), g(3, m, [4]), \ldots, be the terms of the Goodstein sequence G(m) for m over the domain N of the natural numbers, where [i+1] is the base in which the hereditary representation of the i‘th term of the sequence is expressed.

Some properties of Goodstein’s sequence over the natural numbers

We note that, for any natural number m, R. L. Goodstein uses the properties of the hereditary representation of m to construct a sequence G(m) \equiv \{g(1, m, [2]),\ g(2, m, [3]), \ldots\} of natural numbers by an unusual, but valid, algorithm.

Hereditary representation: The representation of a number as a sum of powers of a base b, followed by expression of each of the exponents as a sum of powers of b, etc., until the process stops. For example, we may express the hereditary representations of 266 in base 2 and base 3 as follows:

226_{[2]} \equiv 2^{8_{[2]}}+2^{3_{[2]}}+2 \equiv 2^{2^{(2^{2^{0}}+2^{0})}}+2^{2^{2^{0}}+2^{2^{0}}}+2^{2^{0}}

226_{[3]} \equiv 2.3^{4_{[3]}}+2.3^{3_{[3]}}+3^{2_{[3]}}+1 \equiv 2.3^{(3^{3^{0}}+3^{0})}+2.3^{3^{3^{0}}}+3^{2.3^{0}}+3^{0}

We shall ignore the peculiar manner of constructing the individual members of the Goodstein sequence, since these are not germane to understanding the essence of Goodstein’s argument. We need simply accept for now that G(m) is well-defined over the structure N of the natural numbers, and has, for instance, the following properties:

g(1, 226, [2]) \equiv 2^{2^{2+1}}+2^{2+1}+2

g(2, 226, [3]) \equiv (3^{3^{3+1}}+3^{3+1}+3)-1

g(2, 226, [3]) \equiv 3^{3^{3+1}}+3^{3+1}+2

g(3, 226, [4]) \equiv (4^{4^{4+1}}+4^{4+1}+2)-1

g(3, 226, [4]) \equiv 4^{4^{4+1}}+4^{4+1}+1

If we replace the base [i+1] in each term g(i, m, [i+1]) of the sequence G(m) by [n], we arrive at a corresponding sequence of, say, Goodstein’s functions for m over the domain N of the natural numbers.

Where, for instance:

g(1, 226, [n]) \equiv n^{n^{n+1}}+n^{n+1}+n

g(2, 226, [n]) \equiv n^{n^{n+1}}+n^{n+1}+2

g(3, 226, [n]) \equiv n^{n^{n+1}}+n^{n+1}+1

It is fairly straightforward (see here) to show that, for all i \geq 1:

Either g(i, m, [n]) > g(i+1, m, [n]), or g(i, m, [n]) = 0.

Clearly G(m) terminates in N if, and only if, there is a natural number k > 0 such that, for any i > 0, we have either that g(i, m, [k]) > g(i+1, m, [k]) or that g(i, m, [k]) = 0.

However, since we cannot, equally clearly, immediately conclude from the axioms of the first-order Peano Arithmetic PA that such a k must exist merely from the definition of the G(m) sequence in N, we cannot immediately conclude from the above argument that G(m) must terminate finitely in N.

2. The Goodstein sequence over the finite ordinal numbers

Second, let g_{o}(1, m, [2_{o}]), g_{o}(2, m, [3_{o}]), g_{o}(3, m, [4_{o}]), \ldots, be the terms of the Goodstein sequence G_{o}(m) over the domain \omega of the finite ordinal numbers 0_{o}, 1_{o}, 2_{o}, \ldots, where \omega is Cantor’s least transfinite ordinal.

If we replace the base [(i+1)_{o}] in each term g_{o}(i, m, [(i+1)_{o}]) of the sequence G_{o}(m) by [c], where c ranges over all ordinals upto \varepsilon_{0}, it is again fairly straightforward to show that:

Either g_{o}(i, m, [c]) >_{o} g_{o}(i+1, m, [c]), or g_{o}(i, m, [c]) = 0_{o}.

Clearly, in this case too, G_{o}(m) terminates in \omega if, and only if, there is an ordinal k_{o}>_{o} 0_{o} such that, for all finite i > 0, we have either that g_{o}(i, m, [k_{o}]) >_{o} g_{o}(i+1, m, [k_{o}]), or that g_{o}(i, m, [k_{o}]) =_{o} 0_{o}.

3. Goodstein’s argument over the transfinite ordinal numbers

If we, however, let c =_{o} \omega then—since the ZF axioms do not admit an infinite descending set of ordinals—it now immediately follows that we cannot have:

g_{o}(i, m, [\omega]) >_{o} g_{o}(i+1, m, [\omega]) for all i > 0.

Hence G_{o}(m) must terminate finitely in \omega, since we must have that g(i, m, [\omega]) =_{o} 0_{o} for some finite i > 0.

4. The intuitive justification for Goodstein’s Theorem

The intuitive justification—which must implicitly underlie any formal argument—for Goodstein’s Theorem then is that, since the finite ordinals can be meta-mathematically seen to be in a 1-1 correspondence with the natural numbers, we can conclude from (2) above that every Goodstein sequence over the natural numbers must also terminate finitely.

5. The fallacy in Goodstein’s argument

The fallacy in this conclusion is exposed if we note that, by (2), G_{o}(m) must terminate finitely in \omega even if G(m) did not terminate in N!

6. Why we need to heed Skolem’s cautionary remarks

Clearly, if we heed Skolem’s cautionary remarks (reproduced here) about unrestrictedly corresponding conclusions concerning elements of different formal systems, then we can validly only conclude that the relationship of ‘terminating finitely’ with respect to the ordinal inequality ‘>_{o}‘ over an infinite set S_{0} of finite ordinals in any putative interpretation of a first order Ordinal Arithmetic cannot be obviously corresponded to the relationship of ‘terminating finitely’ with respect to the natural number inequality ‘>‘ over an infinite set S of natural numbers in any interpretation of PA.

7. The significance of Skolem’s qualification

The significance of Skolem’s qualification is highlighted if we note that we cannot force PA to admit a constant denoting a ‘completed infinity’, such as Cantor’s least ordinal \omega, into either PA or into any interpretation of PA without inviting inconsistency.

(The proof is detailed in Theorem 4.1 on p.7 of this preprint. See also this blogpage).

8. PA is finitarily consistent

Moreover, the following paper, due to appear in the December 2016 issue of Cognitive Systems Research, gives a finitary proof of consistency for the first-order Peano Arithmetic PA:

The truth assignments that differentiate human reasoning from mechanistic reasoning: The evidence-based argument for Lucas’ Gödelian thesis.

9. Why ZF cannot have an evidence-based interpretation

It also follows from the above-cited CSR paper that ZF axiomatically postulates the existence of an infinite set which cannot be evidenced as true even under any putative interpretation of ZF.

10. The appropriate conclusion of Goodstein’s argument

So, if a ‘completed infinity’ cannot be introduced as a constant into PA, or as an element into the domain of any interpretation of PA, without inviting inconsistency, it would follow in Russell’s colourful phraseology that the appropriate conclusion to be drawn from Goodstein’s argument is that:

(i) In the first-order Peano Arithmetic PA we always know what we are talking about, even though we may not always know whether it is true or not;

(ii) In the first-order Set Theory we never know what we are talking about, so the question of whether or not it is true is only of notional interest.

Which raises the issue not only of whether we can think about the different sizes of infinity in a consistent way, but also to what extent we may need to justify that such a concept is helpful to an emerging student of mathematics.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

The Unexplained Intellect: Complexity, Time, and the Metaphysics of Embodied Thought

Christopher Mole is an associate professor of philosophy at the University of British Columbia, Vancouver. He is the author of Attention is Cognitive Unison: An Essay in Philosophical Psychology (OUP, 2011), and The Unexplained Intellect: Complexity, Time, and the Metaphysics of Embodied Thought (Routledge, 2016).

In his preface to The Unexplained Intellect, Mole emphasises that his book is an attempt to provide arguments for (amongst others) the three theses that:

(i) “Intelligence might become explicable if we treat intelligence thought as if it were some sort of computation”;

(ii) “The importance of the rapport between an organism and its environment must \ldots be understood from a broadly computational perspective”;

(iii) “\ldots our difficulties in accounting for our psychological orientation with respect to time are indications of the need to shift our philosophical focus away from mental states—which are altogether too static—and towards a theory of the mind in which it is dynamic mental entities that are taken to be metaphysically foundational”.

The Brains blog

Mole explains at length his main claims in The Unexplained Intellect—and the cause that those claims serve—in a lucid and penetrating, VI-part, series of invited posts in The Brains blog (a leading forum for work in the philosophy and science of mind that was founded in 2005 by Gualtiero Piccinini, and has been administered by John Schwenkler since late 2011).

In these posts, Mole seeks to make the following points.

I: The Unexplained Intellect: The mind is not a hoard of sentences

We do not currently have a satisfactory account of how minds could be had by material creatures. If such an account is to be given then every mental phenomenon will need to find a place within it. Many will be accounted for by relating them to other things that are mental, but there must come a point at which we break out of the mental domain, and account for some things that are mental by reference to some that are not. It is unclear where this break out point will be. In that sense it is unclear which mental entities are, metaphysically speaking, the most fundamental.

At some point in the twentieth century, philosophers fell into the habit of writing as if the most fundamental things in the mental domain are mental states (where these are thought of as states having objective features of the world as their truth-evaluable contents). This led to a picture in which the mind was regarded as something like a hoard of sentences. The philosophers and cognitive scientists who have operated with this picture have taken their job to be telling us what sort of content these mental sentences have, how that content is structured, how the sentences come to have it, how they get put into and taken out of storage, how they interact with one another, how they influence behaviour, and so on.

This emphasis on states has caused us to underestimate the importance of non-static mental entities, such as inferences, actions, and encounters with the world. If we take these dynamic entities to be among the most fundamental of the items in the mental domain, then — I argue — we can avoid a number of philosophical problems. Most importantly, we can avoid a picture in which intelligent thought would be beyond the capacities of any physically implementable system.

II: The Unexplained Intellect: Computation and the explanation of intelligence

A lot of philosophers think that consciousness is what makes the mind/body problem interesting, perhaps because they think that consciousness is the only part of that problem that remains wholly philosophical. Other aspects of the mind are taken to be explicable by scientific means, even if explanatorily adequate theories of them remain to be specified.

\ldots I’ll remind the reader of computability theory’s power, with a view to indicating how it is that the discoveries of theoretical computer scientists place constraints on our understanding of what intelligence is, and of how it is possible.

III: The Unexplained Intellect: The importance of computability

If we found that we had been conceiving of intelligence in such a way that intelligence could not be modelled by a Turing Machine, our response should not be to conclude that some alternative must be found to a ‘Classically Computational Theory of the Mind’. To think only that would be to underestimate the scope of the theory of computability. We should instead conclude that, on the conception in question, intelligence would (be) absolutely inexplicable. This need to avoid making intelligence inexplicable places constraints on our conception of what intelligence is.

IV: The Unexplained Intellect: Consequences of imperfection

The lesson to be drawn is that, if we think of intelligence as involving the maintenance of satisfiable beliefs, and if we think of our beliefs as corresponding to a set of representational states, then our intelligence would depend on a run of good luck the chances of which are unknown.

My suggestion is that we can reach a more explanatorily satisfactory conception of intelligence if we adopt a dynamic picture of the mind’s metaphysical foundations.

V: The Unexplained Intellect: The importance of rapport

I suggest that something roughly similar is true of us. We are not guaranteed to have satisfiable beliefs, and sometimes we are rather bad at avoiding unsatisfiability, but such intelligence as we have is to be explained by reference to the rapport between our minds and the world.

Rather than starting from a set of belief states, and then supposing that there is some internal process operating on these states that enables us to update our beliefs rationally, we should start out by accounting for the dynamic processes through which the world is epistemically encountered. Much as the three-colourable map generator reliably produces three-colourable maps because it is essential to his map-making procedure that borders appear only where they will allow for three colorability, so it is essential to what it is for a state to be a belief that beliefs will appear only if there is some rapport between the believer and the world. And this rapport — rather than any internal processing considered in isolation from it — can explain the tendency for our beliefs to respect the demands of intelligence.

VI: The Unexplained Intellect: The mind’s dynamic foundations

\ldots memory is essentially a form of epistemic retentiveness: One’s present knowledge counts as an instance of memory when and only when it was attained on the basis of an epistemic encounter that lies in one’s past. One can epistemically encounter a proposition as the conclusion of an argument, and so can encounter it before the occurrence of any event to which it pertains, but one cannot encounter an event in that way. In the resulting explanation of memory’s temporal asymmetry, it is the dynamic events of epistemic encountering to which we must make reference. These encounters, and not the knowledge states to which they lead, do the lion’s share of the explanatory work.

A: Simplifying Mole’s perspective

It may help simplify Mole’s thought-provoking perspective if we make an arbitrary distinction between:

(i) The mind of an applied scientist, whose primary concern is our sensory observations of a ‘common’ external world;

(ii) The mind of a philosopher, whose primary concern is abstracting a coherent perspective of the external world from our sensory observations; and

(iii) The mind of a mathematician, whose primary concern is adequately expressing such abstractions in a formal language of unambiguous communication.

My understanding of Mole’s thesis, then, is that:

(a) although a mathematician’s mind may be capable of defining the ‘truth’ value of some logical and mathematical propositions without reference to the external world,

(b) the ‘truth’ value of any logical or mathematical proposition that purports to represent any aspect of the real world must be capable of being evidenced objectively to the mind of an applied scientist; and that,

(c) of the latter ‘truths’, what should interest the mind of a philosopher is whether there are some that are ‘knowable’ completely independently of the passage of time, and some that are ‘knowable’ only partially, or incrementally, with the passage of time.

B. Support for Mole’s thesis

It also seems to me that Mole’s thesis implicitly subsumes, or at the very least echoes, the belief expressed by Chetan R. Murthy (‘An Evaluation Semantics for Classical Proofs‘, Proceedings of Sixth IEEE Symposium on Logic in Computer Science, pp. 96-109, 1991; also Cornell TR 91-1213):

“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 …”

If so, the thesis seems significantly supported by the following paper that is due to appear in the December 2016 issue of ‘Cognitive Systems Research’:

The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas’ Goedelian Thesis

The CSR paper implicitly suggests that there are, indeed, (only?) two ways of assigning ‘true’ or ‘false’ values to any mathematical description of real-world events.

C. Algorithmic computability

First, a number theoretical relation F(x) is algorithmically computable if, and only if, there is an algorithm AL_{F} that can provide objective evidence (cf. ibid Murthy 91) for deciding the truth/falsity of each proposition in the denumerable sequence \{F(1), F(2), \ldots\}.

(We note that the concept of `algorithmic computability’ is essentially an expression of the more rigorously defined concept of `realizability’ on p.503 of Stephen Cole Kleene’s ‘Introduction to Metamathematics‘, North Holland Publishing Company, Amsterdam.)

D. Algorithmic verifiability

Second, 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 for deciding the truth/falsity of each proposition in the finite sequence \{F(1), F(2), \ldots, F(n)\}.

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 propositions, 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 theorem (Theorem 2.1, p.37 of the CSR paper) shows that although every algorithmically computable relation is algorithmically verifiable, the converse is not true:

Theorem: There are number theoretic functions that are algorithmically verifiable but not algorithmically computable.

E. The significance of algorithmic ‘truth’ assignments for Mole’s theses

The significance of such algorithmic ‘truth’ assignments for Mole’s theses is that:

Algorithmic computability—reflecting the ambit of classical Newtonian mechanics—characterises natural phenomena that are determinate and predictable.

Such phenomena are describable by mathematical propositions that can be termed as ‘knowable completely’, since at any point of time they are algorithmically computable as ‘true’ or ‘false’.

Hence both their past and future behaviour is completely computable, and their ‘truth’ values are therefore ‘knowable’ independent of the passage of time.

Algorithmic verifiability—reflecting the ambit of Quantum mechanics—characterises natural phenomena that are determinate but unpredictable.

Such phenomena are describable by mathematical propositions that can only be termed as ‘knowable incompletely’, since at any point of time they are only algorithmically verifiable, but not algorithmically computable, as ‘true’ or ‘false’

Hence, although their past behaviour is completely computable, their future behaviour is not completely predictable, and their ‘truth’ values are not independent of the passage of time.

F. Where Mole’s implicit faith in the adequacy of set theoretical representations of natural phenomena may be misplaced

It also seems to me that, although Mole’s analysis justifiably holds that the:

\ldots importance of the rapport between an organism and its environment”

has been underacknowledged, or even overlooked, by existing theories of the mind and intelligence, it does not seem to mistrust, and therefore ascribe such underacknowledgement to any lacuna in, the mathematical and epistemic foundations of the formal language in which almost all descriptions of real-world events are currently sought to be expressed, which is the language of the set theory ZF.

G. Any claim to a physically manifestable ‘truth’ must be objectively accountable

Now, so far as applied science is concerned, history teaches us that the ‘truth’ of any mathematical proposition that purports to represent any aspect of the external world must be capable of being evidenced objectively; and that such ‘truths’ must not be only of a subjective and/or revelationary nature which may require truth-certification by evolutionarily selected prophets.

(Not necessarily religious—see, for instance, Melvyn B. Nathanson’s remarks, “Desperately Seeking Mathematical Truth“, in the Opinion piece in the August 2008 Notices of the American Mathematical Society, Vol. 55, Issue 7.)

The broader significance of seeking objective accountability is that it admits the following (admittedly iconoclastic) distinction between the two fundamental mathematical languages:

1. The first-order Peano Arithmetic PA as the language of science; and

2. The first-order Set Theory ZF as the language of science fiction.

It is a distinction that is faintly reflected in Stephen G. Simpson’s more conservative perspective in his paper ‘Partial Realizations of Hilbert’s Program‘ (#6.4, p.15):

“Finitistic reasoning (my read: ‘First-order Peano Arithmetic PA’) is unique because of its clear real-world meaning and its indispensability for all scientific thought. Nonfinitistic reasoning (my read: ‘First-order Set Theory ZF’) can be accused of referring not to anything in reality but only to arbitrary mental constructions. Hence nonfinitistic mathematics can be accused of being not science but merely a mental game played for the amusement of mathematicians.”

The distinction is supported by the formal argument (detailed in the above-cited CSR paper) that:

(i) PA has two, hitherto unsuspected, evidence-based interpretations, the first of which can be treated as circumscribing the ambit of human reasoning about ‘true’ arithmetical propositions; and the second can be treated as circumscribing the ambit of mechanistic reasoning about ‘true’ arithmetical propositions.

What this means is that the language of arithmetic—formally expressed as PA—can provide all the foundational needs for all practical applications of mathematics in the physical sciences. This was was the point that I sought to make—in a limited way, with respect to quantum phenomena—in the following paper presented at Unilog 2015, Istanbul last year:

Algorithmically Verifiable Logic vis `a 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.)

(ii) Since ZF axiomatically postulates the existence of an infinite set that cannot be evidenced (and which cannot be introduced as a constant into PA, or as an element into the domain of any interpretation of PA, without inviting inconsistency—see Theorem 1 in \S4 of this post), it can have no evidence-based interpretation that could be treated as circumscribing the ambit of either human reasoning about ‘true’ set-theoretical propositions, or that of mechanistic reasoning about ‘true’ set-theoretical propositions.

The language of set theory—formally expressed as ZF—thus provides the foundation for abstract structures that—although of possible interest to philosophers of science—are only mentally conceivable by mathematicians subjectively, and have no verifiable physical counterparts, or immediately practical applications of mathematics, that can materially impact on the study of physical phenomena.

The significance of this distinction can be expressed more vividly in Russell’s phraseology as:

(iii) In the first-order Peano Arithmetic PA we always know what we are talking about, even though we may not always know whether it is true or not;

(iv) In the first-order Set Theory we never know what we are talking about, so the question of whether or not it is true is only of fictional interest.

H. The importance of Mole’s ‘rapport’

Accordingly, I see it as axiomatic that the relationship between an evidence-based mathematical language and the physical phenomena that it purports to describe, must be in what Mole terms as ‘rapport’, if we view mathematics as a set of linguistic tools that have evolved:

(a) to adequately abstract and precisely express through human reasoning our observations of physical phenomena in the world in which we live and work; and

(b) unambiguously communicate such abstractions and their expression to others through objectively evidenced reasoning in order to function to the maximum of our co-operative potential in acieving a better understanding of physical phenomena.

This is the perspective that I sought to make in the following paper presented at Epsilon 2015, Montpellier, last June, where I argue against the introduction of ‘unspecifiable’ elements (such as completed infinities) into either a formal language or any of its evidence-based interpretations (in support of the argument that since a completed infinity cannot be evidence-based, it must therefore be dispensible in any purported description of reality):

Why Hilbert’s and Brouwer’s interpretations of quantification are complementary and not contradictory.’

(Presented 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.)

I. Why mathematical reasoning must reflect an ‘agnostic’ perspective

Moreover, from a non-mathematician’s perspective, a Propertarian like Curt Doolittle would seem justified in his critique (comment of June 2, 2016 in this Quanta review) of the seemingly ‘mystical’ and ‘irrelevant’ direction in which conventional interpretations of Hilbert’s ‘theistic’ and Brouwer’s ‘atheistic’ reasoning appear to have pointed mainstream mathematics for, as I argue informally in an earlier post, the ‘truths’ of any mathematical reasoning must reflect an ‘agnostic’ perspective.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

A new proof?

An interesting review by Natalie Wolchover on May 24, 2016, in the on-line magazine Quanta, reports that:

“With a surprising new proof, two young mathematicians have found a bridge across the finite-infinite divide, helping at the same time to map this strange boundary.

The boundary does not pass between some huge finite number and the next, infinitely large one. Rather, it separates two kinds of mathematical statements: ‘finitistic’ ones, which can be proved without invoking the concept of infinity, and ‘infinitistic’ ones, which rest on the assumption — not evident in nature — that infinite objects exist.”

More concretely:

“In the new proof, Keita Yokoyama, 34, a mathematician at the Japan Advanced Institute of Science and Technology, and Ludovic Patey, 27, a computer scientist from Paris Diderot University, pin down the logical strength of RT_{2}^{2} — but not at a level most people expected. The theorem is ostensibly a statement about infinite objects. And yet, Yokoyama and Patey found that it is ‘finitistically reducible’: It’s equivalent in strength to a system of logic that does not invoke infinity. This result means that the infinite apparatus in RT_{2}^{2} can be wielded to prove new facts in finitistic mathematics, forming a surprising bridge between the finite and the infinite.”

The proof appeals to properties of transfinite ordinals

My immediate reservation—after a brief glance at the formal definitions in \S1.6 on p.6 of the Yokoyama-Patey paper—was that the domain of the structure in which the formal result is proved necessarily contains at least Cantor’s smallest transfinite ordinal \omega, whereas the result is apparently sought to be ‘finitistically reducible’ (as considered by Stephen G. Simpson in an absorbing survey of Partial Realizations of Hilbert’s Program), in the sense of being not only finitarily provable, but interpretable in, and applicable to, finite structures (such as that of the natural numbers) whose domains may not contain (nor, in some cases, even admit—see Theorem 1 in \S4.1 of this post) an infinite ‘number’.

Prima facie, the implicit assumption here (see also this post) seems to reflect, for instance, the conventional wisdom that every proposition which is formally provable about the finite, set-theoretically defined ordinals (necessarily assumed consistent with an axiom of infinity), must necessarily interpret as a true proposition about the natural numbers.

Why we cannot ignore Skolem’s cautionary remarks

In this conventional wisdom—by terming it as Skolem’s Paradox—both accepts and implicitly justifies ignoring Thoraf Skolem’s cautionary remarks about unrestrictedly corresponding putative mathematical relations and entities across domains of different axiom systems.

(Thoralf Skolem. 1922. Some remarks on axiomatized set theory. Text of an address delivered in Helsinki before the Fifth Congress of Scandinavian Mathematicians, 4-7 August 1922. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts.)

However, that the assumption is fragile is seen since, without such an assumption, we can only conclude from, say, Goodstein’s argument that a Goodstein sequence defined over the finite ZF ordinals must terminate finitely even if the corresponding Goodstein sequence over the natural numbers does not terminate (see Theorem 2 of this unpublished investigation)!

(R. L. Goodstein. 1944. On the Restricted Ordinal Theorem. In the Journal of Symbolic Logic 9, 33-41.)

A remarkable exposition of Ramsey’s Theorem

The Yokoyama-Patey proof invites other reservations too.

In a comment—remarkable for its clarity of exposition—academically minded ‘Peter’ illustrates Ramsey’s Theorem as follows:

Something that might help to understand what’s going on here is to start one level lower: Ramsey’s theorem for singletons (RT^1_2) says that however you colour the integers with two colours (say red and blue), you are guaranteed to find an infinite monochromatic subset. To see this is true, simply go along the integers starting from 1 and put them into the red or the blue bag according to their colour. Since in each step you increase the size of one or the other bag, without removing anything, you end up with an infinite set. This is a finitistic proof: it never really uses infinity, but it tells you how to construct the first part of the ‘infinite set’.

Now let’s try the standard proof for RT^2_2, pairs. This time we will go along the integers twice, and we will throw away a lot as we go.

The first time, we start at 1. Because there are infinitely many numbers bigger than 1, each of which makes a pair with 1 and each of which pairs is coloured either red or blue, there are either infinitely many red pairs with 1 or infinitely many blue pairs (note: this is really using RT_2^1). I write down under 1 ‘red’ or ‘blue’ depending on which it turned out to be (in case both sets of pairs are infinite, I’ll write red just to break a tie), then I cross out all the numbers bigger than 1 which make the ‘wrong colour’ pair with 1.

Now I move on to the next number, say s, I didn’t cross out, and I look at all the pairs it makes with the un-crossed-out numbers bigger than it. There are still infinitely many, so either the red pairs or the blue pairs form an infinite set (or both). I write down red or blue below s as before, and again cross out all the number bigger than s which make a wrong colour pair with s. And I keep going like this; because everything stays infinite I never get stuck.

After an infinitely long time, I can go back and look at all the numbers which I did not cross out – there is an infinite list of them. Under each is written either ‘red’ or ‘blue’, and if under (say) number t the word ‘red’ is written, then t forms red pairs with all the un-crossed-out numbers bigger than t. Now (using RT^1_2 again) either the word ‘red’ or the word ‘blue’ was written infinitely often, so I can pick an infinite set of numbers under which I wrote either always ‘red’ or always ‘blue’. Suppose it was always ‘red’; then if s and t are any two numbers in the collection I picked, the pair st will be red – this is because one of s and t, say s, is smaller, and by construction all the pairs from s to bigger un-crossed-out numbers, including t, are red. If it were always blue, by the same argument I get an infinite set where all pairs are blue.

What is different here to the first case? The difference is that in order to say whether I should write ‘red’ or ‘blue’ under 1 (or any other number) in the first step, I have to ‘see’ the whole infinite set. I could look at a lot of these numbers and make a guess – but if the guess turns out to be wrong then it means I made a mistake at all the later steps of the process too; everything falls apart. This is not a finitistic proof – according to some logicians, you should be worried that it might somehow be wrong. Most mathematicians will say it is perfectly fine though.

Moving up to RT^3_2, the usual proof is an argument that looks quite a lot like the RT^2_2 argument, except that instead of using RT^1_2 in the ‘first pass’ it uses RT^2_2. All fine; we believe RT^2_2, so no problem. But now, when you want to write down ‘red’ or ‘blue under 1 in this ‘first pass’ you have to know something more complicated about all the triples using 1; you want to know if you can find an infinite set S such that any pair s,t in S forms a red triple with 1. If not, RT^2_2 tells you that you can find an infinite set S such that any pair s,t in S forms a _blue_ triple with 1. Then you would cross off everything not in S, and keep going as with RT^2_2. The proof doesn’t really get any harder for the general case RT^k_2 (or indeed changing the number of colours to something bigger than 2). If you’re happy with infinity, there’s nothing new to see here. If not – well, these proofs have you recursively using more and infinitely more appeals to something infinite as you increase k, which is not a happy place to be in if you don’t like infinity.

Implicit assumptions in Yokoyama-Patey’s argument

Peter’s clarity of exposition makes it easier to see that, in order to support the conclusion that their proof of Ramsey’s Theorem for pairs is ‘finitistically reducible’, Yokoyama-Patey must assume:

(i) that ZFC is consistent, and therefore has a Tarskian interpretation in which the ‘truth’ of a ZFC formula can be evidenced;

(ii) that their result must be capable of an evidence-based Tarskian interpretation over the ‘finitist’ structure of the natural numbers.

As to (i), Peter has already pointed out in his final sentence that there are (serious?) reservations to accepting that the ZF axiom of infinity can have any evidence-based interpretation.

As to (ii), Ramsey’s Theorem is an existensial ZFC formula of the form (\exists x)F(x) (whose proof must appeal to an axiom of choice).

Now in ZF (as in any first-order theory that appeals to the standard first-order logic FOL) the formula (\exists x)F(x) is merely an abbreviation for the formula \neg(\forall x)\neg(F(x).

So, under any consistent ‘finitistically reducible’ interpretation of such a formula, there must be a unique, unequivocal, evidence-based Tarskian interpretation of (\forall x)F(x) over the domain of the natural numbers.

Now, if we are to avoid intuitionistic objections to the admitting of ‘unspecified’ natural numbers in the definition of quantification under any evidence-based Tarskian interpretation of a formal system of arithmetic, we are faced with the ambiguity where the questions arise:

(a) Is the (\forall x)F(x)] to be interpreted constructively as:

For any natural number n, there is an algorithm T_n (say, a deterministic Turing machine) which evidences that \{F(1), F(2), \ldots, F(n)\} are all true; or,

(b) is the formula (\forall x)F(x) to be interpreted finitarily as:

There is a single algorithm T (say, a deterministic Turing machine) which evidences that, for any natural number n, F(n) is true, i.e., each of \{F(1), F(2), \ldots\} is true?

As Peter has pointed out in his analysis of Ramsey’s Theorem RT_2^2 for pairs, the proof of the Theorem necessitates that:

“I have to ‘see’ the whole infinite set. I could look at a lot of these numbers and make a guess – but if the guess turns out to be wrong then it means I made a mistake at all the later steps of the process too; everything falls apart. This is not a finitistic proof – according to some logicians, you should be worried that it might somehow be wrong.”

In other words, Yokoyama-Patey’s conclusion (that their new proof is ‘finitistically reducible’) would only hold if they have established (b) somewhere in their proof; but a cursory reading of their paper does not suggest this to be the case.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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

In a recent paper A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory, authors Adam Yedidia and Scott Aaronson argue upfront in their Introduction that:

Like any axiomatic system capable of encoding arithmetic, ZFC is constrained by Gödel’s two incompleteness theorems. The first incompleteness theorem states that if ZFC is consistent (it never proves both a statement and its opposite), then ZFC cannot also be complete (able to prove every true statement). The second incompleteness theorem states that if ZFC is consistent, then ZFC cannot prove its own consistency. Because we have built modern mathematics on top of ZFC, we can reasonably be said to have assumed ZFC’s consistency.

The question arises:

How reasonable is it to build modern mathematics on top of a Set Theory such as ZF?

Some immediate points to ponder upon (see also reservations expressed by Stephen G. Simpson in Logic and Mathematics and in Partial Realizations of Hilbert’s Program):

1. “Like any axiomatic system capable of encoding arithmetic, …”

The implicit assumption here that every ZF formula which is provable about the finite ZF ordinals must necessarily interpret as a true proposition about the natural numbers is fragile since, without such an assumption, we can only conclude from Goodstein’s argument (see Theorem 1.1 here) that a Goodstein sequence defined over the finite ZF ordinals must terminate even if the corresponding Goodstein sequence over the natural numbers does not terminate!

2. “ZFC is constrained by Gödel’s two incompleteness theorems. The first incompleteness theorem states that if ZFC is consistent (it never proves both a statement and its opposite), then ZFC cannot also be complete (able to prove every true statement). The second incompleteness theorem states that if ZFC is consistent, then ZFC cannot prove its own consistency.”

The implicit assumption here is that ZF is \omega-consistent, which implies that ZF is consistent and must therefore have an interpretation over some mathematically definable structure in which ZF theorems interpret as ‘true’.

The question arises: Must such ‘truth’ be capable of being evidenced objectively, or is it only of a subjective, revelationary, nature (which may require truth-certification by evolutionarily selected prophets—see Nathanson’s remarks as cited in this post)?

The significance of seeking objective accountbility is that in a paper, “The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas’ Gödelian Thesis“, which is due to appear in the December 2016 issue of Cognitive Systems Research, we show (see also this post) that the first-order Peano Arithmetic PA:

(i) is finitarily consistent; but

(ii) is not \omega-consistent; and

(iii) has no ‘undecidable’ arithmetical proposition (whence both of Gödel’s Incompleteness Theorems hold vacuously so far as the arithmetic of the natural numbers is concerned).

3. “Because we have built modern mathematics on top of ZFC, we can reasonably be said to have assumed ZFC’s consistency.”

Now, one justification for such an assumption (without which it may be difficult to justify building modern mathematics on top of ZF) could be the belief that acquisition of set-theoretical knowledge by students of mathematics has some essential educational dimension.

If so, one should take into account not only the motivations of such a student for the learning of mathematics, but also those of a mathematician for teaching it.

This, in turn, means that both the content of the mathematics which is to be learnt (or taught), as well as the putative utility of such learning (or teaching) for a student (or teacher), merit consideration.

Considering content, I would iconoclastically submit that the least one may then need to accomodate is the following distinction between the two fundamental mathematical languages:

1. The first-order Peano Arithmetic PA, which is the language of science; and

2. The first-order Set Theory ZF, which is the language of science fiction.

A distinction that is reflected in Stephen G. Simpson’s more conservative perspective in Partial Realizations of Hilbert’s Program (\S6.4, p.15):

Finitistic reasoning (read ‘First-order Peano Arithmetic PA’) is unique because of its clear real-world meaning and its indispensability for all scientific thought. Nonfinitistic reasoning (read ‘First-order Set Thyeory ZF’) can be accused of referring not to anything in reality but only to arbitrary mental constructions. Hence nonfinitistic mathematics can be accused of being not science but merely a mental game played for the amusement of mathematicians.

Reason:

(i) PA has two, hitherto unsuspected, evidence-based interpretations (see this post), the first of which can be treated as circumscribing the ambit of human reasoning about `true’ arithmetical propositions; and the second can be treated as circumscribing the ambit of mechanistic reasoning about `true’ arithmetical propositions.

It is this language of arithmetic—formally expressed as PA—that provides the foundation for all practical applications of mathematics where the latter could be argued as having an essential educational dimension.

(ii) Since ZF axiomatically postulates the existence of an infinite set that cannot be evidenced (and which cannot be introduced as a constant into PA, or as an element into the domain of any interpretation of PA, without inviting inconsistency—see paragraph 4.2 of this post), it can have no evidence-based interpretation that could be treated as circumscribing the ambit of either human reasoning about `true’ set-theoretical propositions, or that of mechanistic reasoning about `true’ set-theoretical propositions.

The language of set theory—formally expressed as ZF—thus provides the foundation for abstract structures that are only mentally conceivable by mathematicians (subjectively?), and have no physical counterparts, or immediately practical applications of mathematics, which could meaningfully be argued as having an essential educational dimension.

The significance of this distinction can be expressed more vividly in Russell’s phraseology as:

(iii) In the first-order Peano Arithmetic PA we always know what we are talking about, even though we may not always know whether it is true or not;

(iv) In the first-order Set Theory we never know what we are talking about, so the question of whether or not it is true is only of fictional interest.

The distinction is lost when—as seems to be the case currently—we treat the acquisition of mathematical knowledge as necessarily including the body of essentially set-theoretic theorems—to the detriment, I would argue, of the larger body of aspiring students of mathematics whose flagging interest in acquiring such a wider knowledge in universities around the world reflects the fact that, for most students, their interests seem to lie primarily in how a study of mathematics can enable them to:

(a) adequately abstract and precisely express through human reasoning their experiences of the world in which they live and work; and

(b) unambiguously communicate such abstractions and their expression to others through objectively evidenced reasoning in order to function to the maximum of their latent potential in acieving their personal real-world goals.

In other words, it is not obvious how how any study of mathematics that has the limited goals (a) and (b) can have any essentially educational dimension that justifies the assumption that ZF is consistent.

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

Three perspectives of FOL: Hibertian Theism, Brouwerian Atheism, and Finitary Agnosticism

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

What essentially differentiate our approach of finitary agnosticism to mathematical reasoning in these investigations from both the classical Hilbertian, and the intuitionistic Brouwerian, perspectives is how we address two significant dogmas of the classical first order logic FOL.

Hilbertian Theism

In a 1925 address[1] Hilbert had shown that the axiomatisation \mathcal{L}_{\varepsilon} of classical Aristotlean predicate logic proposed by him as a formal first-order \varepsilon-predicate calculus[2] in which he used a primitive choice-function[3] symbol, `\varepsilon‘, for defining the quantifiers `\forall‘ and `\exists‘ would adequately express—and yield, under a suitable interpretation—Aristotle’s logic of predicates if the \varepsilon-function was interpreted to yield Aristotlean particularisation[4].

Classical approaches to mathematics—essentially following Hilbert—can be labelled `theistic’ in that they implicitly assume—without providing adequate objective criteria—[5] both that:

\bullet The standard first order logic FOL is consistent;

and that:

\bullet The standard interpretation of FOL is finitarily sound (which implies Aristotle’s particularisation).

The significance of the label `theistic’ is that conventional wisdom tacitly believes that Aristotle’s particularisation remains valid—without qualification—even over infinite domains; a belief that is not unequivocally self-evident, but must be appealed to as an article of faith.

Although intended to highlight an entirely different distinction, that the choice of such a label may not be totally inappropriate is suggested by Tarski’s point of view[6] to the effect:

“… that Hilbert’s alleged hope that meta-mathematics would usher in a `feeling of absolute security’ was a `kind of theology’ that `lay far beyond the reach of any normal human science’ …”.

Brouwerian Atheism

We note second that, in sharp contrast, constructive approaches to mathematics—such as Intuitionism[6a]—can be labelled `atheistic’ since they deny both that:

\bullet FOL is consistent (since they deny the Law of The Excluded Middle[7];

and that:

\bullet The standard interpretation of FOL is sound (since they deny Aristotle’s particularisation).

The significance of the label `atheistic’ is that whereas constructive approaches to mathematics deny the faith-based belief in the unqualified validity of Aristotle’s particularisation over infinite domains, their denial of the Law of the Excluded Middle is itself a belief—in the inconsistency of FOL—that is also not unequivocally self-evident, and must also be appealed to as an article of faith since it does not take into consideration the objective criteria for the consistency of FOL that follows from the Birmingham paper.

Although Brouwer’s explicitly stated objection appeared to be to the Law of the Excluded Middle as expressed and interpreted at the time[8], some of Kleene’s remarks[9], some of Hilbert’s remarks[10] and, more particularly, Kolmogorov’s remarks[11] suggest that the intent of Brouwer’s fundamental objection (see this post) can also be viewed today as being limited only to the yet prevailing belief—as an article of faith—that the validity of Aristotle’s particularisation can be extended without qualification to infinite domains.

Finitary Agnosticism

In our investigations, however, we adopt what may be labelled a finitarily `agnostic’ perspective[12] by noting that although, if Aristotle’s particularisation holds in an interpretation then the Law of the Excluded Middle must also hold in the interpretation, the converse is not true.

We thus follow a middle path by explicitly assuming on the basis of the Birmingham paper (reproduced in this post) that:

\bullet FOL is finitarily consistent;

and by explicitly stating when an argument appeals to the postulation that:

\bullet The standard interpretation of FOL is sound.

The significance of the label `agnostic’ is that we neither hold FOL to be inconsistent, nor hold that Aristotle’s particularisation can be applied—without qualification—over infinite domains.

The two seemingly contradictory but perhaps complementary perspectives

It follows from the above that the Brouwerian Atheistic perspective is merely a restricted perspective within the Finitary Agnostic perspective, whilst the Hilbertian Theistic perspective contradicts the Finitary Agnostic perspective.

Curiously, a case can be made (as is attempted in this post, and in this paper that I presented on 10th June 2015 at the Epsilon 2015 workshop on Hilberts Epsilon and Tau in Logic, Informatics and Linguistics, University of Montpellier, France, June 10-11-12) that the Hilbertian perspective formalises the concept of `algorithmically verifiable truth’ in the non-finitary—hence essentially subjective—reasoning that circumscribes human intelligences (which, by the Anthropic principle, can be taken to reflect the ‘truths’ of natural laws), whilst the Finitary perspective formalises the concept of ‘algorithmically computable truth’ in the finitary—hence essentially objective—reasoning that circumscribes mechanical intelligences; and that the two are complementary, not contradictory, perspectives on the nature and scope of quantification.

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.

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.

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.

Br23 L. E. J. Brouwer. 1923. On the significance of the principle of the excluded middle in mathematics, especially in function theory. Address delivered on 21 September 1923 at the annual convention of the Deutsche Mathematiker-Vereinigung in Marburg an der Lahn. In Jean van Heijenoort. 1967. Ed. From Frege to G&oumldel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts.

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.

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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.

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.

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.

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.

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Ko25 Andrei Nikolaevich Kolmogorov. 1925. On the principle of excluded middle. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts.

Li64 A. H. Lightstone. 1964. The Axiomatic Method. Prentice Hall, NJ.

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

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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.

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.<i. MIT Press, Cambridge, Massachusetts.

Ro53 J. Barkley Rosser. 1953. Logic for Mathematicians 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.

An12 Bhupinder Singh Anand. 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 1: Hi25.

Return to 2: Hi27, pp.465-466; see also this post.

Return to 3: Hi25, p.382.

Return to 4: Hi25, pp.382-383; Hi27, p.466(1).

Return to 5: See for instance: 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.

Return to 6: As commented upon in Fr09, p.3.

Return to 6a: But see also Ma09 and MS05.

Return to 7: “The formula \forall x(A(x)\vee \neg A(x)) is classically provable, and hence under classical interpretation true. But it is unrealizable. So if realizability is accepted as a necessary condition for intuitionistic truth, it is untrue intuitionistically, and therefore unprovable not only in the present intuitionistic formal system, but by any intuitionistic methods whatsoever”. Kl52, p.513.

Return to 8: Br23, p.335-336; Kl52, p.47; Hi27, p.475.

Return to 9: Kl52, p.49.

Return to 10: For instance in Hi27, p.474.

Return to 11: In Ko25, fn. p.419; p.432.

Return to 12: See, for instance An12 (reproduced in this post).

Author’s working archives & abstracts of investigations

Bhupinder Singh Anand

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:

(i) The concept of algorithmic verifiability is well-defined under the standard interpretation \mathcal{I}_{PA(\mathbb{N},\ Standard)} of PA over the structure \mathbb{N} of the natural numbers; and

(ii) The concept of algorithmic computability too is well-defined under the algorithmic interpretation \mathcal{I}_{PA(\mathbb{N},\ Algorithmic)} of PA over the structure \mathbb{N} of the natural numbers; and

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<m, \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<m

(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.

An07 Bhupinder Singh Anand. 2007. Why we shouldn’t fault Lucas and Penrose for continuing to believe in the Gödelian argument against computationalism – II. In The Reasoner, Vol(1)7 p2-3.

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 1: cf. Me64, p.237.

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

Return to 11: cf. Me64, p.227.

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 5: cf. BBJ03, p.33.

Return to 6: See Si97.

Return to 7: Go51.

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 11: BBJ03, p. 37.

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 13: BBJ03, p.37.

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 15: The incongruity of this is addressed by Parikh in Pa71.

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 17: By Kurt Gödel; see Go31, Theorem VII.

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 22: Me64, p.118.

Return to 23: cf. Me64, p.131, proposition 3.21.

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 25: cf. Me64, p.132; Go31, p.30(2).

Return to 26: cf. Go31, p.31(2); Me64}, p.132.

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).

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