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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‘:

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

(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.

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‘:

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:

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

(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:

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

(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 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:

(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 $\S$4 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):

(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

Preamble

I find a frustrating equivocity[*] in standard literature between the intent behind the definitions of the classes $P$ and $NP$ as explained informally in terms of ‘verifiability’ and ‘decidability’, and the formal definitions of these two classes.

I call it the $PvNP$ Separation Problem.

My understanding of the intent commonly expressed in standard literature is that:

Verifiability: A number-theoretical formula $F(x)$ is verifiable under an interpretation (and should by intent be defined in $NP$) if, and only if, for any given natural number $n$, there is a deterministic algorithm $AL_{(F,\ n)}$ which can provide objective evidence for deciding the value of each formula in the finite sequence $\{F(1), F(2), \ldots, F(n)\}$.

Decidability: A number theoretical formula $F(x)$ is decidable under an interpretation (and should by intent be defined in $P$) if, and only if, there is a deterministic algorithm $AL_{F}$ that can provide objective evidence for deciding the value of each formula in the denumerable sequence $\{F(1), F(2), \ldots\}$.

In other words, for me ‘decidability’ should imply the existence of a deterministic algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions; whereas ‘verifiability’ need not imply the existence of a deterministic algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions.

For reasons and in ways that I cannot quite grasp as yet, it seems to me that standard definitions of the two classes $P$ and $NP$ only muddy the issue by associating the two concepts with terms such as ‘quickly’, ‘polynomial-time’ and ‘non-deterministic’; and by requiring introduction of some corresponding measure/s of computational complexity.

In other words, standard definitions of the two classes assert that:

Verifiability (standard): A number-theoretical formula $F(x)$ is verifiable under an interpretation (and is defined in $NP$) if, and only if, for any given natural number $n$, there is a deterministic polynomial-time algorithm $AL_{(F,\ n)}$ which can provide objective evidence for deciding the value of each formula in the finite sequence $\{F(1), F(2), \ldots, F(n)\}$.

Decidability (standard): A number theoretical formula $F(x)$ is decidable under an interpretation (and is defined in $P$) if, and only if, there is a deterministic polynomial-time algorithm $AL_{F}$ that can provide objective evidence for deciding the value of each formula in the denumerable sequence $\{F(1), F(2), \ldots\}.$

The following argument attempts to view the consequence for the $PvNP$ problem if we separate the two classes $P$ and $NP$ essentially as intended in the first case above.

The justification for this perspective is the formal distinction between the two concepts of ‘algorithmic verifiability’ and ‘algorithmic computability’ introduced in the Birmingham paper.

The significance that such a distinction may have for the first-order Peano Arithmetic $PA$ in particular—and for the foundations of mathematics, logic and computability in general—has already been suggested in this post.

Abstract

In this investigation we now consider the consequences of separating the classes $P$ and $NP$ logically from a similar—finitary arithmetical rather than computational—perspective.

This perspective too is based on explicitly highlighting that the assignment of satisfaction and truth values to number-theoretic formulas under an interpretation can be constructively defined in two distinctly different ways:

(a) in terms of algorithmic verifiability, and

(b) in terms of algorithmic computability.

From this perspective it can be seen that standard definitions of the classes $P$ and $NP$ implicitly assume without proof that:

Cook’s Thesis: If a propositional formula is algorithmically verifiable in polynomial time, then it is algorithmically computable.

This imples that the differentiation between the two classes is only quantitative, and can therefore be adequately expressed in terms of computational complexity.

However, we argue that there are classically defined arithmetic formulas which are algorithmically verifiable but not algorithmically computable; and so the differentiation between the classes $P$ and $NP$ can also be defined qualitatively, and need not be expressed only in terms of computational complexity.

Specifically, we show how Gödel’s $\beta$-function uniquely corresponds some real numbers to algorithmically verifiable arithmetical formulas that are not algorithmically computable.[**]

Introduction

“The computability precursors of the classes $P$ and $NP$ are the classes of decidable and c.e. (computably enumerable) languages, respectively. We say that a language $L$ is c.e. i.e. (or semi-decidable) iff $L = L(M)$ for some Turing machine $M$. We say that $L$ is decidable iff $L = L(M)$ for some Turing machine $M$ which satisfies the condition that $M$ halts on all input strings $w$. …

Thus the problem Satisfiability is: Given a propositional formula $F$, determine whether $F$ is satisfiable. To show that this is in $NP$ we define the polynomial-time checking relation $R(x, y)$, which holds iff $x$ codes a propositional formula $F$ and $y$ codes a truth assignment to the variables of $F$ which makes $F$ true.”

… Stephen Cook, The P versus NP Problem, 2000, Clay Mathematical Institute, Providence, USA.

In this investigation we shall argue that standard interpretations—such as the above[1]—of the formal definitions of the classes $P$ and $NP$ leave room for ambiguity.

Specifically, such interpretations fail to explicitly highlight that the assignment of satisfaction and truth values to number-theoretic formulas under an interpretation can be constructively defined in two distinctly different ways[1a]:

(a) in terms of algorithmic verifiability (Definition 1 below);

It immediately follows from this definition that a number-theoretical formula $F$ is algorithmically verifiable under an interpretation (and should therefore be defined in $NP$) if, and only if, we can define a checking relation $R(x, y)$[2]—where $x$ codes a propositional formula $F$ and $y$ codes a truth assignment to the variables of $F$—such that, for any given natural number values $(m, n)$, there is a deterministic algorithm which will finitarily decide whether or not $R(m, n)$ holds over the domain $\mathbb{N}$ of the natural numbers.

(b) in terms of algorithmic computability (Definition 2 below).

It immediately follows from this definition that a number-theoretical formula $F$ is algorithmically computable under an interpretation (and should therefore be defined in $P$) if, and only if, we can define a checking relation $R(x, y)$[3]—where $x$ codes a propositional formula $F$ and $y$ codes a truth assignment to the variables of $F$—such that there is a deterministic algorithm which, for any given natural number values $(m, n)$, will finitarily decide whether or not $R(m, n)$ holds over the domain $\mathbb{N}$ of the natural numbers.

In other words, standard interpretations of the classes $P$ and $NP$ implicitly assume without proof that:

Cook’s Thesis: If a propositional formula is algorithmically verifiable in polynomial-time, then it is algorithmically computable.

It would follow from such a perspective that the differentiation between the classes $P$ and $NP$ is only quantitative, and can therefore be adequately expressed in terms of computational complexity.

However, we shall argue that—since the two concepts are well-defined[3a]—there are classically defined arithmetic formulas which are algorithmically verifiable but not algorithmically computable; and so the differentiation between the classes $P$ and $NP$ is also qualitative, and cannot be adequately expressed in terms of only computational complexity.

The $PvNP$ Separation Problem

Accordingly, in this investigation we shall ignore both Cook’s Thesis and polynomial-time computations, and address instead the following non-standard (NS) formulation of the $PvNP$ Separation Problem in arithmetic:

Non-standard $PvNP\ (NS)$: Is there an arithmetical formula $F$ that is algorithmically verifiable but not algorithmically computable?

We shall first show how Gödel’s $\beta$-function uniquely corresponds each classically defined real number to an algorithmically verifiable arithmetical formula.

Since classical theory admits the existence of real numbers that are not algorithmically computable[4], we shall conclude that classical theory must also admit the existence of arithmetical formulas that are algorithmically verifiable but not algorithmically computable.

Algorithmically verifiable and algorithmically computable formulas

Definition 1: A number-theoretical formula $F(x)$ is algorithmically verifiable under an interpretation if, and only if, for any given natural number $n$, there is a deterministic algorithm $AL_{(F,\ n)}$ which can provide objective evidence[5] for deciding the value of each formula in the finite sequence $\{F(1), F(2), \ldots, F(n)\}$.

The concept is well-defined in the sense that, first, every atomic number-theoretical formula is algorithmically verifiable as above[6]; further, by Tarski’s definitions[7], the algorithmic verifiability of the formulas of a formal language which contain logical constants can be inductively defined under an interpretation in terms of the algorithmic verifiability of the interpretations of the atomic formulas of the language.

In particular[8], the formulas of the first order Peano Arithmetic PA are constructively decidable under the standard interpretation of PA over the domain $\mathbb{N}$ of the natural numbers if, and only if, they are algorithmically verifiable under the interpretation.[9]

Definition 2: A number theoretical formula $F(x)$ is algorithmically computable under an interpretation if, and only if, there is a deterministic algorithm $AL_{F}$ that can provide objective evidence for deciding the value of each formula in the denumerable sequence $\{F(1), F(2), \ldots\}$.

This concept too is well-defined in the sense that, first, every atomic number-theoretical formula is algorithmically computable as above[10]; further, by Tarski’s definitions, the algorithmic computability of the formulas of a formal language which contain logical constants can also be inductively defined under an interpretation in terms of the algorithmic computability of the interpretations of the atomic formulas of the language.

In particular[11], the PA-formulas are constructively decidable under an algorithmic interpretation of PA over $\mathbb{N}$ if, and only if, they are algorithmically computable under the interpretation.[12]

We note that:

Lemma 1: There are algorithmically verifiable number theoretical formulas which are not algorithmically computable.

Proof: Let $r(n)$ denote the $n^{th}$ digit in the decimal expression $\sum_{i=1}^{\infty}r(i).10^{-i}$ of a putatively given real number $\mathbb{R}$ in the interval $0 < \mathbb{R} \leq 1$.

By the definition of the real number $\mathbb{R}$ as the limit of the Cauchy sequence of rationals $\{\sum_{i=1}^{k}r(i).10^{-i}: k = 1, 2, \ldots\}$, it follows that $r(n)$ is an algorithmically verifiable number-theoretic function.

Since the set of algorithmically computable reals is countable[13], Cantor’s diagonal argument[14] shows that there are real numbers that are not algorithmically computable.

The Lemma follows. $\Box$

We note that algorithmic computability implies the existence of a deterministic 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 a deterministic algorithm that can decide the truth/falsity of each proposition in a well-defined denumerable sequence of propositions.[15]

Gödel’s $\beta$-function

We note next that Gödel’s $\beta$-function is the primitive recursive number-theoretic function defined by[16]:

Definition 3: $\beta (x_{1}, x_{2}, x_{3}) = rm(1+(x_{3}+ 1) \star x_{2}, x_{1})$, where $rm(x_{1}, x_{2})$ denotes the remainder obtained on dividing $x_{2}$ by $x_{1}$.

We also note that:

Lemma 2: For any non-terminating sequence of values $f(0), f(1), \ldots$, we can construct natural numbers $b_{k},\ c_{k}$ such that:

(i) $j_{k} = max(k, f(0), f(1), \ldots, f(k))$;

(ii) $c_{k} = j_{k}$!;

(iii) $\beta(b_{k}, c_{k}, i) = f(i)$ for $0 \leq i \leq k$.

Proof: This is a standard result[17]. $\Box$

Now we have the standard definition[18]:

Definition 4: A number-theoretic function $f(x_{1}, \ldots, x_{n})$ is said to be representable in the first order Peano Arithmetic PA if, and only if, there is a PA formula $[F(x_{1}, \dots, x_{n+1})]$[19] 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: $[(\exists_{1} x_{n+1})F(x_{1}, \ldots, x_{n}, x_{n+1})]$

We also have that:

Lemma 3: $\beta(x_{1}, x_{2}, x_{3})$ 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[20]. $\Box$

An arithmetical perspective on the $PvNP$ Separation Problem

We finally argue that:

Theorem 1: There is an arithmetical formula that is algorithmically verifiable but not algorithmically computable under any sound interpretation of PA.

Proof: Let $\{r(n)\}$ be the denumerable sequence defined by the denumerable sequence of digits in the decimal expansion $\sum_{i=1}^{\infty}r(i).10^{-i}$ of a putatively given real number $\mathbb{R}$ in the interval $0 < \mathbb{R} \leq 1$.

By Lemma 2, for any given natural number $k$, there are natural numbers $b_{k}, c_{k}$ such that, for any $1 \leq n \leq k$:

$\beta(b_{k}, c_{k}, n) = r(n)$.

By Lemma 3, $\beta(x_{1}, x_{2}, x_{3})$ is strongly represented in PA by $[Bt(x_{1}, x_{2}, x_{3}, x_{4})]$ such that, for any $1 \leq n \leq k$:

If $\beta(b_{k}, c_{k}, n) = r(n)$ then PA proves $[Bt(b_{k}, c_{k}, n, r(n))]$.

We now define the arithmetical formula $[R(b_{k}, c_{k}, n)]$ for any $1 \leq n \leq k$ by:

$[R(b_{k}, c_{k}, n) = r(n)]$ if, and only if, PA proves $[Bt(b_{k}, c_{k}, n, r(n))]$.

Hence every putatively given real number $\mathbb{R}$ in the interval $0 < \mathbb{R} \leq 1$ uniquely corresponds to an algorithmically verifiable arithmetical formula $[R(x)]$ since:

For any $k$, the primitive recursivity of $\beta(b_{k}, c_{k}, n)$ yields a deterministic algorithm AL$_{(\beta, \mathbb{R}, k)}$ that can provide objective evidence for deciding the unique value of each formula in the finite sequence $\{[R(1), R(2), \ldots, R(k)]\}$ by evidencing the truth under a sound interpretation of PA for:

$[R(1) = R(b_{k}, c_{k}, 1)]$
$[R(b_{k}, c_{k}, 1) = r(1)]$

$[R(2) = R(b_{k}, c_{k}, 2)]$
$[R(b_{k}, c_{k}, 2) = r(2)]$

$[R(k) = R(b_{k}, c_{k}, k)]$
$[R(b_{k}, c_{k}, k) = r(k)]$.

The correspondence is unique because, if $\mathbb{R}$ and $\mathbb{S}$ are two different putatively given reals in the interval $0 < \mathbb{R},\ \mathbb{S} \leq 1$, then there is always some $m$ for which:

$r(m) \neq s(m)$.

Hence the corresponding arithmetical formulas $[R(n)]$ and $[S(n)]$ are such that:

$[R(n) = r(n)]$ for all $1 \leq n \leq m$.

$[S(n) = s(n)]$ for all $1 \leq n \leq m$.

$[R(m) \neq S(m)]$.

By Lemma 1, there is an algorithmically uncomputable real number $\mathbb{R}$ such that the corresponding PA formula $[(\exists y)(R(x) = y)]$ is also algorithmically uncomputable, but algorithmically verifiable, under a sound interpretation of PA.

The theorem follows. $\Box$

An arithmetical perspective on the $PvNP$ problem

We conclude that if we were to unambiguously define the classes $P\ (NS)$ and $NP\ (NS)$ as in Section 1(a) and Section 1(b) respectively, then it would follow that:

Corollary 1: $NP \neq P\ (NS).$

References

Kl52 Stephen Cole Kleene. 1952. Introduction to Metamathematics. North Holland Publishing Company, Amsterdam.

Me64 Elliott Mendelson. 1964. Introduction to Mathematical Logic. Van Norstrand. pp.145-146.

Mo12 Cristopher Moore. 2012. P vs. NP, Phase Transitions, and Incomputability in the Wild. Presentation given on 18th June 2012 at the Turing Centenary Workshop on The Incomputable at the Kavli Royal Society International Center at Chicheley Hall, Chichely, Buckinghamshire, UK.

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.

Ta33 Alfred Tarski. 1933. The concept of truth in the languages of the deductive sciences. In Logic, Semantics, Metamathematics, papers from 1923 to 1938. (p152-378). 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.230-365; corrections, Ibid, vol 43 (1937) pp. 544-546.

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.

An13 … 2013. A suggested mathematical perspective for the $EPR$ argument. Presented on 7’th April at the workshop on Logical Quantum Structures‘ at UNILOG’2013, 4’th World Congress and School on Universal Logic, 29’th March 2013 – 7’th April 2013, Rio de Janeiro, Brazil.\

Notes

Return to *: I am not alone! See for instance this blogpost of Timothy Gower’s representing one extreme; and this plaintive appeal representing the other.

Return to **: For a broader perspective on the relation of Gödel’s $\beta$-function to this distinction, see this post.

Return to 1a: The distinction is explicitly introduced—and its significance in establishing a finitary proof of consistency for the first order Peano Arithmetic PA highlighted—in An12.

Return to 2: If $F$ is a formula of the first order Peano Arithmetic PA, the existence of such a checking relation is assured by An12, Theorem 2.

Return to 3: If $F$ is a PA formula, the existence of such a checking relation is assured by An12, Theorem 3.

Return to 3a: See here. We further note informally here how the distinction between the two concepts may have far-reaching and significant consequences not only for the foundations of mathematics, logic and computability, but also for our perspective on the underlying structure of the laws of nature.

Return to 5: “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 …”; Mu91.

Return to 7: On the inductive assignment of satisfaction and truth values to the formulas of a formal language under an interpretation; Ta33.

Return to 9: We note that the Axiom Schema of Finite Induction can be finitarily verified as true under the standard interpretation of PA with respect to ‘truth’ as defined by the algorithmically verifiable formulas of PA.

Return to 12: We note that the Axiom Schema of Finite Induction can be finitarily verified as true under the standard interpretation of PA with respect to ‘truth’ as defined by the algorithmically computable formulas of PA.

Return to 15: From the point of view of a finitary mathematical philosophy, the significant difference between the two concepts could be expressed (An13) 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.

Return to 19: We use square brackets simply as a convenience in informal argumentation to differentiate between a formal expression and its interpretation over some domain when there is a possibility of confusion between the two.

(Paper Presented on 7th April at the workshop on Logical Quantum Structures‘ at UNILOG’2013, 4th World Congress and School on Universal Logic, 29th March 2013 – 7th April 2013, Rio de Janeiro, Brazil. See also the paper Algorithmically Verifiable Logic vis a vis Algorithmically Computable Logic: Could resolving EPR need two complementary Logics? and presentation on 26th 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.)

Some disturbing features of the standard Copenhagen interpretation of Quantum Theory

Amongst the philosophically disturbing features of the standard Copenhagen interpretation of Quantum Theory[1] are its:

$\bullet$ Essential indeterminateness [2];

$\bullet$ Essential separation of the world into system’ and observer’ [3].

$\bullet$ In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen noted [4] that accepting Quantum Theory, but denying these features of the Copenhagen interpretation, logically entails accepting either that the world is non-local [5] (thus contradicting Special Relativity), or that there are hidden variables [6] that would eliminate the need for accepting these features as necesary to any sound interpretation of Quantum Theory.

$\bullet$ In 1952 David Bohm proposed [7] an alternative mathematical development of the existing Quantum Theory (essentially equivalent to it, but based on Louis de Broglie’s pilot wave theory) whose interpretation appealed to hidden variables [8] (presumably hidden natural laws that were implicitly assumed by Bohm to be representable by well-defined classically computable mathematical functions [9]) that eliminated the need for indeterminism and the separation of the world into system’ and observer’.

$\bullet$ In 1964 John Stewart Bell showed [10] that any interpretation of Quantum Theory which appeals to (presumably classically computable) hidden variables in the above sense is necessarily non-local.

However, our foundational investigations into an apparently unrelated area [11] suggest that if the above presumption—concerning an appeal by Bohm and Bell to functions that are implicitly assumed to be classically computable—is correct, then the hidden variables in the Bohm-de Broglie interpretation of Quantum Theory could as well be presumed to involve natural laws that are mathematically representable only by functions that are algorithmically verifiable but not algorithmically computable [12] —in which case the interpretation might avoid being held as appealing to non-locality’.

The underlying perspective of this thesis

The underlying perspective of this thesis is that:

(1) Classical physics assumes that all the observable laws of nature can be mathematically represented in terms of well-defined functions that are algorithmically computable (as defined in the next section).

Since the functions are well-defined, their values are pre-existing and pre-determined as mappings that are capable of being known in their infinite totalities to an omniscient intelligence, such as Laplace’s intellect [13].

(2) However, the overwhelming experimental verification of the mathematical predictions of Quantum Theory suggests that the actual behavior of the real world cannot be assumed as pre-existing and pre-determined in this sense.

In other words, the consequences of some experimental interactions are theoretically incapable of being completely known in advance even to an omniscient intelligence, such as Laplace’s intellect.

So all the observable laws of nature cannot be represented mathematically in terms of functions that are algorithmically computable (as defined in the next section).

(3) Hence, either there is no way of representing all the observable laws of nature mathematically in a deterministic model, or all the observable laws of nature can be represented mathematically in a deterministic model—but in terms of functions that are computable’ in a non-predictable sense (we define these in the next section as algorithmically verifiable functions).

(4) The Copenhagen interpretation appears to opt for the first option in (3) above, and hold that there is no way of representing all the observable laws of nature mathematically in a deterministic model.

Hence the interpretation is not overly concerned with the seemingly essential non-locality of Quantum Theory, and its conflict with the deterministic mathematical representation of the laws of Special Relativity.

(5) The Bohm-de Broglie interpretation appears to reject the first option in (3) above, and to propose a way of representing all the observable laws of nature mathematically in terms of functions that are presumably taken implicitly to be algorithmically computable.

However, the Bohm-de Broglie interpretation has not so far been viewed as being capable of mathematically representing the seemingly essential non-local feature of Quantum Theory.

Let us therefore, for the moment, consider the second option in (3) above from the perspective suggested by the Birmingham paper; i.e., that the apparently non-local feature of Quantum Theory may actually be indicative of a non-constructive and counter intuitive-to-human-intelligence’ phenomena in nature that can, however, be mathematically represented by functions that are algorithmically verifiable, but not algorithmically computable.

Algorithmic verifiabilty and algorithmic computability

We define the two concepts [14]:

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

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 $\{F(1), F(2), \ldots\}$.

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 [16], 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 [17]—and not only to a mathematically definable limit—if and only if such representation is definable by an algorithmically computable function.

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

Theorem 1: There are mathematical functions that are algorithmically verifiable but not algorithmically computable.

Proof: (a) Since any real number is mathematically definable as the limit of a Cauchy sequence of rational numbers:

Let $R(n)$ denote the $n^{th}$ digit in the decimal expression of the real number $R$ in binary notation.

Then, for any given natural number $n$, there is an algorithm $AL_{(R,\ n)}$ that can decide the truth/falsity of each proposition in the finite sequence:

$\{R(1)=0, R(2)=0, \ldots, R(n)=0\}$.

Hence, for any real number $R$, the relation $R(x)=0$ is algorithmically verifiable trivially.

(b) Since it follows from Alan Turing’s Halting argument [18] that there are algorithmically uncomputable real numbers:

Let $[R(n)]$ denote the $n^{th}$ digit in the decimal expression of an algorithmically uncomputable real number $R$ in binary notation.

By (a), the relation $[R(x)=0]$ is algorithmically verifiable trivially.

However, by definition there is no algorithm $AL_{R}$ that can decide the truth/falsity of each proposition in the denumerable sequence:

$\{[R(1)=0], [R(2)=0], \ldots\}$.

Hence the relation $[R(x)=0]$ is algorithmically verifiable but not algorithmically computable. $\Box$

Some mathematical constants are definable only by algorithmically verifiable but not algorithmically computable functions

We note that:

(i) All the mathematically defined functions known to, and used by, science are algorithmically computable, including those that define transcendental numbers such as $\pi,\ e$, etc.

They can be computed algorithmically as they are all definable as the limit of some well-defined infinite series of rationals.

(ii) The existence of mathematical constants that are defined by functions which are algorithmically verifiable but not algorithmically computable—suggested most famously by Georg Cantor’s diagonal argument—has been a philosophically debatable deduction.

Such existential deductions have been viewed with both suspicion and scepticism by scientists such as Henri Poincaré, L. E. J. Brouwer, etc., and disputed most vociferously on philosophical grounds by Ludwig Wittgenstein [19].

(iii) A constructive definition of an arithmetical Boolean function $[R(x)]$ that is true (hence algorithmically verifiable [20]) but not provable in Peano Arithmetic (hence algorithmically not computable [21]) was given by Kurt Gödel in his 1931 paper on formally undecidable arithmetical propositions [22].

It was also disputed vociferously by Wittgenstein [23].

(iv) The definition of a number-theoretic function that is algorithmically verifiable but not algorithmically computable was also given by Alan Turing in his 1936 paper on computable numbers [24].

He defined a halting function, say $H(n)$, that is $0$ if, and only if, the Turing machine with code number $n$ halts on input $n$. Such a function is mathematically well-defined, but assuming that it defines an algorithmically computable real number leads to a contradiction.

Turing thereupon concluded the mathematical existence of algorithmically uncomputable real numbers.

(v) A definition of a number-theoretic function that is algorithmically verifiable but not algorithmically computable was given by Gregory Chaitin [25].

He defined a class of constants—denoted by $\Omega$—which is such that if $C(n)$ is the $n^{th}$ digit in the decimal expression of an $\Omega$ constant, then the function $C(x)$ is algorithmically verifiable but not algorithmically computable.

Some physical constants may be representable by real numbers that are definable only by algorithmically verifiable but not algorithmically computable functions

Now, we note that:

“… the numerical values of dimensionless physical constants are independent of the units used. These constants cannot be eliminated by any choice of a system of units. Such constants include:

$\bullet\ \alpha$, the fine structure constant, the coupling constant for the electromagnetic interaction ($\approx 1/137.036$). Also the square of the electron charge, expressed in Planck units. This defines the scale of charge of elementary particles with charge.

$\bullet\ \mu$ or $\beta$, the proton-to-electron mass ratio, the rest mass of the proton divided by that of the electron ($\approx 1836.15$). More generally, the rest masses of all elementary particles relative to that of the electron.

$\bullet\ \alpha_{s}$, the coupling constant for the strong force ($\approx 1$)

$\bullet\ \alpha{G}$, the gravitational coupling constant ($\approx 10^{-38}$) which is the square of the electron mass, expressed in Planck units. This defines the scale of the mass of elementary particles.

At the present time, the values of the dimensionless physical constants cannot be calculated; they are determined only by physical measurement. This is one of the unsolved problems of physics. …

The list of fundamental dimensionless constants decreases when advances in physics show how some previously known constant can be computed in terms of others. A long-sought goal of theoretical physics is to find first principles from which all of the fundamental dimensionless constants can be calculated and compared to the measured values. A successful Theory of Everything’ would allow such a calculation, but so far, this goal has remained elusive.”

This suggests that:

Thesis 1: Some of the dimensionless physical constants are only representable in a mathematical language as real numbers that are defined by functions which are algorithmically verifiable, but not algorithmically computable.

In other words, we cannot treat such constants as denoting—even in principle—a measurable limit, as we could a constant that is representable mathematically by a real number that is definable by algorithmically computable functions.

From the point of view of mathematical philosophy, this distinction would be expressed by saying that the sequence of digits in the decimal representation of an unmeasurable’ physical constant cannot be treated in a mathematical language as a completed’ infinite sequence, whilst the corresponding sequence in the decimal representation of a measurable’ physical constant can be treated as a completed’ infinite sequence.

Zeno’s argument: The dichotomy between a continuous physical reality and its discretely representable mathematical theory

Zeno’s paradoxical arguments highlight the philosophical and theological dichotomy between our essentially continuous’ perception of the physical reality that we seek to capture with our measurements, and the essential discreteness’ of any mathematical language of Arithmetic in which we seek to express such measurements.

The distinction between algorithmic verifiability and algorithmic computability of Arithmetical functions can be seen as reflecting the dichotomy mathematically.

Classical laws of nature

For instance, the distinction suggests that classical mechanics can be held as complete with respect to the algorithmically computable representation of the physical world, in the sense that:

Thesis 2: Classical laws of nature determine the nature and behaviour of all those properties of the physical world which are mathematically describable completely at any moment of time $t(n)$ by algorithmically computable functions from a given initial state at time $t(0)$.

Neo-classical laws of nature

On the other hand, the distinction also suggests that:

Thesis 3: Neo-classical laws of nature determine the nature and behaviour of those properties of the physical world which are describable completely at any moment of time $t(n)$ by algorithmically verifiable functions; however such properties are not completely describable by algorithmically computable functions from any given initial state at time $t(0)$.

Since such behaviour [26] follows fixed laws and is determinate (even if not algorithmically predictable by classical laws), Albert Einstein could have been justified in his belief that:

“… God doesn’t play dice with the world”

and in holding that:

“I like to think that the moon is there even if I am not looking at it”.

Incompleteness: Arithmetical analogy

The distinction also suggests that neither classical mechanics nor neo-classical quantum mechanics can be described as mathematically complete’ with respect to the algorithmically verifiable behaviour of the physical world.

The analogy here is that Gödel showed in 1931 [27] that any formal arithmetic is not mathematically complete with respect to the algorithmically verifiable nature and behaviour of the natural numbers [28].

However it can be argued that the first-order Peano Arithmetic PA is complete [29] with respect to the algorithmically computable nature and behaviour of the natural numbers.

In this sense, the $EPR$ paper may not be entirely wrong in holding that:

“We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.”

Conjugate properties

The above also suggests that:

Thesis 4: The nature and behaviour of two conjugate properties $F_{1}$ and $F_{2}$ of a particle $P$ that are determined by neo-classical laws are described mathematically at any time $t(n)$ by two algorithmically verifiable, but not algorithmically computable, functions $f_{1}$ and $f_{2}$.

In other words, it is the very essence of the neo-classical laws determining the nature and behaviour of the particle that—at any time $t(n)$—we can only determine either $f_{1}(n)$ or $f_{2}(n)$, but not both.

Hence measuring either one makes the other indeterminate as we cannot go back in time.

However, this does not contradict the assumption that any property of an object must obey some deterministic natural law for any possible measurement that is made at any time.

Entangled particles

The above similarly suggests that:

Thesis 5: The nature and behaviour of an entangled property of two particles $P$ and $Q$ is determined by neo-classical laws and are describable mathematically at any time $t(n)$ by two algorithmically verifiable—but not algorithmically computable—functions $f_{1}$ and $g_{1}$.

In other words, it is the very essence of the neo-classical laws determining the nature and behaviour of the entangled properties of two particles that—at any time $t(n)$—determining one immediately gives the state of the other without measurement since the properties are entangled.

Again, this does not contradict the assumption that any property of an object must obey some deterministic natural law for any possible measurement that is made at any time.

Nor does it require any information to travel from one particle to another consequent to a measurement.[29a]

Schrödinger’s cat

If $[F(x)]$ is an algorithmically verifiable but not algorithmically computable Boolean function, we can take the query:

Is $F(n)=0$ for all natural numbers?

as corresponding mathematically to the Schrödinger question:

Is the cat dead or alive at any given time $t$?

We can then argue that there is no mathematical paradox involved in Schrödinger‘s assertion that the cat is both dead and alive, if we take this to mean that:

I may either assume the cat to be alive until a given time $t$ (in the future), or assume the cat to be dead until the time $t$, without arriving at any logical contradiction in my existing Quantum description of nature.

In other words:

Once we accept Quantum Theory as a valid description of nature, then there is no paradox in stating that the theory essentially cannot predict the state of the cat at any moment of future time (so the inability to predict does not arise out of a lack of sufficient information about the laws of the system that Quantum theory is describing, but stems from the very nature of these laws).

The mathematical analogy for the above would be:

Once we accept that Peano Arithmetic is consistent [30] and categorical [31] then we cannot deduce from the axioms of PA whether $F(n)=0$ for all natural numbers, or whether $F(n)=1$ for some natural number.

Conclusion
To sum up, we suggest that the paradoxical element in the $EPR$ argument may disappear if we could argue that:

(i) All properties of physical reality can be deterministic—in the sense that any physical property can have one, and only one, value at any time $t(n)$—where the value is completely determined by some natural law which need not, however, be algorithmic.

(ii) There are elements of such a physical reality whose properties at any time $t(n)$ are determined completely in terms of their putative properties at some earlier time $t(0)$.

Such properties are predictable mathematically since they are representable mathematically by algorithmically computable functions.

The Laws of Classical Mechanics describe the nature and behaviour of such physical reality only.

(iii) There can be elements of such a physical reality whose properties at any time $t(n)$ cannot be theoretically determined completely from their putative properties at some earlier time $t(0)$.

Such properties are unpredictable mathematically since they are only representable mathematically by algorithmically verifiable, but not algorithmically computable, functions.

The Laws of Quantum Mechanics describe the nature and behaviour of such physical reality.

The need for constructive mathematical foundations

The following perspective [32] emphasises the need for a universally common, constructive, foundation for the mathematical representation of elements of reality such as those considered above:

“Our investigations lead us to consider the possibilities for reuniting the antipodes’. The antipodes being classical mathematics (CLASS) and intuitionism (INT).

… It therefore seems worthwhile to explore the formal’ common ground of classical and intuitionistic mathematics. If systematically developed, many intuitionistic results would be seen to hold classically as well, and thus offer a way to develop a strong constructive theory which is still consistent with the rest of classical mathematics.

Such a constructive theory can form a conceptual framework for applied mathematics and information technology.

These sciences now use an ad-hoc approach to reality since the classical framework is inadequate. … and can easily use the richness of ideas already present in classical mathematics, if classical mathematics were to be systematically developed along the common grounds before the unconstructive elements are brought in.”

Frank Waaldijk [33]

“… we propose that Laplacian determinism be seen in the light of constructive mathematics and Church’s Thesis.

This means amongst other things that infinite sequences (of natural numbers; a real number is then given by such an infinite sequence) are never finished’, instead we see them developing in the course of time.

Now a very consequent, therefore elegant interpretation of Laplacian determinism runs as follows.

Suppose that there is in the real world a developing-infinite sequence of natural numbers, say $\alpha$. Then how to interpret the statement that this sequence is uniquely determined’ by the state of the world at time zero?

At time zero we can have at most finite information since, according to our constructive viewpoint, infinity is never attained. So this finite information about $\alpha$ supposedly enables us to uniquely determine’ $\alpha$ in its course of time.

It is now hard to see another interpretation of this last statement, than the one given by Church’s Thesis, namely that this finite information must be a (Turing-)algorithm that we can use to compute $\alpha (n)$ for any $n \in \mathbb(N)$.

With classical logic and omniscience, the previous can be stated thus:

for every (potentially infinite) sequence of numbers $(a_{n})_{n \in \mathbb{N}}$ taken from reality there is a recursive algorithm $\alpha$ such that $\alpha (n) = a_{n}$ for each $n \in \mathbb{N}$. This statement is sometimes denoted as CT$_{phys}$‘,

… this classical omniscient interpretation is easily seen to fail in real life.

Therefore we adopt the constructive viewpoint.

The statement the real world is deterministic’ can then best be interpreted as:

a (potentially infinite) sequence of numbers $(a_{n})_{n \in \mathbb{N}}$ taken from reality cannot be apart from every recursive algorithm $\alpha$ (in symbols: $\neg \forall \alpha \in \sigma_{\omega REC} \exists n \in \mathbb{N}\ [\alpha(n) \neq a_{n}]$)’.”

Frank Waaldijk [34]

References

Be64 John Stewart Bell. 1964. On the Einstein Podolsky Rosen Paradox Physics Vol 1, No. 3, pp.195-200, 1964. Reprinted in J. S. Bell, Speakable and unspeakable in quantum mechanics, Cambridge, 2004, p. 14-21.

Bo52 David Bohm. 1952. A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables’, I. Physical Review 85 n.2, pp.166-179 (1952) and A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables’, II. Physical Review 85 n.2, pp.180-193 (1952).

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

Ch82 Gregory J. Chaitin. 1982. Gödel’s Theorem and Information. International Journal of Theoretical Physics 22 (1982), pp. 941-954.

EPR35 Albert Einstein, Boris Yakovlevich Podolsky, Nathan Rosen. 1935. Can Quantum-Mechanical Description of Physical Reality be Considered Complete? InPhysical Review PHYS REV X – vol. 47, no. 10, pp.777-780. Bibcode 1935PhRv\ldots 47…777E. doi:10.1103/PhysRev.47.777.

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. pp.5-38.

Kl52 Stephen Cole Kleene. 1952. Introduction to Metamathematics. North Holland Publishing Company, Amsterdam.

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.

Sc35 Erwin Schrödinger. Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaftern. 23: pp. 807-812; 823-823, 844-849. (1935). English translation: John D. Trimmer, The present situation in Quantum Mechanics, Proceedings of the American Philosophical Society 124, pp. 323-38 (1980), reprinted in Quantum Theory and Measurement, p. 152 (1983).

Sh11 Sheldon Goldstein et al. 2011. Bell’s Theorem, Scholarpedia, 6(10):8378, doi:10.4249/scholarpedia.8378.

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.

Wa03 Frank Waaldijk. 2003. On the foundations of constructive mathematics. Web paper.

Wi78 Ludwig Wittgenstein. 1937. Remarks on the Foundations of Mathematics. 1978 ed., MIT Press, Cambridge.

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.

An12a Bhupinder Singh Anand. 2012. Some consequences of interpreting the associated logic of the first-order Peano Arithmetic PA finitarily. Draft.

An13 … 2013. A suggested mathematical perspective for the EPR argument. Paper Presented on 7th April at the workshop on Logical Quantum Structures‘ at UNILOG’2013, 4th World Congress and School on Universal Logic, 29th March 2013 – 7th April 2013, Rio de Janeiro, Brazil.

Notes

Return to 1: “Because it consists of the views developed by a number of scientists and philosophers during the second quarter of the 20th Century, there is no definitive statement of the Copenhagen interpretation”, Wikipedia; cf., Copenhagen Interpretation of Quantum Mechanics, Stanford Encyclopedia of Philosophy; also The Copenhagen Interpretation of Quantum Mechanics by Ben Best.

Return to 2: “It is a general principle of orthodox formulations of quantum theory that measurements of physical quantities do not simply reveal pre-existing or pre-determined values, the way they do in classical theories. Instead, the particular outcome of the measurement somehow “emerges” from the dynamical interaction of the system being measured with the measuring device, so that even someone who was omniscient about the states of the system and device prior to the interaction couldn’t have predicted in advance which outcome would be realized.” … Sh11.

Return to 2: Highlighted famously by Erwin Schrödinger’s caustic observation on the dubious condition of his Platonic pet:

“One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The $\psi$-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.” … Sc35, §5.

Return to 5: “Non-local’ … means that there exist interactions between events that are too far apart in space and too close together in time for the events to be connected even by signals moving at the speed of light.” … Sh11.

Return to 6: “Traditionally, the phrase hidden variables’ is used to characterize any elements supplementing the wave function of orthodox quantum theory.” … Sh11.

Return to 8: “This terminology is, however, particularly unfortunate in the case of the de Broglie-Bohm theory, where it is in the supplementary variables—definite particle positions—that one finds an image of the manifest world of ordinary experience.” … Sh11.

Return to 9: Which could be considered as having pre-existing or pre-determined mathematical values over the domain over which the functions are well-defined.

Return to 11: Concerning evidence-based and finitary interpretations of the first order Peano Arithmetic PA, in An12.

Return to 13: “We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all the forces that animate Nature and the mutual positions of the beings that comprise it, if this intellect were vast enough to submit its data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom: for such an intellect nothing could be uncertain; and the future just like the past would be present before its eyes.” … Pierre Simon Laplace, A Philosophical Essay on Probabilities.

Return to 15: 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 …” … Mu91.

Return to 16: We note that the concept of algorithmic computability’ is essentially an expression of the more rigorously defined concept of realizability’ in Kl52, p.503.

Return to 17: In the sense of a physically completable’ infinite sequence (as needed to resolve Zeno’s paradox).

Return to 26: A putative model for such behaviour is given in Wa03, §1.5, p.5:

“The second way to model our real world is to assume that it is deterministic. …

It would be worthwhile to explore the consequences of a deterministic world with incomplete information (since under the assumption of determinancy in the author’s eyes this comes closest to real life).

That is a world in which each infinite sequence is given by an algorithm, which in most cases is completely unknown.

We can model such a world by introducing two players, where player I picks algorithms and hands out the computed values of these algorithms to player II, one at a time.

Sometimes player I discloses (partial) information about the algorithms themselves.

Player II can of course construct her or his own algorithms, but still is confronted with recursive elements of player I about which she/he has incomplete information.”

Return to 28: Which—as shown in An12—is the behaviour sought to be captured by the Standard interpretation of the first order Peano Arithmetic PA.

Return to 29a: For a model of widely separated outputs that are correlated but don’t allow communication, which demonstrates that non-locality doesn’t imply communication, see this paper Wim van Dam: Implausible Consequences of Superstrong Nonlocality and this explanatory blog A Neighborhood of Infinity: Distributed computing with alien technology.

Return to 31: An12a establishes the `categoricity’—which means that any two models of the Arithmetic are isomorphic.

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Critical Perspectives on U.S. Foreign Policy

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Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao

Quanta Magazine

Reviewing classical interpretations of Cantor's, Gödel's, Tarski's, and Turing's reasoning and addressing some grey areas in the foundations of mathematics, logic and computability

The Brains Blog

Since 2005, a leading forum for work in the philosophy and science of mind

Logic Matters

Reviewing classical interpretations of Cantor's, Gödel's, Tarski's, and Turing's reasoning and addressing some grey areas in the foundations of mathematics, logic and computability

A Neighborhood of Infinity

Reviewing classical interpretations of Cantor's, Gödel's, Tarski's, and Turing's reasoning and addressing some grey areas in the foundations of mathematics, logic and computability

Combinatorics and more

Gil Kalai's blog

Mathematics and Computation

Reviewing classical interpretations of Cantor's, Gödel's, Tarski's, and Turing's reasoning and addressing some grey areas in the foundations of mathematics, logic and computability

Foundations of Mathematics, Logic & Computability

Reviewing classical interpretations of Cantor's, Gödel's, Tarski's, and Turing's reasoning and addressing some grey areas in the foundations of mathematics, logic and computability

John D. Cook

Reviewing classical interpretations of Cantor's, Gödel's, Tarski's, and Turing's reasoning and addressing some grey areas in the foundations of mathematics, logic and computability

Shtetl-Optimized

Reviewing classical interpretations of Cantor's, Gödel's, Tarski's, and Turing's reasoning and addressing some grey areas in the foundations of mathematics, logic and computability

Nanoexplanations

the blog of Aaron Sterling

Eric Cavalcanti

Quantum physicist

East Asia Forum

Reviewing classical interpretations of Cantor's, Gödel's, Tarski's, and Turing's reasoning and addressing some grey areas in the foundations of mathematics, logic and computability

Tanya Khovanova's Math Blog

Reviewing classical interpretations of Cantor's, Gödel's, Tarski's, and Turing's reasoning and addressing some grey areas in the foundations of mathematics, logic and computability

The polymath blog

Massively collaborative mathematical projects

Gowers's Weblog

Mathematics related discussions