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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.)

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

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

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

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

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

3. The significance of such accountability for mathematics

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

(i) Algorithmic verifiability

A number-theoretical relation $F(x)$ is algorithmically verifiable if, and only if, for any given natural number $n$, there is an algorithm $AL_{(F,\ n)}$ which can provide objective evidence for deciding the truth/falsity of each proposition in the finite sequence $\{F(1), F(2), \ldots, F(n)\}$.

(ii) Algorithmic computability

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

(iii) Algorithmic verifiability vis à vis algorithmic computability

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

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

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

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

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

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

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

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

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

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

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

By considering the properties of Gödel’s $\beta$ function (see $\S$4.1 on p.8 of this preprint)—which allows us to strongly represent any non-terminating sequence of natural numbers by an arithmetical function—it would follow that, since any projection of the future values of a quantum-phenomena-associated, algorithmically verifiable, function is consistent with an infinity of algorithmically computable functions, all of whose past values are identical to the algorithmically verifiable past values of the function, the phenomena itself would be essentially unpredicatable if it cannot be represented by an algorithmically computable function.

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

7. Such representation does not need to admit multiverses

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

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

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

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

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

10. Two prime probabilities

We consider the two probabilities:

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

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

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

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

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

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

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

12. The prime divisors of a natural number are independent

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

Why Integer Factorising cannot be polynomial time

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

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

13. The distribution of primes is a quantum phenomena

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

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

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

14. Why the universe may be algorithmically computable

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

Author’s working archives & abstracts of investigations

(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

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

A new proof?

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

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

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

More concretely:

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

The proof appeals to properties of transfinite ordinals

My immediate reservation—after a brief glance at the formal definitions in $\S$1.6 on p.6 of the Yokoyama-Patey paper—was that the domain of the structure in which the formal result is proved necessarily contains at least Cantor’s smallest transfinite ordinal $\omega$, whereas the result is apparently sought to be ‘finitistically reducible’ (as considered by Stephen G. Simpson in an absorbing survey of Partial Realizations of Hilbert’s Program), in the sense of being not only finitarily provable, but interpretable in, and applicable to, finite structures (such as that of the natural numbers) whose domains may not contain (nor, in some cases, even admit—see Theorem 1 in $\S$4.1 of this post) an infinite ‘number’.

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

Why we cannot ignore Skolem’s cautionary remarks

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

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

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

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

A remarkable exposition of Ramsey’s Theorem

The Yokoyama-Patey proof invites other reservations too.

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

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

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

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

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

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

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

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

Implicit assumptions in Yokoyama-Patey’s argument

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Author’s working archives & abstracts of investigations

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

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

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

The question arises:

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

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

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

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

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

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

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

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

(i) is finitarily consistent; but

(ii) is not $\omega$-consistent; and

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

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

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

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

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

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

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

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

A distinction that is reflected in Stephen G. Simpson’s more conservative perspective in Partial Realizations of Hilbert’s Program ($\S$6.4, p.15):

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

Reason:

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

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

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

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

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

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

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

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

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

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

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

Author’s working archives & abstracts of investigations

There is indeed cause for disquiet

Jan Pedersen introduces hmself in his LinkedIn profile as a software developer and an artificial intelligence thinker with a passion for philosophy.

So, from a lay person’s perspective, I can empathise with the sense of professional disquietitude that perhaps motivated him recently to invite the following discussion (alive and kicking as of this moment) in the LinkedIn group History and Philosophy of Science’:

“Does mathematics need a new foundation?

Reading George Lakoff’s & Rafael Nunez excellent book Where Mathematics comes from’ leaves me with that impression.

The view that mathematics deals with Platonic like objects in some higher realm is too naïve and does not go well along with several paradoxes and meta-mathematical theories.”

Jan is not alone in his desire to ground his lifes work upon a firmer mathematical, logical and philosophical foundation.

Melvyn B. Nathanson

For instance, the point was forcibly made some time back by Melvyn B. Nathanson that:

“ … many great and important theorems don’t actually have proofs. They have sketches of proofs, outlines of arguments, hints and intuitions that were obvious to the author (at least, at the time of writing) and that, hopefully, are understood and believed by some part of the mathematical community.

But the community itself is tiny. In most fields of mathematics there are few experts. Indeed, there are very few active research mathematicians in the world, and many important problems, so the ratio of the number of mathematicians to the number of problems is small. In every field, there are “bosses” who proclaim the correctness or incorrectness of a new result, and its importance or unimportance.

Sometimes they disagree, like gang leaders fighting over turf. In any case, there is a web of semi-proved theorems throughout mathematics. Our knowledge of the truth of a theorem depends on the correctness of its proof and on the correctness of all of the theorems used in its proof. It is a shaky foundation.”

… “Desperately Seeking Mathematical Truth,“, Opinion piece in the August 2008 Notices of the American Mathematical Society, Vol. 55, Issue 7.

Elliott Mendelson

A similar concern was reflected earlier in another context by Elliott Mendelson:

“Here is the main conclusion I wish to draw: it is completely unwarranted to say that CT is unprovable just because it states an equivalence between a vague, imprecise notion (effectively computable function) and a precise mathematical notion (partial-recursive function).

… The concepts and assumptions that support the notion of partial-recursive function are, in an essential way, no less vague and imprecise than the notion of effectively computable function; the former are just more familiar and are part of a respectable theory with connections to other parts of logic and mathematics. (The notion of effectively computable function could have been incorporated into an axiomatic presentation of classical mathematics, but the acceptance of CT made this unnecessary.) … Functions are defined in terms of sets, but the concept of set is no clearer than that of function and a foundation of mathematics can be based on a theory using function as primitive notion instead of set. Tarski’s definition of truth is formulated in set-theoretic terms, but the notion of set is no clearer, than that of truth. The model-theoretic definition of logical validity is based ultimately on set theory, the foundations of which are no clearer than our intuitive understanding of logical validity.

… The notion of Turing-computable function is no clearer than, nor more mathematically useful (foundationally speaking) than, the notion of an effectively computable function …”

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

Albert G. Dragalin

Another perspective towards the inadequacy of current foundational assumptions is that of Albert G. Dragalin, for whom the central question is:

“… what is the correspondence between real facts and these counterparts in mathematics? This is very deep and difficult philosophical problem, we are very far from solving it completely …

We note three sorts of such difficulties:

1) The impredicativity, i.e. the circular character of definitions for our objects of investigation. This impredicativity is characteristic for the set theoretic theories, it closely concerns using the abstraction of alienation, when a person formulates some thoughts and then uses the thoughts as the subject of investigation, as a separate thing to be considered. In the extreme form this abstraction leads to paradoxes like the famous Liar’s paradox. But even in the more modest fortm of the type theory comprehension axiom we get somje unsurmountable difficulties.

Strictly speaking, even the notion of natural number contains an impredicativity (in the form of an unrestricted arithmetical induction principle) and there were some attempts to construct a predicative arithmetic, but most of mathematicians adopt the thesis that the natural numbers can be considered as given for granted entities (the so-called Kroneker’s thesis).

2) The lack of constructivity. It means that usual theories use the classical logic and, hence, some non-constructive proofs. In combination with the set theory it leads to complicated abstract constructs which have no computational or even no nominalistic ontological sense.

3) The lack of feasibility. It means that even in our constructive proofs we usually do not distinct feasible and not feasible calculations. So our (even ‘practically looking’) results are often principally not feasible, and so they are beyond human ability.”

Geoffrey Hellman and John L. Bell

From a slightly different perspective, Geoffrey Hellman and John L. Bell note that:

“Contrary to the popular (mis)conception of mathematics as a cut-and dried body of universally agreed upon truths and methods, as soon as one examines the foundations of mathematics, one encounters divergences of viewpoint and failures of communication that can easily remind one of religious, schismatic controversy. While there is indeed universal agreement on a substantial body of mathematical results, and while classical methods overwhelmingly dominate actual practice, as soon as one asks questions concerning fundamentals, such as “What is mathematics about?”, “What makes mathematical truths true?”, “What axioms can we accept as unproblematic?”, and notoriously, even “What are the acceptable logical rules by which mathematical proofs can proceed?”, we ﬁnd we have entered a mine-ﬁeld of contentiousness.”

Pluralism and the Foundations of Mathematics. In Itekellersetal:Sp (2006).

William P. Thurston

A similar disquiet is echoed by William P. Thurston:

On the most fundamental level, the foundations of mathematics are much shakier than the mathematics that we do. Most mathematicians adhere to foundational principles that are known to be polite ﬁctions. For example, it is a theorem that there does not exist any way to ever actually construct or even deﬁne a well-ordering of the real numbers. There is considerable evidence (but no proof) that we can get away with these polite ﬁctions without being caught out, but that doesn’t make them right. Set theorists construct many alternate and mutually contradictory “mathematical universes” such that if one is consistent, the others are too. This leaves very little conﬁdence that one or the other is the right choice or the natural choice.

On Proof And Progress In Mathematics. In the Bulletin of the American Mathematical Society. Volume 30, Number 2, April 1994, Pages 161-177.

W. T. Gowers

Whilst introducing this blog I noted two troubling philosophical issues in the foundations of mathematics that seemed to need addressing if mathematicians, logicians and computationalists were to build upon their fundamental assumptions with a reasonably comfortable sense of security.

These were the ones identified and highlighted by W. T. Gowers:

“If you ask a philosopher what the main problems are in the philosophy of mathematics, then the following two are likely to come up: what is the status of mathematical truth, and what is the nature of mathematical objects? That is, what gives mathematical statements their aura of infallibility, and what on earth are these statements about?”

… “Does mathematics need a philosophy?“, presented before the Cambridge University Society for the Philosophy of Mathematics and Mathematical Sciences, 2002.

Gila Sher

In a recent thought-provoking survey of the foundational problem of logic, Gila Sher obliquely refers to the problem of satisfactorily defining both mathematical and logical ‘truth’, whilst concluding that the issue is essentially one of inadequate methodology:

“It is an interesting fact that, with a small number of exceptions, a systematic philosophical foundation for logic, a foundation for logic rather than for mathematics or language, has rarely been attempted. …

By a philosophical foundation for logic I mean in this paper a substantive philosophical theory that critically examines and explains the basic features of logic, the tasks logic performs in our theoretical and practical life, the veridicality of logic—including the source of the truth and falsehood of both logical and meta-logical claims, the grounds on which logical theories should be accepted (rejected, or revised), the ways logical theories are constrained and enabled by the mind and the world, the relations between logic and related theories (e.g., mathematics), the source of the normativity of logic, and so on. The list is in principle open-ended since new interests and concerns may be raised by different persons and communities at present and in the future. In addition, the investigation itself is likely to raise new questions (whether logic is similar to other disciplines in requiring a grounding in reality, what the distinctive characteristics of logical operators are, etc.).

A foundational theory of this kind … should provide us with tools for criticizing, justifying, evaluating, constructing, and improving specific theories. These elements—critical examination, veridical justification, epistemic evaluation, and so on, as well as creating theoretical tools for these tasks—are the main elements of what I call “grounding” in this paper. …

Logic, however, is a very broad discipline, and the present investigation does not purport to apply to all its branches. Instead, it focuses on that branch which in our time is often referred to as “mathematical logic” and in earlier times took the forms of syllogistic logic, Fregean logic, and type-theoretic logic. And even here it is largely concerned with finitistic versions of this branch. These and other self-imposed restrictions will enable us to be more specific on the questions we address in this paper and will give us the space to discuss several issues of interest to mathematical as well as philosophical logicians. These include, for example, Feferman’s criticism of what he called the Tarski–Sher thesis (Feferman [1999], [2010]), the relation between logic and mathematics, the possibility of extending structuralism from mathematics to logic, and topics of relevance to model-theoretic logic.

I have said that systematic attempts to construct a philosophical foundation for logic have been rare. But was not the period between the late 19th-century and the early 20th-century a period of “foundational studies in logic and mathematics”, indeed a period of extraordinary growth and remarkable breakthroughs in this area? The answer is “Yes” with a caveat.

Yes, there were foundational investigations and groundbreaking developments, but for the most part they aimed at a foundation for mathematics, with logic playing a mostly instrumental, if crucial, role. Frege, for example, developed a logical system that would provide a foundation for mathematics, but aside from a few hints, did not attempt to provide a systematic philosophical foundation for logic itself. Russell improved and further developed Frege’s logicism, but although he appreciated the need to provide a systematic philosophical explanation of logic itself—one that would answer such questions as: “In virtue of what are logical propositions true?” he despaired of accomplishing this task. Thus he says:

The fundamental characteristic of logic, obviously, is that which is indicated when we say that logical propositions are true in virtue of their form. . . . I confess, however, that I am unable to give any clear account of what is meant by saying that a proposition is “true in virtue of its form”.

… Bertrand Russell, The Principles of Mathematics, In his introduction to the second edition, 1938, p.xii, Norton, New York.

Indeed, many of the momentous discoveries in meta-logic (by Hilbert, Gödel, Turing, and others) are commonly designated as contributions to “meta-mathematics”. These epochal achievements, however, are not irrelevant to the foundational problemof logic. On the contrary, by giving rise to a sophisticated logical framework and establishing its mathematical properties they created a fertile ground for a theoretical foundation for logic. It is all the more surprising, therefore, that few 20th- and 21st-century philosophers have taken up the challenge. Many have believed that a substantive, theoretical foundation for logic is impossible, some have considered it superfluous, quite a few have been content to simply say that logic is obvious, others have viewed logic as conventional, hence not in need of a foundation, and so on.

This tendency to avoid a philosophical engagement with the foundational problem of logic is not limited to the recent past. We can see it in the great philosophical systems of the 17th and 18th century. Take Kant, for example. Without purporting to offer scholarly exegesis of Kant’s philosophy of logic, we may note that Kant’s approach to logic is quite different from his approach to other disciplines. While Kant set out to provide a foundation for human knowledge in its entirety, he took formal logic largely as given. Logic, Kant emphasizes, has not required a major revision since Aristotle, and although there is room for clarifications and adjustments, there is no need for establishing the “certainty” of logic:

That logic has already, from the earliest times, proceeded upon this sure path is evidenced by the fact that since Aristotle it has not required to retrace a single step, unless, indeed, we care to count as improvements the removal of certain needless subtleties or the clearer exposition of its recognised teaching, features which concern the elegance rather than the certainty of the science.

… Immanuel Kant, Critique of Pure Reason,, 1781/7, Bviii, Macmillan, London, translated by N. Kemp Smith, 1929.

The scarcity of attempts to provide a theoretical foundation for logic is especially notable in light of epistemologists’ recognition that logic has a special standing in knowledge. Compare logic and physics, for example. It is quite common to say that physics is bound by the laws of logic but logic is not bound by the laws of physics, that a serious error in logic might undermine our physical theory, but a serious error in physics would not undermine our logical theory. And it stands to reason that the more general, basic and normative a given field of knowledge is, the more important it is to provide it with a foundation. Nevertheless a theoretical foundation for logic, and in particular a non-trivializing foundation, has rarely been attempted. Why?

Clearly, the failure to attempt such a foundation is not due to neglect, oversight, or intellectual limitations. The extraordinary advances in logic and meta-logic on the one hand, and the wealth of attempts to construct a philosophical foundation for mathematics and science on the other, suggest that neither neglect nor intellectual handicaps are the problem. In my view, the source of the problem is methodological. Certain features of the customary foundational methodology make it very problematic to construct a philosophical foundation for logic, and the first step in confronting the foundational problem of logic is, therefore, dealing with the methodological difficulty.”

… “The Foundational Problem of Logic“, Submitted draft of the paper in The Bulletin of Symbolic Logic, Volume 19, Issue 2 (2013), pp.145-198.

What mathematics does need is a greater awareness of the nature of its foundations

So where do we go from here?

In the following pages we shall argue that what mathematics needs is not a new foundation, but a greater awareness of the nature of its existing foundations.

Author’s working archives & abstracts of investigations

Hilbert’s interpretation of quantification

Hilbert interpreted quantification in terms of his $\varepsilon$-function as follows [1]:

“IV. The logical $\varepsilon$-axiom

13. $A(a) \rightarrow A(\varepsilon (A))$

Here $\varepsilon (A)$ stands for an object of which the proposition $A(a)$ certainly holds if it holds of any object at all; let us call $\varepsilon$ the logical $\varepsilon$-function.

1. By means of $\varepsilon$, “all” and “there exists” can be defined, namely, as follows:

(i) $(\forall a) A(a) \leftrightarrow A(\varepsilon(\neg A))$

(ii) $(\exists a) A(a) \leftrightarrow A(\varepsilon(A)) \ldots$

On the basis of this definition the $\varepsilon$-axiom IV(13) yields the logical relations that hold for the universal and the existential quantifier, such as:

$(\forall a) A(a) \rightarrow A(b)$ … (Aristotle’s dictum),

and:

$\neg((\forall a) A(a)) \rightarrow (\exists a)(\neg A(a))$ … (principle of excluded middle).”

Thus, Hilbert’s interpretation of universal quantification— defined in (i)—is that the sentence $(\forall x)F(x)$ holds (under a consistent interpretation $\mathcal{I}$) if, and only if, $F(a)$ holds whenever $\neg F(a)$ holds for any given $a$ (in $\mathcal{I}$); hence $\neg F(a)$ does not hold for any $a$ (since $I$ is consistent), and so $F(a)$ holds for any given $a$ (in $\mathcal{I}$).

Further, Hilbert’s interpretation of existential quantification—defined in (ii)—is that $(\exists x)F(x)$ holds (in $\mathcal{I}$) if, and only if, $F(a)$ holds for some $a$ (in $\mathcal{I}$).

Brouwer’s objection to such an unqualified interpretation of the existential quantifier was that, for the interpretation to be considered sound when the domain of the quantifiers under an interpretation is infinite, the decidability of the quantification under the interpretation must be constructively verifiable in some intuitively and mathematically acceptable sense of the term constructive’ [2].

Two questions arise:

(a) Is Brouwer’s objection relevant today?

(b) If so, can we interpret quantification constructively’?

The standard interpretation of PA

We consider the structure $[\mathbb{N}]$, defined as:

$\mathbb{N}$ … the set of natural numbers;

$=$ … equality;

$S$ … the successor function;

$+$ … the addition function;

$\ast$ … the product function;

$0$ … the null element,

that serves for a definition of today’s standard interpretation, say $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$, of the first-order Peano Arithmetic PA.

Now, if $[(\forall x)F(x)]$ and $[(\exists x)F(x)]$ are PA-formulas, and the relation $F(x)$ is the interpretation in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ of the PA-formula $[F(x)]$ then, in current literature:

(1a) $[(\forall x)F(x)]$ is defined true in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ if, and only if, for any given natural number $n$, the sentence $F(n)$ holds in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$;

(1b) $[(\exists x)F(x)]$ is an abbreviation of $[\neg (\forall x)\neg F(x)]$, and is defined as true in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ if, and only if, it is not the case that, for any given natural number $n$, the sentence $\neg F(n)$ holds in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$;

(1c) $F(n)$ holds in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ for some natural number $n$ if, and only if, it is not the case that, for any given natural number $n$, the sentence $\neg F(n)$ holds in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$.

Since (1a), (1b) and (1c) together interpret $[(\forall x)F(x)]$ and $[(\exists x)F(x)]$ in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ as intended by Hilbert’s $\varepsilon$-function, they attract Brouwer’s objection.

A finitary model of PA

Clearly, the specific target of Brouwer’s objection is (1c), which appeals to Platonically non-constructive, rather than intuitively constructive, plausibility.

We can thus re-phrase question (b) more specifically: Can we define an interpretation of PA over $[\mathbb{N}]$ that does not appeal to (1c)?

Now, it follows from Turing’s seminal 1936 paper on computable numbers that every quantifier-free arithmetical function (or relation, when interpreted as a Boolean function) $F$ defines a Turing-machine $TM_{F}$ [3].

We can thus define another interpretation $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ over the structure $[\mathbb{N}]$ [4] where:

(2a) $[(\forall x)F(x)]$ is defined as true in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ if, and only if, the Turing-machine $TM_{F}$ evidences [5] the assertion denoted by $F(n)$ as always true (i.e., as true for any given natural number $n$) in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$;

(2b) $[(\exists x)F(x)]$ is an abbreviation of $[\neg (\forall x)\neg F(x)]$, and is defined as true in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ if, and only if, it is not the case that the Turing-machine $TM_{F}$ evidences the assertion denoted by $F(n)$ as always false (i.e., as false for any given natural number $n$) in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$.

We note that $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ is a finitary model of PA since—when interpreted suitably [6]—all theorems of first-order PA are constructively true in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$.

Are both interpretations of PA over the structure $[\mathbb{N}]$ sound?

The structure $[\mathbb{N}]$ can thus be used to define both the standard interpretation $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ and a finitary model $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ for PA.

However, in the finitary model, from the PA-provability of $[\neg (\forall x)F(x)]$, we may only conclude that $TM_{F}$ does not algorithmically compute $F(n)$ as always true in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$.

We may not conclude further that $TM_{F}$ must algorithmically compute $F(n)$ as false in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ for some natural number $n$, since $F(x)$ may be a Halting-type of function that is not algorithmically computable [7].

In other words, we may not conclude from the PA-provability of $[\neg (\forall x)F(x)]$ that $F(n)$ does not hold in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ for some natural number $n$.

The question arises: Are both the interpretations $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ and $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ of PA over the structure $[\mathbb{N}]$ sound?

PA is $\omega$inconsistent

Now, Gödel has shown [8] how to construct an arithmetical formula $[R(x)]$ such that, if PA is assumed simply consistent, then $[R(n)]$ is PA-provable for any given numeral $[n]$, but $[(\forall x)R(x)]$ is not PA-provable.

He further showed [9] it would follow that if PA is additionally assumed to be $\omega$-consistent, then $[\neg (\forall x)R(x)]$ too is not PA-provable.

Now, Gödel also defined [10] a formal language $P$ as $\omega$-consistent if, and only if, there is no $P$-formula $[F(x)]$ for which:

(i) $[\neg (\forall x)F(x)]$ is $P$-provable,

and:

(ii) $[F(n)]$ is $P$-provable for any given numeral $[n]$ of $P$.

However, a significant consequence of the finitary proof of the consistency of PA in this paper on evidence-based interpretations of PA [11] is that the formula $[\neg (\forall x)R(x)]$ is PA-provable, and so PA is $\omega$inconsistent!

The interpretation $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ of PA over the structure $[\mathbb{N}]$ is not sound

Now, $R(n)$ holds for any given natural number $n$ since Gödel has defined $R(x)$ [12] such that $R(n)$ is instantiationally equivalent to a primitive recursive relation $Q(n)$ which is algorithmically computable as true in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ for any given natural number $n$ by the Turing-machine $TM_{Q}$.

It follows that we cannot admit the standard Hilbertian interpretion of $[\neg (\forall x)R(x)]$ in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ as:

$R(n)$ is false for some natural number $n$.

In other words, the interpretation $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ of PA over the structure $[\mathbb{N}]$ is not sound.

However, we can interpret $[\neg (\forall x)R(x)]$ in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ as:

It is not the case that the Turing-machine $TM_{R}$ algorithmically computes $R(n)$ as true in $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ for any given natural number $n$.

Moreover, the $\omega$-inconsistent PA is consistent with the finitary interpretation of quantification in (2a) and (2b), since the interpretation $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ of PA over the structure $[\mathbb{N}]$ is sound [13].

Why the interpretation $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ of PA over $[\mathbb{N}]$ is not sound

The reason why the interpretation $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ of PA over the structure $[\mathbb{N}]$ is not sound lies in the fact that, whereas (1b) and (2b) preserve the logical properties of formal PA-negation under interpretation in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ and $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ respectively, the further non-constructive inference in (1c) from (1b)— to the effect that $F(n)$ must hold in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ for some natural number $n$—does not, and is the one objected to by Brouwer.

Conclusion

If we assume only simple consistency for Hilbert’s system, then we cannot unconditionally define:

$[(\exists x)F(x)]$ is true in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ if, and only if, $F(n)$ holds for some natural number $n$ in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$.

This follows since, if $[(\exists x)F(x)]$ is true in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ if, and only if, $F(n)$ holds for some natural number $n$ in $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$, then PA is necessarily $\omega$-consistent — which is not the case.

Thus the interpretation $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ is not a model of PA, and Brouwer was justified in his objection to Hilbert’s unqualified interpretation of quantification.

References

Br08 L. E. J. Brouwer. 1908. The Unreliability of the Logical Principles. English translation in A. Heyting, Ed. L. E. J. Brouwer: Collected Works 1: Philosophy and Foundations of Mathematics. Amsterdam: North Holland / New York: American Elsevier (1975): pp.107-111.

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.

Hi27 David Hilbert. 1927. The Foundations of Mathematics. In The Emergence of Logical Empiricism. 1996. Garland Publishing Inc.

Hi30 David Hilbert. 1930. Die Grundlegung der elementaren Zahlenlehre. Mathematische Annalen. Vol. 104 (1930), pp.485-494.

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.

Tu36 Alan Turing. 1936. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp.544-546. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. pp.116-154.

An07 Bhupinder Singh Anand. 2007. Why we shouldn’t fault Lucas and Penrose for continuing to believe in the Gödelian argument against computationalism – I. The Reasoner, Vol(1)6 p3-4.

An12 Bhupinder Singh Anand. 2012. Evidence-Based Interpretations of PA. In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.

Notes

Return to 5: We note that we can, in principle, define the classical satisfaction’ and truth’ of the formulas of a first order arithmetical language such as PA constructively under an interpretation using as evidence the computations of a simple functional language (in the sense of Mu91).

When is the concept of a completed infinity consistent with a formal language?

The second issue is when, and whether, the concept of a completed infinity is consistent with the interpretation of a formal language.

Clearly, the consistency of the concept would follow immediately in any sound interpretation [1] of the axioms (and rules of inference) of a set theory such as ZF, but whether such an interpretation exists at all is, of course, another question.

In view of the perceived power of ZF [2] as an unsurpassed language of rich and adequate expression of mathematically expressible abstract concepts precisely, it is not surprising that many of the semantic and logical paradoxes depend on the implicit assumption that:

(i) such an interpretation may be assumed to exist, and that

(ii) the domain over which the paradox quantifies can always be assumed to be a well-defined mathematical object that can be formalised in ZF, even if this domain is not explicitly defined (or definable) set-theoretically.

This assumption is rooted in the questionable belief that ZF can express all mathematical truths’ [3].

From this it is but a short step to the non-constructive argument—rooted in Gödel’s Platonic interpretation of his own formal reasoning in his seminal 1931 paper on undecidable’ arithmetical propositions [4]—that PA must have non-standard models.

However, it is our contention that both of the above foundational issues need to be reviewed carefully, and that we need to recognize explicitly:

(a) the limitations on the ability of highly expressive mathematical languages such as ZF to communicate effectively; and

(b) the limitations on the ability of effectively communicating mathematical languages such as PA to adequately express abstract concepts—such as those involving Cantor’s first limit ordinal $\omega$ [5].

Prima facie, the semantic and logical paradoxes—as also the seeming paradoxes associated with fractal constructions such as the Cantor ternary set and the constructions described below—seem to arise out of a blurring of this distinction, and an attempt to ask of a language more than it is designed to deliver.

For instance, consider the claim [6] that fractal constructions’—such as the Cantor ternary set’ [7]—yield (presumably in some interpretable sense) valid mathematical objects (sets) in the limit’.

We consider an equilateral triangle $BAC$ of height $h$ and side $s$.

Divide the base $BC$ in half and construct two isosceles triangles of height $h.d$ and base $s/2$ on $BC$, where $1 \geq d>0$.

Iterate the construction on each constructed triangle ad infinitum (see Figs 1-3 below).

Thus, the height of each of the $2^{n}$ triangles on the base $BC$ at the $n$‘th construction is $h.d^n$, and the base of each triangle $s/2^{n}$.

Hence, the total area of all these triangles subtended by the base $BC$ is $s.h.d^{n}/2$.

Now, if $d=1$, the total area of all the constructed triangles after each iteration remains constant at $s.h/2$, although the total length of all the sides opposing the base $BC$ increases monotonically.

Moreover, if $1>d>0$, it would appear that the base $BC$ of the original equilateral triangle will always be the limiting’ configuration of the sides opposing the base $BC$.

Fig 1

This is indeed so if $0, since (see fig. 1 above) the total length of all the sides opposing the base $BC$ at the $n$‘th iteration—say $l_{n}$—yields a Cauchy sequence whose limiting value is, indeed, the length $s$ of the base $BC$.

Fig 2

However, if $d=1/2$, the total length (see fig. 2 above) of all the sides opposing their base on $BC$ is always $2s$!

Fig. 3

Finally, if $1>d>1/2$, the total length (see fig. 3 above) of all the sides opposing their base on $BC$ is a monotonically increasing value.

Consider now:

Case 1 Treating the above iteration as the fragmentation of a land-holding, how would one interpret the limit’ of such an interpretation (which is postulated as existing in a putative completion’ of Euclidean space)?

Case 2 Let $s$ be one light-year and consider how long it would take a light signal to travel from $B$ to $C$ along the sides opposing the base in each of the above cases.

Case 3 Let the area $BAC$ denote the population size of a virus cluster, where each virus cell has a virulence’ measure $h/s$. Let each triangle at the $n$‘th iteration denote a virus cluster—with a virulence factor $h.d^n/(s/2^{n})$—that reacts to the next generation anti-virus by splitting into two smaller clusters with inherited virulence $h.d^{n+1}/(s/2^{n+1})$.

If $d<1/2$, the effects of the virus can—in a sense—be contained and eventually eliminated’. If $d=1/2$, the effects of the virus can be contained’, but never eliminated’. However, if $d>1/2$, the effects of the virus can neither be contained’ nor eliminated’.

Case 4 Let the base $BC$ denote an elastic string, stretched iteratively into the above configurations. If $d<1/2$, the elastic will, in principle, eventually return to its original state. If $d>1/2$, then the elastic must break at some point. However, what if $d=1/2$?

If $d=1/2$, we then arrive at a two-dimensional version of Zeno’s paradox; one way of resolving which is by admitting the possibility that such an elastic length’ undergoes a phase change in the limit that need not correspond to the limit of its associated Cauchy sequence!

The question arises: Since the raison d’etre of a mathematical language is—or should be [8]—to express our abstractions of natural phenomena precisely and communicate them unequivocally, in what sense can we sensibly admit an interpretation of a mathematical language that asserts all the above cases as having limiting’ configurations in a putative completion’ of Euclidean Space?

References

Ba88 Michael Barnsley. 1988. Fractals Everywhere. Academic Press, Inc., San Diego.

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

Br08 L. E. J. Brouwer. 1908. The Unreliability of the Logical Principles. English translation in A. Heyting, Ed. L. E. J. Brouwer: Collected Works 1: Philosophy and Foundations of Mathematics. Amsterdam: North Holland, New York: American Elsevier (1975): pp. 107-111.

Fe85 Richard P. Feynman. 1985. Judging Books by Their Covers in Surely You’re Joking, Mr. Feynman! (Adventures of a curious character). Norton, New York. (Extracts: Judging Books by Their Covers.)

Ff02 Solomon Feferman. 2002. Predicativity. Source: http://math.stanford.edu/~feferman/papers/predicativity.pdf.

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.

He04 Catherine Christer-Hennix. 2004. Some remarks on Finitistic Model Theory, Ultra-Intuitionism and the main problem of the Foundation of Mathematics. ILLC Seminar, 2nd April 2004, Amsterdam.

Hi27 David Hilbert. 1927. The Foundations of Mathematics. Text of an address delivered in July 1927 at the Hamburg Mathematical Seminar. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts.

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

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.

Na08 Melvyn B. Nathanson. 2008. Desperately Seeking Mathematical Truth. Opinion in the August 2008 Notices of the American Mathematical Society, Vol. 55, Issue 7.

Pa95 Charles Parsons. 1995. Platonism and Mathematical Intuition in Kurt Gödel’s Thought. The Bulletin of Symbolic Logic, Volume 1, Number 1, March 1995, pp. 44-74.

Ru53 Walter Rudin. 1953. Principles of Mathematical Analysis. McGraw Hill, New York.

Sh67 Joseph R. Shoenfield. 1967. Mathematical Logic. Reprinted 2001. A. K. Peters Ltd., Massachusetts.

Ta33 Alfred Tarski. 1933. The concept of truth in the languages of the deductive sciences. In Logic, Semantics, Metamathematics, papers from 1923 to 1938 (p152-278). ed. John Corcoran. 1983. Hackett Publishing Company, Indianapolis.

Tu36 Alan Turing. 1936. On computable numbers, with an application to the Entscheidungsproblem. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

Tu39 Alan Turing. 1939. Systems of logic based on ordinals. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 45 (1939), pp.161-228.

Ya93 Stephen Yablo. 1993. Paradox without self-reference. Analysis, 53(4), pp. 251-252.

An07 Bhupinder Singh Anand. 2007. Why we shouldn’t fault Lucas and Penrose for continuing to believe in the Gödelian argument against computationalism – I. The Reasoner, Vol(1)6 p3-4.

An12 Bhupinder Singh Anand. 2012. Evidence-Based Interpretations of PA. In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.

Notes

Return to 1: We define an interpretation $I_{S}$ of a formal system $S$ as sound if, and only if, every provable formula $[F]$ of $S$ translates as a true proposition $F$ under the interpretation (cf. BBJ03, p.174).

Return to 3: In a series of nine previous posts starting here we have shown how—in the case of Goodstein’s Theorem—such a belief leads to a curious argument.

Return to 5: In this previous post we show why we cannot add a symbol corresponding formally to the concept of an infinite’ mathematical entity (such as is referred to symbolically in the literature by $\aleph$‘ or $\omega$‘) to the first-order Peano Arithmetic PA without inviting inconsistency.

Return to 6: See Ba88, p.35, §2.7, The completeness of the space of fractals.

Return to 7: Defined as a set-theoretical limit of an iterative process in the completion of a metric space; see, for instance, Ru53, p.54; Ba88, p.45.

Return to 8: See, for instance, Richard P. Feynman’s delightful commentary (Fe85) on his horrifying experiences of how the teaching of mathematical languages was actually introduced into the schools curricula in the State of California in 1964.

Is Hilbertian quantification currently interpreted constructively over infinite mathematical domains?

We consider next the first—and more significant—issue of whether the currently accepted interpretations of formal quantification—essentially as defined by David Hilbert in his formalisation of Aristotle’s logic of predicates in terms of his $\varepsilon$-function [1] —can be treated as constructive over an infinite domain.

Classical Theory

Now, classical theory follows the 2000-year old philosophical perspective of Greek philosophers such as Aristotle, and asks only that the core’ axioms of a formal language, and its rules of inference, be interpretable—under Alfred Tarski’s inductive definitions of the satisfiability and truth of the formulas of a formal language under an interpretation [2] —as self-evidently’ sound [3] propositions.

The subjective—and essentially irresolvable—element in the soundness’ of such a viewpoint is self-evident.

Brouwer’s objection

L. E. J. Brouwer [4] emphatically (and, as we shall show, justifiably so far as number theory was concerned) objected to such subjectivity, and asserted that Hilbert’s interpretations of formal quantification were non-constructive.

Although Hilbert’s formalisation of the quantifiers (an integral part of his formalisation of Aristotle’s logic of predicates) appeared adequate, Brouwer rejected Hilbert’s interpretations of them on the grounds that the interpretations were open to ambiguity [5].

However, Brouwer’s rejection of the Law of the Excluded Middle as a resolution of the objection was seen—also justifiably [6] —as unconvincingly rejecting a comfortable interpretation that—despite its Platonic overtones—appeared intuitively plausible to the larger body of academics that was increasingly attracted to, and influenced by, the remarkably expressive powers provided by Cantor-inspired set theories such as ZF.

Since Brouwer’s seminal work preceded that of Alan Turing, it was unable to offer his critics an alternative—and intuitively convincing—constructive definition of quantification based on the view—gaining currency today—that a simple functional language can be used for specifying evidence for propositions in a constructive logic [7].

Moreover, since Brouwer’s objections did not gain much currency amongst mainstream logicians, they were unable to influence Turing who, it is our contention, could easily have provided the necessary constructive interpretations [8] sought by Hilbert for number theory, had Turing not been influenced by Gödel’s powerful presentation—and Gödel’s persuasive Platonic interpretation of his own formal reasoning in his (Gödel’s) seminal 1931 paper on formally undecidable arithmetical propositions [9].

Gödel’s Platonism and $\omega$-consistency

Although meriting a more complete discussion than is appropriate to the intent of this investigation, it is worth noting that the roots of of Gödel’s Platonism can be cogently argued as lying—contrary to generally held opinions—purely in an axiomatic logical, rather than philosophical, presumption.

More specifically in:

Gödel’s presumption that Peano Arithmetic is $\omega$-consistent [10].

The presumption is unwittingly shared universally even by those who [11] accept Gödel’s formal arguments [12] but claim to reject Gödel’s Platonic’ interpretations of them.

Reason: Gödel’s presumption is equivalent to the premise—almost universally unchallenged hitherto in mainstream literature, and excepted to largely by ultra-finitists such as Alexander Yessenin-Volpin [13] —that:

The standard interpretation of first-order PA may be assumed to yield a model for number theory.

Rosser’s implicit presumption

A point worth noting is that common folklore mistakenly perceives J. B. Rosser’s proof of undecidability’ as having avoided Gödel’s explicit presumption of $\omega$-consistency.

Mistakenly, because the perception is illusory.

Rosser’s proof merely replaces an explicit appeal to Gödel’s presumption by an implicit presumption that the standard interpretation of first-order PA may be assumed to yield a model for number theory.

Thus Gödel’s Platonism essentially reflects the fact (as we have detailed here) that PA is $\omega$-consistent if, and only if, Aristotle’s particularisation is presumed to always hold under any interpretation of the associated logic; and that the standard interpretation of PA is a model of PA if, and only if, PA is $\omega$-consistent.

In this respect, Gödel was only endorsing Hilbert’s essentially Platonic formalisation of Aristotle’s Logic of Predicates [14] as valid.

Turing’s perspective

Be that as it may, in this 1939 paper [15] on ordinal-based logics (essentially his 1938 PhD dissertation), Turing applied his computational method—which he had developed in his 1936 paper [16] —in seeking partial completeness in interpretations of Cantor’s ordinal arithmetic (as defined in a set theory such as ZF) rather than in seeking a categorical interpretation of PA [17].

Turing perhaps viewed his 1936 paper as complementing and extending Gödel’s and Cantor’s reasoning.

“The well-known theorem of Gödel shows that every system of logic is in a certain sense incomplete, but at the same time it indicates means whereby from a system $L$ of logic a more complete system $L'$ may be obtained. By repeating the process we get a sequence of $L,\ L_{1}=L',\ L_{2}=L_{1}',\ \ldots$ each more complete than the preceding.

… Proceeding in this way we can associate a system of logic with any constructive ordinal. It may be asked whether a sequence of logics of this kind is complete in the sense that to any problem $A$ there corresponds an ordinal $\alpha$ such that $A$ is solvable by means of the logic $L_{\alpha}$. I propose to investigate this question in a more general case, and to give some other examples of ways in which systems of logic may be associated with constructive ordinals”. [18]

It is our contention that Turing thus overlooked the fact that his 1936 paper actually conflicts with Gödel’s and Cantor’s interpretations of their own, formal, reasoning by admitting an objective definition of satisfaction (as we have detailed here) that yields a sound, finitary, interpretation $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ of PA.

Specifically, whereas Gödel’s and Cantor’s reasoning implicitly presumes that satisfaction under the standard interpretation $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ of PA can only be defined non-constructively in terms of subjectively verifiable truth [19], it can be cogently argued that satisfaction under both $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ and $\mathcal{I}_{PA(\mathcal{N},\ Algorithmic)}$ is definable constructively in terms of objective algorithmic verifiability [20].

As a result, current theory continued—and continues to this day—to essentially follow Hilbert’s Platonically-influenced (hence, subjective) definitions and interpretations of the quantifiers (based on Aristotle’s logic of predicates) when defining them under the standard interpretation [21] $\mathcal{I}_{PA(\mathcal{N},\ Standard)}$ of formal number theory.

Tarski’s definitions of satisfiability and truth

Now, the latter definitions and interpretations [22] are, in turn, founded upon Tarski’s analysis of the inductive definability of the truth of compound expressions of a symbolic language under an interpretation in terms of the satisfaction of the atomic expressions of the language under the interpretation [23].

Tarski defines the formal sentence P as True if and only if p—where p’ is the proposition expressed by P’. In other words, the sentence Snow is white’ may be treated as a semantic truth if, and only if, it is subjectively true in all cases; and it is subjectively true in a particular case if, and only if, it expresses the subjectively verifiable fact that snow is white in that particular case.

Thus, for Tarski the commonality of the satisfaction of the atomic formulas of a language under an interpretation is axiomatic [24].

We highlight the limitations of such subjectivity here and, in the case of the standard’ interpretation of the Peano Arithmetic PA, show how to avoid them by requiring that the core’ axioms of PA, and its rules of inference, be interpretable as algorithmically (and, ipso facto, objectively) verifiable sound’ propositions.

We shall next consider the issue of when, and whether, the concept of a completed infinity is consistent with the interpretation of a formal language.

References

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

Br08 L. E. J. Brouwer. 1908. The Unreliability of the Logical Principles. English translation in A. Heyting, Ed. L. E. J. Brouwer: Collected Works 1: Philosophy and Foundations of Mathematics. Amsterdam: North Holland, New York: American Elsevier (1975): pp. 107-111.

Fe85 Richard P. Feynman. 1985. Judging Books by Their Covers in Surely You’re Joking, Mr. Feynman! (Adventures of a curious character). Norton, New York. (Extracts: Judging Books by Their Covers.)

Ff02 Solomon Feferman. 2002. Predicativity. Source: http://math.stanford.edu/~feferman/papers/predicativity.pdf.

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.

He04 Catherine Christer-Hennix. 2004. Some remarks on Finitistic Model Theory, Ultra-Intuitionism and the main problem of the Foundation of Mathematics. ILLC Seminar, 2nd April 2004, Amsterdam.

Hi27 David Hilbert. 1927. The Foundations of Mathematics. Text of an address delivered in July 1927 at the Hamburg Mathematical Seminar. In Jean van Heijenoort. 1967. Ed. From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931. Harvard University Press, Cambridge, Massachusetts.

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

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.

Na08 Melvyn B. Nathanson. 2008. Desperately Seeking Mathematical Truth. Opinion in the August 2008 Notices of the American Mathematical Society, Vol. 55, Issue 7.

Pa95 Charles Parsons. 1995. Platonism and Mathematical Intuition in Kurt Gödel’s Thought. The Bulletin of Symbolic Logic, Volume 1, Number 1, March 1995, pp. 44-74.

Ru53 Walter Rudin. 1953. Principles of Mathematical Analysis. McGraw Hill, New York.

Sh67 Joseph R. Shoenfield. 1967. Mathematical Logic. Reprinted 2001. A. K. Peters Ltd., Massachusetts.

Ta33 Alfred Tarski. 1933. The concept of truth in the languages of the deductive sciences. In Logic, Semantics, Metamathematics, papers from 1923 to 1938 (p152-278). ed. John Corcoran. 1983. Hackett Publishing Company, Indianapolis.

Tu36 Alan Turing. 1936. On computable numbers, with an application to the Entscheidungsproblem. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 42 (1936-7), pp.230-265; corrections, Ibid, vol 43 (1937) pp. 544-546.

Tu39 Alan Turing. 1939. Systems of logic based on ordinals. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York. Reprinted from the Proceedings of the London Mathematical Society, ser. 2. vol. 45 (1939), pp.161-228.

Ya93 Stephen Yablo. 1993. Paradox without self-reference. Analysis, 53(4), pp. 251-252.

An07 Bhupinder Singh Anand. 2007. Why we shouldn’t fault Lucas and Penrose for continuing to believe in the Gödelian argument against computationalism – I. The Reasoner, Vol(1)6 p3-4.

An12 Bhupinder Singh Anand. 2012. Evidence-Based Interpretations of PA. In Proceedings of the Symposium on Computational Philosophy at the AISB/IACAP World Congress 2012-Alan Turing 2012, 2-6 July 2012, University of Birmingham, Birmingham, UK.

Notes

Return to 3: The word sound’ is occasionally used informally as a synonym for reliable’, but mostly as a technical term of mathematical logic (cf. Me64, p.51). The distinction is assumed to be obvious from the context in which it is used.

Return to 6: Reason: Although Hilbert’s formalisation and interpretation of the quantifiers implies that the Law of the Excluded Middle must hold in any model of the formalisation, the converse is not true.

Return to 17: Perhaps since such a search would involve questioning either the validity of Gödel’s belief that systems of Arithmetic such as PA are $\omega$-consistent, or Gödel’s interpretation of his 1931 undecidability’ argument as having meta-mathematically proven that systems of Arithmetic such as PA are essentially incomplete!

Return to 19: A view (such as here) reflected in the not uncommon interpretation of Tarski’s Theorem (see Me64, p.151) as establishing the formal undefinability of arithmetical truth in arithmetic. We shall show why such an interpretation is untenable.

Return to 21: cf. Me64, p.107; Sh67, p.23, p.209; BBJ03, p.104. For purposes of this investigation, we treat Me64 as a reliable and representative exposition—where cited—of the standard logical concepts and theory that are addressed at that point.

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