You are currently browsing the tag archive for the ‘set theory’ tag.

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

**Ferguson’s and Priest’s thesis**

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

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

Ferguson and Priest conclude their review by remarking that:

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

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

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

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

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

A struggle reflected so eloquently in *this nineteenth century quote*:

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

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

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

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

**Making sense of mathematical propositions about infinite processes**

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

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

**The Liar paradox**

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

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

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

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

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

**Gödel’s influence on Turing’s reasoning**

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

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

**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 of rules is a Logic of a formal mathematical language if, and only if, constructively assigns unique truth-values:

(a) Of provability/unprovability to the formulas of ; and

(b) Of truth/falsity to the sentences of the Theory which is defined semantically by the -interpretation of over a structure .

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

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 of the natural numbers;

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

Interpret such representations unambiguously; and

Conclude further:

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

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.

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

**A Economist: The return of the machinery question**

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

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

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

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

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

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

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

**C Justifying irrationality**

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

**D Differentiating between Human reasoning and Mechanistic reasoning**

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

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

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

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

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

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

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

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

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

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

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

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

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

(*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 be understood from a broadly computational perspective”;

(iii) “ 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”.

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.

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**

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 is algorithmically computable if, and only if, there is an algorithm that can provide objective evidence (cf. ibid Murthy 91) for deciding the truth/falsity of each proposition in the denumerable sequence .

(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 is algorithmically verifiable if, and only if, for any given natural number , there is an algorithm which can provide objective evidence for deciding the truth/falsity of each proposition in the finite sequence .

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:

“ 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 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.

**“If I have seen a little further it is by standing on the shoulders of Giants”**

Prior to Isaac Newton’s tribute (above) to Rene Descartes and Robert Hooke in a letter to the latter, it was reportedly the 12th century theologian and author John of Salisbury who was recorded as having used an even earlier version of this humbling admission—in a treatise on logic called *Metalogicon*, written in Latin in 1159, the gist of which is translatable as:

“Bernard of Chartres used to say that we are like dwarfs on the shoulders of giants, so that we can see more than they, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant size.

(*Dicebat Bernardus Carnotensis nos esse quasi nanos, gigantium humeris insidentes, ut possimus plura eis et remotiora videre, non utique proprii visus acumine, aut eminentia corporis, sed quia in altum subvenimur et extollimur magnitudine gigantea.*)”

Contrary to a contemporary interpretation of the remark:

‘standing on the shoulders of Giants’

as describing:

“building on previous discoveries“,

it seems to me that what Bernard of Chartres apparently intended was to suggest that it doesn’t necessarily take a genius to see farther; only someone both humble and willing to:

first, clamber onto the shoulders of a giant and have the self-belief to see things at first-hand as they appear from a higher perspective (achieved more by the nature of height—and the curvature of our immediate space as implicit in such an analogy—than by the nature of genius); and,

second, avoid trying to see things first through the eyes of the giant upon whose shoulders one stands (for the giant might indeed be a vision-blinding genius)!

It was this latter lesson that I was incidentally taught by—and one of the few that I learnt (probably far too well for better or worse) from—one of my Giants, the late Professor Manohar S. Huzurbazaar, in my final year of graduation in 1964.

The occasion: I protested that the axiom of infinity (in the set theory course that he had just begun to teach us) was not self-evident to me, as (he had explained in his introductory lecture) an axiom should seem if a formal theory were to make any kind of coherent sense under interpretation.

Whilst clarifying that his actual instruction to us had not been that an axiom should necessarily ‘seem’ self-evident, but only that it should ‘be treated’ as self-evident, Professor Huzurbazar further agreed that the set-theoretical axiom of infinity was not really as self-evident as an axiom ideally ought to seem in order to be treated as self-evident.

To my natural response asking him if it seemed at all self-evident to him, he replied in the negative; adding, however, that he believed it to be ‘true’ despite its lack of an unarguable element of ‘self-evidence’.

It was his remarkably candid response to my incredulous—and youthfully indiscreet—query as to how an unimpeachably objective person such as he (which was his defining characteristic) could hold such a subjective belief that has shaped my thinking ever since.

He said that he had ‘had’ to believe the axiom to be ‘true’, since he could not teach us what he did with ‘conviction’ if he did not have such faith!

Although I did not grasp it then, over the years I came to the realisation that committing to such a belief was the price he had willingly paid for a responsibility that he had recognised—and accepted—consciously at a very early age in his life (when he was tutoring his school going nephew, the renowned physicist Jayant V. Narlikar):

**Nature had endowed him with the rare gift shared by great teachers—the capacity to reach out to, and inspire, students to learn beyond their instruction!**

It was a responsibility that he bore unflinchingly and uncompromisingly, eventually becoming one of the most respected and sought after teachers (of his times in India) of Modern Algebra (now Category Theory), Set Theory and Analysis at both the graduate and post-graduate levels.

At the time, however, Professor Huzurbazar pointedly stressed that his belief should not influence me into believing the axiom to be true, nor into holding it as self-evident.

His words—spoken softly as was his wont—were:

“Challenge it”.

Although I chose not to follow an academic career, he never faltered in encouraging me to question the accepted paradigms of the day when I shared the direction of my reading and thinking (particularly on Logic and the Foundations of Mathematics) with him on the few occasions that I met him over the next twenty years.

Moreover, even if the desired self-evident nature of the most fundamental axioms of mathematics (those of first-order Peano Arithmetic and Computability Theory) were to be shown as formally inconsistent with a belief in the ‘self-evident’ truth of the axiom of infinity (a goal that continues to motivate me), I believe that the shades of Professor Huzurbazaar would feel more liberated than bruised by the ‘fall’.

**Hilbert’s formalisation of Aristotle’s logic of predicates**

A fundamental tenet of classical logic—unrestrictedly adopted by formal first-order predicate calculus as axiomatic ^{[1]}—is Aristotlean particularisation.

**Aristotlean particularisation**: This holds that an assertion such as:

`There exists an unspecified such that holds’

—usually denoted symbolically by `‘—can always be validly inferred in the classical, Aristotlean, logic of predicates ^{[2]} from the assertion:

`It is not the case that: for any given , does not hold’

—usually denoted symbolically by `‘.

In a 1927 address, Hilbert reviewed, as part of his `proof theory’, his axiomatisation of classical Aristotle’s logic of predicates as a formal first-order -predicate calculus ^{[3]}.

A specific aim of the axiomatisation appears to have been the introduction of a primitive choice-function symbol, `‘, for formalising the existence of the unspecified object in Aristotle’s particularisation ^{[4]}:

“… stands for an object of which the proposition certainly holds if it holds of any object at all …” ^{[5]}

Hilbert showed, moreover, how the quantifiers `‘ and `‘ are definable using the choice-function `‘ (see this earlier post)—and noted that ^{[6]}:

“… The fundamental idea of my proof theory is none other than than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds.”

More precisely ^{[7]}:

**Lemma** adequately expresses—and yields, under a suitable interpretation—Aristotle’s logic of predicates if the -function is interpreted so as to yield the unspecified object in Aristotlean particularisation.

What came to be known later as Hilbert’s Program ^{[8]} —which was built upon Hilbert’s `proof theory’—can be viewed as, essentially, the subsequent attempt to show that the formalisation was also necessary for communicating Aristotle’s logic of predicates effectively and unambiguously under any interpretation of the formalisation.

This goal is implicit in Hilbert’s remarks ^{[9]}:

“Mathematics in a certain sense develops into a tribunal of arbitration, a supreme court that will decide questions of principle—and on such a concrete basis that universal agreement must be attainable and all assertions can be verified.”

“… a theory by its very nature is such that we do not need to fall back upon intuition or meaning in the midst of some argument.”

**The postulation of an `unspecified’ object in Aristotle’s particularisation is `stronger’ than the Axiom of Choice**

The difficulty in attaining this goal constructively along the lines desired by Hilbert—in the sense of the above quotes—becomes evident from Rudolf Carnap’s analysis in a 1962 paper, “On the use of Hilbert’s -operator in scientific theories” ^{[10]}.

Carnap noted that if we define a formal language ZF by replacing:

with

with

in the Zermelo-Fraenkel set theory ZF, then:

**Lemma** The Axiom of Choice is a theorem of ZF, and therefore true in any sound interpretation of the Zermelo-Fraenkel set theory ZF that admits Aristotle’s logic of predicates.

Thus, the postulation of an `unspecified’ object in Aristotlean particularisation is a stronger postulation than the Axiom of Choice!

**Cohen and The Axiom of Choice**

The significance of this is seen in the accepted interpretation of Cohen’s argument in his 1963-64 papers ^{[11]}; the argument is accepted as definitively establishing that the Axiom of Choice is essentially independent of a set theory such as ZF.

Now, Cohen’s argument—in common with the arguments of many important theorems in standard texts on the foundations of mathematics and logic—appeals to the unspecified object in Aristotle’s particularisation when interpreting the existential axioms of ZF (or statements about ZF ordinals).

This is seen in his proof ^{[12]} and application of the—seemingly paradoxical ^{[13]} —(downwards) Löwenheim-Skolem Theorem.

**(Downwards) Löwenheim-Skolem Theorem** ^{[14]}: If a first-order proposition is satisfied in any domain at all, then it is already satisfied in a denumerably infinite domain.

Cohen appeals to this theorem for legitimising putative models of a language—such as the standard model `M’ of ZF ^{[15]}, and its forced derivative `N’ ^{[16]}—in his argument ^{[17]}.

Now, the significance of Hilbert’s formalisation of Aristotle’s particularisation by means of the -function is seen in Cohen’s following remarks, where he explicitly appeals in the above argument to a semantic—rather than formal—definition of the unspecified object in Aristotle’s particularisation ^{[18]}:

“When we try to construct a model for a collection of sentences, each time we encounter a statement of the form we must invent a symbol and adjoin the statement . … when faced with , we should choose to have it false, unless we have already invented a symbol for which we have strong reason to insist that be true.”

Cohen, then, shows that the the Axiom of Choice is false in N.

**Any interpretation of ZF which appeals to Aristotle’s particularisation is not sound**

Since Hilbert’s -function formalises precisely Cohen’s concept of `‘—more properly, `‘—as , it follows that:

**Theorem** If the underlying logic is classical first order logic in which the quantifiers are interpreted according to Aristotle’s logic of predicates, then any model of ZF is a model of ZF plus the logical -axiom, since the expression must interpret to yield Cohen’s symbol `‘ whenever interprets as true.

(Note that we cannot argue that ZF is a conservative extension of ZF.)

Hence Cohen’s argument is also applicable to ZF plus the logical -axiom.

However, since the Axiom of Choice is true in any sound interpretation of ZF plus the logical -axiom which appeals to Aristotle’s logic of predicates, Cohen’s argument ^{[19]} —when applied to ZF plus the logical -axiom—actually shows that:

**Corollary** ZF plus the logical -axiom has no model that appeals to Aristotle’s logic of predicates.

**Corollary** ZF has no model that appeals to Aristotle’s particularisation.

We cannot, therefore, conclude that the Axiom of Choice is essentially independent of the axioms of ZF, since none of the putative models `forced’ by Cohen (in his argument for such independence) are defined by a sound interpretation of ZF.

**Cohen and the Gödelian Argument**

At the conclusion of his lectures on “Set Theory and the Continuum Hypothesis”, delivered at Harvard University in the spring term of 1965, Cohen remarked ^{[20]}:

“We close with the observation that the problem of CH is not one which can be avoided by not going up in type to sets of real numbers. A similar undecidable problem can be stated using only the real numbers. Namely, consider the statement that every real number is constructible by a countable ordinal. Instead of speaking of countable ordinals we can speak of suitable subsets of . The construction for , where is countable, can be completely described if one merely gives all pairs such that . This in turn can be coded as a real number if one enumerates the ordinals. In this way one only speaks about real numbers and yet has an undecidable statement in ZF. One cannot push this farther and express any of the set-theoretic questions that we have treated as statements about integers alone. Indeed one can postulate as a rather vague article of faith that any statement in arithmetic is decidable in “normal” set theory, i.e., by some recognizable axiom of infinity. This is of course the case with the undecidable statements of Gödel’s theorem which are immediately decidable in higher systems.”

Cohen appears to assert here that if ZF is consistent, then we can `see’ that the Continuum Hypothesis is subjectively true for the integers under some model of ZF, but—along with the Generalised Continuum Hypothesis—we cannot objectively `assert’ it to be true for the integers since it is not provable in ZF, and hence not true in all models of ZF.

However, by this argument, Gödel’s undecidable arithmetical propositions, too, can be `seen’ to be subjectively true for the integers in the standard model of PA, but cannot be `asserted’ to be true for the integers since the statements are not provable in an -consistent PA, and hence they are not true in all models of an -consistent PA!

The latter is, essentially, John Lucas’ well-known Gödelian argument ^{[21]}, forcefully argued by Roger Penrose in his popular expositions, `Shadows of the Mind’^{[22]} and `The Emperor’s New Mind’ ^{[23]}.

As I have argued in The Reasoner ^{[24]}, the argument is plausible, but unsound. It is based on a misinterpretation—of what Gödel actually proved formally in his 1931 paper—for which, moreover, neither Lucas nor Penrose ought to be taken to account.

The distinction sought to be drawn by Cohen is curious, since we have shown that his argument—which assumes that sound interpretations of ZF can appeal to Aristotle’s particularisation—actually establishes that sound interpretations of ZF cannot appeal to Aristotle’s particularisation; just as we shall show that Gödel’s argument ^{[25]} actually establishes that any sound interpretation of PA, too, cannot appeal to Aristotle’s particularisation.

Loosely speaking, the cause of the undecidability of the Continuum Hypothsis—and of the Axiom of Choice—in ZF as shown by Cohen, and that of Gödel’s undecidable proposition in Peano Arithmetic, is common; it is interpretation of the existential quantifier under an interpretation as Aristotlean particularisation.

In Cohen’s case, such interpretation is made explicitly and unrestrictedly in the underlying predicate logic ^{[29]} of ZF, and in its interpretation in Aristotle’s logic of predicates ^{[30]}.

In Gödel’s case it is made explicitly—but formally to avoid attracting intuitionistic objections—through his specification of what he believed to be a `much weaker assumption’ ^{[31]} of -consistency for his formal system P of Peano Arithmetic ^{[32]}.

**References**

**Ca62** Rudolf Carnap. 1962. *On the use of Hilbert’s -operator in scientific theories.* In *Essays on the Foundations of Mathematics.* Edited by Y. Bar-Hillel, E. I. J. Posnanski, M. O. Rabin, and A. Robinson for The Hebrew University of Jerusalem. 1962. North Holland Publishing Company, Amsterdam: pp.156-164.

**Co63** Paul J. Cohen. *The Independence of the Continuum Hypothesis I.* Proceedings of the U. S. National Academy of Sciences. 50, 1143-1148, 1963.

**Co64** Paul J. Cohen. *The Independence of the Continuum Hypothesis II.* Proceedings of the National Academy of Sciences of the United States of America, 51(1), 105 110, 1964.

**Co66** Paul J. Cohen. 1966. *Set Theory and the Continuum Hypothesis.* (Lecture notes given at Harvard University, Spring 1965) W. A. Benjamin, Inc., New York.

**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.

**HA28** David Hilbert & Wilhelm Ackermann. 1928. *Principles of Mathematical Logic.* Translation of the second edition of the *Grundzüge Der Theoretischen Logik.* 1928. Springer, Berlin. 1950. Chelsea Publishing Company, New York.

**Hi25** David Hilbert. 1925. *On the Infinite.* Text of an address delivered in Münster on 4th June 1925 at a meeting of the Westphalian Mathematical Society. In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

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

**Lo15** Leopold Löwenheim. 1915. *On possibilities in the calculus of relatives.* In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

**Lu61** J. R. Lucas. 1961. *Minds, Machines and Gödel.* Philosophy, XXXVI, 1961, pp.112-127; reprinted in *The Modeling of Mind.* Kenneth M. Sayre and Frederick J. Crosson, eds., Notre Dame Press, 1963, pp.269-270; and *Minds and Machines*, ed. Alan Ross Anderson, Prentice-Hall, 1954, pp.43-59.

**Pe90** Roger Penrose. 1990. *The Emperor’s New Mind: Concerning Computers, Minds and the Laws of Physics.* 1990, Vintage edition. Oxford University Press.

**Pe94** Roger Penrose. 1994. *Shadows of the Mind: A Search for the Missing Science of Consciousness.* Oxford University Press.

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

**Sk22** 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.

**Sk28** Thoralf Skolem. 1928. *On Mathematical Logic.* Text of a lecture delivered on 22nd October 1928 before the Norwegian Mathematical Association. In Jean van Heijenoort. 1967. Ed. *From Frege to Gödel: A source book in Mathematical Logic, 1878 – 1931.* Harvard University Press, Cambridge, Massachusetts.

**Wa63** Hao Wang. 1963. *A survey of Mathematical Logic.* North Holland Publishing Company, Amsterdam.

**An07a** Bhupinder Singh Anand. 2007. *The Mechanist’s Challege.* The Reasoner, Vol(1)5 p5-6.

**An07** … 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: See Hi25, p.382; HA28, p.48; Sk28, p.515; Be59, pp.178 & 218; Co66, p.4.

Return to 2: HA28, pp.58-59.

Return to 3: Hi27, pp.465-466.

Return to 4: Hi25, p.382; Ca62, p.156; see also this earlier post.

Return to 5: Note that need not be a `term’ of , since it is a term if, and only if, `holds’ for some term .

Return to 6: Hi27, p.475.

Return to 7: cf. Hi25, pp.382-383; Hi27, p.466(1).

Return to 8: See, for instance, the Stanford Encyclopedia of Philosophy: Hilbert’s Program.

Return to 9: Hi25, p.384; Hi27, p.475. Eighty years down the line, Hilbert’s optimistic vision of `a tribunal of arbitration’ stands in stark contrast to the `tribunal of bosses’ reflected in Melvyn B. Nathanson’s despairing comments here!

Return to 10: Ca62, pp.157-158; see also Wang’s remarks in Wa63, pp.320-321.

Return to 11: Co63 & Co64.

Return to 12: Co66, p.19.

Return to 13: See Skolem’s remarks in Sk22, p295; also Co66, p.19.

Return to 14: Lo15, p.245, Theorem 6; Sk22, p.293.

Return to 15: Co66, p.19 & p.82.

Return to 16: Co66, p.121.

Return to 17: Co66, p.83 & p.112-118.

Return to 18: Co66, p.112; see also p.4.

Return to 19: Co63 & Co64; Co66.

Return to 20: Co66, p.151.

Return to 21: Lu61.

Return to 22: Pe94.

Return to 23: Pe90.

Return to 24: An07a; An07b; An07c.

Return to 28: Go31, p.24, Theorem VI.

Return to 29: Co66, p.4.

Return to 30: Co66 p.112.}

Return to 31: The significance of Gödel’s `much weaker assumption’ is highlighted in An12, where we note that, a Peano Arithmetic has a sound interpretation that appeals to Aristotle’s particularisation—such as the interpretation currently defined as the `standard’ interpretation of the first order Peano Arithmetic PA—if, and only if, the Arithmetic is -consistent.

Return to 32: Go31, p.9 & pp.23-24.

## Recent comments