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(*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.

(*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 — 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 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 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 , 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 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 () 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 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 , 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 . Because there are infinitely many numbers bigger than , each of which makes a pair with and each of which pairs is coloured either red or blue, there are either infinitely many red pairs with or infinitely many blue pairs (note: this is really using ). I write down under ‘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 which make the ‘wrong colour’ pair with .

Now I move on to the next number, say , 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 as before, and again cross out all the number bigger than which make a wrong colour pair with . 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 the word ‘red’ is written, then forms red pairs with all the un-crossed-out numbers bigger than . Now (using 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 and are any two numbers in the collection I picked, the pair will be red – this is because one of and , say , is smaller, and by construction all the pairs from to bigger un-crossed-out numbers, including , 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 (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 , the usual proof is an argument that looks quite a lot like the argument, except that instead of using in the ‘first pass’ it uses . All fine; we believe , so no problem. But now, when you want to write down ‘red’ or ‘blue under in this ‘first pass’ you have to know something more complicated about all the triples using ; you want to know if you can find an infinite set such that any pair in forms a red triple with . If not, tells you that you can find an infinite set such that any pair in forms a _blue_ triple with . Then you would cross off everything not in , and keep going as with . The proof doesn’t really get any harder for the general case (or indeed changing the number of colours to something bigger than ). 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 (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 is merely an abbreviation for the formula .

So, under any consistent ‘finitistically reducible’ interpretation of such a formula, there must be a unique, unequivocal, evidence-based Tarskian interpretation of 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 to be interpreted constructively as:

For any natural number , there is an algorithm (say, a deterministic Turing machine) which evidences that are all true; or,

(b) is the formula to be interpreted finitarily as:

There is a single algorithm (say, a deterministic Turing machine) which evidences that, for any natural number is true, i.e., each of is true?

As Peter has pointed out in his analysis of Ramsey’s Theorem 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.

(*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 -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* -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* (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.

It is, indeed, gratifying that, after over 50 years of pursuing a path which challenged the accepted textbook wisdom that mathematical truth (which is the basis for asserting that any scientific proposition may be treated as true) is not definable objectively, my *contrary contention* has been accepted by the editors of the journal ‘*Cognitive Systems Research*‘ for publication in the December 2016 issue of the Journal. In a sense, this gives closure to the most challenging part of a journey which I have been privileged to afford and endure so far only because of the blessings, indulgence, and support provided by a generation of late elders (my teachers C. B. Nix James and Professor Manohar S. Huzurbazar, parents and mentors), contemporaries, and countless others who gave me the encouragement and strength to continue on such a nebulous path at crucial moments of my life. Defending the thesis promises to be as challenging—albeit far shorter—a journey!

In a paper *The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas’ Gödelian Thesis*, due to appear in the December 2016 issue of **Cognitive Systems Research**, we briefly consider (*Anand [1]*) a philosophical challenge that arises when an intelligence—whether human or mechanistic—accepts arithmetical propositions as true under an interpretation—either axiomatically or on the basis of subjective self-evidence—*without* any specified methodology for evidencing such acceptance (for a brief, relatively recent, review of such challenges, see *Feferman [2], Feferman [3]*).

**The ambiguity in the standard interpretation of the Peano Arithmetic PA**

For instance conventional wisdom, whilst accepting Tarski’s classical definitions of the satisfiability and truth of the formulas of a formal language under an interpretation (*Anand [1]*, p.44), *postulates* that under the classical standard interpretation (we shall refer to this henceforth as ) of the first-order Peano Arithmetic PA (we take this to be the first-order theory defined in any standard text corresponding to the theory S in *Mendelson [4]*, p.102) over the domain of the natural numbers:

(i) The satisfiability/truth of the atomic formulas of PA can be assumed as *uniquely* decidable under ;

(ii) The PA axioms can be assumed to *uniquely* interpret as satisfied/true under ;

(iii) The PA rules of inference—Generalisation and Modus Ponens—can be assumed to *uniquely* preserve such satisfaction/truth under ;

(iv) Aristotle’s particularisation can be assumed to hold under .

We define Aristotle’s particularisation as the *non-finitary* assumption that an assertion such as, `There exists an such that holds’—usually denoted symbolically by `‘—can always be validly inferred in the classical logic of predicates from the assertion, `It is not the case that: for any given , does not hold’—usually denoted symbolically by `‘ (see also *Hilbert & Ackermann [5]*, pp.58-59).

We argue that the seemingly innocent and self-evident assumptions of *uniqueness* in (i) to (iii)—as also the seemingly innocent assumption in (iv) which, despite being obviously *non-finitary*, is unquestioningly accepted in classical literature as equally self-evident under any logically unexceptionable interpretation of the classical first-order logic FOL—conceal an ambiguity with far-reaching consequences.

**The two, hitherto unsuspected and essentially different, interpretations of PA**

The ambiguity is revealed if we note that Tarski’s classic definitions permit both human and mechanistic intelligences to admit *finitary* evidence-based definitions of the satisfaction and truth of the *atomic* formulas of PA over the domain of the natural numbers in two, hitherto unsuspected and essentially different, ways:

(1a) In terms of *classical* algorithmic verifiabilty; and

(1b) In terms of *finitary* algorithmic computability.

By ‘finitary’ we mean that (for a brief review of ‘finitism’ and ‘constructivity’ in the context of this paper see *Feferman [3]*):

“… there should be an algorithm for deciding the truth or falsity of any mathematical statement”

… http://en.wikipedia.org/wiki/Hilbert’s\_program.

We show that:

(2a) The two definitions correspond to two distinctly different assignments of satisfaction and truth to the *compound* formulas of PA over —say and (we shall refer to these henceforth as and respectively); where

(2b) The PA axioms are true over , and the PA rules of inference preserve truth over , under both and .

**A finitary proof of consistency for Arithmetic: The solution to Hilbert’s Second Millenium Problem**

We then show that:

(3a) If we assume the satisfaction and truth of the compound formulas of PA are always *non-finitarily* decidable under the assignment , then this assignment defines a *non-finitary* interpretation of PA in which Aristotle’s particularisation always holds over ; and which corresponds to the classical *non-finitary* standard interpretation of PA over the domain —from which only a human intelligence may *non-finitarily* conclude (as Gentzen’s argument does) that PA is consistent; whilst

(3b) The satisfaction and truth of the compound formulas of PA are always *finitarily* decidable under the assignment , which thus defines a *finitary* interpretation of PA—from which both intelligences may *finitarily* conclude that PA is consistent (as sought by David Hilbert for the second of the twenty three problems that he highlighted at the International Congress of Mathematicians in Paris in 1900; see *Hilbert [6]*).

**PA is categorical and has no non-standard models**

We show further that both intelligences would logically conclude that:

(4a) The assignment defines a subset of PA formulas that are algorithmically computable as true under the standard interpretation if, and only if, the formulas are PA provable;

(4b) PA is categorical (and so has no non-standard model, as argued in *Anand [7]*);

We note that the standard argument to the contrary—as detailed, for instance, in *Kaye [8]* (pp.10-11)—violates finitarity by adding a new constant to the language of PA that is not definable in and, ipso facto, by adding an atomic formula to PA whose satisfaction under any interpretation of PA is not algorithmically verifiable.

However, since the atomic formulas of PA are algorithmically verifiable under the standard interpretation (Theorem 5.1, p. 38, in *Anand [1]*), the above argument invalidly postulates precisely that which it seeks to prove (as also do arguments in: *Boolos, Burgess & Jeffrey [9]*, p.306, Corollary 25.3; *Luna [10]*, p.7)!

**There are no ‘undecidable’ arithmetical propositions: Gödel’s Theorems hold vacuously**

Both intelligences would also logically conclude that:

(4c) PA is not -consistent.

(5a) Since PA is not -consistent, Gödel’s argument in *Gödel [11]* (p.28(2))—that “ is not -PROVABLE”—does not yield a `formally undecidable proposition’ in PA;

The reason we prefer to consider Gödel’s original argument (rather than any of its subsequent avatars) is that, for a purist, Gödel’s remarkably self-contained 1931 paper—it neither contained, nor needed, any formal citations—remains unsurpassed in mathematical literature for thoroughness, clarity, transparency and soundness of exposition—*from first principles* (thus avoiding any implicit mathematical or philosophical assumptions)—of his notion of arithmetical `undecidability’ as based on his Theorems VI and XI and their logical consequences.

We also note that if PA is not -consistent, then Aristotle’s particularisation does not hold in any finitary interpretation of PA over .

Now, J. Barkeley Rosser’s ‘undecidable’ arithmetical proposition in *Rosser [12]* is of the form .

Thus his ‘extension’ of Gödel’s proof of undecidability too does not yield a ‘formally undecidable proposition’ in PA, since it implicitly presumes that Aristotle’s particularisation holds when interpreting under a finitary interpretation over (*Rosser [12]*, Theorem II, pp.233-234; *Kleene [13]*, Theorem 29, pp.208-209; *Mendelson [4]*, Proposition 3.32, pp.145-146).

(5b) The appropriate conclusion to be drawn from Gödel’s argument (in *Gödel [11]*, p.27(1))—that “ is not -PROVABLE”—is thus not that there is a ‘formally undecidable arithmetical proposition’ (see also *Feferman [4]* for an interesting perspective on how he—as well as, reportedly, both Gödel and Hilbert—informally viewed the concept of ‘formally undecidable arithmetical propositions’) but that any such putatively ‘undecidable arithmetical proposition’ is an instantiation of the argument (corresponding to Cantor’s diagonal argument and Turing’s halting argument) that we can define number-theoretic formulas which are algorithmically verifiable as always true, but not algorithmically computable as always true.

**The argument for Lucas’ Gödelian Thesis**

We conclude from this that Lucas’ Gödelian argument can validly claim:

**Thesis**: There can be no mechanist model of human reasoning if the assignment can be treated as circumscribing the ambit of human reasoning about ‘true’ arithmetical propositions, and the assignment can be treated as circumscribing the ambit of mechanistic reasoning about ‘true’ arithmetical propositions.

Although Lucas’ original 1961 thesis (*Lucas [14]*):

“… we cannot hope ever to produce a machine that will be able to do all that a mind can do: we can never not even in principle, have a mechanical model of the mind.”

deserves consideration that lies beyond the immediate scope of this investigation, we draw attention to his informal 1996 defence of it from a philosophical perspective in *Lucas [15]*, where he concludes with the argument that:

“Thus, though the Gödelian formula is not a very interesting formula to enunciate, the Gödelian argument argues strongly for creativity, first in ruling out any reductionist account of the mind that would show us to be, au fond, necessarily unoriginal automata, and secondly by proving that the conceptual space exists in which it is intelligible to speak of someone’s being creative, without having to hold that he must be either acting at random or else in accordance with an antecedently specifiable rule”.

**Argument**: Gödel has shown how to construct an arithmetical formula with a single variable—say (Gödel refers to this formula only by its Gödel number (*Gödel [11]*, p.25(12)))—such that is not PA-provable, but is instantiationally PA-provable for any given PA numeral .

Hence, for any given numeral , Gödel’s primitive recursive relation must hold for some natural number (where denotes Gödel’s primitive recursive relation ‘ is the Gödel-number of a proof sequence in PA whose last term is the PA formula with Gödel-number ‘ (*Gödel [11]*, p.22(45)); and denotes the Gödel-number of the PA formula ).

If we assume that any mechanical witness can only reason *finitarily* then although, for any given numeral , a mechanical witness can give evidence under the assignment that the PA formula holds in , no mechanical witness can conclude *finitarily* under the assignment that, for any given numeral , the PA formula holds in .

However, if we assume that a human witness can also reason *non-finitarily*, then a human witness *can* conclude under the assignment that, for any given numeral , the PA formula holds in .

**Bibliography**

[1] Bhupinder Singh Anand. 2016. *The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas’ Gödelian Thesis*. To appear in *Cognitive Systems Research*. Volume 40, December 2016, Pages 35-45, doi:10.1016/j.cogsys.2016.02.004.

[2] Solomon Feferman. 2006. *Are There Absolutely Unsolvable Problems? Gödel’s Dichotomy*. In *Philosophia Mathematica* (2006) 14 (2): 134-152.

[3] Solomon Feferman. 2008. *Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert’s program*. In the *Special Issue: Gödel’s dialectica Interpretation* of *Dialectica*, Volume 62, Issue 2, June 2008, pp. 245-290.

[4] Elliott Mendelson. 1964. *Introduction to Mathematical Logic*. Van Norstrand, Princeton.

[5] 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.

[6] David Hilbert. 1900. *Mathematical Problems*. An address delivered before the *International Congress of Mathematicians* at Paris in 1900. Dr. Maby Winton Newson translated this address into English with the author’s permission for the *Bulletin of the American Mathematical Society*, 8 (1902), 437-479. An HTML version is accessible at http://aleph0.clarku.edu/~djoyce/hilbert/problems.html.

[7] Bhupinder Singh Anand. 2008. *Can we really falsify truth by dictat?*. In *The Reasoner*, Vol(2)1 pp. 7-8.

[8] Richard Kaye. 1991. *Models of Peano Arithmetic*. Oxford Logic Guides, 15. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1991.

[9] George S. Boolos, John P. Burgess, & Richard C. Jeffrey. 2003. *Computability and Logic* (4th ed). Cambridge University Press, Cambridge.

[10] Laureano Luna. 2008. *On non-standard models of Peano Arithmetic*. In *The Reasoner*, Vol(2)2 p. 7.

[11] 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.

[12] J. Barkley Rosser. 1936. *Extensions of some Theorems of Gödel and Church*. In M. Davis (ed.). 1965. *The Undecidable*. Raven Press, New York. Reprinted from *The Journal of Symbolic Logic*, Vol.1, pp.87-91.

[13] Stephen Cole Kleene. 1952. *Introduction to Metamathematics*. North Holland Publishing Company, Amsterdam.

[14] J. R. Lucas. 1961. *Minds, Machines and Gödel*. In *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 in *Minds and Machines*, ed. Alan Ross Anderson, Prentice-Hall, 1954, pp.43-59.

[15] J. R. Lucas. 1996. The Gödelian Argument: Turn Over the Page. A paper read at a BSPS conference in Oxford. Reproduced in 2003 as *Series/Report no.: Etica & Politica / Ethics & Politics V* (2003) 1, EUT Edizioni Università di Trieste, URI: http://hdl.handle.net/10077/5477.

** Are there still unsolved problems about the numbers **

In a 2005 Clay Math Institute invited large lecture to a popular audience at MIT ‘*Are there still unsolved problems about the numbers *’, co-author with William Stein of *Prime Numbers and the Riemann Hypothesis*, and Gerhard Gade Harvard University Professor, *Barry Mazur* remarked that (p.6):

“If I give you a number , say = one million, and ask you for the first number after that is prime, is there a method that answers that question without, in some form or other, running through each of the successive numbers after rejecting the nonprimes until the first prime is encountered? **Answer: unknown**.”

Although this may represent conventional wisdom, it is not, however, strictly true!

** Reason**: The following

*1964 Theorem*yields two algorithms (

*Trim/Compact*), each of which affirmatively answers the questions:

(i) Do we necessarily discover the primes in a Platonic domain of the natural numbers, as is suggested by the sieve of Eratosthenes, or can we also generate them sequentially in a finitarily definable domain that builds into them the property of primality?

(ii) In other words, given the first primes, can we generate the prime algorithmically, without testing any number larger than for primality?

**I: A PRIME NUMBER GENERATING THEOREM**

**Theorem**: For all , let denote the 'th prime, and define such that:

,

and:

.

Let be such that, for all , there is some such that:

.

Then:

.

**Proof**: For all , there is some such that:

.

Since :

.

Hence:

,

and so it is the prime .

**II: TRIM NUMBERS**

The significance of the Prime Number Generating Theorem is seen in the following algorithm.

We define Trim numbers recursively by , and , where is the ‘th prime and:

(1) , and is the only element in the nd array;

(2) is the smallest even integer that does not occur in the ‘th array ;

(3) is selected so that:

for all .

It follows that the Trim number is, thus, a prime unless all its prime divisors are less than .

*The Trim Number Algorithm*

n

1 **2** **2** **3** **5** **7** **11** **13** **17** **19** **23** *27* **29** **31** **37**

2 **3** 1 **5**

3 **5** 1 1 **7**

4 **7** 1 2 3 **11**

5 **11** 1 1 4 3 **13**

6 **13** 1 2 2 1 9 **17**

7 **17** 1 1 3 4 5 9 **19**

8 **19** 1 2 1 2 3 7 15 **23**

9 **23** 1 1 2 5 10 3 11 15 *27*

10 *27* 1 0 3 1 6 12 7 11 19 **29**

11 **29** 1 1 1 6 4 10 5 9 17 **31**

n …

**NOTE**: The first Trim Numbers upto consist of the first primes and the composites:

(i) Trim composite :

(ii) Trim composite :

(iii) Trim composite :

(iv) Trim composite :

**Theorem**: For all

**II A: -TRIM NUMBERS**

For any given natural number , we define -Trim numbers recursively by , and , where is the ‘th prime and:

(1) , and is the only element in the nd array;

(2) is the smallest even integer that does not occur in the ‘th array ;

(3) is selected so that:

for all .

It follows that the -Trim number is, thus, not divisible by any prime unless the prime is smaller than .

**III: COMPACT NUMBERS**

Compact numbers are defined recursively by , and , where is the ‘th prime and:

(1) , and is the only element in the nd array;

(2) is the smallest even integer that does not occur in the nth array ;

(3) is selected so that:

for all ;

(4) is selected so that:

;

(5) if .

It follows that the compact number is either a prime, or a prime square, unless all, except a maximum of , prime divisors of the number are less than .

*The Compact Number Algorithm*

n

1 **2** **2** **3** **5** **7** **11** **13** **17** **19** **23** **29** **31** **37** **41**

2 **3** 1 **5**

3 **5** 1 **7**

4 **7** 1 *9*

5 *9* 1 0 **11**

6 **11** 1 1 **13**

7 **13** 1 2 **17**

8 **17** 1 1 **19**

9 **19** 1 2 **23**

10 **23** 1 1 *25*

11 *25* 1 2 0 **29**

12 **29** 1 1 1 **31**

13 **31** 1 2 4 **37**

14 **37** 1 2 3 **41**

15 **41** 1 1 4 **43**

16 **43** 1 2 2 **47**

17 **47** 1 1 3 *49*

18 *49* 1 2 1 0 **53**

19 **53** 1 1 2 3 *57*

20 *57* 1 0 3 6 **59**

21 **59** 1 1 1 4 **61**

22 **61** 1 2 4 2 **67**

23 **67** 1 2 3 3 **71**

24 **71** 1 1 4 6 **73**

25 **73** 1 2 2 4 **79**

26 **79** 1 2 1 5 **83**

27 **83** 1 1 2 1 *87*

28 *87* 1 0 3 4 **89**

29 **89** 1 1 1 2 *93*

30 *93* 1 0 2 5 **97**

31 **97** 1 2 3 1 **101**

32 **101** 1 1 4 4 **103**

33 **103** 1 2 2 2 **107**

34 **107** 1 1 3 5 **109**

35 **109** 1 2 1 3 **113**

36 **113** 1 1 2 6 *117*

37 *117* 1 0 3 2 *121*

38 *121* 1 2 4 5 0 **127**

39 **127** 1 2 3 6 5 **131**

40 **131** 1 1 4 2 1 **137**

41 **137** 1 1 3 3 6 **139**

42 **139** 1 2 1 1 4 *145*

43 *145* 1 2 0 2 9 **149**

44 **149** 1 1 1 5 5 **151**

45 **151** 1 2 4 5 0 **157**

46 **157** 1 2 3 4 8 **163**

47 **163** 1 2 2 5 2 **167**

48 **167** 1 1 3 1 9 *169*

49 *169* 1 2 1 6 7 0 **173**

50 **173** 1 1 2 2 3 9 *177*

51 *177* 1 0 3 5 10 5 **179**

52 **179** 1 1 1 3 8 3 **181**

53 **181** 1 2 4 1 6 1 *189*

54 *189* 1 0 1 0 9 6 **191**

55 **191** 1 1 4 5 7 4 **193**

56 **193** 1 2 2 3 5 2 **197**

57 **197** 1 1 3 6 1 11 **199**

58 **199** 1 2 1 4 10 9 *205*

59 *205* 1 2 0 5 4 3 **211**

60 **211** 1 2 4 6 9 10 *219*

61 *219* 1 0 1 5 1 2 **223**

62 **223** 1 2 2 1 8 11 **227**

63 **227** 1 1 3 4 4 7 **229**

64 **229** 1 2 1 2 2 5 **233**

65 **233** 1 1 2 5 9 14 *237*

66 *237* 1 0 3 1 5 10 **239**

67 **239** 1 1 1 6 3 8 **241**

68 **241** 1 2 4 4 1 6 *249*

69 *249* 1 0 1 3 4 11 **251**

70 **251** 1 1 4 1 2 9 **257**

71 **257** 1 1 3 2 7 3 *261*

72 *261* 1 0 4 5 3 12 **263**

73 **263** 1 1 2 3 1 10 *267*

74 *267* 1 0 3 6 8 6 **269**

75 **269** 1 1 1 4 6 4 **271**

76 **271** 1 2 4 2 4 2 **277**

77 **277** 1 2 3 3 9 9 **281**

78 **281** 1 1 4 6 5 5 **283**

79 **283** 1 2 2 4 3 3 *289*

n …

**NOTE**: The first Compact Numbers upto consist of the first primes, prime squares, and composites.

**Theorem 1**: There is always a Compact Number such that .

**Theorem 2**: For sufficiently large .

** A 2-dimensional Eratosthenes sieve**

A *later investigation* (see also *this post*) shows why the usual, linearly displayed, Eratosthenes sieve argument reveals the structure of divisibility (and, ipso facto, of primality) more transparently when displayed as the 1964, -dimensional matrix, representation of the residues , defined for all and all by:

, where .

‘**Density**‘: For instance, the residues can be defined for all as the values of the non-terminating sequences , defined for all (as illustrated below in Fig.1).

*Fig.1*

Sequence …

n=1 1 2 3 4 5 6 7 8 9 10 … n-1

n=2 0 1 2 3 4 5 6 7 8 9 … n-2

n=3 0 1 1 2 3 4 5 6 7 8 … n-3

n=4 0 0 2 1 2 3 4 5 6 7 … n-4

n=5 0 1 1 3 1 2 3 4 5 6 … n-5

n=6 0 0 0 2 4 1 2 3 4 5 … n-6

n=7 0 1 2 1 3 5 1 2 3 4 … n-7

n=8 0 0 1 0 2 4 6 1 2 3 … n-8

n=9 0 1 0 3 1 3 5 7 1 2 … n-9

n=10 0 0 2 2 0 2 4 6 8 1 … n-10

n=11 0 1 1 1 4 1 3 5 7 9 … n-11

n …

We note that:

For any , the non-terminating sequence cycles through the values with period ;

For any the ‘density’—over the set of natural numbers—of the set of integers that are divisible by is ; and the ‘density’ of integers that are not divisible by is .

**Primality**: The residues can be alternatively defined for all as values of the non-terminating sequences, , defined for all (as illustrated below in Fig.2).

*Fig.2*

Sequence …

1 2 3 4 5 6 7 8 9 10 … n-1

0 1 2 3 4 5 6 7 8 9 … n-2

0 1 1 2 3 4 5 6 7 8 … n-3

0 0 2 1 2 3 4 5 6 7 … n-4

0 1 1 3 1 2 3 4 5 6 … n-5

0 0 0 2 4 1 2 3 4 5 … n-6

0 1 2 1 3 5 1 2 3 4 … n-7

0 0 1 0 2 4 6 1 2 3 … n-8

0 1 0 3 1 3 5 7 1 2 … n-9

0 0 2 2 0 2 4 6 8 1 … n-10

0 1 1 1 4 1 3 5 7 9 … n-11

…

We note that:

The non-terminating sequences highlighted in bold correspond to a prime (since for any ) in the usual, linearly displayed, Eratosthenes sieve:

The non-terminating sequences highlighted in italics identify a crossed out composite (since for some ) in the usual, linearly displayed, Eratosthenes sieve.

**The significance of expressing Eratosthenes sieve as a -dimensional matrix**

Fig.1 illustrates that although the probability of selecting a number that has the property of being prime from a given set of numbers is definable if the precise proportion of primes to non-primes in is definable, if is the set of all integers, and we cannot define a precise ratio of primes to composites in , but only an order of magnitude such as , then equally obviously cannot be defined in (see Chapter 2, p.9, Theorem 2.1, *here*).

**The probability of determining a proper factor of a given number **

Fig.2 illustrates, however, that the probability of determining a proper factor of a given number is , since *this paper* shows that whether or not a prime divides a given integer is independent of whether or not a prime divides .

We thus have that , with a binomial standard deviation.

**The putative non-heuristic probability that a given is a prime**

Hence, even though we cannot define the probability of selecting a number from the set of all natural numbers that has the property of being prime, can be treated as the putative non-heuristic probability that a given is a prime.

** “What** *sort* **of Hypothesis is the Riemann Hypothesis?”**

The significance of the above perspective is that (see *this investigation*) it admits non-heuristic approximations of prime counting functions with what appear to be (see Fig.15 of *the investigation*) square root accurate errors, where:

“It has long been known that that for any real number the number of prime numbers less than (denoted is approximately in the sense that the *ratio* tends to as goes to infinity. The *Riemann Hypothesis* would give us a much more accurate “count” for in that it will offer a specific smooth function (hardly any more difficult to describe than ) and then conjecture that is an *essentially square root accurate approximation* to ; i.e., for any given exponent greater than (you choose it: , for example) and for large enough where the phrase “large enough” depends on your choice of exponent the error term i.e., the difference between and the number of primes less than in absolute value is less than raised to that exponent (e.g. , etc.)"

… Barry Mazur and William Stein, ‘What is Riemann’s Hypothesis‘

*This argument laid the foundation for this investigation. See also this arXiv preprint, and this broader update.*

Abstract We define the residues for all and all such that if, and only if, is a divisor of . We then show that the joint probability of two unequal primes dividing any integer is the product . We conclude that the prime divisors of any integer are independent; and that the probability of being a prime is . The number of primes less than or equal to is thus given by . We further show that , and conclude that does not oscillate.

** The residues **

We begin by defining the residues for all and all as below:

**Definition 1:** where .

Since each residue cycles over the values , these values are all incongruent and form a complete system of residues ^{[1]} .

It immediately follows that:

**Lemma 1:** if, and only if, is a divisor of .

** The probability **

By the standard definition of the probability of an event , we conclude that:

**Lemma 2:** For any and any given integer , the probability that is , and the probability that is .

We note the standard definition:

**Definition 2:** Two events and are mutually independent for if, and only if, .

** The prime divisors of any integer are mutually independent**

We then have that:

**Lemma 3:** If and are two primes where then, for any , we have:

where and .

**Proof:** The numbers , where and , are all incongruent and form a complete system of residues ^{[2]} . Hence:

.

By Lemma 2:

.

The lemma follows.

If and in Lemma 3, so that both and are prime divisors of , we conclude by Definition 2 that:

**Corollary 1:** .

**Corollary 2:** .

**Theorem 1:** The prime divisors of any integer are mutually independent. ^{[3]}

** The probability that is a prime**

Since is a prime if, and only if, it is not divisible by any prime , it follows immediately from Lemma 2 and Lemma 3 that:

**Lemma 4:** For any , the probability of an integer being a prime is the probability that for any if .

**Lemma 5:** ^{[4]}.

** The Prime Number Theorem**

The number of primes less than or equal to is thus given by:

**Lemma 6:** .

This now yields the Prime Number Theorem:

**Theorem 2:** .

**Proof:** From Lemma 6 and Mertens’ Theorem that

^{[5]}:

it follows that:

The behaviour of as is then seen by differentiating the right hand side, where we note that :

Hence does not oscillate as .

**Acknowledgements**

I am indebted to my erstwhile classmate, Professor Chetan Mehta, for his unqualified encouragement and support for my scholarly pursuits over the past fifty years; most pertinently for his patiently critical insight into the required rigour without which the argument of this 1964 investigation would have remained in the informal universe of seemingly self-evident truths.

**References**

**HW60** G. H. Hardy and E. M. Wright. 1960. *An Introduction to the Theory of Numbers* 4th edition. Clarendon Press, Oxford.

**Ti51** E. C. Titchmarsh. 1951. *The Theory of the Riemann Zeta-Function*. Clarendon Press, Oxford.

**Notes**

Return to 1: HW60, p.49.

Return to 2: HW60, p.52, Theorem 59.

Return to 3: In the previous post we have shown how it immediately follows from Theorem 1 that integer factorising is necessarily of order ; from which we conclude that integer factorising cannot be in the class of polynomial-time algorithms.

Return to 4: By HW60, p.351, Theorem 429, Mertens’ Theorem.

Return to 5: By the argument in Ti51, p.59, eqn.(3.15.2).

Return to 6: HW60, p.9, Theorem 7.

**Finitarily consistent mechanist reasoning and non-finitarily consistent human reasoning: Mutually inconsistent yet complementary!**

We now consider the following (tentatively expressed) conclusions suggested by our previous post, which we shall aim to investigate from various perspectives in these pages.

**Structures**

The Birmingham paper suggests that we may need to distinguish much more sharply than we do at present between:

Mathematical structures that are built upon only finitary reasoning, and

Mathematical structures that admit non-finitary reasoning.

**Interpretations**

For instance the Birmingham paper provides:

An example of a mathematical structure based on finitary reasoning, namely the finitarily sound algorithmic interpretation of the first order Peano Arithmetic PA.

An example of a mathematical structure based on non-finitary reasoning, namely the non-finitarily sound standard interpretation of the first order Peano Arithmetic PA.

The Birmingham paper suggests that the roots of the distinction between these two structures lies in the fact that:

Finitary reasoning does not assume that Aristotle’s particularisation is always true over infinite domains.

Non-finitary reasoning assumes that Aristotle’s particularisation is always true over infinite domains.

**Consistency of Arithmetic**

In the Birmingham paper we also show that:

Finitary reasoning proves that PA is consistent finitarily (as demanded by the second of Hilbert’s celebrated twenty three problems).

Non-finitary reasoning proves that PA is consistent non-finitarily (a consequence of Gentzen’s non-finitary proof of consistency for PA).

**FOL is consistent; FOL+AP is -consistent**

This suggests that:

Finitary reasoning as formalised in first order logic (FOL) is consistent.

Non-finitary reasoning as formalised in Hilbert’s -calculus (FOL+AP) is -consistent.

**-consistency**

Since the Birmingham paper shows that Aristotle’s particularisation holds over the structure of the natural numbers if, and only if, PA is -consistent, it suggests that:

Finitary reasoning does not admit that PA can be -consistent (see Corollary 4 of this post).

Non-finitary reasoning admits that PA can be -consistent.

**Arithmetical undecidability**

Since proofs of arithmetical undecidability implicitly assume Aristotle’s particularisation, this further suggests that:

Finitary reasoning does not admit undecidable arithmetical propositions (see Corollary 3 of this post).

Non-finitary reasoning admits undecidable arithmetical propositions.

**Completed Infinity**

A significant consequence is that:

Finitary reasoning does not admit an axiom of infinity.

Non-finitary reasoning admits an axiom of infinity.

**Non-standard models of PA**

A further consequence of this is that:

Finitary reasoning does not admit non-standard models of PA.

Non-finitary reasoning too does not admit non-standard models of PA.

**Algorithmically computable truth and algorithmically verifiable truth**

The Birmingham paper also suggests that:

The truths of finitary reasoning are algorithmically computable.

The truths of non-finitary reasoning are algorithmically verifiable, but not necessarily algorithmically computable.

**Categoricity and incompleteness of Arithmetic**

We show in Corollary 1 of this post that it also follows from the Birmingham paper that:

Finitary reasoning proves that PA is categorical with respect to algorithmically computable truth.

Non-finitary reasoning proves that PA is incomplete with respect to algorithmically verifiable truth (a consequence of Gödel’s proof of of the undecidability of some arithmetical propositions in any -consistent system of arithmetic).

**How intelligences reason**

This suggests that:

Finitary reasoning is a shared characteristic of all intelligences, human or non-human.

Non-finitary reasoning is a characteristic of human intelligence that may not be shared by any other intelligence.

**Communication between intelligences: SETI**

It further suggests that the search for extra-terrestrial intelligence may benefit from the argument that:

Finitary reasoning admits effective and unambiguous communication between two intelligences with respect to its (algorithmically computable) arithmetical truths.

Non-finitary reasoning does not admit effective and unambiguous communication between two intelligences with respect to its (algorithmically verifiable) arithmetical truths.

**Determinism, Unpredictability and the EPR paradox**

An unexpected consequence of the arguments of the Birmingham paper is that our perspectives on the relation between determinism and predictability may benefit from the paradigm shift demanded by the argument that:

Finitary reasoning admits the EPR paradox.

Non-finitary reasoning does not admit the EPR paradox.

The arguments of the Birmingham paper also suggest a fresh perspective on the issue of computationalism since:

Finitary reasoning does not admit Lucas’ Gödelian argument.

Non-finitary reasoning admits Lucas’ Gödelian argument.

**Effective computability**

It further suggests that the nature and status of ‘effective computability’ may also need to be assessed afresh since:

Finitary reasoning naturally equates algorithmic computability with effective computability.

Non-finitary reasoning naturally equates algorithmic verifiability with effective computability.

**Church Turing Thesis**

As also the nature of CT, since:

Finitary reasoning admits the Church-Turing Thesis.

Non-finitary reasoning does not admit the Church-Turing Thesis.

Broadly speaking, the two conflicting-but-complementary structures defined in the Birmingham paper suggest that we should be more explicit—in our argumentation—of the structure to which a particular assertion about the natural numbers pertains, since:

Both finitary and non-finitary reasoning do not admit the proof of Goodstein’s Theorem as neither admits a completed infinity.

Set-theoretical reasoning admits the proof of Goodstein’s Theorem as it admits a completed infinity.

**There’s more …**

In the next post we shall consider some further intriguing consequences suggested by the Birmingham paper.

**What do you think?**

Does Goodstein’s sequence over the natural numbers always terminate or not?

**Aristotle’s particularisation: A grey area in our accepted foundational concepts**

We shall now argue that what mathematics needs is not a new foundation, but a greater awareness of the nature of its existing foundations.

In particular, it is the thesis of these investigations that almost all of the unresolved philosophical issues in the foundations of mathematics reflect the fact that the nature and role of Aristotle’s particularisation is left implicit when it is postulated over infinite domains.

Perhaps that is the unintended consequence of ignoring Hilbert’s efforts to integrate the concept formally into first order logic by formally defining universal and existential quantification through the introduction of his -operator.

**Semantic postulation of Aristotle’s particularisation **

Aristotle’s particularisation (AP) is the postulation that from the negation of a universal we may always deduce the existence of a contrafactual.

(*It is necessarily true over finite domains.*)

More formally:

**Aristotle’s particularisation under an interpretation**

If the formula of a first order language interprets as true under a sound interpretation of , then we may always conclude that there must be some object in the domain of the interpretation such that, if the formula interprets as the unary relation in , then the proposition is true under the interpretation.

(*We note that Aristotle’s particularisation is a non-constructive—and logically fragile—semantic deduction rule. It is reflected in classical first order deduction either by some similarly non-constructive syntactic rule of natural deduction—such as Rosser’s Rule C—or by the assumption that FOL is -consistent.*)

**Is the price of Aristotle’s particularisation too high?**

If so, we shall argue that the price being asked for assuming AP implicitly—instead of explicitly as Hilbert had proposed—may be too high!

Partially because the assumption seems to effectively obscure the far-reaching consequences of the non-finitary nature of AP from immediate view in natural and formal deductive chains.

(*And therefore of the first order logic FOL under the implicit assumption of Aristotle’s particularisation.*)

For instance, as Carnap’s deduction of the Axiom of Choice in ZF illustrates, the non-finitary consequences of assuming AP over infinite domains becomes apparent when the underlying logic is taken as Hilbert’s -calculus instead of the classical first order logic FOL.

However, formal deductions apparently prefer to substitute—seemingly arbitrarily—the implicit assumption of AP in the underlying logic by the introduction of `contrived’ formal assumptions such as Gödel’s -consistency or Rosser’s Rule C.

**-consistency**

A formal system S is -consistent if, and only if, there is no S-formula for which, first, is S-provable and, second, is S-provable for any given S-term .

**Rosser’s Rule C**

“Since the rule ‘If , then ‘ corresponds to a hypothetical act of choice, we shall call it the rule of choice, or more briefly, Rule C.”

… J. Barkley Rosser. *Logic for Mathematicians.* 1953. McGraw Hill Book Company Inc., New York.

Similarly natural deduction chains apparently prefer to substitute—again seemingly arbitrarily—the implicit assumption of AP in the underlying logic by admitting a Rule of Infinite Induction (transfinite induction).

More importantly, the price may be too high because the implicit assumption of Aristotle’s particularisation in the underlying logic has masked the fact that, without such assumption, FOL is finitarily consistent; and—as we note below—that mathematically significant finitary structures can be built upon it without assuming AP.

**Evidence-Based Interpretations of PA**

Some consequences of making the assumption of Aristotle’s particularisation explicit are highlighted in `Evidence-Based Interpretations of PA’ that was presented at the AISB/IACAP Turing 2012 conference in Birmingham last year.

We showed there that Tarski’s inductive definitions admit evidence-based interpretations of the first-order Peano Arithmetic PA which allow us to define the satisfaction and truth of the quantified formulas of PA *constructively* over the domain of the natural numbers in *two* essentially different ways:

In terms of algorithmic verifiabilty; and

In terms of algorithmic computability.

That there can be even *one*, let alone *two*, logically sound (one finitary and one non-finitary) assignments of satisfaction and truth certificates to both the atomic and compound formulas of PA had hitherto been unsuspected!

**Definition: Algorithmically verifiable arithmetical truth**

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 .

“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 …”.

… 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.

We show in the Birmingham paper (as we shall refer to it hereafter) that the `algorithmic verifiability’ of the formulas of a formal language which contain logical constants can be inductively defined under an interpretation in terms of the `algorithmic verifiability’ of the interpretations of the atomic formulas of the language; further, that the PA-formulas are decidable under the standard interpretation of PA over if, and only if, they are algorithmically verifiable under the interpretation.

**Definition: Algorithmically computable arithmetical truth**

A number theoretical relation is algorithmically computable if, and only if, there is an algorithm that can provide objective evidence for deciding the truth/falsity of each proposition in the denumerable sequence .

We show in the Birmingham paper that the `algorithmic computability’ of the formulas of a formal language which contain logical constants can also be inductively defined under an interpretation in terms of the `algorithmic computability’ of the interpretations of the atomic formulas of the language; further, that the PA-formulas are decidable under an algorithmic interpretation of PA over if, and only if, they are algorithmically computable under the interpretation.

**Algorithmic verifiability vis à vis algorithmic computability**

We show in the Birmingham paper that the concepts of Algorithmic verifiability and Algorithmic computability are both well-defined under the standard interpretation of PA over ; moreover they identify distinctly different subsets of the well-defined PA formulas.

We show in this paper that although every algorithmically computable relation is algorithmically verifiable, the converse is not true.

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 (as addressed in more detail in this paper) 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.

**The finitarily sound algorithmic interpretation of PA over **

We argued from the above distinction that the algorithmically computable PA-formulas *can* provide a finitarily sound algorithmic interpretation of PA over the domain of the natural numbers.

We showed, moreover, that this yields a finitary proof of consistency for PA—as demanded by the Second of Hilbert’s celebrated Twenty Three Problems.

**The non-finitarily sound standard interpretation of PA over **

On the other hand, the distinction also suggests that Gerhard Gentzen’s transfinite proof of consistency for PA corresponds to the argument that the algorithmically verifiable PA-formulas of PA provide a non-finitarily sound standard interpretation of PA over .

Moreover—as has been generally suspected (perhaps for the reason noted towards the end of this post)—the distinction also suggests why the standard interpretation cannot yield the finitary proof of consistency for PA as demanded by Hilbert.

**The distinction between finitary and non-finitary arithmetical reasoning introduced in the Birmingham paper has far reaching consequences**

In these pages we shall argue that the power of this simple distinction actually goes far beyond the immediate conclusions drawn in the Birmingham paper.

Reason: We can further constructively define an unambiguous distinction between finitary and non-finitary reasoning, at the level of first order logic itself, which shows that the two are both mutually inconsistent yet complementary!

As can be expected, such a distinction could have far-reaching consequences for the foundations of mathematics, logic and computabiity (which form the focus of the investigations in these pages).

We shall consider some of these in the next post.

**“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’.

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