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

Michael Atiyah’s reasoning on the, possible, formal undecidability of the Riemann Hypothesis RH (even though presented as a non-finitary proof), and his main paper on the fine structure constant, are intriguing not so much for their argumentation per se, but because they raise four foundational issues that need to be addressed beyond the context of his reasoning.

1. Relation of $\zeta (\sigma + it)$ to the distribution of primes is valid only for $\sigma > 1$

It is significant that Atiyah does not relate RH to the the distribution of primes, but only to the zeros of the Riemann Zeta-function $\zeta(\sigma + it)$; even though the conventionally perceived significance of RH is essentially that the behavior of $\zeta (\sigma + it)$, using analytic continuation for defining the function for $\sigma \leq 1$, may be treated as yielding the most reliable estimates for the number $\pi (n)$ of primes less than $n$ for all natural numbers $n > 2$.

Perhaps he too is uncomfortable (as I am) with unrestrictedly deducing such conclusions about the behaviour of the primes from the behaviour of $\zeta (\sigma + it)$, using analytic contin- uation, since the relationship of $\zeta (\sigma + it)$ to the primes is well-defined only for $\sigma >1$.

Although continuing a function analytically may satisfy the mathematician’s obsession to extend to its maximum the domain over which a function can be theoretically defined, it tends to obscure, under an interpretation, that which the function was originally defined to represent.

In other words, if an event (say, for instance the emergence of primes in an Eratosthenes sieve operation) is described by some mathematical function which is defined only for points along the $x$-axis for $x>0$, then how can we deduce any putative property or behaviour of the event from the behaviour of the function when $x<0$; i.e., in areas where the function might, but the event does not, exist?

To see this note that, classically (E. C. Titchmarsh: The Theory of the Riemann Zeta-Function), the Zeta-function $\zeta(\sigma + it)$ relates the distribution of the primes through the Euler product to a fundamental Dirichlet series, where both are convergent for $\sigma > 1$, by the argument that:

$[1]\ \ldots\ \zeta(\sigma + it) = \prod_{_{p}}(1-\frac{1}{p^{\sigma + it}})^{-1} = \sum_{n=1}^{\infty}(\frac{1}{n^{\sigma + it}})$

where the Euler product ranges over all primes $p$.

Now, if $\sigma > 1$, we can re-arrange the terms of the Dirichlet series to yield (see p.145 in Derbyshire, J. 2002. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Published by Plume. 2004. Penguin Group (USA) Inc., NY):

$[2]\ \ldots\ \{\sum_{n=1}^{\infty}(\frac{1}{n^{\sigma + it}})\}(1-\frac{1}{2^{\sigma -1 + it}})= \sum_{n=1}^{\infty}((-1^{n+1})\frac{1}{n^{\sigma + it}})$

using the fact that, if the infinite sum $a+b+c+d+e+f+ ...$ is absolutely convergent, then:

$a-b+c-d+e-f+ ... = (a+b+c+d+e+f+ ...) - 2(b+d+f+ ...)$

However, we can now view $\zeta'(\sigma + it)$ as an analytic continuation of $\zeta(\sigma + it)$, defined implicitly for $\sigma > 0,\ \sigma \neq 1$ by:

$[3]\ \ldots\ \zeta'(\sigma + it)(1-\frac{1}{2^{\sigma -1 + it}})= \sum_{n=1}^{\infty}((-1^{n+1})\frac{1}{n^{\sigma + it}})$

In such case, though, since the Euler product diverges everywhere for $\sigma \leq 1$, whilst $(1-\frac{1}{2^{\sigma -1 + it}})$ converges for $\sigma < 1$, it is not obvious whether the re-arrangement of terms in the implicit definition of $\zeta'(\sigma + it)$ admits the claim that:

$[4]\ \ldots\ \prod_{_{p}}(1-\frac{1}{p^{\sigma + it}})^{-1} = \zeta'(\sigma + it) = \{\sum_{n=1}^{\infty}((-1^{n+1})\frac{1}{n^{\sigma + it}})\}(1-\frac{1}{2^{\sigma -1 + it}})^{-1}$

for $\sigma > 1$, and that the Euler product can be analytically continued through this identity for $0 < \sigma < 1$.

Reason: Since the Euler product is defined only for $\sigma > 1$, it is not obvious how the behaviour of $\zeta'(\sigma + it)$ in any region $\sigma \leq 1$ can be related to the distribution of the primes merely because of some incidental identification of the Euler product with $\zeta'(\sigma + it)$ outside this region.

Incidental, since the distribution of primes is related to $\zeta'(\sigma + it)$ by the identity, for $\sigma > 1$:

$[5]\ \ldots\ log_{e} \{\prod_{_{p}}(1-\frac{1}{p^{\sigma + it}})^{-1}\} = s \int_{_{2}}^{^{\infty}}(\frac{\pi(x)}{x(x^{\sigma + it}-1)})dx$

where the emergence of $\pi(x)$ also involves the re-arrangement of an infinite series which is contingent upon the condition that $\sigma > 1$.

(Just as some behaviour towards me of my brother-in-law is conditional upon, and antecedent to, my marriage on a specific date.)

The above suggests that it is perhaps worth reflecting, for a moment, upon the implications of conventional reasoning for mathematically modelling the behavior of real world events as exemplified, for instance, by a light signal $S$ which is emitted from some origin along an $x$-axis, so its distance from the origin at time $t>0$ is $x=ct$, where $c$ is the speed of light.

Obviously, the function $x=ct$ is well-defined for values of $t \leq 0$.

However, no such value of the function can describe any property of the signal $S$, since $S$ did not exist for $t \leq 0$.

In other words, the behaviour of a mathematical expression at limiting points may not be unique—as is necessary and, prima facie, implicitly assumed, in any analytic continuation—but may be contingent on that which such an expression is intended to represent under an evidence-based interpretation.

The following para in Section 5, p.223, of this intriguing paper by Yuriy Zayko seems to make a similar point:

“… the $\zeta$-function calculations for real arguments are connected by certain contradictions that are resolved if we assume that the geometry of the numerical continuum is different from the Euclidean one. Note that the distinctive feature of the results obtained is the presentation of calculation as the process (as it really is) and the introduction of the concept of time, what is not typical for traditional mathematics, that claims its statements are timeless.”

Namely that, in order to analytically continue $\zeta(\sigma + it)$ uniquely outside $\sigma > 1$, we may need to define additional parameters—such as, say, the geometry over which the function is defined—which would allow us to describe the behaviour of $\zeta(\sigma + it)$ at $\sigma = 1$ uniquely.

(Prima facie, conventional wisdom does not feel the need to establish such uniqueness for the analytic continuation of a function.)

In other words, divergence of a function at a point ought not to be accepted as necessarily implying that the function cannot be defined at the point, but only that the function may not have been well-defined at that point due to some implicit assumptions (as was shown in the case of parallel lines by the recognition of non-Euclidean geometry).

This, of course, reflects the thesis of my investigations into evidence-based reasoning; which is that the properties of any mathematical expression are determined not only by what can be formally proven within the formal language in which the expression is well-defined, but also by the evidence-based truth assignments to the expression under a well-defined, evidence-based, interpretation.

2. Limit of infinite iterations needs justification; which is why physicists resort to renormalisation

In case this seems far-fetched, Sections 4 to 6 of this abstract geometrically represent and analyse a two-dimensional Zeno-type gedanken where—in striking contrast to the classical, one-dimensional, Zeno argument—even the theoretical limiting behaviour of a putative virus cluster, or the putative behaviour of an elastic string, under a specified iterative transform- ation, does not correspond to the putative Cauchy limit of the function that seeks to describe a property of the iteration!

In other words, admitting such a limit in these gedanken can only be described as admitting a misleading mathematical myth into our putative representation of the corresponding natural phenomena.

The significance of this is that Hadamard and de la Vallée Poussin’s proof of the Prime Number Theorem deduces the behaviour of the prime counting function $\pi (n)$, as $n \rightarrow \infty$, from the limiting behaviour of $\zeta' (\sigma + it)$ along the line $\sigma = 1$.

However, as these graphs (see also this post) of evidence-based, non-heuristic, estimations of $\pi (n)$ illustrate, rather than unrestrictedly assuming that the existence of a Cauchy limit for a Cauchy sequence that represents a real world event implies that the event must tend to a limit corresponding to the Cauchy limit of its representation, we may need to seriously consider the possibility that $\frac{\pi (x)}{x/log_{e}(x)}$ may be discontinuous in the limit $x \rightarrow \infty$.

(Or, in the jargon of physicists, $\pi (x)$ too may need renormalization as $x \rightarrow \infty$.)

3. Fine structure constant: Algorithmically verifiable but not algorithmically computable.

Curiously, Professor Atiyah’s evaluation of the fine structure constant $\alpha$ in this paper also seems to involve deductions from the limiting behaviour of the function defined by his eqn.7.1 on p.8 of the paper.

Whether such a mathematical limit, which involves constructing `an infinite sequence of iterated exponentials, followed by taking the weak closure in Hilbert space’ can be proven to correspond to the value of the fine structure constant $\alpha$ that it seeks to evaluate is not obvious.

Reason: It can be reasonably argued (see p.273 of Section 29.6 in Chapter 29 of this thesis) that the fine structure constant $\alpha$ can only be represented mathematically by a function that is algorithmically verifiable, but not algorithmically computable (see Definitions 5.2 and 5.3 on p.38, Section 5.1 of this thesis).

That Atiyah too—albeit implicitly—views the digits in the decimal representation of $\alpha$ to be defined by a function that is algorithmically verifiable, but not algorithmically computable, in the above sense is evidenced by his comments on p.10 of his discussion on the computation of the initial values of the fine structure constant:

7.7. The two formulae (1.1) and (7.1) now follow by mimicry from the corresponding formulae for $\pi$ and $\lambda$. More precisely, each truncated version, for fixed $n$ is analytic. The mimicry is then applied before we pass to the limit.

7.8. The next section on numerics will carry out the computations and check the answers at every level $n$. This provides a proof analogous to those of Archimedes and Euler and is sufficient for physicists. But, as mentioned in 6.6, mathematical logicians since Gödel have raised the bar for the notion of proof. It is probable that such computations will never fail, i.e. a counterexample will never be found, but that it is not possible to prove this without additional axioms.

7.9. The logical issues raised in 7.8 appear in a different form in the size of the steps needed for the effective computation of further decimals in the expansion of Ж. I asserted, for example, that 16 should be sufficient to produce 9 figure accuracy. There is no algorithm that will confirm this fact, though general theory will guarantee that some (unknown) large number would be sufficient. But someone doing the calculation is likely to stumble on a sufficient number, probably finding that 16 is enough. But 12 figure accuracy might well require 32. So, the logical qualms that hard-nosed physicists ignore at the conceptual level, come back to haunt them at the computational level. Gödel can only be ignored at your peril.”

Should it turn out that Atiyah has, indeed, given a valid mathematical definition of the fine structure constant $\alpha$ which—like Chaitin’s family of $\Omega$ constants—is algorithmically verifiable, even if not algorithmically computable, that in itself could be viewed as a major achievement (which the title and introductory paragraphs of his main paper implicitly indicate as his goal), far over-shadowing in significance whether or not his perceived perspective of its implications for RH is, or is not, eventually vindicated.

4. Mapping infinite real dimensions as finite integer dimensions violates Skolem’s dictum

Mapping infinite real dimensions as finite integer dimensions, as Atiyah apparently does in Sections 2.1 to 2.6 of his paper, seems to implicitly violate Skolem’s dictum cautioning about the paradoxes inherent in corresponding entities and entailments across the domains of different formal systems.

For instance, Section 22.4 on p.189 of this thesis analyses how ignoring Skolem’s dictum leads to the seriously misleading conclusions usually drawn from Goodstein’s theorem, if we do not distinguish between the axiomatic constraints within which the properties of arithmetically defined numerals of a first-order Peano Arithmetic are defined, and the axiomatic constraints within which the properties of the corresponding, intuitively isomorphic, set-theoretically defined finite ordinals of a first-order set theory such as ZF are defined.

The significance of this for Atiyah’s following reasoning is not obvious:

“In this section I will indicate the main strategy and formulate the precise statements which will be proved. The key question is how to extend the Archimedes-Euler approach to $\pi$, from the commutative world of $\mathbb{C}$ to the non-commutative world of Hamilton’s quaternions $\mathbb{H}$, constructed by an infinite iteration of exponentials, discrete or continuous. The key idea goes back to von Neumann as I now explain.

2.1 The von Neumann hyperfinite factor $A$ of type II. The hyperfinite type II-1 factor A is unique up to isomorphism. It is constructed by an infinite sequence of iterated exponentials, followed by taking the weak closure in Hilbert space. This process converts type I, with integer dimensions, to type II with real dimensions. A dimension which is formally infinite in type I becomes finite in type II, so I call the process renormalization. All factors have an invariant trace mapping the factor to its centre. Any inner automorphism gives an isomorphic but different trace. The comparison between two such automorphisms is what leads to the Todd map T (see 3.4).

2.2 The Hirzebruch Formalism. Hirzebruch [1], following in the footsteps of Euler and Riemann, introduced a formal algebraic process of multiplicative sequences. In such processes he defined exponentials over $\mathbb{Q}$. He showed that any such exponential has a generating function, and he focused on the Todd exponential, whose generating function is the Bernoulli function $\frac{x}{1-e^{-x}}$. The fact that this function is analytic implies that the Hirzebruch process extends from $\mathbb{Q}$ to $\mathbb{R}$, implementing the weak closure of 2.1.”

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