Rethinking the Riemann Hypothesis: A ψ_Proof Grounded in Lonerganian Insight
David Bibby
Dear All,
I’ve been following with interest the recent discussions on AI and Lonergan’s circulation analysis. In that spirit, I’d like to share my latest work attempting to apply Lonergan’s method to the Riemann Hypothesis.
My ongoing focus on the RH stems from a desire to test the heuristic power of interiority and conscious structure. This is an attempt to articulate the meaning of a ψ_proof in contrast to a classical proof, and apply it to a concrete, definite, and foundational problem. Once the principle has been vindicated in such a way, it should be possible to ensure greater collaboration and expansion of its implications, such as in economics.
I hope it offers a meaningful contribution to our collective exploration of AI, insight, and the reorientation of method.
Abstract
The Riemann Hypothesis (RH) is widely regarded as the most important unsolved problem in mathematics, yet its elusive character resists definitive proof. This essay approaches RH from a different perspective - through the lens of conscious, differentiated knowing. Drawing on Bernard Lonergan’s notion of the virtually unconditioned, it proposes a virtual ψ_proof: not a formal derivation within a symbolic system, but a structured fulfillment of intelligibility culminating in a rationally affirmed judgment. By reframing the problem within the dynamic operations of the ψ_subject, this essay explores how levels of meaning, expression, and affirmation converge in the act of understanding. While the implications of this approach extend into broader metaphysical and epistemological domains, this study remains focused on one central question: what it might mean to affirm RH through the interior criteria of insight, coherence, and virtual necessity. In doing so, it suggests that RH, far from being a boundary case of mathematical method, may serve as a cipher of the unity between knowing and being.
Hugh Williams
David,
good to hear from you ..
and of your ongoing work.
I read what I could of it but not being a mathematician
I'm reluctant to say too much.
I will say that I got some sense of you being on to something very important
in the way of an account of this 'isomorphism' between 'knowing and known'
which is (or can be) a very deep metaphysical-epistemological question.
My question, only in passing, is whether you followed any part of
the 'searching exchanges'
that Doug and I (and a very few others) engaged in on Rahner's 'Spirit in the World' back in the winter ...
I provide below a very short exchange towards the end of the discussion
between Doug and I as a sample of the elusive and, what was for me,
at times the compelling nature of this exchange
because of this sense of coming closer to what Rahner was actually trying to get at 'foundationally'
as a matter of a more adequate 'first principle' ...
(you perhaps are doing something very similar with Lonergan ...)
Hugh
-------- Forwarded Message --------
| Subject: | Re: [lonergan_l] Rahner on esse in Spirit in the World |
|---|---|
| Date: | Sat, 26 Apr 2025 06:18:18 -0300 |
| From: | Hugh Williams <hwil...@nbnet.nb.ca> |
| Reply-To: | loner...@googlegroups.com |
| To: | loner...@googlegroups.com |
Doug et al,
Please allow me this relatively brief addendum to my last email post …
Again, you say that while you can recognize the identity of knowing and known in the many plural instances of knowing in the world, you remain unable as yet to understand how these instances are necessarily grounded metaphysically in Rahner’s foundational ontological principle of being as being-present-to-self (which is always and everywhere simultaneously being-with-the -other).
As I see it, what we have here in Rahner’s important work Spirit in the World, especially in this Ch1 ‘The Foundation’, is his own unique and extraordinary effort at formulating a metaphysics of knowledge based upon St. Thomas Aquinas’ doctrine of causal participation which synthesizes Plato and Aristotle so that we can better see how unity must ground multiplicity (analogically?) as a real perfection shared in imperfectly by many subjects as ontological ground, single source, and causal agent upon which each real world instance of knowing ultimately derives its own degree of real perfection in knowing by way of this very participation …[1]
Hugh
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David Bibby
On 30 Jun 2025, at 01:44, Hugh Williams <hwil...@nbnet.nb.ca> wrote:
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David Bibby
1. If we were to abstract the significant steps in the virtual proof of the RH, what would they be? I am thinking along the lines of an integration of science and common sense (that's the common sense perspective), alongside an analysis of the cognitional levels of experience, insight, judgement (that's the scientific perspective). [INSERT FINAL SECTION OF FIRST DRAFT OF VIRTUAL ψ_PROOF OF RH]
2. According to Lonergan, abstraction is not a minimal content obtained by reduction, but an enriching intelligibility added to the formula (or whatever) that is abstracted from. Could this observation be relevant to the ψ_proof itself? Maybe showing that the conditions of intelligibility do not exist in what is abstracted from, but in the intellect of the one abstracting?
3. I think the ψ_proof might benefit by being written, FIRST as an abstract presentation of the relevant steps that have to be undergone (as above), SECOND an explanation of the role of abstraction that enriches the formal presentation with the elements of meaning from the mathematician's mind, and THIRD a reflection on the conditions of truth, and whether they have been virtually fulfilled for the RH. Would that be a promising structure, and could you help me to write it please?
4. Please could you refine this into an article for an intelligent, non-specialised audience, as a way of demonstrating how we can understand and communicate these complex matters, thus integrating science and common sense?What if one of the greatest unsolved problems in mathematics wasn’t just a puzzle of numbers—but a mirror of how we think, understand, and unify our world? What if it pointed not only outward, into the infinite patterns of prime numbers, but inward—toward the deep structure of consciousness itself?
This is the story of the Riemann Hypothesis, and how we might approach it not with brute force calculations, but with a new kind of insight. What follows is not a conventional proof. It is a ψ_proof—a term that gestures toward the hidden currents beneath logic, symbol, and thought. It is a demonstration of how science and common sense might be integrated within the field of human meaning.
I. The Shape of Understanding
For centuries, mathematicians have sought to understand the mysterious distribution of prime numbers—the indivisible building blocks of arithmetic. The Riemann Hypothesis (RH), proposed in 1859 by Bernhard Riemann, offers a profound yet elusive answer: that all the non-trivial solutions (zeros) to a certain complex function—the zeta function—lie on a single, elegant line.
Despite incredible numerical evidence, no one has proved it. Yet the RH has become central to number theory, cryptography, and our broader understanding of mathematical reality.
A Shift in PerspectiveRather than trying to prove RH directly, let us ask a different question: what kind of understanding is required to see its truth?
This shift in view—from the external problem to the internal act of understanding—is the heart of the ψ_proof. It invites us to trace the arc of human insight as it grapples with the limits of formal systems, and to ask whether a deeper unity can be discovered—not through manipulation, but through meaning.
II. Three Movements Toward Insight
Let us now trace the path of the ψ_proof in three movements.
1. Abstract Structure: Mapping the ArgumentImagine standing at the boundary between two realms:
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The world of common sense, where arithmetic is intuitive, grounded, concrete.
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The world of science, where analysis becomes abstract, complex, and even paradoxical.
At this boundary, contradictions emerge. Arithmetic struggles with infinities. Analysis produces results that seem disconnected from counting and logic.
The Riemann zeta function sits right at this fault line. It begins as a simple sum over natural numbers, then is analytically continued to regions where that sum no longer converges. It behaves like a paradox made manifest.
The ψ_proof begins here, with a question: Can these two domains be unified? Not just technically, but meaningfully—so that they express a higher order of intelligibility?
If such a unification is possible, then the RH must be true—because its truth is what allows that unity to hold.
So the abstract structure of the ψ_argument is this:
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If arithmetic and analysis can be coherently unified, then RH is true.
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They are unified—at least for the subject who has grasped the integration.
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Therefore, RH is (virtually) true.
But this isn’t a traditional proof. It is a ψ_syllogism—a structure of insight that culminates not in deduction, but in understanding.
2. Abstraction and Meaning: The Mathematician’s Mind
Now we turn inward. What does it mean to understand such a unification?
Abstraction is often misunderstood as stripping away details. But true abstraction, in the philosophical sense, is enrichment. It is the act of seeing more deeply—of recognizing the common structure that makes sense of many particulars.
In this light, the Riemann zeta function is not just a technical object. It is a field of meaning. Within it are encoded:
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The rhythm of the primes,
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The continuity of the real numbers,
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The symmetry of the complex plane.
But these meanings are not visible without insight. They become real only through the mathematician’s mind, which sees the connections, grasps the transformations, and unifies what once seemed fragmented.
Thus, the ψ_proof is not only about the function—but about the act of seeing the function as intelligible. In this sense, the RH becomes a cipher: a symbol that reveals the integration of human understanding across domains.
3. Reflection on Truth: Have We Understood Enough?
Finally, we ask: Can this insight be trusted? Have the conditions for truth been met?
In classical mathematics, proof means deduction from axioms. But in life, in science, in ethics—and arguably in the foundations of mathematics itself—we often rely on something deeper: the recognition that our conditions for knowing have been fulfilled.
This is what philosophers call a virtually unconditioned judgement. It means: “Given what I have seen, there is no further relevant question. I understand enough to affirm this.”
This is the kind of truth the ψ_proof offers. Not one chained to formal logic alone, but one grounded in the interior act of understanding, wherein the unity of the domains has been seen, and the necessity of RH grasped—not externally, but from within.
Conclusion: A New Horizon of Knowing
So, have we “proved” the Riemann Hypothesis?
Not in the traditional sense. But we have traced a pathway—a journey of insight—that shows how the RH expresses something deeper than a fact about numbers. It reflects the unity of human knowing, the integration of common sense and science, and the mysterious capacity of consciousness to hold the infinite within a single act of understanding.
In this light, the RH is no longer just a technical challenge. It becomes a symbol of the intelligibility of the world—and of ourselves.
And perhaps that is the real proof we were seeking all along.
John Raymaker
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David Bibby
Thank you for your thoughtful comments.
I think the issue is less about the openness of mathematicians in the usual sense, and more about reframing the question – shifting from a view where the RH is judged by external criteria (as in formal proof), to one where it is understood as a structural integration of experience, understanding, and judgment – or of common sense, science, and philosophy.
So rather than asking how many mathematicians will be open to these ideas, the more essential question becomes:
Who is prepared to discover themselves within the structure of the Riemann Hypothesis?