Er is geen enkel verschil tussen wetenschap (incl wiskunde),kunst als articulatie van de persoonlijkheid ,een mensenleven. of een samenleving hoe groot en divers die ook lijkt
Ze ontmoeten elkaar op een knoop in het gote visnet van het universum.
J.Konstapel Leiden,28-8-2026.
Short Summary
The MAZE is a system designed to store the world’s “knowledge” efficiently.
It uses trits as its fundamental unit of information.
Knowledge is organized as interconnected structures rather than isolated documents.
This makes relationships between pieces of knowledge explicit and searchable.
The system aims to reduce redundancy while preserving meaning and context.
In essence, the MAZE treats knowledge storage as a structured, navigable machine.
Reason for this Blog:
Today I received the “Leiden Declaration on Artificial Intelligence and Mathematics“
It is a excellent opportunity to try out the MAZE
TThe MAZE is an experimental operational system for effectively storing all the world’s “knowledge” using ‘Trits.
Introduction: From the Proof to the Place of the Proof
The arrival of artificial intelligence in mathematics is often described as a problem of correctness. Can a machine produce a valid proof? Can a human mathematician trust an AI-generated argument? Can formal verification establish that the derivation is logically sound?
These are important questions, but they may not be the deepest ones.
The more fundamental problem is what happens when a mathematical result can be correct without having an intelligible origin. A proof can be logically valid while its authorship is uncertain, its novelty difficult to establish, its intellectual ancestry invisible, and its existence dependent upon a proprietary computational system to which the mathematical community has no access. Correctness, in other words, does not automatically provide provenance.
This is the central philosophical proposition developed in The Address of a Proof. The essay places the Leiden Declaration on Artificial Intelligence and Mathematics beside the MAZE, a public implementation of a self-addressing knowledge architecture. Its argument is that the four dilemmas identified by the Leiden Declaration — responsibility, credit, novelty, and access — are not four independent ethical questions. They are manifestations of a single infrastructural problem: mathematics historically relied on the fact that a proof had a human signature, while machine-generated mathematics can detach mathematical content from the social and institutional mechanisms that made that signature meaningful.
The official Leiden Declaration, published on 2 June 2026, approaches the problem primarily through responsibility, transparency, attribution, open science, human authorship, and community governance. It explicitly calls upon mathematicians to disclose AI use, retain responsibility for correctness, acknowledge prior work, preserve human authorship, and participate in shaping the future of the discipline.
The MAZE essay accepts the seriousness of these concerns but proposes a different level of intervention. Its claim is architectural rather than primarily normative: instead of asking participants in the mathematical system to behave properly, construct a substrate in which provenance, attribution and priority become properties of the system itself.
This difference is more than a technical disagreement. It represents two different philosophies of knowledge.
One philosophy says that trustworthy mathematics depends upon trustworthy agents following appropriate norms.
The other asks whether some of those norms can be transformed into properties of the knowledge infrastructure.
The distinction matters because artificial intelligence changes the scale and speed at which mathematical artifacts can be generated. A world in which millions of machine-generated proofs can be produced cannot rely indefinitely on the same mechanisms of attribution and priority that evolved for a world in which proofs were written by identifiable human beings and published sequentially in journals.
The question, therefore, is not simply whether machines can prove theorems.
It is whether mathematics can know where its proofs are.
1. The Leiden Problem: Four Dilemmas That May Be One
The Leiden Declaration emerged from a 2025 workshop at the Lorentz Center in Leiden involving mathematicians, computer scientists, philosophers, historians and social scientists. The resulting declaration addresses the increasing use of AI in mathematical research and asks the mathematical community to preserve its fundamental values while adapting to new technologies.
Its concerns can be understood through four questions.
Who is responsible when an AI system produces an erroneous mathematical result?
Who deserves credit when a machine produces a correct result?
How can mathematicians determine whether an apparently new result is genuinely new rather than an undisclosed reformulation of previous mathematics?
And what happens to mathematical autonomy when important mathematical systems are developed inside proprietary companies whose models and training processes are inaccessible to researchers?
These questions appear to belong to different categories. Responsibility is ethical. Credit is social. Novelty is epistemological. Access is institutional and political.
The argument of The Address of a Proof is that this classification is misleading.
All four questions concern the relationship between a mathematical object and its provenance.
A proof is not merely a sequence of logical transformations. Within mathematics, it is also an event in an historical knowledge system. Someone produced it. Something preceded it. It was communicated at a particular time. It may have been derived from earlier work. It may have corrected another result. It may have introduced a new idea. It may have been independently discovered. It may have been generated by a machine. All these relationships belong to the identity of the mathematical artifact.
The traditional mathematical publication system stores this information indirectly. The author is on the paper. The journal records the publication date. Citations indicate ancestry. Peer review establishes a social mechanism of validation. Institutional affiliations provide additional context.
In this sense, the mathematical proof historically came with an address.
The address was the author’s name, the manuscript, the journal, the date and the citation network.
The AI problem begins when the proof and the address become separable.
The PDF expresses this transformation particularly sharply: for twenty-five centuries, the provenance of a proof was effectively a person. Machine-generated mathematics dissolves that relationship.
This is why the problem cannot be reduced to hallucination, reliability or formal correctness.
A perfectly correct proof can still be problematic if nobody can establish who generated it, when it first appeared, what previous work it depends upon, whether another researcher independently produced it, or whether its creator has the authority to modify or withdraw it.
The problem is therefore not merely epistemic.
It is ontological.
What kind of object is a mathematical result when its provenance is no longer inherent in its mode of production?
2. The Difference Between Correctness and Provenance
Formal verification represents one of the great achievements of contemporary mathematical practice. Systems such as Lean and Coq can check formal derivations against precisely defined logical foundations. The PDF rightly emphasizes the historical significance of machine-verified mathematics, including the Four Color Theorem, the Feit–Thompson theorem, the Flyspeck formalization of the Kepler conjecture, and developments in Lean’s mathematical library.
The Leiden Declaration also treats formalization and computer-assisted methods as important parts of the changing mathematical landscape.
But formal verification answers a particular question:
Is this derivation valid within the specified formal system?
That question is extraordinarily powerful.
It is not, however, identical to:
Who produced this result?
Was it produced before?
Who should receive credit?
What intellectual lineage does it belong to?
Was it generated independently?
Is it publicly accessible?
Formal verification therefore solves a correctness problem without necessarily solving a provenance problem.
This distinction resembles the difference between verifying a physical object and establishing its ownership history. A laboratory can establish that a painting is authentic in a technical sense without thereby establishing who first discovered it, who owned it, or whether its historical provenance has been falsified.
Mathematics has traditionally combined these functions because the human author served as a common anchor.
Artificial intelligence separates them.
An AI system can generate a proof. A proof checker can verify it. But the verifier does not thereby know its intellectual history.
This is the philosophical importance of the distinction.
Correctness concerns the internal relation between premises, rules and conclusion.
Provenance concerns the external relation between an artifact and the history of its production, transmission and recognition.
A proof checker examines the first relation.
A knowledge registry must preserve the second.
The PDF therefore describes formal verification as an “oracle without a library.”
The metaphor is exact.
An oracle can answer whether something is correct. A library can tell us where something belongs.
Mathematics needs both.
3. From Library to Land Registry
The most important conceptual move in the MAZE proposal is the replacement of the metaphor of a library with that of a land registry.
A library organizes objects according to a catalogue. A land registry does something more fundamental: it assigns a legally meaningful place to an object or property and records its relation to an identifiable owner or claimant.
The distinction becomes crucial when knowledge is generated at machine speed.
In a traditional library, a new book has to be catalogued. The catalogue is external to the book.
In the MAZE architecture, by contrast, the address is derived from the content itself.
The PDF describes the MAZE as a balanced-ternary address space in which an address is a finite route consisting of +1, 0 and −1 movements from a common origin. Content is mapped into this space, and the resulting address is intended to provide a computationally derived place for the material.
This creates an important philosophical inversion.
The conventional question is:
“Where should we put this piece of knowledge?”
The MAZE asks:
“Where does this piece of knowledge belong by virtue of what it is?”
That is a profound change.
Classification normally precedes placement. In the MAZE conception, placement becomes part of classification itself.
The address is not merely a label attached by an external librarian.
The address is an expression of the content.
This makes the knowledge space self-addressing.
The idea has intellectual precedents. The PDF connects it to Gödel numbering, Spencer-Brown’s concept of the first distinction, Borges’s imaginary Library of Babel, and the balanced-ternary logic developed in the theoretical program underlying the MAZE.
But the philosophical ambition goes further than any one of these analogies.
The goal is to create an epistemic geography.
A mathematical result does not merely exist.
It occupies a place.
And if the place is computationally determined, then questions of novelty and priority can potentially become questions of location.
4. The Address as an Epistemic Primitive
The most interesting idea in the MAZE architecture may therefore not be the database, the cryptography or even the retrieval mechanism.
It is the concept of address.
An address normally identifies a location independently of what occupies it. The address of a house remains the same whether the house is empty, demolished or occupied.
The MAZE reverses this relation.
The content determines the address.
That means that the knowledge space can exist before the knowledge itself is discovered.
An empty address is therefore not merely an absence.
It is a structured absence.
The PDF calls the unknown a “first-class object.” An empty location has a depth, neighbors, an opposite and a route through the larger address space. The system can consequently record not only what is known but also where its architecture indicates that something is missing.
This introduces an unusual philosophy of ignorance.
Modern scientific databases usually represent ignorance negatively: there is no record.
The MAZE attempts to represent ignorance positively: there is a place whose occupation is absent.
The difference is substantial.
If the unknown has a coordinate, it becomes possible to discuss the geography of ignorance.
The scientific question changes from:
“What do we not know?”
to:
“Where are the structured absences in our current map of knowledge?”
That is potentially significant for artificial intelligence.
A machine that merely generates answers can produce an enormous number of locally plausible results.
A machine operating against a structured map of vacancies can instead be directed toward regions where the knowledge system itself indicates that something is missing.
The unknown becomes a research program.
5. Provenance by Construction
The central institutional innovation proposed by the MAZE is the receipt principle.
According to the PDF, every admitted piece of material carries a source identifier, a provenance trail and a cryptographic source key. The right to delete the material is connected to the same key.
The conceptual significance is that provenance is no longer merely an obligation imposed upon the author.
It becomes a condition of admission.
This distinction between provenance by declaration and provenance by construction is fundamental.
A declaration says:
“Please disclose where this came from.”
An architectural system says:
“Nothing enters unless its origin is recorded.”
The first depends upon compliance.
The second depends upon system design.
The difference is analogous to the historical development of digital signatures and version control. The PDF argues that scholarly infrastructure became more reliable when certain disputes could be resolved mechanically rather than through appeals to individual virtue or institutional authority.
This does not mean that governance is unnecessary.
Architecture cannot decide every normative question.
But architecture can change the cost and possibility of deception.
The philosophical principle is therefore simple:
A system should not ask human beings to remember or voluntarily perform a function that can safely and transparently be performed by the infrastructure.
For mathematical provenance, this means that attribution should not depend entirely on a reader reconstructing history from papers, citations and institutional memory.
The history should travel with the mathematical object.
6. Responsibility, Credit, Novelty and Access
The architectural proposal becomes most interesting when applied directly to the four Leiden dilemmas.
Responsibility
If a result is admitted under a cryptographic source key, responsibility can become associated with the identity under which the result entered the registry.
The PDF proposes that the deletion right should hang on the same key. Responsibility and provenance therefore become structurally related.
This does not eliminate responsibility in the philosophical sense.
It makes responsibility traceable.
Credit
Credit becomes a property of the receipt.
A human-authored proof can be associated with its author. An AI-generated proof can be registered under an identifier representing the model or system that produced it, together with the human or organization responsible for submitting it.
The point is not to give an AI “honor” in the human sense.
The point is to distinguish origin from authorship and ensure that the origin cannot disappear.
This complements the Leiden Declaration’s insistence that human authorship and responsibility remain central.
Novelty
Novelty is perhaps the most difficult problem.
A theorem can be expressed in different languages. Two proofs can use different terminology while being mathematically equivalent. Semantic equivalence can be undecidable in general.
The MAZE does not claim to solve this problem completely.
Its more modest proposition is that a computationally derived address can turn certain novelty questions into lookup questions.
If the same content, or sufficiently equivalent content at the system’s resolution, already occupies an address, its receipt can reveal that the territory was already occupied.
The PDF explicitly limits the claim: the current system detects placement collisions rather than semantic equivalence in full generality.
This limitation is important.
The value of the architecture does not depend upon solving the philosophical problem of identity once and for all.
It is sufficient to make some previously opaque relationships mechanically visible.
Access
Access is the most political of the four questions.
If mathematical discovery increasingly depends on proprietary models, then the mathematical community risks becoming dependent upon institutions that control important parts of the production process.
The Leiden Declaration explicitly raises questions about proprietary AI systems, open science and the autonomy of mathematics.
The MAZE offers a different definition of openness.
A company may retain a closed model.
But if a mathematical result is to participate in the public mathematical knowledge system, its result can be required to enter a public registry with a provenance record.
The distinction is between access to the machine and access to its mathematical consequences.
This is potentially a powerful institutional principle.
Open mathematics does not necessarily require open weights.
It may require open results, open provenance and open priority.
7. The Political Philosophy of Mathematical Infrastructure
The argument therefore has implications beyond software architecture.
Infrastructure is never neutral.
The design of a knowledge system determines which actions are easy, which are difficult, which are visible and which disappear.
A publication system makes certain forms of scholarship possible.
A citation system creates a particular conception of intellectual ancestry.
A peer-review system establishes a particular relationship between expertise and institutional authority.
A digital registry creates another kind of epistemic order.
The MAZE proposal can consequently be understood as a political philosophy of mathematical knowledge.
It asks who should control the map.
If private AI laboratories become the principal producers of machine-generated mathematics, then they may acquire de facto authority over what mathematical objects are generated, which ones become visible and which ones remain inaccessible.
A public provenance layer changes that balance.
The model can remain private.
The public record does not have to be.
This is where the MAZE and the Leiden Declaration are closer than they initially appear.
The Declaration argues that mathematicians should retain agency over the direction of their discipline and should carefully consider their relationship with AI companies and proprietary tools.
The MAZE translates part of that aspiration into infrastructure.
The philosophical question becomes:
Can intellectual autonomy be encoded?
If it can, then autonomy is no longer only a matter of professional culture.
It becomes partly a property of the technical system through which mathematical knowledge circulates.
8. The Deeper Meaning of the “Signature”
The metaphor of the signature deserves closer attention.
A signature does more than identify an author.
It connects an artifact to an agent capable of standing behind it.
When a mathematician signs a paper, the signature implicitly carries several commitments:
“I produced or endorse this.”
“I accept responsibility for it.”
“I claim priority where appropriate.”
“I can be questioned about it.”
“I belong to a community in which these claims have consequences.”
Machine-generated mathematics destabilizes this bundle.
The machine can generate the proof.
The researcher can submit it.
The company may own the model.
The training corpus may contain millions of human contributions.
The formal verifier may certify the result.
Who, then, is the author?
The question is difficult partly because “authorship” has traditionally combined several different relationships.
There is causal authorship: who produced the artifact?
There is intellectual authorship: who contributed the idea?
There is legal authorship: who possesses rights?
There is institutional authorship: who stands behind the publication?
There is epistemic responsibility: who can answer for its correctness?
AI makes it increasingly difficult to assume that one name can represent all these relations.
The appropriate response may therefore not be to find a new singular “author.”
It may be to record a richer provenance structure.
The receipt is valuable precisely because it can separate these dimensions rather than forcing them into one name.
The result can have a computational origin, a human sponsor, a formal verifier, a publication event, a sequence of revisions and a network of antecedent results.
The mathematical object becomes an object with history.
9. Borges, Gödel and the Philosophy of the Complete Space
The MAZE’s philosophical background becomes clearer when viewed through Borges and Gödel.
Borges’s Library of Babel imagines a library containing every possible book. The paradox is that completeness does not produce knowledge. If everything exists, meaningful information becomes indistinguishable from noise.
The MAZE takes the same intuition and makes one crucial alteration.
The catalogue becomes the shelf.
In the Library of Babel, the problem is not the existence of books but the impossibility of locating the meaningful one.
In the MAZE, the proposed address is computed from the content itself. The possibility space therefore becomes navigable through the structure of the thing being sought.
Gödel provides another conceptual precedent.
Gödel numbering demonstrated that formal expressions can be assigned numerical identities through construction rather than through arbitrary cataloguing. The MAZE generalizes the intuition: knowledge can potentially be given a computationally derived location.
The philosophical importance is not the numerical representation itself.
It is the transition from external naming to internal addressability.
A knowledge object can, in principle, carry the means of locating itself within the larger system.
This is what makes the MAZE more than another database.
Its ambition is to turn knowledge from a collection of documents into a structured space.
10. The Census of Ignorance
The MAZE’s occupancy census gives this philosophical argument an empirical dimension.
The PDF reports that, as of 28 August 2026, the public network contained 6,691 inhabited windings carrying approximately 3.6 million pieces of material. It then measures occupancy across different rings of the address space.
The striking feature of the census is the rapid decline in occupancy at deeper levels of refinement.
At ring 4, 40 of 81 addresses are occupied.
At ring 5, 122 of 243.
At ring 12, 1,673 of 531,441.
At ring 24, 516 of more than 282 billion.
The philosophical interpretation proposed by the essay is that human knowledge occupies only a tiny filament of the possible space of knowledge.
This should not be mistaken for a census of reality.
It is a census of the system’s sources, and the PDF explicitly acknowledges that limitation.
But even with that qualification, the concept is valuable.
Science traditionally measures what it has discovered.
A self-addressing knowledge system can additionally measure how much of its own possibility space it occupies.
That creates a new epistemic metric:
not knowledge alone, but knowledge density.
The scientific landscape becomes a topology.
Dense regions represent mature bodies of knowledge.
Sparse regions represent underdeveloped domains.
Empty regions may represent either ignorance, inadequate coverage, or potentially unexplored conceptual territory.
The distinction between “unknown” and “not yet represented” becomes visible.
11. From Search Engines to Scientific Cartography
The difference between an ordinary search engine and a knowledge registry is therefore fundamental.
A search engine retrieves documents containing relevant words.
A registry attempts to establish the location and provenance of knowledge objects themselves.
Search is fundamentally retrospective.
A registry can be prospective.
If the address space exists before a result is discovered, then an empty address can become an invitation to research.
This transforms artificial intelligence from an answer-generating technology into a space-exploring technology.
Instead of asking:
“Can the AI solve this problem?”
we can ask:
“Which regions of the knowledge space remain structurally unoccupied, and can the AI explore them?”
This is a much more interesting conception of machine intelligence.
It changes the machine from an oracle into an explorer.
The distinction also matters for scientific institutions.
If research agendas are determined entirely by existing literature, researchers tend to move toward already visible questions.
A map containing explicit vacancies could make absence itself actionable.
The research program becomes:
locate the vacancy;
formulate the distinction;
attempt the proof;
verify the result;
register the result;
attach its receipt;
and thereby transform an empty coordinate into an inhabited one.
Scientific discovery becomes a process of changing the topology of the knowledge space.
12. The Self-Test and the Philosophy of Failure
Perhaps the most philosophically convincing part of the PDF is its postscript.
The essay was itself submitted to the machinery it describes. The system assigned it a signature and address, but the resulting classification also exposed a defect. The system incorrectly associated aspects of the text with “art” and “cities and construction” because it interpreted metaphors such as “architectural” and “by construction” too literally.
The authors do not hide this failure.
They publish it.
That is significant.
An infrastructure for provenance should not merely record successful outputs. It should record failures of its own mechanisms.
This is a deeper principle than conventional software testing.
A trustworthy epistemic instrument must be capable of representing its own limitations.
The MAZE therefore contains, in miniature, a philosophy of scientific instrumentation:
the instrument must be testable;
its failures must be observable;
its limitations must be recorded;
and its claims must remain proportional to what it can actually establish.
This is why the limitations section of the PDF is important. The authors explicitly state that the current semantic coding is too coarse to serve as a definitive fingerprint for mathematical proofs and that full semantic equivalence is not solved.
The argument becomes stronger because it does not depend upon pretending that the system is finished.
The claim is narrower:
provenance-by-construction is technically realizable;
it can already be implemented in a working system;
and its limitations can themselves be measured.
That is a defensible philosophical position.
13. What the MAZE Does Not Solve
A serious architectural proposal must also identify what it cannot solve.
First, an address derived from content does not automatically establish semantic identity.
Two proofs can be mathematically equivalent while being syntactically different. Conversely, similar formulations may represent genuinely different mathematical claims.
Second, a provenance record cannot establish intellectual responsibility by itself.
A person may submit material under a legitimate key without having understood it.
Third, a public registry cannot guarantee that all relevant knowledge enters the registry.
The PDF’s occupancy figures therefore describe the knowledge represented in the system, not the totality of human mathematical knowledge.
Fourth, cryptographic identity is not equivalent to moral responsibility.
A key can establish who controlled an identity. It cannot by itself determine whether that person behaved ethically.
These limitations matter because they establish the proper boundary between infrastructure and governance.
Architecture can make provenance visible.
It cannot make human beings wise.
It can establish priority records.
It cannot decide what deserves intellectual credit in every philosophical sense.
It can expose an absence.
It cannot guarantee that the absence is scientifically meaningful.
The correct conclusion is therefore not that infrastructure replaces ethics.
It is that ethics becomes more effective when infrastructure preserves the facts upon which ethical judgment depends.
14. The Synthesis: Verification Plus Registration
The most constructive interpretation of the relationship between the Leiden Declaration and the MAZE is therefore not opposition but synthesis.
The Leiden program and the MAZE solve different halves of the same problem.
Formal verification provides a mechanism for establishing correctness.
The registry provides a mechanism for establishing place and provenance.
Together they create a potentially stronger mathematical infrastructure:
the proof is checked;
the result is addressed;
the origin is receipted;
the publication is timestamped;
the relation to existing results is visible;
and the object becomes part of a public mathematical memory.
The PDF proposes precisely this combination: a proof submitted to the MAZE would carry a machine-checkable certificate, and admission to the registry would require successful verification. The address would then provide a novelty mechanism, while the receipt would preserve credit and responsibility.
This is where the “wrong horse” metaphor becomes more subtle.
The issue is not governance versus verification.
Nor is it MAZE versus formal proof.
The deeper architectural possibility is verification plus registration.
The verifier establishes:
“This proof is valid.”
The registry establishes:
“This proof has this place, this origin and this history.”
Together they approach something mathematics has never previously needed at this scale:
a public provenance layer for machine-generated mathematical knowledge.
15. A New Philosophy of Mathematical Memory
The deepest consequence may concern memory.
Human mathematics has always been partly a memory system.
Libraries remember books.
Journals remember publications.
Citation networks remember intellectual ancestry.
Universities remember institutional affiliation.
Researchers remember arguments and traditions.
But these systems were designed for a human-scale rate of knowledge production.
AI changes the rate.
If machines can produce mathematical artifacts faster than human communities can read, classify, compare and attribute them, then the bottleneck moves from generation to memory.
The future mathematical problem may not be:
“How do we produce more proofs?”
It may be:
“How do we remember the proofs we produce?”
This is why provenance becomes infrastructural.
Without a memory architecture, machine-generated mathematics risks becoming epistemic noise.
With such an architecture, machine generation could instead expand the mathematical landscape while preserving historical continuity.
The essential technological achievement would therefore not be artificial proof production.
It would be artificial mathematical memory.
Conclusion: The Address of a Proof
The Leiden Declaration is fundamentally concerned with preserving the values of mathematics in an era of artificial intelligence. Its recommendations emphasize responsibility, transparency, attribution, open science, human authorship, community participation and autonomy. These principles are necessary.
But the MAZE argument suggests that principles alone may be insufficient.
The arrival of machine-generated mathematics changes not merely how proofs are produced but how mathematical objects acquire identity.
For centuries, a proof came with a human signature.
That signature connected correctness to responsibility, discovery to credit, novelty to priority and publication to access.
AI can break those connections.
The response should therefore not be limited to asking humans to behave better after the fact. Some of the missing relationships can potentially be reconstructed in the architecture through which mathematical knowledge is stored and exchanged.
A self-addressing knowledge network offers one such possibility.
Its central idea is simple:
every result should have a place;
every occupied place should have a receipt;
every receipt should preserve provenance;
every formal proof should be verifiable;
every meaningful absence should be representable;
and every claim should remain within the limits of what the system can actually establish.
The philosophical shift is from knowledge as a collection of statements to knowledge as a structured geography.
In such a geography, novelty is not merely a declaration.
It is a location.
Credit is not merely a citation.
It is a receipt.
Ignorance is not merely an absence.
It has coordinates.
And a proof is not merely a derivation.
It is an event in the history of a knowledge space.
The most important question raised by artificial intelligence in mathematics may therefore not be whether machines will replace mathematicians.
It may be whether mathematics will build an infrastructure capable of remembering what machines, humans and mathematical communities discover together.
The Leiden Declaration identifies the danger.
The MAZE proposes a substrate.
Formal verification provides the oracle.
The registry provides the library.
The combination offers a possible foundation for a new mathematical order in which correctness, provenance and access are not competing values but mutually reinforcing properties of the same infrastructure.
The central principle can be stated in one sentence:
A mathematical result should not merely be correct; it should be locatable, attributable, verifiable and historically situated.
That is the philosophical meaning of giving a proof an address.
Annotated References
[1] Alper, J., Barany, M. J., Chavarri Villarello, A., Dahmen, S., Dean, W., Ganapathy, K., Harris, M., Holmes, D., Jamnik, M., Kelk, S., Kra, B., Martin, U., Naskręcki, B., Ochigame, R., Portegies, J., & Schmitt, J. (2026). Leiden Declaration on Artificial Intelligence and Mathematics. DOI: 10.5281/zenodo.20302944.
The primary external source for the discussion of AI and mathematics. The Declaration frames the problem in terms of responsibility, attribution, transparency, open science, human authorship, research autonomy and appropriate use of AI tools. It is deliberately normative: it calls upon mathematicians, institutions, governments and industry to act responsibly.
[2] Konstapel, H. & Claude. (2026). The Address of a Proof. MAZE Essay, August 2026.
The primary source for the architectural argument developed in this essay. It proposes that the four dilemmas identified by the Leiden Declaration are manifestations of a provenance problem and presents the MAZE as a working implementation of a self-addressing knowledge registry. The essay also supplies the occupancy census, system measurements and discussion of the MAZE’s current limitations.
[3] Dahmen, S. (2026). “De Leiden Declaration: AI zet kernwaarden van de wiskunde onder druk.” Lecture announcement and Vici research programme.
The PDF identifies this lecture as the immediate occasion for its engagement with the Leiden Declaration and with the formal checking of difficult mathematical proofs. The reference is particularly important because it represents the formal-verification side of the argument that the MAZE essay seeks to complement. The date described in the PDF should be checked against the publication timeline before formal publication of the essay.
[4] Borges, J. L. (1941). “The Library of Babel.”
Borges supplies the central philosophical metaphor for a complete possibility space whose completeness alone does not make knowledge accessible. The MAZE modifies the metaphor by making the address of an item computationally derivable from the item itself.
[5] Gödel, K. (1931). “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.”
Gödel numbering provides an important conceptual precedent for self-addressing symbolic structures: formal expressions can receive computational identities by construction rather than by external cataloguing. The MAZE extends this intuition from formulas toward a general knowledge space.
[6] Spencer-Brown, G. (1969). Laws of Form. Allen & Unwin.
Spencer-Brown’s concept of the first distinction is used in the MAZE theory as the conceptual origin of the address space. The significance here is philosophical: a structured universe begins with a distinction between marked and unmarked states, from which increasingly refined distinctions can emerge.
[7] Konstapel, H. (1993–2026). Paths to the Knot and the self-addressing knowledge net. constable.blog.
This is the theoretical programme underlying the MAZE implementation. According to the PDF, it develops balanced-ternary addressing, the 65 initial fields of knowledge, receipt discipline and the treatment of vacancies within a knowledge network.
[8] Rowlands, P. (2007). Zero to Infinity: The Foundations of Physics. World Scientific.
The reference provides the theoretical background for the nilpotent rewrite-system interpretation used in the MAZE framework. Its relevance is the idea that new structure can repeatedly emerge from a constrained generative mechanism while preserving an invariant.
[9] Hurwitz, A. (1898). “Über die Composition der quadratischen Formen von beliebig vielen Variablen.”
The PDF invokes Hurwitz’s work in connection with normed division algebras and the finite alphabet underlying the rewrite ladder. Its philosophical role is to support the distinction between a finite generative mechanism and the potentially unbounded mathematical structures it can produce.
[10] Henshilwood, C. S. et al. (2002). “Emergence of modern human behavior: Middle Stone Age engravings from South Africa.” Science, 295, 1278–1280.
This source is used to support the essay’s long historical perspective on symbolic behavior. It provides the background for the claim that symbolic cognition is vastly older than formal mathematics, making mathematics itself a relatively recent refinement within a much older symbolic capacity.
[11] Shannon, C. E. (1948). “A Mathematical Theory of Communication.” Bell System Technical Journal, 27, 379–423, 623–656.
Shannon is presented as an example of a scientific field that emerged suddenly once an appropriate conceptual distinction had been formulated. The reference supports the PDF’s proposition that the history of science does not justify assuming that the current inventory of disciplines is complete.
[12] Gonthier, G. (2008). “Formal Proof — The Four-Color Theorem.” Notices of the AMS, 55(11), 1382–1393.
This work represents a landmark in machine-verified mathematics and illustrates the power of formal proof systems. Within the argument of The Address of a Proof, it exemplifies the “oracle”: verification can establish correctness with extraordinary precision, but correctness does not itself establish provenance.
[13] Hales, T. et al. (2017). “A Formal Proof of the Kepler Conjecture.” Forum of Mathematics, Pi, 5, e2.
The Flyspeck project demonstrates how formal verification can resolve extraordinarily complex mathematical arguments. Its significance for the MAZE thesis is precisely that verification can settle correctness while leaving questions of historical place, attribution and novelty conceptually separate.
[14] The mathlib Community. (2020). “The Lean Mathematical Library.” Proceedings of CPP, 367–381.
Mathlib represents a large-scale living repository of formal mathematics. The PDF uses it as evidence of the maturation of formal verification while also pointing toward the unresolved provenance question: knowing what is already in a formal library and under whose name remains partly dependent upon search and convention.
[15] Castelvecchi, D. (2021), and the Liquid Tensor Experiment (2022).
These sources document the formal verification of central results in condensed mathematics associated with Peter Scholze. The PDF treats the episode as evidence that formalization can expose genuine gaps in mathematical reasoning and thereby alter mathematical practice, while remaining primarily a mechanism for establishing correctness.
[16] Buzzard, K. (2024–). The Fermat’s Last Theorem Project. Imperial College London / Lean community.
The project represents a transition from formalizing established mathematics toward formalizing frontier mathematics. The PDF uses it to anticipate a future provenance problem: once major mathematical results are formalized, determining exactly what a new formal result contributes beyond existing mathematics becomes increasingly important.
[17] Trinh, T. et al. (2024). “Solving olympiad geometry without human demonstrations.” Nature, 625, 476–482.
This work is cited as evidence of machine-generated mathematical reasoning reaching high competitive levels. It represents the emerging class of artifacts that can be correct, abundant and machine-produced — precisely the kind of mathematical object for which provenance infrastructure becomes increasingly important.
[18] Google DeepMind. (2024). “AI achieves silver-medal standard solving International Mathematical Olympiad problems.”
The AlphaProof result illustrates the acceleration of AI-generated mathematical reasoning and the significance of proprietary development environments. In the argument of the PDF, it represents the access problem in concentrated form: machine-generated mathematical ability can advance inside systems whose internal workings are not publicly available.
[19] Portegies, J. et al. / Leiden Declaration community (2026). Subsequent commentary on AI and mathematics.
Subsequent discussion reinforces that the Declaration is not simply an anti-AI statement. Its purpose is to establish conditions under which AI can be integrated without weakening mathematical integrity, autonomy and attribution. Nature has characterized the Declaration as a model for wider scientific communities, while the European Mathematical Society has emphasized that the long-term implications remain open and that opportunities as well as risks require continued discussion.
Final Note
The central proposition of this essay is deliberately narrower than the claim that the MAZE has solved mathematical provenance. It has not. The source itself explicitly acknowledges that its current semantic coding is too coarse for definitive proof fingerprinting and that the occupancy census measures the system’s sources rather than all human knowledge.
The stronger and more defensible proposition is that the problem identified by the Leiden Declaration can be reformulated as an infrastructure problem, and that a public, self-addressing knowledge registry offers a concrete way to investigate that problem. The significance of the MAZE therefore lies not in claiming completion, but in demonstrating that provenance can be treated as an architectural property of a knowledge system rather than solely as a matter of professional conduct.
