THE ROAD NOT TAKEN

A Chronology of Available Choices

Who knew what, and when? What choices were made — and what choices were actually available?

Version 0.7

INTRODUCTION: THE QUESTION OF WHAT WAS POSSIBLE

Every version of “what might have been” in the history of science must answer a prior question: was the alternative actually available? It is one thing to say, with the benefit of hindsight, that a different choice would have led to a better outcome. It is another to establish that the choice was genuinely present at the time — that the knowledge, the tools, and the conceptual framework existed to make it.

The previous versions of this document have explored what quantum mechanics would look like if it had been built on the ultrametric geometry of the p-adic numbers — a tree rather than a line. But those versions left a critical question partially unexamined: who could have known what, and when? What was actually published, where, and by whom? Who had access to it? What choices were made — explicitly or implicitly — and what choices were not made but could have been, given the state of knowledge at each historical moment?

This version addresses those questions directly. It is organized as a chronology of available choices. At each checkpoint, it examines what was known and by whom, what explicit and implicit choices were made, what alternatives were actually available at that moment, and whether the connection between ultrametric mathematics and quantum physics could reasonably have been made. The analysis draws on published biographical and historical sources. Where the record is clear, conclusions are stated. Where it is not, uncertainty is acknowledged.

1. 1900: WHAT PLANCK COULD HAVE KNOWN

The Situation

In December 1900, Max Planck presented his derivation of the blackbody radiation law to the German Physical Society in Berlin. He was forty-two years old and a well-established figure. He had been a professor of theoretical physics at the University of Berlin since 1889, when he was appointed to succeed Gustav Kirchhoff. He was at the center of German physics.

Planck’s derivation introduced the quantum of action and the idea that energy is exchanged in discrete packets. The derivation used Boltzmann’s statistical methods: Planck counted the number of ways to distribute energy elements among oscillators. Implicit in this counting was a metric. Planck used the ordinary Archimedean metric: the absolute difference between two energies.

What Was Available

Three years earlier, in 1897, a paper had appeared in the Jahresbericht der Deutschen Mathematiker-Vereinigung — the Annual Report of the German Mathematical Society. The paper was titled “Uber eine neue Begrundung der Theorie der algebraischen Zahlen” and its author was Kurt Hensel. It occupied six pages (Volume 6, Issue 3, pages 83-88). In it, Hensel introduced the p-adic numbers: a new number system in which distance is measured by divisibility rather than magnitude. Under this metric, for the prime 2, the number 16 is closer to 0 than the number 1 is. The geometry of these numbers is not a line but a tree.

Who was Hensel? He was a mathematician, thirty-six years old in 1897, who had earned his doctorate at the University of Berlin in 1884 under Leopold Kronecker. After his habilitation in 1886, Hensel became a Privatdozent at Berlin: an unsalaried lecturer whose income depended on student fees. He was not a professor. He was a junior, unestablished member of the mathematics faculty, working in pure number theory.

Hensel would not become a full professor until 1901, when he was appointed to a chair at the University of Marburg, where he remained until retirement in 1930.

Where They Stood

In 1897-1900, Planck and Hensel were both at the University of Berlin. Planck was a senior professor of physics. Hensel was a junior Privatdozent in mathematics. They were at the same institution, but in different departments, at different career stages, embedded in different intellectual communities, reading different journals, and attending different seminars. There is no documentary evidence they knew each other.

Hensel’s 1897 paper was published in the Jahresbericht der DMV — a mathematics society journal. Theoretical physicists in 1900 did not routinely read such publications. The paper was a brief, six-page announcement with no mention of physical applications.

What Choices Were Made

Explicit choices by Planck: to quantize oscillator energies; to use Boltzmann’s statistical methods; to present the derivation as a solution to the ultraviolet catastrophe.

Implicit choices: to use the Archimedean metric. This was not experienced as a choice. It was the only metric Planck knew. The idea that distance could be defined differently — and that this choice might have physical consequences — was not part of the conceptual framework of physics in 1900.

Choices not made: Planck did not consider the p-adic metric. He did not read Hensel’s paper. Even if he had, the physical relevance would not have been apparent.

Assessment

Could Planck, in 1900, reasonably have connected Hensel’s p-adic numbers to his blackbody problem? Almost certainly no. The conceptual distance between abstract number theory and thermal radiation physics was enormous. No bridge between these fields existed. Planck had no reason to look at the Jahresbericht der DMV. The leap from “numbers organized by divisibility” to “energy states organized on a tree” would have required an insight for which no precedent existed.

The choice was not made because it was not visible as a choice. In 1900, the road did not exist. It could only be seen in retrospect.

2. 1897-1925: MATHEMATICS AND PHYSICS IN PARALLEL

After Hensel’s 1897 paper, the p-adic numbers developed steadily within pure number theory. They proved powerful for studying Diophantine equations and algebraic number fields. By the 1920s, p-adic analysis was a mature subfield, with contributions from Hasse, Ostrowski, and others. Hensel published a book-length treatment in 1908 and a textbook in 1913. But the p-adic numbers remained entirely within mathematics, with no physical interpretation.

The same period saw the development of quantum mechanics: Einstein’s photon hypothesis (1905), Bohr’s atomic model (1913), de Broglie’s matter waves (1924), and the full formalism of 1925-1927. Throughout, physics and number theory had no interaction. They were different disciplines with different journals, conferences, and intellectual cultures. The Archimedean metric remained the unquestioned foundation of all physical theory.

Between 1897 and 1925, the road not taken remained invisible. The mathematics was developing in isolation. Physicists had no access to it — not because it was hidden, but because disciplinary boundaries made it functionally inaccessible.

3. 1925-1927: THE EXPLICIT CHOICES

The years 1925-1927 saw the construction of quantum mechanics as we know it. This period embedded choices that would shape the next century.

Heisenberg (1925) chose to represent observables as non-commuting matrices, with [x, p] = i-hbar as the fundamental postulate. Schrodinger (1926) chose to represent states as complex-valued wavefunctions. Born (1926) chose to interpret the squared magnitude as probability — a postulate, not a derivation. Heisenberg (1927) chose to interpret uncertainty as a fundamental limitation on reality. Bohr (1927) codified these into the Copenhagen interpretation.

Beneath all these explicit choices lay an implicit one: Archimedean geometry. Every formalism assumed it. Hilbert space is Archimedean. The possibility of ultrametric geometry was never considered — not rejected, but never asked.

By 1925, Hensel’s p-adic numbers were nearly three decades old. But the core geometric tools — the Bruhat-Tits tree (1972), the Monna map (1968), Shapiro’s lemma (1983) — were decades away. What existed was the metric itself. An unusually insightful physicist might have noticed the strong triangle inequality produces tree-like geometry, but no physicist was looking.

The explicit choices of 1925-1927 have been debated ever since. But the deepest choice — Archimedean geometry — was not visible as a choice. The tools for the alternative were decades away.

4. 1935-1964: LOCK-IN

By 1935, quantum mechanics was the most successful physical theory in history. The EPR paper challenged its completeness; Bohr defended Copenhagen. In 1964, Bell proved no local hidden-variable theory could reproduce quantum correlations. Both debates operated entirely within the Archimedean framework.

By 1964, p-adic analysis was mature. The adele ring had been constructed (Chevalley, Weil, 1930s-1940s). The Langlands program was beginning. But none of this had crossed into physics.

Between 1935 and 1964, the Archimedean framework became locked in. Quantum mechanics was too successful to question at its deepest level. The puzzles it produced were seen as evidence that the quantum world is strange, not that the framework might be incomplete.

5. 1968-1983: THE MISSING MATHEMATICS ARRIVES

In a remarkable convergence, the mathematical tools for an ultrametric quantum mechanics were developed during this period — entirely within pure mathematics.

In 1968, A. F. Monna published the Monna map in Indagationes Mathematicae: a function that converts p-adic numbers into ordinary real numbers by reversing the digit expansion. It is a projection from the tree-structured p-adic world to the linear Archimedean world.

In 1972, Francois Bruhat and Jacques Tits published the Bruhat-Tits tree in the Publications Mathematiques de l’IHES. This was the geometric realization of the p-adic numbers — the explicit tree structure implicit in Hensel’s metric. For a prime p, it is an infinite regular tree where each vertex connects to p+1 other vertices.

In 1983, H. N. Shapiro published the proof that the Monna map is an isometry for the shift metric. The projection faithfully preserves the tree structure when measured with the correct metric — the shift metric (distance by first differing digit) — rather than the ordinary Archimedean metric (absolute difference).

Together, these three developments supply the complete machinery for ultrametric quantum mechanics: the tree as state space (replacing Hilbert space), the p-adic metric as distance (replacing the Archimedean inner product), the Monna map as the connection to observed outcomes (replacing the Born rule), and Shapiro’s lemma as the proof of faithful projection (replacing the assumption of irreducible probability).

None of this was noticed by physicists. The Monna paper was in a Dutch mathematics journal. The Bruhat-Tits paper was in a French algebraic geometry publication. Shapiro’s lemma was embedded in a number theory textbook. By 1983, quantum mechanics had been established for over half a century. Generations of physicists had been trained in Hilbert spaces and the Copenhagen interpretation. The idea that the geometric foundation might be questioned was literally unthinkable — not because it was wrong, but because the question had never been raised.

This is the most significant missed connection in the entire chronology. By 1983, all the mathematical pieces were in place. The mathematics existed. The physics community was not looking at it.

6. 1980s-PRESENT: BEGINNINGS OF RECOGNITION

Beginning in the 1980s, Vladimirov, Volovich, and Zelenov began exploring p-adic models of quantum mechanics, string theory, and quantum field theory. Their work, collected in p-adic Analysis and Mathematical Physics (1994), demonstrated structural parallels between p-adic and conventional quantum theories. Subsequent work connected the Bruhat-Tits tree to anti-de Sitter space, linking tree geometry to holography and quantum gravity.

Why did this work remain marginal? Several factors. Path dependency: the Archimedean framework is embedded in every textbook and curriculum. Perceived strangeness: the idea that 16 is closer to 0 than 1 is strikes most physicists as absurd — but this is Archimedean intuition, not a law of nature. Lack of a killer application: the paradigm resolves interpretive problems but does not yet produce a number differing from standard quantum mechanics at accessible energy scales. Institutional inertia: funding, journals, and hiring favor incremental progress within established frameworks.

Recognition is beginning but slow. The mathematics exists. The geometric picture is clear. The testable predictions are formulated. What remains is persuading a community that its deepest assumption is worth questioning.

7. THE CHOICE WE FACE NOW

Unlike Planck in 1900, we have all the pieces: the p-adic numbers (Hensel, 1897), the tree (Bruhat-Tits, 1972), the projection map (Monna, 1968), the isometry proof (Shapiro, 1983), the adele ring (Chevalley, Weil, 1930s-1940s), the physical explorations (Vladimirov et al., 1980s-1990s), and the connection to holography (multiple authors, 2000s-present).

We also have something Planck did not: a clear understanding of the puzzles the Archimedean framework generates. The measurement problem. The probability puzzle. The nonlocality question. The fragmentation of physics. These are symptoms of measuring a tree with a ruler.

The Archimedean metric was not chosen in 1900. It was inherited. No physicist since has explicitly decided to retain it. It has simply persisted as the unquestioned geometry of physics. We now know an alternative exists — one that resolves the puzzles the Archimedean framework generates.

If the ultrametric paradigm is correct, the measurement problem never existed (it was a projection artifact). Quantum probability is geometric counting, not fundamental randomness. Entanglement is shared ancestry on a tree, not nonlocal signaling. The Born rule is projective geometry, not a postulate. Spacetime is the large-scale appearance of the tree. The fragmentation of physics is a sign that we have been using the wrong space.

Planck could not have made this choice in 1900. The conceptual framework did not exist. But we can make it now. The mathematics is on the shelf. The geometry is clear. The questions are formulated. The experiments are waiting.

The road not taken in 1900 is open now. The question is whether we will walk it.

APPENDIX: KEY CONCEPTS

Archimedean metric. The ordinary way of measuring distance: absolute difference of two numbers. Geometry is a line. Default metric of physics since Newton.

Ultrametric (p-adic metric). Distance by divisibility. For p = 2, the number 16 is closer to 0 than 1 is. Geometry is a tree. Introduced by Kurt Hensel (1897) in the Jahresbericht der Deutschen Mathematiker-Vereinigung, Volume 6, Issue 3, pages 83-88.

Bruhat-Tits tree. Geometric realization of the p-adic numbers. Infinite regular tree, p+1 edges per vertex. Constructed by Bruhat and Tits (1972) in Publications Mathematiques de l’IHES, Volume 41.

Monna map. Converts p-adic numbers to ordinary real numbers in [0,1] by reversing the digit expansion. Published by Monna (1968) in Indagationes Mathematicae, Volume 71. Preserves all information; scrambles proximity relationships when read with the ordinary Archimedean metric.

Shapiro’s lemma. Proof (Shapiro, 1983, Introduction to the Theory of Numbers) that the Monna map is an isometry under the shift metric (distance measured by first differing digit rather than absolute difference).

Threshold principle. In an ultrametric space, every point is bounded by a threshold. Disturbances smaller than the threshold cannot move a state out of its container. Basis for geometric fault tolerance.

Adele ring. Mathematical structure unifying ordinary real numbers with all p-adic number fields into a single object. Constructed by Chevalley and Weil (1930s-1940s).

REFERENCES

Mathematics:

Hensel, K. (1897). “Uber eine neue Begrundung der Theorie der algebraischen Zahlen.” Jahresbericht der Deutschen Mathematiker-Vereinigung, 6(3), 83-88. [The paper introducing p-adic numbers. Six pages. Published while Hensel was a Privatdozent at the University of Berlin.]

Bruhat, F. and Tits, J. (1972). “Groupes reductifs sur un corps local: I. Donnees radicielles valuees.” Publications Mathematiques de l’IHES, 41, 5-251. [Construction of the Bruhat-Tits tree.]

Monna, A. F. (1968). “Sur une transformation simple des nombres p-adiques en nombres reels.” Indagationes Mathematicae, 71, 225-231. [The Monna map.]

Shapiro, H. N. (1983). Introduction to the Theory of Numbers. Dover Publications. [Shapiro’s lemma: the Monna map is an isometry for the shift metric.]

Serre, J.-P. (1980). Trees. Springer-Verlag. [Mathematical treatment of trees and their groups.]

Physics:

Vladimirov, V. S., Volovich, I. V., and Zelenov, E. I. (1994). p-adic Analysis and Mathematical Physics. World Scientific. [Foundational text on p-adic quantum physics.]

Historical sources (quantum mechanics):

Planck, M. (1900). “Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum.” Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 237-245.

Einstein, A. (1905). “Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt.” Annalen der Physik, 322(6), 132-148.

Bohr, N. (1913). “On the Constitution of Atoms and Molecules.” Philosophical Magazine, 26(151), 1-25.

Heisenberg, W. (1925). “Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen.” Zeitschrift fur Physik, 33, 879-893.

Schrodinger, E. (1926). “Quantisierung als Eigenwertproblem.” Annalen der Physik, 79, 361-376.

Born, M. (1926). “Zur Quantenmechanik der Stossvorgange.” Zeitschrift fur Physik, 37, 863-867.

Einstein, A., Podolsky, B., and Rosen, N. (1935). “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Physical Review, 47, 777-780.

Bell, J. S. (1964). “On the Einstein Podolsky Rosen Paradox.” Physics, 1(3), 195-200.

Biographical sources:

MacTutor History of Mathematics Archive, University of St Andrews (entries for Max Planck and Kurt Hensel). Wikipedia (entries for Max Planck and Kurt Hensel).

The ultrametric paradigm:

Quni-Gudzinas, R. B. (2026). “The Ultrametric Paradigm: How the Choice of Geometry Determines Everything.” Version 0.9.

Version 0.7 of “The Road Not Taken.” Fact-checked chronological analysis of available choices in the history of quantum mechanics. Key factual corrections from previous versions: (1) Hensel’s 1897 paper was published in the Jahresbericht der Deutschen Mathematiker-Vereinigung (Vol. 6, Issue 3, pp. 83-88), not Crelle’s Journal; (2) Hensel was a Privatdozent (junior unsalaried lecturer) at the University of Berlin in 1897-1900, not a professor; he became a full professor at Marburg in 1901; (3) Planck and Hensel were at the same university (Berlin) simultaneously from 1889 to 1901, but in different departments, at different career stages, with minimal intellectual proximity. The spatial proximity was real; the intellectual and social proximity was not. Dated 2026-05-03.