The Absolute Zero: Saburou Saitoh’s Crusade to Redefine Mathematics

Overview

The "Fun Person" of the Day: Prof. Saburou Saitoh (Gunma University).

The Phenomenon: A legitimate mathematician with a background in Reproducing Kernels who has seemingly pivoted his entire career toward proving that $z/0 = 0$.

This isn't your standard crankery. Saitoh isn't an outsider scribbling in green ink; he publishes via SCIRP (a known quantity in the predatory/low-tier world) and viXra, yet somehow managed to slip his "Division by Zero" manifesto into a Springer proceeding. This puts him in a league comparable to the late Myron W. Evans—academic credentials used to push a theory that breaks fundamental axioms.

Speculation on Intent:

Is this an elaborate academic troll? Unlikely. The tone is far too earnest. It appears to be a classic case of the "Golden Hammer." Saitoh seems to have taken the concept of the Moore-Penrose pseudoinverse (where the "inverse" of a zero matrix can be defined as zero) or Tikhonov regularization and elevated it from a computational tool to a fundamental law of the universe. By declaring that singularities are simply "zero," he eliminates all poles, singularities, and infinities. It is a "brute force" solution to complex analysis: if a function blows up, just define it as zero. The result is a mathematical system where the point at infinity collapses into the origin—a topological nightmare presented as "elementary and fundamental mathematics."

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Detailed Analysis of the Texts

I. The Manifesto: Matrices and Division by Zero z/0 = 0

Source: Matsuura, T. and Saitoh, S. (2016). Matrices and Division by Zero z/0 = 0. Advances in Linear Algebra & Matrix Theory, 6, 51-58. https://doi.org/10.4236/alamt.2016.62007

This paper lays the groundwork for the "Yamada Field" (Y-Field), a structure where division by zero is permitted and defined as zero.

• The Core Axiom: Saitoh asserts that for any complex number $b$, the fraction $b/0 = 0$. This is justified "incidentally" by Tikhonov regularization.

• **The Geometric Consequence (The Collapse):**In standard geometry, parallel lines meet at infinity.

• In Saitoh’s geometry, the "point at infinity" is represented by $z=0$.

• Therefore, parallel lines intersect at the origin $(0,0)$.

• Critique: This implies that if you have two parallel lines anywhere in space, their intersection point is the origin. The logic effectively folds the entire Euclidean plane such that the "horizon" touches the center.

**Cramer's Law "Fix":**Standard Cramer's Law fails when the determinant is zero ($D=0$, singular matrix). Saitoh claims that by defining $z/0=0$, the solution to a system with a singular matrix is simply the zero vector."By the division by zero, we can understand that if $a_1b_2 - a_2b_1 = 0$, then the common point is always given by $(0,0)$ even when the two lines are the same."

Physical Interpretations:Hooke's Law: For springs in series $\frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2}$. If $k_1=0$, normally the system breaks. Saitoh argues $1/0 = 0$, so $1/k = 1/k_2$, meaning $k=k_2$.

Einstein Misquote: He invokes Einstein to support his theory, though the quote is likely apocryphal or misinterpreted. "Blackholes are where God divided by zero."

II. The Escalation: log 0 = log ∞ = 0 and Applications

Source: Michiwaki, H., Matuura, T., Saitoh, S. (2018). $\log 0 = \log \infty = 0$ and Applications. In: Pinelas, S., Caraballo, T., Kloeden, P., Graef, J. (eds) Differential and Difference Equations with Applications. ICDDEA 2017. Springer Proceedings in Mathematics & Statistics, vol 230. Springer, Cham. https://doi.org/10.1007/978-3-319-75647-9_24

Here, the theory expands to logarithmic functions and complex analysis, creating even more startling discontinuities.

• Redefining Logarithms: Since $\log(z)$ has a singularity at $z=0$ and a pole at $z=\infty$, and since the "point at infinity" is now the "origin" (via $1/0=0$), Saitoh concludes:"In this paper, we will show that $\log 0 = \log \infty = 0$ by the division by zero $z/0 = 0$."

• The $1/z$ Mapping: He relies heavily on the mapping $W = 1/z$. In standard analysis, $z \to 0$ implies $W \to \infty$. Saitoh argues that the image of $z=0$ is exactly $W=0$."The image of $z = 0$ is $W = 0$ (should be defined)... The division by zero will give great impacts to complex analysis and to our ideas for the space and universe."

• Schrödinger's Euler Identity: This leads to bizarre multivalued definitions.$e^0 = 1$ (standard definition).

• $e^0 = 0$ (via the "division by zero" logic where the value at a singularity is zero).

He acknowledges this "strong discontinuity" but accepts it as a "natural way" the universe works.

Applications to Differential Equations: He solves differential equations by simply ignoring the singular parts. If a term contains a $1/x$ and $x=0$, that term vanishes."For the differential equation... we have the solution... Then, if $\mu = 0$, we obtain, immediately, by the division by zero [a solution where the singular term vanishes]."

The "Zero" Constant: He posits that $0^0 = 1$ is convenient for Taylor series, but $0^0 = 0$ is also valid depending on context. He also claims $\cos 0 = 1$ and $\cos 0 = 0$ are both "valid" in different contexts (reflection of infinity to the origin).

Selected Quotes

• On the ambition of the project: "The division by zero has a long and mysterious story over the world... however, Sin-Ei Takahasi established a simple and decisive interpretation."

• On the redefinition of infinity: "The point at infinity is represented by zero, that is, the coincidence of the point at infinity and the origin."

• On the "Zero" value of functions: "We wish to consider also the value $\cos 0 = 0$. The values $e^0 = 0$ and $\cos 0 = 0$ may be considered that the values at the point at infinity are reflected to the origin."

A Serious Critique

Mathematics, as a deductive system, stands on the bedrock of consistent axioms—propositions whose truth is assumed to build coherent, predictive, and applicable frameworks. To redefine a foundational operation like division by zero is not an act of "revolutionary insight," but a violation of this coherence, as Saitoh’s work demonstrates through its cascading logical contradictions and detachment from both theoretical rigor and empirical reality.

The core flaw of Saitoh’s theory lies in its confusion of computational convenience with ontological truth. Tools like the Moore-Penrose pseudoinverse or Tikhonov regularization are valuable precisely because they are contextual: they resolve ambiguities in specific computational tasks (e.g., solving underdetermined systems) while acknowledging their status as practical adaptations, not universal laws. By elevating these tools to axiomatic status, Saitoh erases the distinction between "what works for a problem" and "what is true of the mathematical structure itself." The result is not a "new field" but a house of cards: parallel lines intersecting at the origin dismantles Euclidean geometry, conflicting values of $e^0$ and $\cos 0$ render analysis meaningless, and "solving" differential equations by ignoring singularities abandons the very purpose of such equations—to model continuous change accurately.

Equally troubling is the way Saitoh’s academic credentials have lent a veneer of legitimacy to a theory that would otherwise be dismissed. His ability to publish in a Springer proceeding—an imprint associated with rigorous peer review—raises critical questions about the integrity of academic publishing, particularly in specialized fields where gatekeeping relies on expert familiarity with niche topics. This is not a trivial concern: bad mathematics, when cloaked in academic authority, can mislead early-career researchers, waste resources on unfruitful inquiries, and erode public trust in the discipline’s ability to distinguish fact from fancy.

To be clear, academic progress often emerges from challenging orthodoxy—but meaningful challenges adhere to a basic contract: they either show the existing framework is incomplete (e.g., non-Euclidean geometry extending, not rejecting, Euclid’s axioms) or propose a new system that is more consistent and explanatory than the old. Saitoh’s work does neither. It replaces a coherent system with one that is deliberately contradictory, then claims this contradiction as a "natural" feature of the universe—a rhetorical sleight of hand.