不在 Absence

① Absence of evidence is not evidence of absence.

-- Carl Sagan (1934-1996), an American astronomer and one of the leading science communicators of the 20th century.

証拠の不在は非存在の証拠ではない。

This is profound. According to Wikipedia, however

② An absence of proof is a challenge; an absence of definition is deadly.

-- Pierre Deligne (1944-), a Belgian mathematician, best known for work on the Weil conjectures on the top of various contributions.
He was awarded the Fields Medal in 1978 when I just started studying Algebraic Topology with particular interest in homotopy theory but felt that Algebraic Geometry was far beyond my capacity at that time.

証明がないことは挑戦に繋がるが、定義の不在は致命的である。

I'm unable to locate this quote by Pierre Deligne, and I only learned it in "Stacks and Moduli" (by Jarod Alper). Even though Pierre Deligne is a superlatively prominent mathematician, I would not say he is well-known like Carl Sagan.
The Belgian post office has issued a postage stamp in honor of his achievements in fundamental mathematics. The following depicts the Ramanujan conjecture he proved as a consequence of his proof of the Weil conjectures:

I like this quote very much except for the good-sounding word "challenge", which was the fundamental keyword in the accounting fraud scandal of Tōshiba in 2015. In fact, this was a case in which management forced each department to meet absurd profit targets under the guise of "challenges," and the pressure caused profits to be padded. I don't have the faintest doubt that such corporate culture is not unique to this particular company, but that's only relevant to this article in terms of "absence of moral".

Back to mathematics, one can find no+e articles based on ambiguous or incorrect definitions, or lack of intention for definition, virulent perversion of theories (most conspicuously on Gödel's incompleteness theorems), et alia.

③ On the Non-Existence of Elements of Hopf Invariant One

When I started writing this article, it never occurred to me to refer to non-existence, but suddenly I remembered that I had chosen the following paper by J.F. Adams as my research topic for my fourth year at an institute of technology:

On the same year 1960, Michel Kervaire constructed a 10-dimensional manifold which does not allow a differentiable structure. In other words, no differentiable structure exists on a Kervaire manifold of dimension 10. In May 2024, there was an announcement that Kervaire invariant problem for dimension n = 126 was settled with existence of a manifold of Kervaire invariant 1 in dimension 126.
https://en.wikipedia.org/wiki/Exotic_sphere

The worst example of "absence"

is an attempt for peace talks initiated by Felon47 and its brain-dead puppet envoys denying "Nothing about Ukraine without Ukraine".
https://www.bbc.com/news/articles/ckg051z21zno

This suggests the envoys are amateurs ignorant of diplomacy, but are professionals when it comes to diplomacy by threats and intimidation, nothing more than Путин 2.0.
Those who appreciate such peace talks as pragmatic effort under the situation with no way out are nothing different from MAGA (禍) except for the mask of intelligence.

Header Image Credit: Cultura Inquieta via Pinterest
(Una mujer de 91 años rememora su vida en una inteligente y surrealista serie fotográfica)
■