Chapter 1: The Intersection of Madness and Rationality

Mathematics is frequently associated with madness. We hear tales of luminaries like Gödel, Cantor, Nash, and Grothendieck, whose brilliance often bordered on insanity as they delved deeper into the realm of mathematics. This notion has some merit; a psychologist once remarked that "A paranoid person is irrationally rational... Paranoid thinking is marked by a distorted logic, an unrestrained form of reasoning."

Mathematics exemplifies rational thought, yet when this rationality permeates every facet of life, one might question the mental wellbeing of the individual. However, let us examine a contrasting narrative: the story of a mathematician who embodied reason and sanity in a chaotic world, whose mathematical principles mirrored his social philosophy. Introducing Alexander Yessenin-Volpin (1924–2016).

Section 1.1: A Poet's Legacy

Born amidst the political turmoil of the USSR and raised under Stalin's regime, young Alexander witnessed the emergence of a repressive political order. Initially, many Russian intellectuals fervently supported Bolshevik ideals, believing in the possibility of a brighter future. Volpin's father, the esteemed poet Sergei Yesenin (1895–1925), was a fervent advocate for the revolution. Though they never met, the ambiance of collaboration between Russian intelligentsia and the communist government likely influenced Alexander's formative years.

As he later recounted in his "Free Philosophical Tractate," written at age twenty, he experienced an "adolescent crisis" in April 1939, where he resolved to prioritize reason over the commonly embraced "emotion." This emotional perspective, promoted by the era's communist ideologues, was viewed as a counter to the bourgeois, abstract philosophies of anti-Marxists. In contrast, Volpin sought liberation through authenticity and precision in language, aspiring for a mathematically-inspired clarity in ethics and law.

Section 1.2: The Need for Clarity in Law

Volpin passionately believed that without a clear and unambiguous language, our thoughts cannot be trusted. He endeavored to refine legal language using modal logic, which proved useful in theoretical contexts, though it faced challenges in practical applications. He argued that traditional interpretations of laws should adhere strictly to their written form, rather than the interpretations of governing bodies.

Volpin famously praised the 1936 "Stalinist" constitution for its provisions of civic rights, challenging those around him to recognize the laws that existed, despite the government's failure to uphold them. His insistence on a literal interpretation of laws forced the authorities to allow political opposition to gather publicly, illustrating the effectiveness of his concrete approach.

Chapter 2: Mathematical Foundations Reconsidered

Section 2.1: A Radical Revision of Mathematics

Yessenin-Volpin considered the prevailing methods in mathematics to be as flawed as the legal system in the USSR. He argued that the indiscriminate inclusion of infinity in mathematical discourse undermined its precision. He called for a reevaluation of mathematical foundations, asserting that both potential and actual infinities were nonsensical. Thus, he restricted mathematical inquiry to finite objects.

His stance, which he termed "ultrafinitism," closely resembled the views of finitists like Hilbert or Skolem but diverged significantly. Unlike Hilbert, who acknowledged the existence of infinite mathematics, Volpin dismissed operations involving infinity altogether. For him, concepts such as f(n) were meaningless without a definitive understanding of n, as he did not accept the existence of a complete set of natural numbers.

Section 2.2: The Essence of Numbers

Volpin's perspective was radical; he rejected even the notion of large numbers that could not be physically represented. This challenging viewpoint resonated within the mathematical community, as evidenced by an encounter with Harvey Friedman. In a discussion, Volpin's hesitation to affirm the reality of larger numbers illustrated his belief that comprehension of larger concepts required a foundational understanding of smaller ones.

He viewed mathematics as a structure built from concrete "atoms," emphasizing a rejection of metaphysical assumptions. His desire for clarity sought to liberate mathematics from what he perceived as unjustified dogmas regarding infinity.

Section 2.3: The Fight Against Demagogy

Volpin's life exemplified a struggle against intellectual dishonesty in both mathematics and civic rights. His advocacy for civil liberties in the USSR resulted in multiple stints in psychiatric institutions and a five-year exile in Siberia. Ironically, his "diagnosis" of pathological honesty led to his institutionalization, highlighting the dissonance between his principles and the oppressive society he navigated.

His legacy includes inspiring others through his commitment to freedom. Volpin's involvement in the Glasnost movement and his influence on a generation of political dissidents underscore his pivotal role in advocating for transparency and rights.

In conclusion, Yessenin-Volpin's life was not only a quest for mathematical and ethical clarity but also a testament to the interconnectedness of abstract reasoning and practical application. He exemplified a profound commitment to integrity in thought, reminding us that logic and ethics are ultimately intertwined.