BBC Lê – “Por que alguns matemáticos querem acabar com o infinito: 'É uma ilusão'”

Episode Date: October 11, 2025

Host: BBC Brasil

Overview

This episode, narrated in the BBC Lê format where a journalist reads selected articles, dives into a thought-provoking debate in modern mathematics: Should the concept of infinity be eliminated? Inspired by an article from New Scientist, the episode presents both the philosophical and practical arguments surrounding the idea of infinity, focusing on the movement known as ultrafinitism—a radical perspective gaining traction among certain mathematicians, philosophers, computer scientists, and physicists.

Key Discussion Points & Insights

1. The Eternal Fascination with Infinity

• The narrator expresses a personal fascination with infinity, describing it as “liberdade criativa, intelectual e emocional” and marveling at humanity’s ability to conceptualize such an “assombroso” idea from a young age.
  • [01:18] “Para mim, o infinito é liberdade criativa, intelectual e emocional. Também fico maravilhado quando penso que podemos conceber um conceito tão assombroso desde pequenos.”
• The episode references classic thinkers like Zeno of Elea, Archimedes, Newton, Leibniz, and later, Georg Cantor, who demonstrated that there are multiple sizes of infinity.

2. The Modern Revolt: Ultrafinists vs. Infinity

• The episode introduces Doron Zeilberger, a prominent Rutgers mathematician and leading ultrafinitist, who bluntly describes infinity as “uma ilusão.”
  • [03:32] “O infinito não é mais do que uma ilusão” — prof. Doron Zeilberger.
• Ultrafinists argue that even unimaginably large finite numbers—like 10^90—are meaningless in practice or even in principle, given physical limits like the number of atoms in the observable universe.
  • [05:08] “[Até] números finitos, mas enormes, como 10 elevado à nonagésima potência, talvez sejam insignificantes... nunca atingiríamos esse número.”
• Zeilberger analogizes to pre-Copernican notions: Embracing infinity, he says, was a fundamental misstep for mathematics, akin to believing the Earth is flat out of optical illusion.

3. The Proposal: Mathematics Without Infinity

• Zeilberger draws a parallel to Einstein’s speed-of-light boundary, arguing that if mathematics had established a largest possible number, things would be simpler.
• When asked what happens if you add 1 to the largest number, Zeilberger responds, “voltaríamos ao zero”—circular like circumnavigating the globe indefinitely instead of extending it forever.
  • [07:25] “Simplesmente... voltaríamos ao zero. O que defendo é algo análogo à revolução de Albert Einstein... Eu não tenho ideia de qual seja esse número maior, mas é irrelevante... você pode recriar toda a matemática e torná-la muito mais simples.”
• Zeilberger admits this reinterpretation would render mathematics more “tedioso,” but also more practical.

4. Feasibility and Practicality of Numbers

• Rohit Parikh (City University of New York) brought formality to ultrafinitist philosophy in the 1970s, focusing on “números factíveis”—those usable in human or physical contexts.
  • [09:40] “Se um número não pode ser nomeado, calculado, armazenado, transmitido... será que ele realmente existe como objeto matemático?”
• The episode uses the “número de Skews”—a number so large its digits can't fit in the universe—as an example of mathematical constructs ultrafinitists reject as meaningless.

5. The Volpin Anecdote: Defining Feasible Limits

• The show relays a story, via mathematician Harvey Friedman, about Alexander Esenin-Volpin, a founding ultrafinitist, to illustrate escalation of feasibility for larger numbers.
  • [11:30] As numbers increase (like 2^1, 2^2, 2^3...), Esenin-Volpin would affirm their existence, but take increasingly longer to answer, demonstrating the physical and cognitive impracticality of extremely large numbers.

6. Modern Mathematics and Finitism in Practice

• Doron Zeilberger observes that much of modern mathematical work, such as cryptography and algorithm design, is already inherently finite.
  • [12:18] “Grande parte do trabalho moderno com a matemática já reside no finito, como a criptografia... os algoritmos aleatórios.”
• In physics, Max Tegmark argues infinity is aesthetically pleasing but impractical: our best computational simulations always use finite resources.

7. The Philosophical Implications

• The narrator questions whether removing infinity would limit human creativity or the “adventure” of mathematical thought.
• Zeilberger’s final reflection likens infinity (in math) to belief in God—neither is necessary for mathematics to function:
  • [12:58] “O infinito pode ou não existir. Deus pode ou não existir. Mas nenhum dos dois é necessário na matemática.”

Notable Quotes & Memorable Moments

• “O infinito não é mais do que uma ilusão.” — Prof. Doron Zeilberger [03:32]
• “As pessoas acreditaram que o universo é infinito, e alguns ainda acreditam nisso, mas outras pensam que é finito. Não é limitado, porque sempre podemos seguir adiante, mas é finito, como o nosso planeta…” — Prof. Doron Zeilberger [05:50]
• “Se um número não pode ser nomeado, calculado, armazenado, transmitido... será que ele realmente existe como objeto matemático?” — Rohit Parikh [09:40]
• “Mas, usando computadores, você consegue chegar a uma ótima aproximação, suficiente para todos os propósitos práticos.” — Prof. Doron Zeilberger [12:08]
• “O infinito pode ou não existir. Deus pode ou não existir. Mas nenhum dos dois é necessário na matemática.” — Prof. Doron Zeilberger [12:58]

Timestamps for Key Segments

• [01:00–03:30] — Introduction to infinity’s role in mathematics and history
• [03:30–06:00] — Introduction of ultrafinitism and Zeilberger’s arguments
• [06:00–08:30] — Comparison to Einstein, the idea of the “largest number,” circular arithmetic
• [08:30–10:00] — Rohit Parikh and “números factíveis”
• [10:00–11:30] — The Skews number and ultrafinitist objections
• [11:30–12:18] — The Volpin anecdote and modern practical mathematics
• [12:18–12:58] — Zeilberger, Max Tegmark, and finitism in computational physics
• [12:58–13:14] — Conclusion, limitations of mathematics without infinity, and philosophical take

Conclusion

The podcast serves as an accessible, balanced, and sometimes poetic exploration of the controversy around infinity in mathematics. By weaving in historical, philosophical, practical, and even storytelling elements, it challenges listeners to reconsider something most have taken for granted: Is infinity truly a necessary component of our math and science, or just a grand illusion? The episode closes without definitive answers, reflecting the ongoing debate and the mystery that still surrounds the infinite.