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Project ideas from Hacker News discussions.

The most famous transcendental numbers

Original Article

Hacker News Discussion

📝 Discussion Summary (Click to expand)

1. Rigor Required for Claiming Transcendentality

Debate over including unproven cases like Euler-Mascheroni constant (γ) in lists of transcendentals, insisting on mathematical proof over belief.

"If it’s not proven transcendental, it’s not to be considered such." - loloquwowndueo
"Math assumes that a claim is proven. Math is much stricter compared to most natural... sciences." - senfiaj
"Not proven to be transcendental, but generally believed to be by mathematicians." - auggierose (defending inclusion)

2. Definability, Computability, and "Almost All" Numbers

Philosophical discussion on uncountably many transcendentals being undefinable/computable, with debates on reality, models, and paradoxes like Skolem's.

"Almost all numbers are transcendental... Finding new transcendental numbers is trivial." - mg
"Most reals cannot be described in any human sense." - wiml
"it is possible for all real numbers... to be definable under ZFC." - dwohnitmok (challenging undefinability)

3. Fame, Utility, and Practical Importance of Constants

Critiques of list's "fame" (e.g., manufactured numbers) and downplaying e's role vs. alternatives like ln(2) or 2π in applications.

"The number e... does not have any importance in practice." - adrian_b
"Chapernowne's number... occur in nature, or was it just manufactured?" - drob518
"e^(iπ) = -1... d/dx e^x = e^x." - qnleigh (defending e's utility)

🚀 Project Ideas

MathArgue: The Rigorous Proof Tracker

Summary

  • A community-driven platform to track the status of mathematical conjectures, specifically addressing the "proven vs. believed" frustration seen in the discussion.
  • It provides a definitive "Proof Status" (Proven, Disproven, Open, or Heuristic Evidence) for mathematical constants and theorems to prevent the spread of "pop-math clickbait."

Details

Key Value
Target Audience Math students, researchers, and science communicators.
Core Feature Status dashboard with peer-reviewed proof citations for famous constants.
Tech Stack Next.js, PostgreSQL, LaTeX support.
Difficulty Medium
Monetization Hobby

Notes

  • Direct response to the debate over Euler's constant: "If you say 'A list of 15 transcendental numbers' a mathematician will assume all 15 are proven... Math assumes that a claim is proven."
  • Resolves the "clickbait" frustration by providing a source of truth for labels like "transcendental" or "irrational."

Real-Value Randomness Explorer (RVRE)

Summary

  • A simulator and educational tool that explores the "scrambled" nature of real numbers and the density of transcendental numbers.
  • It allows users to "interact" with Cantor’s diagonalization and Skolem's Paradox through visual algorithmic representations.

Details

Key Value
Target Audience Educators and math enthusiasts.
Core Feature Interactive infinite-precision bit-flipping and set-unscrambling animations.
Tech Stack React, Three.js, WebAssembly.
Difficulty Medium
Monetization Hobby

Notes

  • Solves the confusion expressed by users like testaccount28 ("how can i pick a real number at random?") and helps visualize the complex "definability" arguments made by dwohnitmok.
  • Provides practical utility for the "ants argument" and Zeno's Paradox discussions mentioned in the thread.

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