Scott Aaronson introduces "Square Theory," a playful yet insightful analogy for theoretical computer science research. He compares the field to exploring a vast grid of squares, each representing a possible computational model or problem. Some squares are "brightly lit," representing well-understood areas like classical computation. Others are shrouded in darkness, symbolizing open questions like P vs. NP or the nature of quantum computation. Researchers "shine flashlights" into the darkness, sometimes illuminating adjacent squares and revealing connections, other times stumbling upon entirely new, unexpected landscapes. The central idea is that progress is often made incrementally, expanding our understanding outward from established knowledge, and that even seemingly small advances can illuminate larger swaths of the unknown.
Scott Aaronson's blog post, entitled "Square Theory," introduces a novel, albeit whimsical, theoretical framework he playfully dubs "Square Theory." This theory posits the existence of a fundamental, irreducible unit of area, conceptualized as a discrete, indivisible square. Aaronson meticulously explores the ramifications of this seemingly simple premise, delving into how such a quantized area would impact various aspects of geometry, physics, and even computer science.
The core tenet of Square Theory is that all geometric figures are ultimately constructed from these indivisible square units. Consequently, concepts like continuous lines and smoothly curved shapes become approximations, represented by jagged, stepwise arrangements of these fundamental squares. This quantization of area has profound implications for the calculation of quantities like pi, which, in Square Theory, would necessarily deviate from its conventional value due to the inherent limitations of approximating a circle using discrete squares.
Aaronson further elaborates on the potential consequences of Square Theory within the realm of physics. He contemplates how the discrete nature of space, as dictated by the square units, might manifest itself in physical phenomena. He raises questions about the impact on the behavior of particles and fields, suggesting that their interactions might be governed by the underlying grid of squares. The possibility of observable differences between Square Theory and continuous space-time is also explored, with the implication that sufficiently precise measurements might one day reveal the granular nature of reality predicted by the theory.
Furthermore, Aaronson touches upon the computational implications of Square Theory. He discusses the potential for simplifying certain computational problems, particularly those related to geometry and graphics, by reducing them to operations on discrete squares. Conversely, he also acknowledges that the inherent limitations of Square Theory might introduce complexities in other areas of computation.
The post maintains a lighthearted tone throughout, acknowledging the speculative and somewhat absurd nature of the proposed theory. Nevertheless, Aaronson uses Square Theory as a thought-provoking platform to explore fundamental questions about the nature of space, continuity, and the potential limits of our understanding of the universe. He ultimately concludes by inviting readers to consider the broader philosophical implications of a reality built upon discrete foundations, even if those foundations are as seemingly simple as indivisible squares.
Summary of Comments ( 77 )
https://news.ycombinator.com/item?id=44107942
Hacker News users discuss Aaronson's "Square Theory" post, mostly focusing on its playful, philosophical nature. Several commenters appreciate the thought-provoking, though admittedly "silly," premise and its exploration of mathematical and computational concepts through a simplified lens. Some highlight the parallels to Conway's Game of Life and cellular automata, while others delve into the implications for computational complexity and the potential universality of such a system. A few find the concept less engaging, describing it as trivial or underdeveloped. There's also a thread discussing the possibility of implementing Square Theory in various programming languages.
The Hacker News post titled "Square Theory," linking to Scott Aaronson's blog post of the same name, has generated a moderate number of comments discussing various aspects of the theory and its implications.
Several commenters engage with the core idea of "squarability," exploring its potential connection to other mathematical concepts. One commenter questions the relationship between squarability and concepts like Kolmogorov complexity and Shannon entropy, wondering if a non-squarable sequence could be considered random. Another commenter delves into the computational aspects, pondering the complexity class of deciding whether a given string is squarable. They speculate about the possibility of it being undecidable or residing in a complexity class beyond NP.
A few comments touch upon the philosophical implications of the theory. One commenter draws a parallel between the search for non-squarable sequences and the search for non-trivial zeros of the Riemann zeta function, highlighting the allure of exploring seemingly simple mathematical questions with potentially profound consequences. Another reflects on the nature of mathematical discovery, musing on how seemingly playful explorations like "Square Theory" can lead to unexpected insights.
Some commenters offer alternative perspectives or extensions to Aaronson's ideas. One suggests a potential connection to combinatorics on words, proposing the use of tools from that field to analyze squarability. Another explores variations on the definition of "square," considering different ways to partition and compare substrings.
A few commenters provide more practical or technical observations. One notes the potential for using squarable sequences in compression algorithms, albeit acknowledging the likely impracticality due to the computational cost of determining squarability. Another points out a minor typo in Aaronson's blog post.
While not a highly active discussion, the comments on the Hacker News post demonstrate a genuine engagement with Aaronson's "Square Theory." They explore the mathematical, computational, and philosophical facets of the concept, offering diverse perspectives and potential avenues for further exploration. The discussion, while not reaching definitive conclusions, showcases the thought-provoking nature of the theory and its ability to spark intellectual curiosity.