John Baez's post "Surprises in Logic" explores counterintuitive results within mathematical logic. It highlights the unexpected power of first-order logic, capable of expressing sophisticated concepts like finiteness and the natural numbers despite its seemingly simple structure. Conversely, it demonstrates limitations, such as the inability of first-order theories of the natural numbers to capture all true statements about them (Gödel's incompleteness theorem). The post emphasizes the surprising disconnect between a theory's ability to define a concept and its ability to characterize it completely, using examples like Peano arithmetic. This leads to the exploration of second-order logic and its increased expressive power, though at the cost of losing the completeness and compactness theorems enjoyed by first-order logic. The overall message is that even seemingly basic logical systems can harbor deep and often unintuitive complexities.
Mathematicians are exploring the boundaries of provability using large language models (LLMs) and other automated theorem provers. While these tools can generate novel and valid proofs, they often rely on techniques too complex for human comprehension. This raises questions about the nature of mathematical truth and understanding. If a proof is too long or intricate for any human to verify, does it truly constitute "knowledge"? Researchers are investigating ways to make these computer-generated proofs more accessible and understandable, potentially revealing new insights into mathematical structures along the way. The ultimate goal is to find a balance between the power of automated proving and the human need for comprehensible explanations.
HN commenters discuss the implications of Gödel's incompleteness theorems and the article's exploration of concrete examples in Ramsey theory and Diophantine equations. Some debate the philosophical significance of undecidable statements, questioning whether they represent "true" mathematical facts or merely artifacts of formal systems. Others highlight the practical limitations of proof assistants and the ongoing search for more powerful automated theorem provers. The connection between computability and the physical universe is also raised, with some suggesting that undecidable statements could have physical implications, while others argue for a separation between abstract mathematics and the concrete world. Several commenters express appreciation for the article's clarity in explaining complex mathematical concepts to a lay audience.
This article dissects the structure of a formal mathematical proof, illustrating it with a simple example about even and odd numbers. It emphasizes the distinction between informal proofs aimed at human understanding and formal proofs designed for automated verification. Formal proofs meticulously lay out every logical step, referencing specific axioms and inference rules within a chosen formal system. This detailed approach, while tedious for humans, enables computer-assisted verification and eliminates ambiguity, ensuring absolute rigor. The article highlights the importance of choosing appropriate axioms and the role of proof assistants in constructing and checking these complex formal structures, ultimately increasing confidence in mathematical results.
HN commenters discuss the accessibility of formal proof systems, particularly referencing Lean. Some express excitement about the potential of formal proofs to revolutionize mathematics, while others are more skeptical, citing the steep learning curve and questioning the practical benefits for most mathematicians. Several commenters debate the role of intuition versus rigor in mathematical practice, with some arguing that formalization can enhance understanding and others suggesting it might stifle creativity. The feasibility of formalizing existing mathematical knowledge is also discussed, with varying opinions on the timescale and resources required for such a project. Some users highlight the potential of AI in assisting with formalization efforts, while others remain cautious about its current capabilities. The overall tone is one of cautious optimism, acknowledging the challenges but also recognizing the potential transformative power of formal proof systems.
Summary of Comments ( 6 )
https://news.ycombinator.com/item?id=43763291
Hacker News users discuss various aspects of the surprises in mathematical logic presented in the linked article. Several commenters delve into the implications of Gödel's incompleteness theorems, with some highlighting the distinction between truth and provability. The concept of "surprising" itself is debated, with some arguing that the listed examples are well-known within the field and therefore not surprising to experts. Others point out the connection between logic and computation, referencing Turing machines and the halting problem. The role of axioms in shaping mathematical systems is also mentioned, alongside the challenge of finding "natural" axioms that accurately reflect our intuitive understanding of mathematics. A few commenters express appreciation for the article's clear explanations of complex topics.
The Hacker News post titled "Surprises in Logic (2016)" linking to John Baez's article has generated several comments discussing various aspects of logic, set theory, and their implications.
One commenter highlights the significance of Löwenheim-Skolem theorem, which states that if a first-order theory has an infinite model, then it has a model of every infinite cardinality. They explain how this theorem can be counterintuitive, especially when applied to theories intended to describe a unique structure, like the real numbers. They suggest it implies there are "countable models of set theory that think they contain uncountable sets," a concept they find fascinating and paradoxical.
Another comment dives into the implications of Gödel's incompleteness theorems, specifically focusing on their impact on Hilbert's program. They mention how Gödel's work demonstrated the inherent limitations of formal systems in proving all true statements within a given set of axioms. This commenter further connects this to the concept of truth being "larger" than provability, emphasizing that there will always be true statements that are unprovable within a given formal system.
Further discussion revolves around the nature of infinity and its various interpretations within set theory. One comment clarifies the distinction between countable and uncountable infinities, using the analogy of integers versus real numbers. They point out that while both sets are infinite, the real numbers are "denser" than the integers, leading to the concept of uncountability.
The conversation also touches upon the implications of the Axiom of Choice, a fundamental principle in set theory that allows for making infinitely many arbitrary choices. Some comments express how counterintuitive this axiom can be, even though it's necessary for many important mathematical theorems. They mention how it can lead to seemingly paradoxical results, like the Banach-Tarski paradox, which demonstrates how a sphere can be decomposed and reassembled into two spheres identical to the original.
A few commenters also delve into the philosophical implications of these mathematical concepts, questioning the nature of mathematical truth and its relationship to reality. They discuss whether mathematical structures are discovered or invented, and whether the limitations of formal systems reflect limitations on our ability to understand the universe.
Finally, some comments offer additional resources for exploring these topics further, including links to relevant Wikipedia pages, books, and online lectures. These recommendations provide avenues for those interested in gaining a deeper understanding of the discussed concepts.