# Math and Logic A computing-foundations discipline covering mathematical and logical concepts used in computer science. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Solving a Double-Sided Wooden Puzzle with Coordinate Systems and Search Pruning DevFeed: [Solving a Double-Sided Wooden Puzzle with Coordinate Systems and Search Pruning]() Original publisher: [Read original article]() Author: Khan Academy Published: 2015-07-06T22:00:00Z Content type: article Language: en Sources: [Khan Academy]() Topics: [Development](), [Math and Logic]() Tags: [dev](), [engineering](), [logic](), [news](), [program](), [puzzle](), [puzzles](), [web-frontend]() ### AI overview John Sullivan describes how five Khan Academy developers wrote solvers for a double-sided wooden puzzle. The article covers representing piece positions, handling rotations and reflections, and pruning the search space to improve placement search. ### Source excerpt By John Sullivan Last week a mysterious double-sided puzzle appeared at Khan Academy. To solve the puzzle you ... Read more ## Some thoughts on the Navier-Stokes Millennium Prize Problem DevFeed: [Some thoughts on the Navier-Stokes Millennium Prize Problem]() Original publisher: [Read original article]() Author: Simon Willison Published: 2026-09-08T23:55:12Z Content type: opinion Language: en Sources: [Simon Willison](), [Simon Willison's Weblog]() Topics: [OpenAI](), [Mathematics](), [Claude](), [codex](), [Math and Logic](), [Lean]() Tags: [agents](), [ai](), [ai-2-235](), [ai-ethics](), [ai-ethics-342](), [claude](), [codex](), [generative-ai](), [generative-ai-1-981](), [llms](), [llms-1-947](), [mathematics](), [mathematics-22](), [openai](), [openai-463](), [paper](), [training-data](), [training-data-68]() ### AI overview This commentary examines OpenAI's reported resolution of the Navier-Stokes existence and smoothness problem with an unreleased model, alongside accusations that the effort may have drawn on information from related work by mathematicians using Claude and Codex. It also describes questions about timing, data access, authorship, and OpenAI's subsequent use of agents and Lean formalization. ### Source excerpt On the Navier-Stokes Millennium Prize Problem introduces an impressive result from OpenAI, who used an unreleased model to produce a resolution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems that have been subject to a $1,000,000 prize since May 24th, 2000. The discovery is somewhat overshadowed by accusations of skulduggery from Tristan Buckmaster, an NYU mathematics professor who was collaborating on related problems with Levent Alpöge, an accomplished mathematician who currently works for Anthropic. Tristan's complaint accompanied a hastily published version of their own results. Here's the PDF describing what happened. The very short version is that Tristan and Levent worked on the problem for almost a year, making extensive use of Claude and Codex (mainly GPT-5.6 Sol), then had a breakthrough on August 15th. The mathematical rumour mill kicked into gear and Tristan and Levent heard that OpenAI had heard that Anthropic had resolved "a major open problem", so they reached out and learned that OpenAI had a team working on a related problem, with a similar approach. Quoting Tristan: I asked when the first prompt had been sent by them. This question was not answered directly by OpenAI for some time. Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI. I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer. It gets more complicated from there. The OpenAI team offered to wait for Tristan to publish, or to have him author a paper about their result, but were clear that Levent would not be invited as a co-author due to OpenAI's competitive relationship with his employer. Here's how OpenAI described their work: On Tuesday, September 1, we h ## On the Navier-Stokes Millennium Prize Problem DevFeed: [On the Navier-Stokes Millennium Prize Problem]() Original publisher: [Read original article]() Published: 2026-09-08T10:00:00Z Content type: article Language: en Sources: [OpenAI News]() Topics: [Math and Logic](), [Lean]() Tags: [ai](), [gpt](), [model](), [openai](), [research]() ### AI overview OpenAI shares an AI-generated solution to the Navier-Stokes existence and smoothness problem, with a proof writeup and a Lean formalization. ### Source excerpt We're sharing an AI-generated solution to the Navier-Stokes Millennium Prize Problem, including a writeup and a formal proof in Lean. ## OpenAI's Claimed Navier-Stokes Result Covered Alternatives Fefferman Included in the Official Problem DevFeed: [OpenAI's Claimed Navier-Stokes Result Covered Alternatives Fefferman Included in the Official Problem]() Original publisher: [Read original article]() Published: 2026-09-08T00:00:00Z Content type: opinion Language: en Sources: [Alex Korbonits]() Topics: [OpenAI](), [Mathematics](), [Math and Logic]() Tags: [mathematics](), [openai](), [statement]() ### AI overview The article argues that OpenAI did not prove the standard Navier-Stokes existence and smoothness problem, but did prove alternatives (C) and (D) included in Charles Fefferman's official 2000 problem statement. It distinguishes that result from the unresolved alternatives (A) and (B), which concern existence and smoothness on ℝ³ and the torus with zero force. ### Source excerpt OpenAI did not prove Navier-Stokes, and it also did not solve the wrong problem. It proved alternatives (C) and (D), which Fefferman put in the official statement on purpose in 2000, using theorem statements DeepMind had already written. ## Anthropic Formalized Fermat's Last Theorem in Lean, Shifting the Verification Challenge DevFeed: [Anthropic Formalized Fermat's Last Theorem in Lean, Shifting the Verification Challenge]() Original publisher: [Read original article]() Published: 2026-09-06T00:00:00Z Content type: opinion Language: en Sources: [Alex Korbonits]() Topics: [Lean](), [anthropic](), [Math and Logic]() Tags: [anthropic](), [compiler](), [formalized](), [research](), [statement](), [trust]() ### AI overview The article argues that Anthropic's machine-generated Lean proof of Fermat's Last Theorem changes the challenge from formalizing the theorem's statement to checking a very large proof artifact. It describes public build checks and an independent Rust-based Lean kernel re-check, while noting that the author has not read the patches used to complete that re-check. ### Source excerpt Anthropic formalized Fermat's Last Theorem in Lean in eleven days. It is the frontier I said in May was untouched, and the first result in this series where nobody had to trust the statement. The problem that replaced it is that 13 million lines is more than anyone can read. ## New Post: A Crash Course in Predicate Logic DevFeed: [New Post: A Crash Course in Predicate Logic]() Original publisher: [Read original article]() Author: Hillel Wayne Published: 2026-09-01T18:30:01Z Content type: release Language: en Sources: [Newsletter feed for Hillel Wayne's Newsletter]() Topics: [Math and Logic]() Tags: [blog](), [course](), [crash](), [free]() ### AI overview The author announces the release of a free second chapter from Logic for Programmers, titled "A Crash Course in Logic," on their blog. The chapter focuses on predicate logic. ### Source excerpt Logic for Programmers has now been out for a month! To celebrate, I'm releasing the entire second chapter, "A Crash Course in Logic", for free on my blog. Read it here! ## A decade of mathematical certainty: Reflections on the Automated Reasoning Group DevFeed: [A decade of mathematical certainty: Reflections on the Automated Reasoning Group]() Original publisher: [Read original article]() Author: Byron Cook Published: 2026-08-11T16:22:19Z Content type: article Language: en Sources: [Amazon Science homepage]() Topics: [Automated reasoning](), [Formal verification](), [Math and Logic](), [systems](), [Amazon Web Services](), [VPC](), [network security](), [Security](), [AWS IAM](), [Amazon S3](), [Cloud](), [amazon]() Tags: [agentic-ai](), [amazon](), [amazon-web-services-aws](), [automated-reasoning](), [aws](), [compliance](), [formal-verification](), [network-security](), [security](), [security-privacy-and-abuse-prevention](), [vpc](), [vulnerabilities]() ### AI overview Amazon's Automated Reasoning Group reflects on a decade of applying mathematical logic, formal verification, and program analysis to AWS security and reliability. The article describes how research projects became production systems, including Tiros for VPC and network analysis and Zelkova for analyzing policies, S3 Block Public Access, and IAM Access Analyzer. ### Source excerpt Ten years after we founded the Automated Reasoning Group, mathematical logic has moved from academic research into production services that secure millions of customer workloads -- demonstrating that systems can be provably correct, not just probably correct. ## Logic for Programmers extra credits DevFeed: [Logic for Programmers extra credits]() Original publisher: [Read original article]() Author: Hillel Wayne Published: 2026-06-02T14:48:48Z Content type: article Language: en Sources: [Newsletter feed for Hillel Wayne's Newsletter]() Topics: [Math and Logic](), [Concurrent Programming]() Tags: [book](), [concurrent](), [math](), [writing]() ### AI overview The author announces four supplementary pieces for Logic for Programmers covering concurrent-process orderings, first-order logic and sets of functions, Liskov's history rule in subtyping, and orders on sets. The supplements contain roughly 2,000-3,000 words of math content and may be less polished than the book. ### Source excerpt So I said there wasn't a proper newsletter this week, since I'm in Budapest prepping for a conference. But I still got a thing for y'all. There's a lot of interesting topics I wanted to cover for Logic for Programmers, but the book is dense enough as it is and many of these were too tangential or technical to fit in well. So I've been writing some supplements and uploading them here. I've got four so far: How we compute the number of orderings of multiple concurrent processes How first-order logic can quantify over "a set of functions", what a "set of functions" looks like, and how functions can be defined in terms of sets (plus a bit on currying and type theory) Barbara Liskov's "history rule" in subtyping Total and partial orders on sets. Now I'm going to caveat that these were written off the cuff and haven't gone through the obsessive editing of the book itself, so they may be rough and there might be errors in them. Still, it's like 2-3000 words of math content, so hopefully covers not having a proper newsletter this time. Seeya next week! ## Who Verifies the Verifier DevFeed: [Who Verifies the Verifier]() Original publisher: [Read original article]() Published: 2026-05-28T00:00:00Z Content type: opinion Language: en Sources: [Alex Korbonits]() Topics: [Artificial Intelligence](), [Formal verification](), [Compiler](), [Lean](), [Math and Logic](), [Google](), [Inference](), [Language models]() Tags: [ai](), [compiler](), [formal-verification](), [google](), [inference](), [model](), [paper](), [research]() ### AI overview The article examines whether formal verification can make AI-generated mathematical proofs scalable. It contrasts human review of natural-language proofs with Google DeepMind's approach of generating proofs directly in Lean and using the Lean compiler to verify them, while noting that the system's ability to read existing mathematics remains limited. ### Source excerpt An AI built the machine I said mathematics needed -- a compiler that verifies proofs for cents instead of expert weekends. The catch is what it still can't read. ## Assumptions weaken properties DevFeed: [Assumptions weaken properties]() Original publisher: [Read original article]() Author: Hillel Wayne Published: 2026-05-20T15:13:16Z Content type: tutorial Language: en Sources: [Newsletter feed for Hillel Wayne's Newsletter]() Topics: [Formal methods](), [Math and Logic](), [Parser](), [JSON]() Tags: [bug](), [formal-methods](), [json](), [math](), [test]() ### AI overview The article explains, using logical implication, why adding assumptions weakens a formal property. It illustrates the idea with tests, formal specifications, fairness constraints, and a JSON parser verified only for ASCII input. ### Source excerpt In some tests are stronger than others, I defined STRONG => WEAK to mean "any system passing test STRONG is also guaranteed to pass WEAK". This uses the logical implication operator, defined as P => Q = !P || (P && Q). Implication may be the most overworked operator in logic. Among other things, it's also used in formal specification, where Spec => Prop means "any system satisfying Spec has property Prop" and ASSUME => Spec means "The assumption ASSUME must hold in order for the system to satisfy Spec." Now let's mush these all together and do some math. To start, "the system has property Prop" is the same as "the system passes the test that checks Prop", so test strength is also property strength. Now let "ASSUME => Prop" mean "the system passes Prop assuming ASSUME is true." In classic logic, if P is true, then obviously !Q || P is true. Further, that is equivalent (just draw the truth table!) to !Q || (P && Q). So for any propositions P and Q, P => (Q => P). In other words, Prop => (ASSUME => Prop). In other other words, "the system passes Prop" is a stronger property than "the system passes Prop whenever our assumptions hold." In other other other words, any assumption added makes a property weaker. This makes intuitive sense to me. A JSON parser that's only been verified with ASCII strings has the property "input only uses ASCII && is valid json => correctly parsed". A better JSON parser that works for all Unicode will have the property "is valid json => correctly parsed", which has fewer assumptions, meaning it's guaranteed to work in a strict superset of cases. It also matches the intuition that "more assumptions means more likely to go wrong". We have a bug whenever Prop is false. The only way for Spec => Prop to be true and Prop be false is if Spec is false, eg our system doesn't satisfy the specification we intended to implement. On the other hand, Spec => (ASSUME => Prop) && !Prop is true whenever Spec and/or ASSUME is false, meaning a correctly-implement ## The Problem of Pedagogy in Advanced Mathematics DevFeed: [The Problem of Pedagogy in Advanced Mathematics]() Original publisher: [Read original article]() Published: 2026-05-11T00:00:00Z Content type: opinion Language: en Sources: [Susam Pal]() Topics: [Mathematics](), [Math and Logic]() Tags: [advanced](), [mathematics](), [opinion](), [students](), [theory]() ### AI overview The article argues that pedagogy remains a serious problem in advanced mathematics. It focuses on graduate-level textbooks whose proofs are often presented as high-level outlines, leaving students and even professional mathematicians to reconstruct omitted intermediate steps. It advocates explanations that are correct, complete, and accessible to reasonably motivated students. ### Source excerpt It is a commonly held opinion that educational institutions could do more to improve the pedagogy of mathematics. This is especially applicable to primary and secondary schools, where students are first exposed to mathematics as a formal subject, along with other new subjects. Poor exposition can turn students away from mathematics for a lifetime. Only the highly motivated ones continue to engage with the subject. This is very unfortunate because mathematics is a beautiful subject and it is filled with wonder. It also teaches rigour in reasoning, clarity of thought and the discipline of constructing arguments from first principles to obtain intricate and often beautiful results. What is perhaps less known is that pedagogy is a problem even for graduate-level mathematics students and professional mathematicians. The proofs in many graduate-level mathematics textbooks are, in my humble opinion, not really proofs at all. They are closer to high-level outlines of proofs. The authors simply do not show their work. The student then has to put in an extraordinary amount of effort to understand and justify each line. Sometimes a 10-line argument in a textbook might expand into a 10-page proof if the student really wants to convince themselves that the argument works. I am not a mathematician, but out of personal interest, I have worked with professional mathematicians in the past to help refine notes that explain certain intermediate steps in textbooks (for example, Galois Theory by Stewart, in a specific case). I was surprised to find that it was not just me who found the intermediate steps of certain proofs obscure. Even professional mathematicians who had studied the subject for much of their lives found them obscure. It took us two days of working together to untangle a complicated argument and present it in a way that satisfied three properties: correctness, completeness and accessibility to a reasonably motivated student. There is a reason why jokes like 'proof by obv ## Proofs and Essays Are Paths: An LLM ↔ Prover Loop for Falsifying Hallucinations DevFeed: [Proofs and Essays Are Paths: An LLM ↔ Prover Loop for Falsifying Hallucinations]() Original publisher: [Read original article]() Published: 2026-04-30T00:00:00Z Content type: opinion Language: en Sources: [Alex Korbonits]() Topics: [Large Language Model](), [Automated reasoning](), [Graphs](), [Artificial Intelligence](), [Math and Logic]() Tags: [grounding](), [hallucinations](), [knowledge-graph](), [llm](), [reasoning](), [smt](), [verification](), [verify]() ### AI overview This essay proposes a bidirectional loop between a large language model and a theorem prover. LLM outputs become propositions and graph connections that the prover checks, while verification results feed back into the model. In this framework, hallucinations appear as knowledge-graph paths with edges that fail formal verification. ### Source excerpt If a proof is a path through a formal space and an essay is a path through a semantic one, can we close the loop -- translate LLM outputs into a theorem prover, build a knowledge graph of verifiable propositions, and use the gaps as a signal for hallucination? Notebook ideas from March 2024 expanding on the SMT-grounding piece. ## CKKS -- Polynomials, the Canonical Embedding, and Encoding DevFeed: [CKKS -- Polynomials, the Canonical Embedding, and Encoding]() Original publisher: [Read original article]() Published: 2026-04-29T12:25:44Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [homomorphic encryption](), [Cryptography](), [Mathematics](), [Math and Logic]() Tags: [ckks](), [ckks-tutorial](), [cryptography](), [encoding](), [encryption](), [homomorphic-encryption](), [mathematics](), [neural-network](), [polynomial-ring](), [polynomials](), [programming](), [python](), [technical](), [tutorial]() ### AI overview This tutorial introduces the CKKS homomorphic encryption scheme and develops mathematical background on the polynomial ring used in its basic formulation and the canonical embedding used to encode cleartext messages as plaintexts. It also outlines CKKS's history, including its support for approximate arithmetic and later bootstrapping improvements. ### Source excerpt Table of Contents In this tutorial series, I will introduce the CKKS homomorphic encryption scheme from the ground up, in rather intricate detail. Each article in this series corresponds to a pull request on a GitHub repository. The code for this article is in this pull request. Follow along by cloning the repository and checking out the code at the relevant commit. This first article will cover some of the mathematical background necessary in the formulation of the CKKS encryption scheme, specifically the polynomial ring used in the most basic version of CKKS, and the canonical embedding used to encode cleartext messages as plaintexts. ## The Philosophy of Machine Learning, or: What Comes After Hegel? DevFeed: [The Philosophy of Machine Learning, or: What Comes After Hegel?]() Original publisher: [Read original article]() Published: 2026-04-09T00:00:00Z Content type: opinion Language: en Sources: [Alex Korbonits]() Topics: [Automated reasoning](), [Artificial Intelligence](), [LLMs](), [Math and Logic](), [pattern matching]() Tags: [ai](), [automated-reasoning](), [conditional-probability](), [llms](), [logic](), [machine-learning](), [pattern-matching](), [reasoning]() ### AI overview This opinion essay compares large language models with Kant's distinction between a priori structures and a posteriori experience. It argues that LLMs learn patterns from human text but lack reliable logical scaffolding, and presents automated reasoning systems such as SMT solvers and theorem provers as a possible complement. ### Source excerpt A notebook entry that maps AI paradigms onto the history of Western philosophy -- from scholasticism to Hegel -- and asks what comes next when the current moment exhausts itself. ## Leanstral: Open-Source foundation for trustworthy vibe-coding DevFeed: [Leanstral: Open-Source foundation for trustworthy vibe-coding]() Original publisher: [Read original article]() Published: 2026-03-16T16:00:00Z Content type: release Language: en Sources: [Mistral AI Blog]() Topics: [Lean](), [AI-assisted coding](), [Vibe coding](), [Code generation](), [Math and Logic](), [Open Source](), [Benchmark](), [Model Context Protocol](), [Rust]() Tags: [agents](), [ai](), [benchmark](), [code-generation](), [math](), [mcp](), [models](), [open-source](), [release](), [rust](), [vibe-coding]() ### AI overview Mistral introduces Leanstral, an open-source coding agent for Lean 4 designed to generate formally verified code and proofs in realistic repositories. The article describes its sparse architecture, 6B active parameters, Apache 2.0 release, MCP support, and evaluation against commercial and open-source coding agents. ### Source excerpt The most powerful AI platform for enterprises. Customize, fine-tune, and deploy AI assistants, autonomous agents, and multimodal AI with open models. ## Free Books DevFeed: [Free Books]() Original publisher: [Read original article]() Author: Hillel Wayne Published: 2026-03-03T16:34:33Z Content type: news Language: en Sources: [Newsletter feed for Hillel Wayne's Newsletter]() Topics: [Math and Logic](), [bug]() Tags: [books](), [bug](), [europe](), [free]() ### AI overview The newsletter is skipped for the week, with ten free copies of Logic for Programmers offered as an apology. The giveaway timing was intended to give people in Europe a better chance to claim copies, but a Leanpub bug prevented the release from working properly. ### Source excerpt Spinning a lot of plates this week so skipping the newsletter. As an apology, have ten free copies of Logic for Programmers. These five are available now. These five should be available at 10:30 AM CEST tomorrow, so people in Europe have a better chance of nabbing one. Nevermind Leanpub had a bug that made this not work properly ## New Blog Post: Some Silly Z3 Scripts I Wrote DevFeed: [New Blog Post: Some Silly Z3 Scripts I Wrote]() Original publisher: [Read original article]() Author: Hillel Wayne Published: 2026-02-23T16:49:10Z Content type: article Language: en Sources: [Newsletter feed for Hillel Wayne's Newsletter]() Topics: [Code](), [Math and Logic](), [Script]() Tags: [blog-post](), [code](), [idea]() ### AI overview The author announces a new blog post containing Z3 scripts and discusses its mathematical examples, including total operations, arrays, quantifiers, and an unsuccessful attempt to encode Goldbach's conjecture as an SMT problem. ### Source excerpt Now that I'm not spending all my time on Logic for Programmers, I have time to update my website again! So here's the first blog post in five months: Some Silly Z3 Scripts I Wrote. Normally I'd also put a link to the Patreon notes but I've decided I don't like publishing gated content and am going to wind that whole thing down. So some quick notes about this post: Part of the point is admittedly to hype up the eventual release of LfP. I want to start marketing the book, but don't want the marketing material to be devoid of interest, so tangentially-related-but-independent blog posts are a good place to start. The post discusses the concept of "chaff", the enormous quantity of material (both code samples and prose) that didn't make it into the book. The book is about 50,000 words... and considerably shorter than the total volume of chaff! I don't think most of it can be turned into useful public posts, but I'm not entirely opposed to the idea. Maybe some of the old chapters could be made into something? Coming up with a conditioned mathematical property to prove was a struggle. I had two candidates: a == b * c => a / b == c, which would have required a long tangent on how division must be total in Z3, and a != 0 => some b: b * a == 1, which would have required introducing a quantifier (SMT is real weird about quantifiers). Division by zero has already caused me enough grief so I went with the latter. This did mean I had to reintroduce "operations must be total" when talking about arrays. I have no idea why the array example returns 2 for the max profit and not 99999999. I'm guessing there's some short circuiting logic in the optimizer when the problem is ill-defined? One example I could not get working, which is unfortunate, was a demonstration of how SMT solvers are undecidable via encoding Goldbach's conjecture as an SMT problem. Anything with multiple nested quantifiers is a pain. ## Gemini 3 Deep Think: Advancing science, research and engineering DevFeed: [Gemini 3 Deep Think: Advancing science, research and engineering]() Original publisher: [Read original article]() Author: The Deep Think team Published: 2026-02-12T16:15:09Z Content type: article Language: en Sources: [Google DeepMind News]() Topics: [Google AI](), [Artificial Intelligence](), [Benchmark](), [API](), [Programming](), [Math and Logic]() Tags: [ai](), [api](), [benchmark](), [benchmarks](), [chemistry](), [gemini](), [google](), [math](), [none](), [physics](), [programming](), [reasoning]() ### AI overview Google DeepMind announces an upgraded Gemini 3 Deep Think reasoning mode for challenging science, research, and engineering problems. The article describes its availability in the Gemini app and Gemini API, along with reported results across mathematical, programming, chemistry, physics, and other academic benchmarks. ### Source excerpt Our most specialized reasoning mode is now updated to solve modern science, research and engineering challenges. ## Proving What's Possible DevFeed: [Proving What's Possible]() Original publisher: [Read original article]() Author: Hillel Wayne Published: 2026-02-11T18:36:53Z Content type: opinion Language: en Sources: [Newsletter feed for Hillel Wayne's Newsletter]() Topics: [Formal methods](), [Specifications](), [Math and Logic](), [systems]() Tags: [flow](), [formal-methods](), [specifications](), [state](), [statement]() ### AI overview A formal methods consultant introduces possibility properties for reasoning about what can happen in a system, distinguishing them from safety and liveness properties. Using temporal-logic notation and examples involving databases and state machines, the article describes possibility and reachability properties and combinations such as always possible and eventually possible. ### Source excerpt As a formal methods consultant I have to mathematically express properties of systems. I generally do this with two "temporal operators": A(x) means that x is always true. For example, a database table always satisfies all record-level constraints, and a state machine always makes valid transitions between states. If x is a statement about an individual state (as in the database but not state machine example), we further call it an invariant. E(x) means that x is "eventually" true, conventionally meaning "guaranteed true at some point in the future". A database transaction eventually completes or rolls back, a state machine eventually reaches the "done" state, etc. These come from linear temporal logic, which is the mainstream notation for expressing system properties. 1 We like these operators because they elegantly cover safety and liveness properties, and because we can combine them. A(E(x)) means x is true an infinite number of times, while A(x => E(y) means that x being true guarantees y true in the future. There's a third class of properties, that I will call possibility properties: P(x) is "can x happen in this model"? Is it possible for a table to have more than ten records? Can a state machine transition from "Done" to "Retry", even if it doesn't? Importantly, P(x) does not need to be possible immediately, just at some point in the future. It's possible to lose 100 dollars betting on slot machines, even if you only bet one dollar at a time. If x is a statement about an individual state, we can further call it a reachability property. I'm going to use the two interchangeably for flow. A(P(x)) says that x is always possible. No matter what we've done in our system, we can make x happen again. There's no way to do this with just A and E. Other meaningful combinations include: P(A(x)): there is a reachable state from which x is always true. A(x => P(y)): y is possible from any state where x is true. E(x && P(y)): There is always a future state where x is true a ## Analytical Skills for Data Professionals: Estimation, Baselines, Root Cause Analysis, and Metrics DevFeed: [Analytical Skills for Data Professionals: Estimation, Baselines, Root Cause Analysis, and Metrics]() Original publisher: [Read original article]() Author: SeattleDataGuy Published: 2026-01-23T16:49:29Z Content type: tutorial Language: en Sources: [SeattleDataGuy's Newsletter]() Topics: [Data Science](), [Math and Logic](), [data](), [dataset]() Tags: [analysis](), [critical-thinking](), [example](), [framework](), [how-to](), [metrics](), [root-cause-analysis]() ### AI overview This article discusses analytical skills that data professionals often develop on the job, including analytical intuition, estimation with limited information, baseline reasoning, critical thinking, root cause analysis, and selecting meaningful metrics. ### Source excerpt Estimation, Baselines, Root Cause Analysis, and Metrics That Actually Matter ## One more week to the Logic for Programmers Food Drive DevFeed: [One more week to the Logic for Programmers Food Drive]() Original publisher: [Read original article]() Author: Hillel Wayne Published: 2025-11-24T18:21:49Z Content type: opinion Language: en Sources: [Newsletter feed for Hillel Wayne's Newsletter]() Topics: [cobol](), [Math and Logic]() Tags: [article](), [charity](), [cobol]() ### AI overview A newsletter update promotes a charity fundraiser for Logic for Programmers and previews research into "wild goto" statements. It argues that BASIC had wild gotos, FORTRAN had tame gotos, and COBOL's ALTER statement could change a goto target at runtime. ### Source excerpt A couple of weeks ago I started a fundraiser for the Greater Chicago Food Depository: get Logic for Programmers 50% off and all the royalties will go to charity.1 Since then, we've raised a bit over $1600. Y'all are great! The fundraiser is going on until the end of November, so you still have one more week to get the book real cheap. I feel a bit weird about doing two newsletter adverts without raw content, so here's a teaser from a old project I really need to get back to. Notes on structured concurrency argues that old languages had a "old-testament fire-and-brimstone goto" that could send control flow anywhere, like from the body of one function into the body of another function. This "wild goto", the article claims, what Dijkstra was railing against in Go To Statement Considered Harmful, and that modern goto statements are much more limited, "tame" if you will, and wouldn't invoke Dijkstra's ire. I've shared this historical fact about Dijkstra many times, but recently two separate people have told me it doesn't makes sense: Dijkstra used ALGOL-60, which already had tame gotos. All of the problems he raises with goto hold even for tame ones, none are exclusive to wild gotos. So This got me looking to see which languages, if any, ever had the wild goto. I define this as any goto which lets you jump from outside to into a loop or function scope. Turns out, FORTRAN had tame gotos from the start, BASIC has wild gotos, and COBOL is a nonsense language intentionally designed to horrify me. I mean, look at this: The COBOL ALTER statement changes a goto's target at runtime. (Early COBOL has tame gotos but only on a technicality: there are no nested scopes in COBOL so no jumping from outside and into a nested scope.) Anyway I need to write up the full story (and complain about COBOL more) but this is pretty neat! Reminder, fundraiser here. Let's get it to 2k. Royalties are 80% so if you already have the book you get a bit more bang for your buck by donating to the GCFD d ## GPT-5 and the future of mathematical discovery DevFeed: [GPT-5 and the future of mathematical discovery]() Original publisher: [Read original article]() Published: 2025-11-24T00:00:00Z Content type: article Language: en Sources: [OpenAI News]() Topics: [Artificial Intelligence](), [Optimization](), [Large Language Model](), [Algorithm](), [Math and Logic](), [OpenAI](), [ChatGPT]() Tags: [ai](), [future](), [gpt](), [large-language-model](), [llm](), [math](), [math-and-logic](), [openai](), [optimization](), [speed]() ### AI overview The article describes how UCLA mathematician Ernest Ryu used GPT-5 to investigate a longstanding open problem in optimization theory involving the Nesterov Accelerated Gradient method. GPT-5 helped surface mathematical ideas and techniques quickly, contributing to an explanation of why NAG can accelerate optimization while remaining stable. ### Source excerpt UCLA Professor Ernest Ryu and GPT-5 solved a key question in optimization theory, showcasing AI's role in accelerating mathematical discovery. ## Get Logic for Programmers 50% off & Support Chicago Foodbanks DevFeed: [Get Logic for Programmers 50% off & Support Chicago Foodbanks]() Original publisher: [Read original article]() Author: Hillel Wayne Published: 2025-11-10T16:31:11Z Content type: article Language: en Sources: [Newsletter feed for Hillel Wayne's Newsletter]() Topics: [Math and Logic]() ### AI overview Logic for Programmers is offered at half price through the end of the month using the coupon code feedchicago. Royalties from the coupon will be donated to the Greater Chicago Food Depository. ### Source excerpt From now until the end of the month, you can get Logic for Programmers at half price with the coupon feedchicago. All royalties from that coupon will go to the Greater Chicago Food Depository. Thank you! ## Design for Amiability: Lessons from Vienna DevFeed: [Design for Amiability: Lessons from Vienna]() Original publisher: [Read original article]() Author: by Mark Bernstein Published: 2025-10-15T15:35:00Z Content type: article Language: en Sources: [A List Apart: The Full Feed]() Topics: [Web](), [Computer science](), [Math and Logic]() Tags: [community](), [community-industry-state-of-the-web-web-strategy](), [design](), [history](), [industry](), [math](), [research](), [state-of-the-web](), [web](), [web-strategy]() ### AI overview The article examines how amiable design can improve web environments and uses the Vienna Circle and the origins of computer science in Vienna as a historical case study. It connects the quality of interaction in research communities with the design of welcoming online spaces. ### Source excerpt Today's web is not always an amiable place. Sites greet you with a popover that demands assent to their cookie policy, and leave you with Taboola ads promising "One Weird Trick!" to cure your ailments. Social media sites are tuned for engagement, and few things are more engaging than a fight. Today it seems that people want to quarrel; I have seen flame wars among birders. These tensions are often at odds with a site's goals. If we are providing support and advice to customers, we don't want those customers to wrangle with each other. If we offer news about the latest research, we want readers to feel at ease; if we promote upcoming marches, we want our core supporters to feel comfortable and we want curious newcomers to feel welcome. In a study for a conference on the History of the Web, I looked to the origins of Computer Science in Vienna (1928-1934) for a case study of the importance of amiability in a research community and the disastrous consequences of its loss. That story has interesting implications for web environments that promote amiable interaction among disparate, difficult (and sometimes disagreeable) people. The Vienna Circle Though people had been thinking about calculating engines and thinking machines from antiquity, Computing really got going in Depression-era Vienna. The people who worked out the theory had no interest in building machines; they wanted to puzzle out the limits of reason in the absence of divine authority. If we could not rely on God or Aristotle to tell us how to think, could we instead build arguments that were self-contained and demonstrably correct? Can we be sure that mathematics is consistent? Are there things that are true but that cannot be expressed in language? The core ideas were worked out in the weekly meetings (Thursdays at 6) of a group remembered as the Vienna Circle. They got together in the office of Professor Moritz Schlick at the University of Vienna to discuss problems in philosophy, math, and language. The i [Next page]()