The mathematical guarantees arise from a different source. First of all, the form of the mathematical guarantees is that either the predictor or the agentic version will have an exponentially small probability of achieving what I call a 'challenging and harmful' goal.
这一观点提出了数学保证的形式，即预测器或代理版本实现有害目标的概率呈指数级减小，这是AI安全理论的重要突破。
Erdős also noticed that the score drops if all of a set's numbers are large—the larger the numbers, the less large the score could become. He guessed that as the set's numbers approached infinity, the maximum score would drop to exactly one.
这个数据点提供了具体的数学预测值'1'，这是一个精确的量化结果。当数字趋近于无穷大时，分数降至1的预测展示了数学中的极限概念，这是AI可能帮助验证的精确数学命题。'exactly one'的表述强调了数学的精确性。
Erdős also came up with the Erdős sum, a 'score' you can calculate for any primitive set. He showed that the sum had a maximum possible value—and conjectured that this value must hold only for the set of all prime numbers.
这里提供了数学概念的具体量化指标。'最大可能值'的表述暗示了有明确的数学界限，但文章未提供具体数值。这反映了数学中某些概念虽然可量化，但具体数值可能需要更专业的数学背景才能理解，体现了数学研究的抽象性。
GPT‑5.5 found a proof of a longstanding asymptotic fact about off-diagonal Ramsey numbers, later verified in Lean. The result is a concrete example of GPT‑5.5 contributing not just code or explanation, but a surprising and useful mathematical argument in a core research area.
大多数人认为AI在数学研究中的作用主要是辅助计算和验证，但作者认为GPT-5.5能够独立发现数学证明，这在数学研究领域是革命性的。这一观点挑战了人们对AI在创造性思维和抽象推理领域能力的传统认知，暗示AI可能正在从工具转变为研究伙伴。
Der kurz vor der COP16 zur Biodiversität veröffentlichte Living Planet Index zeigt das Ausmaß des Biodiversitätsverlusts in den vergangenen 50 Jahren, auch wenn an den dabei angewendeten statistischen Verfahren starke Zweifel bestehen. Die Wirbeltier-Populationen haben nach diesem Index um 73% abgenommen, am stärksten in Lateinamerika und der Karibik. Die wichtigste Ursache ist die veränderte Landnutzung. https://www.theguardian.com/environment/2024/oct/10/collapsing-wildlife-populations-points-no-return-living-planet-report-wwf-zsl-warns
"But there's a very famous theorem in topology called the Jordan curve theorem. You have a plane and on it a simple curve that doesn't intersect and closes—in other words, a loop. There's an inside and an outside to the loop." As Riehl draws this, it seems obvious enough, but here's the problem: No matter how much your intuition tells you that there must be an inside and an outside, it's very hard to prove mathematically that this holds true for any loop that can be drawn.
How does one concretely define "inside" and "outside"? This definition is part of the missing space between the intuition and the mathematical proof.
I'm using LaTeX to create my Zettel notes. .t3_158gy35._2FCtq-QzlfuN-SwVMUZMM3 { --postTitle-VisitedLinkColor: #9b9b9b; --postTitleLink-VisitedLinkColor: #9b9b9b; --postBodyLink-VisitedLinkColor: #989898; }
reply to u/AndreSanch at https://www.reddit.com/r/Zettelkasten/comments/158gy35/im_using_latex_to_create_my_zettel_notes/
This sort of thing has certainly been done before by many. Be careful of going overboard.
If you don't already have a list of most of the common LaTeX math symbols, here's a good starter list, but make sure that your assigned meaning to them from an argumentation perspective is either "standard" or you've written it down for later use/memory. (There's nothing worse than a 10 year old note whose symbols you no longer remember.)
If you haven't done a course in philosophy or logic (something along the lines of Elements of Logic), then that may also help you in terms of many of the common uses/meanings, though there are a variety of meanings to various symbols through time, so take care.
Scribes and scholars over time have used a variety of symbols and annotations to mean various things, some of which were standardized in various contexts. For more on this take a look at some of Evina Stein's work and research on historic texts. Some of this might include:
Steinová, Evina. “Nota and Require. The Oldest Western Annotation Symbols and Their Dissemination in the Early Middle Ages.” Scribes and the Presentation of Texts (from Antiquity to c. 1550). Proceedings of the 20th Colloquium of the Comité International de Paléographie Latine, 2021, 473–89. https://doi.org/10.1484/M.BIB-EB.5.124987.<br /> ———. Notam Superponere Studui: The Use of Annotation Symbols in the Early Middle Ages. Brepols, 2019.<br /> Steinova, Evina. “Technical Signs in Early Medieval Manuscripts Copied in Irish Minuscule.” In The Annotated Book in the Early Middle Ages: Practices of Reading and Writing, edited by M. J. Teeuwen and I. Van Renswoude, 37–85. Brepols, 2017.
For those interested in scratching the surface of some possibilities and history, I might recommend:
Scheinerman, Edward R. Mathematical Notation: A Guide for Engineers and Scientists. CreateSpace, 2011.
Your note about Forte, while cute and clever doesn't necessarily mean that he's an old man, however, so take care about your propositions and what you draw from them or else your system won't hold up for long.
God's number is 20. Proven by Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge
Resources and inspiration for math instruction and learning
Welcome to AMS Open Math Notes, a repository of freely downloadable mathematical works hosted by the American Mathematical Society as a service to researchers, faculty and students. Open Math Notes includes: - Draft works including course notes, textbooks, and research expositions. These have not been published elsewhere and are subject to revision. - Items previously published in the Journal of Inquiry-Based Learning in Mathematics, a refereed journal - Refereed publications at the AMS
Visitors are encouraged to download and use any of these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.
An Introduction to Proofs and the Mathematical Vernacular<br /> by Martin Day
My experience is that whatever students learn from that formalism is left by the wayside as soon as they move into a mathematical context of any substance.
I've experienced and seen this myself.
One of the problems in approaching quantum gravity is the choice for how to best represent it mathematically. Most of quantum mechanics is algebraic in nature but gravity has a geometry component which is important. (restatement)
This is similar to the early 20th century problem of how to best represent quantum mechanics: as differential equations or using group theory/Lie algebras?
This prompts the question: what other potential representations might also work?
Could it be better understood/represented using Algebraic geometry or algebraic topology as perspectives?
[handwritten notes from 2023-02-02]
Woit does provide problems, but they are all at the back of the book. It would have been better to see them between chapters. That provides a natural break in the material and gives the student a quick check on his understanding.
Homework problems are pedagogical devices and many (most) authors place them in the text near where they would be profitably be done. They also provide a useful break in the text to prompt more novice students to actually perform them at the end of a section.
More advanced students, however, should have caught on eventually at the need to work out examples for themselves which are presented in a textbook, but they should also be seeking out additional problems where ever they appear in the text, not to mention seeing out any outside additional problems, making up their own, and exploring any additional questions these pose.
In mathematics textbooks this working of problems, expanding on them and seeking out new ones is often a large part of what is lurking behind the sometimes nebulous sounding idea of "mathematical sophistication". The rest of that equation typically includes experience with the various methods and means of proofs and some basic background in logic.
Woit does not, for the most part, follow the death march of proposition, lemma, proof. He writes in the style of a theoretical physicist.
"death march of proposition, lemma, proof"
This is a bit harsh n'cest pas?
My freely downloadable Beginning Mathematical Logic is a Study Guide, suggesting introductory readings beginning at sub-Masters level. Take a look at the main introductory suggestions on First-Order Logic, Computability, Set Theory as useful preparation. Tackling mid-level books will help develop your appreciation of mathematical approaches to logic.
This is a reference to a great book "Beginning Mathematical Logic: A Study Guide [18 Feb 2022]" by Peter Smith on "Teach Yourself Logic A Study Guide (and other Book Notes)". The document itself is called "LogicStudyGuide.pdf".
It focuses on mathematical logic and can be a gateway into understanding Gödel's incompleteness theorems.
I found this some time ago when looking for a way to grasp the difference between first-order and second-order logics. I recall enjoying his style of writing and his commentary on the books he refers to. Both recollections still remain true after rereading some of it.
It both serves as an intro to and recommended reading list for the following: - classical logics - first- & second-order - modal logics - model theory<br /> - non-classical logics - intuitionistic - relevant - free - plural - arithmetic, computability, and incompleteness - set theory (naïve and less naïve) - proof theory - algebras for logic - Boolean - Heyting/pseudo-Boolean - higher-order logics - type theory - homotopy type theory
2 • 3 = 6
Ideally we could use Unicode, 3 2 1 ~ 3 2 1 ≃
Weiss, D. J., & Shanteau, J. (2021). The futility of decision making research. Studies in History and Philosophy of Science Part A, 90, 10–14. https://doi.org/10.1016/j.shpsa.2021.08.018
Shematologist, MD on Twitter: “How it started. How it’s going. Https://t.co/il5DWFm11W” / Twitter. (n.d.). Retrieved October 10, 2021, from https://twitter.com/acweyand/status/1442304094945873922
Impact of Social Sciences. “How Models Change the World – and What We Should Do about It,” August 20, 2021. https://blogs.lse.ac.uk/impactofsocialsciences/2021/08/20/how-models-change-the-world-and-what-we-should-do-about-it/.
Wang, B., Gou, M., Guo, Y., Tanaka, G., & Han, Y. (2020). Network structure-based interventions on spatial spread of epidemics in metapopulation networks. Physical Review E, 102(6), 062306. https://doi.org/10.1103/PhysRevE.102.062306
🔥 Kareem Carr 🔥 on Twitter. (n.d.). Twitter. Retrieved 1 May 2021, from https://twitter.com/kareem_carr/status/1383925269132582912
Holme, Petter, and Jari Saramäki. ‘Temporal Networks as a Modeling Framework’. ArXiv:2103.13586 [Physics], 24 March 2021. http://arxiv.org/abs/2103.13586.
Two of the predominant types of relationships in knowledge-representation systems are predication and the universally quantified conditional.
Ryan, W., Baum, S., & Evers, E. (2021). People Behave as if they Anticipate Regret Conditional on Experiencing a Bad Outcome. PsyArXiv. https://doi.org/10.31234/osf.io/dcgpy
To the consternation of some users, 3.x employed Unicode variable names such as λ, φ, τ and π for a concise representation of mathematical operations. A downside of this approach was that a SyntaxError would occur if you loaded the non-minified D3 using ISO-8859-1 instead of UTF-8. 3.x also used Unicode string literals, such as the SI-prefix µ for 1e-6. 4.0 uses only ASCII variable names and ASCII string literals (see rollup-plugin-ascii), avoiding encoding problems.
Health Nerd. (2021, February 1). The story continues—After @ikashnitsky and I pointed out that this paper was mathematically impossible, and had numerous errors, it was partially corrected Now, the lead author is calling us ‘trolls’ [Tweet]. @GidMK. https://twitter.com/GidMK/status/1356085063998267398
The above errors can be resolved by simply adding the ⚡attribute to the <html> tag like so: <html ⚡ lang="en">
the overloaded operators ¬, =, ≠, and abs are defined
Most of Algol's "special" characters (⊂, ≡, ␣, ×, ÷, ≤, ≥, ≠, ¬, ⊃, ≡, ∨, ∧, →, ↓, ↑, ⌊, ⌈, ⎩, ⎧, ⊥, ⏨, ¢, ○ and □) can be found on the IBM 2741 keyboard with the APL "golf-ball" print head inserted; these became available in the mid-1960s while ALGOL 68 was being drafted. These characters are also part of the Unicode standard and most of them are available in several popular fonts.
Kucharski, A. J., Klepac, P., Conlan, A. J. K., Kissler, S. M., Tang, M. L., Fry, H., Gog, J. R., Edmunds, W. J., Emery, J. C., Medley, G., Munday, J. D., Russell, T. W., Leclerc, Q. J., Diamond, C., Procter, S. R., Gimma, A., Sun, F. Y., Gibbs, H. P., Rosello, A., … Simons, D. (2020). Effectiveness of isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV-2 in different settings: A mathematical modelling study. The Lancet Infectious Diseases, 0(0). https://doi.org/10.1016/S1473-3099(20)30457-6
Velásquez, N., Leahy, R., Restrepo, N. J., Lupu, Y., Sear, R., Gabriel, N., Jha, O., Goldberg, B., & Johnson, N. F. (2020). Hate multiverse spreads malicious COVID-19 content online beyond individual platform control. ArXiv:2004.00673 [Nlin, Physics:Physics]. http://arxiv.org/abs/2004.00673
Once a searchable atlas has been constructed there are fundamentally two approaches that can be used to analyze the data: one visual, the other mathematical.