欧氏几何与非欧几何
Euclidean and non-Euclidean geometries
Marvin J. Greenberg · 初版 1974 年
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Euclidean and Non-Euclidean Geometries 介绍了非欧几里得几何的发现，以及对欧几里得几何基础的重新构建。
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Euclidean and non-Euclidean geometries
Marvin J. Greenberg · 初版 1974 年
Euclidean and Non-Euclidean Geometries 介绍了非欧几里得几何的发现，以及对欧几里得几何基础的重新构建。
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……看着每个部分都在前一部分的基础上层层构建！如今15年多过去了，我依然在跟自己说去读 GEB 吧，但我总怀疑这还有多大意义。一本非常值得与 N&N（内格尔与纽曼的《哥德尔证明》）对照着读的书，是格林伯格（Greenberg）写的《欧几里得与非欧几里得几何学》（Euclidean and Non-Euclidean Geometries）。它更偏教材风格，但依然很简练，而且能让你看到同样的一些主题是如何在几何学以及算术中分别展开的。
…and see each piece build on the last! Now, over 15 years later, I'm still telling myself to read GEB, but I always wonder how much point there is. A very worthwhile book to read in parallel to N&N is Euclidean and Non-Euclidean Geometries, by Greenberg. It's more textbooky, but still short, and it will let you see how some of the same themes played out in geometry as well as arithmetic.
有时候这被称为“数学无理由的有效性”——如果你想 Google 搜搜看的话。我总觉得这世界的秩序隐隐有种奇迹感。你可能也会想去读读欧氏几何与非欧几何，以及平行公设的历史，看看人类过去是如何与这个问题死磕的，以及我们的思维在过去这150年左右里是如何演进的。这是一本绝妙的好书，融合了……
Sometimes this is called "the unreasonable effectiveness of mathematics"---if you want something to Google for. I feel like the order of the world is vaguely miraculous. You might also want to read about Euclidean and Non-Euclidean geometries and the history of the parallel postulate for how humans have grappled with this question in the past, and how our thinking has evolved in the last 150ish years. This is a fantastic book with a mix of…
我又想到了一些“好玩的”：- 查尔斯·佩佐尔德的《图灵的秘密》（The Annotated Turing）- 斯科特·亚伦森的《从德谟克利特到量子计算》（Quantum Computing since Democritus）- 马文·杰伊·格林伯格的《欧几里得与非欧几里得几何学》（Euclidean and Non-Euclidean Geometries，是本教材，但自学读起来很轻松）- 内格尔与纽曼的《哥德尔证明》（Gödel's Proof，最好能找个搭档一起读，互相讨论每一步推导）- 《素数的挣扎》（Prime Obsession），作者是……
I thought of a few others that are "fun": - The Annotated Turing by Charles Petzold - Quantum Computing since Democritus by Scott Aaronson - Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg (a textbook, but an easy one to read on your own) - Gödel's Proof by Nagel and Newman (best if he can read it with a partner and talk through the steps) - Prime Obsession by…
……自洽？”哥德尔也用他的“不完备性定理”回答了这个问题，该定理指出所有自洽的形式公理系统都包含某些不可判定命题。欲知详情，请参阅欧几里得与非欧几何的简史。这说法并不严谨，而且最后一句话完全是风马牛不相及（non sequitur）。
…consistent?" Gödel also answered this problem with his "incompleteness theorem," which states that all consistent axiomatic formulations include some undecidable propositions. For more, see the short history of Euclidean and non-Euclidean geometries. This is imprecise, and the last sentence is a non sequitur.