数学2017arxiv.org
实数有多真实？
How real are real numbers? (2004)
进阶 · 约 40 分钟读完前置知识：可计算性理论、实数与可数性、数学哲学
导读
这篇论文（作者是 Gregory Chaitin）讨论实数作为连续统是否真的“存在”，还是只是一种数学理想化。文章从可定义性与可计算性出发，指出绝大多数实数无法被有限描述或计算，并梳理数学与物理上支持离散性、质疑连续性的论证。
HN 关注点：连续统的哲学与可计算性基础：如果大多数实数不可计算、不可命名，实数作为数学对象的地位是否被动摇。
HN 评论精选
这几条评论里：质疑 1 补充 3
补充介绍直觉主义用自由选择序列处理实数连续统的构造性方案。
对构造性和直觉主义进路感兴趣的人，Dummett 的《Elements of Intuitionism》非常值得一读。直觉主义是一种构造性数学基础：它既注意到否认实数不可数的困难，也注意到在内部定义实数会破坏构造性。它提出的解决办法是假定存在“自由选择序列”——本质上是潜在无限的、不可预测的二元选择序列；由于没有先验理由认为它们可被预测，它们可以比可计算对象“更大”，从而用来刻画实数。基于自由选择实数可以建立表现良好的构造性理解，尽管它会揭示我们经典假设下实数结构的各种怪异之处。这个方案的好处是绕开了难点：自由选择比有限/构造性实数要求更弱，但后者可以透明地编码进自由选择序列，数学仍能运转。
tel查看英文原文
For those interested in constructive and intuitionistic approaches here Dummett's [0] Elements of Intuitionism is an extremely good read. Intuitionism is a form of a constructive foundation for mathematics which (a) notes that any attempt to deny the uncountability of reals leads to difficulties and (b) any attempt to internally define them violates constructivity. The resolution proposed is to posit the existence of "free choice sequences". These are essentially "unpredictable" sequences of potentially infinite binary choices. As there is no a priori reason to believe that they can be predicted they are able to be much larger than what is computable and thus can be used to give a characterization of the reals. Atop this you build a constructive understanding of free choice reals which behaves very nicely (at least foundationally... it points out all kinds of weirdnesses about what we assume classically to be the structure of reals). What's very nice about this solution is that it sidesteps the difficulty. Free choice is a weaker thing to ask for than finite/constructive reals, but finite/constructive reals could be transparently encoded into free choice sequences and all the math would just work. [0] https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-Guides/dp/0198505248
质疑质疑 Borel：可数多个名称不等于大多数实数不存在，混淆了名称与对象。
文中（大意）说，按 Borel 的看法，几乎所有实数都以概率 1 是数学幻想，因为无法唯一指定它们——毕竟可能描述一个数的数学论文只有可数多篇。我认为 Borel 把名字和东西混淆了：我们只能写下可数多个数的表达式，并不意味着存在我们永远无法写出表达式的数；它只意味着在单一符号系统中，我们不能同时拥有所有数的表达式。再者，如果把“存在”和“有用”混为一谈，你可能最终会相信某些随机大整数并不“存在”！
whatshisface查看英文原文
"So, in Borel’s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely." (Paraphrasing, because there are only countably many possible math papers that might describe a number.) I think Borel has confused names with things. The fact that we can only write down only countably many expressions for numbers doesn't mean that there are numbers that we may never write expressions for - only that in a single symbolic system we can't have expressions for all of them at once. Besides, if you confuse extant with useful you might end up believing that some random large integers aren't "there!"
补充把实数视为好用的理论工具，并指出实数到复数的跨越远小于有理数到实数的完备化跨越。
实数的连续性为函数连续性提供了干净的理论基础。我个人把它看作一个相当好用的理论工具，并尽量避开哲学问题。但有一点很重要：从整数/有理数等跳到实数，比从实数跳到复数要剧烈得多。复数只是增加一个维度，而实数则是在很具体的数学意义上对有理数做抽象“完备化”。所以如果你接受实数，那么嘲笑“虚数”就完全没道理。
CogitoCogito查看英文原文
The continuity of real numbers provides a clean theoretical basis for continuity of functions. I personally view it as more of a theoretical tool that seems to do pretty well and instead steer clear of the philosophical questions. One thing that I think is important to note though is that the jump from real numbers to complex numbers is nothing compared to the jump from integers/rational numbers/etc. to real numbers. The complex numbers come about by simply adding one dimension whereas the real numbers come about from an abstract "completion" of (say) the rational numbers in a very specific mathematical sense. My point is that deriding "imaginary numbers" (as many do) is total nonsense if you accept real numbers.
补充补充作者身份与相关背景：Chaitin 常数和 Chaitin-Kolmogorov 复杂度。
这篇论文的作者是 Gregory Chaitin，他以 Chaitin 常数等工作闻名（我之前不知道 Kolmogorov 复杂度也叫 Chaitin-Kolmogorov 复杂度！）。
Sharlin查看英文原文
The author of this paper is Gregory Chaitin of Chaitin's constant fame, among other things (I didn't know that Kolmogorov complexity is also known as Chaitin-Kolmogorov complexity!)
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