(之前寫的一个跟帖)： In principle, can AI or any other software calculate the sum of a convergent infinite series? (Provided that the AI is indefinitely powerful).
For instance, the simple series 1 - 1/3 + 1/5 - 1/7 + ....... = pi /4
It doesn't matter how many terms are to be added up, the partial sum is always a rational number, but the sum (the limit) of the whole series is an irrational number (in fact, a so-called"transcendental number"). How can the AI decide on the irrationality of the sum? Since there are infinitely many terms, the speed of AI doesn't help.
If the general answer to this question is "No", then AI seems to have some serious intrinsic limitations.
Using the infinite series in the post as an analogy, the algorithm defeating human Go players is not required to decide on the "irrationality" of the sum of the whole series from a finite number of its terms that are all rational. That is, no "jump" of that kind of decision is required.
How to decide on the "irrationality" of the sum of such an infinite series without going into infinity. Not the specific value of it, but the type of number it is.
The sample given is of the best known simplest type. AI in the real world in the presumable future will hardly be dealing with anything that simple.
吾不是學計算機或人工智能的，因此衹是也衹能從普通邏輯(即常識常理)去思考一下，但也正因為這樣才合適貼論壇。也希望了解計算機或人工智能的網友能解惑一二。
一，無窮多項问题
(之前寫的一个跟帖)： In principle, can AI or any other software calculate the sum of a convergent infinite series? (Provided that the AI is indefinitely powerful).
For instance, the simple series 1 - 1/3 + 1/5 - 1/7 + ....... = pi /4
It doesn't matter how many terms are to be added up, the partial sum is always a rational number, but the sum (the limit) of the whole series is an irrational number (in fact, a so-called"transcendental number"). How can the AI decide on the irrationality of the sum? Since there are infinitely many terms, the speed of AI doesn't help.
If the general answer to this question is "No", then AI seems to have some serious intrinsic limitations.
二，任意長度问题
譬如，理論上，可以讓人工智能寫一首任意長度的詩(如一千萬行)；無疑，這樣的作品是不成立的(不可能有真實的文學意義)。而這，不正説明了它寫的正常長度的詩(如二十行)本貭上也是不成立的？
周傑倫一些歌的歌詞就給我這個感覺，字面(看似)文雅，實則無先無後，可不斷生成，長度任意。
though I don't play Go 圍棋 .
Using the infinite series in the post as an analogy, the algorithm defeating human Go players is not required to decide on the "irrationality" of the sum of the whole series from a finite number of its terms that are all rational. That is, no "jump" of that kind of decision is required.
上帝造不出搬不动的石头，所以上帝非万能。
看起来聪明，其实是小聪明。以自己能想象到能接受的能为能。
上帝的万能，是即能能，也能不能。
即能造祂搬得动的，也能造祂搬不动的。
简化版：一块一般常见的石头，上帝既能搬动又搬不动，既搬又不搬，既有又无。。。。
能想象这个，才有可能入“看”圣经的门。但离挑战上帝还差着十万八千光年长的一纳米呐。
人工智能也一样。
只能对不能错，只能赢不能输，只能精确不能模糊，这都是小聪明。
不堪大用。
走哪一条路径. 人脑创造力想象力的机理尚未搞清呢
而且這些都還是很简单的问题。。
答案是:能！
否则就不是上帝了。於是就"不是上帝"了。所以，上帝超越了是與不是。
不亦晚乎。
用符号计算软件 Maple 试验了一下， 一点困难也没有。 结果就是超越数 Pi/4
真正困难的是证明数学命题。 机器证明搞了几十年连边都没摸到。
又或，這個數列本身就是被軟件用來定義 pi ，因其简洁。
否则crush system. Ai 是用算法写的程序。这个公式几行code 就能完成. 当然是近似，可以定近似的程度。不是Ai 的范围。
How to decide on the "irrationality" of the sum of such an infinite series without going into infinity. Not the specific value of it, but the type of number it is.
The sample given is of the best known simplest type. AI in the real world in the presumable future will hardly be dealing with anything that simple.
我估计后面的算法是把级数对应到某个反三角函数的泰勒展示，推出Pi/4. 有一批人专门研究这个。
小艾同学的最大长处是搜索， 不难搜出， 这个无限求和是下面这个反正切函数泰勒展式
带入 x=1， 就知道 arctan（1） = Pi/4：
小菜一碟呀。
楼主误以为求和就必须一项一项地加。 实际上求和可以有多种不必相加的算法。
"实际上求和可以有多种不必相加的算法 " Naturally.
最多可用于机器辅助的数学定理证明。这类计算模型不可能完全自主地完成证明，但有潜力从大数据里提出人想不到的猜想。
学习的局限性之一，就是垃圾进，垃圾出。
ChatGPT的成功，很大程度得归功于，训练中用的海量的优质素材，几乎全是英文的。