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href="https://blog.sciencenet.cn/home.php?mod=space&do=blog&view=all&uid=107667&catid=1">科研笔记</a> </div> <div id="blog_article" class="d cl" ><p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">汉语是联合国官方正式使用的 6 种同等有效语言之一。请不要歧视汉语！<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-size: 14px; white-space: normal; text-align: center; line-height: 25px; font-family: tahoma, helvetica, simsun, sans-serif, hei; background-color: rgb(255, 255, 255);">Chinese is one of the six equally effective official languages of the United Nations.<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-size: 14px; white-space: normal; text-align: center; line-height: 25px; font-family: tahoma, helvetica, simsun, sans-serif, hei; background-color: rgb(255, 255, 255);">Not to discriminate against Chinese, please!<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">                           <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">[数学文化，客观派，讨论] 欧几里得、高斯对“算术基本定理”研究的有效性<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">                           <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">数学： mathematics<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">数学基础： foundations of mathematics<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">素数： prime number<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);"><p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">算术基本定理，素数分解定理： fundamental theorem of arithmetic,  theorem of prime factorization, unique factorization theorem<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">归纳逻辑： inductive logic<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">                   <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);"><p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">要点：<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">讨论欧几里得、高斯对“算术基本定理”研究的有效性。<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;"><p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">        <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">           <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: left;">一、讨论：欧几里得、高斯对“算术基本定理”研究的有效性   算术基本定理：任何一个大于 1 的自然数可以分解成一些素数的乘积；并且在不计次序的情况下，这种分解方式是唯一的。<a href="https://www.kepuchina.cn/article/articleinfo?business_type=100&classify=0&ar_id=289929" target="_blank" target="_blank">https://www.kepuchina.cn/article/articleinfo?business_type=100&classify=0&ar_id=289929</a>            1.1  欧几里得的证明<a href="https://www.britannica.com/science/number-theory" target="_blank" target="_blank">https://www.britannica.com/science/number-theory</a>    Second, Euclid gave a version of what is known as the <a class="md-dictionary-link md-dictionary-tt-off eb" href="https://www.britannica.com/dictionary/unique" style="box-sizing: border-box; border-bottom: 2px dotted var(--blue); font-family: Georgia, serif; font-size: 18px; background-color: rgb(255, 255, 255);">unique</a> factorization theorem or the <a href="https://www.britannica.com/science/fundamental-theorem-of-arithmetic" class="md-crosslink " style="box-sizing: border-box; font-family: Georgia, serif; background-color: rgb(255, 255, 255);" target="_blank" target="_blank">fundamental theorem of arithmetic</a>. This says that any <a href="https://www.britannica.com/science/natural-number" class="md-crosslink autoxref " style="box-sizing: border-box; font-family: Georgia, serif; background-color: rgb(255, 255, 255);" target="_blank" target="_blank">whole number</a> can be factored into the product of primes in one and only one way. For example, 1,960 = 2 × 2 × 2 × 5 × 7 × 7 is a decomposition into prime factors, and no other such decomposition exists. Euclid’s discussion of unique factorization is not satisfactory by modern standards, but its essence can be found in Proposition 32 of Book VII and Proposition 14 of Book IX.   【机器翻译】其次，欧几里得给出了所谓的唯一因式分解定理或算术基本定理的一个版本。这意味着任何整数都可以以一种且只有一种方式被分解为素数的乘积。例如，1960=2×2×2×5×7×7是素数分解，不存在其他分解。欧几里得关于唯一因式分解的讨论按现代标准并不令人满意，但它的本质可以在第七卷的命题32和第九卷的命题14中找到。<a href="https://www.britannica.com/science/number-theory" target="_blank" target="_blank">https://www.britannica.com/science/number-theory</a>            1.2  高斯的证明<a href="https://www.britannica.com/science/arithmetic" target="_blank" target="_blank">https://www.britannica.com/science/arithmetic</a>    The <a href="https://www.britannica.com/science/fundamental-theorem-of-arithmetic" class="md-crosslink " style="box-sizing: border-box; font-family: Georgia, serif; background-color: rgb(255, 255, 255); font-size: 20px; text-decoration: underline;" target="_blank" target="_blank">fundamental theorem of arithmetic</a> was proved by <a href="https://www.britannica.com/biography/Carl-Friedrich-Gauss" class="md-crosslink " style="box-sizing: border-box; font-family: Georgia, serif; background-color: rgb(255, 255, 255); font-size: 20px; text-decoration: underline;" target="_blank" target="_blank">Gauss</a> in his <a href="https://www.britannica.com/topic/Disquisitiones-Arithmeticae" class="md-crosslink " style="box-sizing: border-box;" target="_blank" target="_blank">Disquisitiones Arithmeticae</a>. It states that every composite number can be expressed as a product of prime numbers and that, save for the order in which the factors are written, this representation is unique. <a href="https://www.britannica.com/science/Gausss-law" class="md-crosslink autoxref " style="box-sizing: border-box; font-family: Georgia, serif; font-size: 18px; background-color: rgb(255, 255, 255);" target="_blank" target="_blank">Gauss’s theorem</a> follows rather directly from another <a href="https://www.britannica.com/topic/theorem" class="md-crosslink autoxref " style="box-sizing: border-box; font-family: Georgia, serif; background-color: rgb(255, 255, 255); text-decoration: underline; font-size: 20px;" target="_blank" target="_blank">theorem</a> of Euclid to the effect that if a prime divides a product, then it also divides one of the factors in the product; for this reason the fundamental theorem is sometimes <a class="md-dictionary-link md-dictionary-tt-off eb" href="https://www.britannica.com/dictionary/credited" style="box-sizing: border-box; border-bottom: 2px dotted var(--blue); font-family: Georgia, serif; background-color: rgb(255, 255, 255); font-size: 20px; text-decoration: underline;">credited</a> to Euclid.   【机器翻译】高斯在《算术研究》中证明了算术的基本定理。它指出，每个复数都可以表示为素数的乘积，除了因子的书写顺序外，这种表示是唯一的。高斯定理相当直接地遵循欧几里得的另一个定理，即如果一个素数除一个乘积，那么它也除乘积中的一个因子；因此，基本定理有时被归功于欧几里德。<a href="https://www.britannica.com/science/arithmetic" target="_blank" target="_blank">https://www.britannica.com/science/arithmetic</a>            1.3  一些相关出处<a href="https://mathshistory.st-andrews.ac.uk/HistTopics/Prime_numbers/" target="_blank" target="_blank">https://mathshistory.st-andrews.ac.uk/HistTopics/Prime_numbers/</a>    Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.   【机器翻译】欧几里得还证明了算术基本定理：每个整数都可以以一种本质上唯一的方式写成素数的乘积。<a href="https://mathshistory.st-andrews.ac.uk/HistTopics/Prime_numbers/" target="_blank" target="_blank">https://mathshistory.st-andrews.ac.uk/HistTopics/Prime_numbers/</a>         <a href="https://www.britannica.com/science/factor-mathematics" target="_blank" target="_blank">https://www.britannica.com/science/factor-mathematics</a>    By the fundamental theorem of arithmetic, except for the order in which the prime factors are written, every whole number larger than 1 can be uniquely expressed as the product of its prime factors; for example, 60 can be written as the product 2·2·3·5.   【机器翻译】根据算术的基本定理，除了素数的书写顺序外，每个大于1的整数都可以唯一地表示为其素数的乘积；例如，60可以写成乘积2·2·3·5。<a href="https://www.britannica.com/science/factor-mathematics" target="_blank" target="_blank">https://www.britannica.com/science/factor-mathematics</a>         <a href="https://www.britannica.com/science/fundamental-theorem-of-arithmetic" target="_blank" target="_blank">https://www.britannica.com/science/fundamental-theorem-of-arithmetic</a>    fundamental theorem of arithmetic, Fundamental principle of number theory proved by Carl Friedrich Gauss in 1801. It states that any integer greater than 1 can be expressed as the product of prime numbers in only one way.   【机器翻译】算术基本定理，卡尔·弗里德里希·高斯于1801年证明的数论基本原理。它指出，任何大于1的整数都只能以一种方式表示为素数的乘积。<a href="https://www.britannica.com/science/fundamental-theorem-of-arithmetic" target="_blank" target="_blank">https://www.britannica.com/science/fundamental-theorem-of-arithmetic</a>         二、相关的逻辑问题<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">2.1  演绎证明的结论，是前提（假设）蕴含的<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);"><p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   一个现存的解释是，演绎推理的结论早已包含在前提之中了，实际上是已知的，推理过程只不过是把前提中隐含的信息明朗化，是对前提中已有内容的某种重复。因此，演绎推理推不出新知识。<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;">吴炜，程本学，李珍编著. 自然辩证法概论[M]. 2019，第 122-123 页<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">        <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   通俗些：<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   采用演绎推理，您的“前提”假定了什么，经过“严格”推理（以及符号公式推导）之后，就会得到“您假定的结果”。<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   严格说，演绎证明差不多几乎就是“同义反复”。<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">        <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">2.2  完全归纳推理不属于归纳推理范围，属于演绎推理的范围<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   2022-01-20，完全归纳推理/complete inductive inference/张晓芒，中国大百科全书，第三版网络版[DB/OL]<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;"><a href="https://www.zgbk.com/ecph/words?SiteID=1&ID=118651&Type=bkzyb&SubID=104156" target="_blank" target="_blank">https://www.zgbk.com/ecph/words?SiteID=1&ID=118651&Type=bkzyb&SubID=104156</a> <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   归纳推理的极限形式。又称完全归纳法。<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   完全归纳推理由前提到结论之间是一种必然性推理，所以现代逻辑学认为完全归纳推理不属于归纳推理范围，属于演绎推理的范围，是演绎推理的逆向归纳法。<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);"><p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">        <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">2.3  《Encyclopedia of Mathematics》的词条“Proof”<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255); text-align: center;"><a href="https://encyclopediaofmath.org/wiki/Proof" target="_blank" target="_blank">https://encyclopediaofmath.org/wiki/Proof</a> <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   Proof<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   A reasoning conducted according to certain rules in order to demonstrate some proposition (statement, theorem); it is based on initial statements (axioms). In practice, however, it may also be based on previously demonstrated propositions. Any proof is relative, since it is based on certain unprovable assumptions. Rules of conducting a reasoning and methods of proof form a main topic in logic. See <a href="https://encyclopediaofmath.org/wiki/Proof_theory" target="_blank" style="overflow-wrap: break-word; color: rgb(47, 73, 111); text-decoration-line: none;" target="_blank">Proof theory</a>.<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   【机器翻译】证明<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">   根据某些规则进行的推理，以证明某个命题（陈述、定理）；它基于初始陈述（公理）。然而，在实践中，它也可能基于之前证明的命题。任何证明都是相对的，因为它是基于某些未经证明（unprovable）的假设。推理规则和证明方法是逻辑学的一个主要课题。参见<a href="https://encyclopediaofmath.org/wiki/Proof_theory" target="_blank" style="overflow-wrap: break-word; color: rgb(47, 73, 111); text-decoration-line: none;" target="_blank">证明论</a>。<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">           <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">                           <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">参考资料：[1] 科普中国，2021-12-31，算术基本定理 <a href="https://www.kepuchina.cn/article/articleinfo?business_type=100&classify=0&ar_id=289929" target="_blank" target="_blank">https://www.kepuchina.cn/article/articleinfo?business_type=100&classify=0&ar_id=289929</a>    算术基本定理：任何一个大于 1 的自然数可以分解成一些素数的乘积；并且在不计次序的情况下，这种分解方式是唯一的。[2] Prime numbers, MacTutor History of Mathematics<a href="https://mathshistory.st-andrews.ac.uk/HistTopics/Prime_numbers/" target="_blank" target="_blank">https://mathshistory.st-andrews.ac.uk/HistTopics/Prime_numbers/</a>    Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.[3] fundamental theorem of arithmetic, mathematics, britannica<a href="https://www.britannica.com/science/fundamental-theorem-of-arithmetic" target="_blank" target="_blank">https://www.britannica.com/science/fundamental-theorem-of-arithmetic</a> [4] number theory, mathematics, britannica<a href="https://www.britannica.com/science/number-theory" target="_blank" target="_blank">https://www.britannica.com/science/number-theory</a> [5] arithmetic, britannica<a href="https://www.britannica.com/science/arithmetic" target="_blank" target="_blank">https://www.britannica.com/science/arithmetic</a> [6] factor, mathematics, britannica<a href="https://www.britannica.com/science/factor-mathematics" target="_blank" target="_blank">https://www.britannica.com/science/factor-mathematics</a> [7] Weisstein, Eric W. "Fundamental Theorem of Arithmetic." From MathWorld--A Wolfram Web Resource.<a href="https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html" target="_blank" target="_blank">https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html</a> [8] 知乎，2021-07-15，算术基本定理的证明<a href="https://zhuanlan.zhihu.com/p/345488859" target="_blank" target="_blank">https://zhuanlan.zhihu.com/p/345488859</a> [9] A.Göksel Ağargün, E.Mehmet Özkan. A Historical Survey of the Fundamental Theorem of Arithmetic [J]. Historia Mathematica, 2001, 28(3): 207-214.doi:  10.1006/hmat.2001.2318<a href="https://www.sciencedirect.com/science/article/pii/S0315086001923183" target="_blank" target="_blank">https://www.sciencedirect.com/science/article/pii/S0315086001923183</a>            [10] 2022-01-20，归纳逻辑/inductive logic/张家龙撰，任晓明修订，中国大百科全书，第三版网络版[DB/OL]<a href="https://www.zgbk.com/ecph/words?SiteID=1&ID=122326&Type=bkzyb&SubID=104156" target="_blank" target="_blank">https://www.zgbk.com/ecph/words?SiteID=1&ID=122326&Type=bkzyb&SubID=104156</a> [11] 2023-08-22,数学基础/foundations of mathematics/何浩平，中国大百科全书，第三版网络版[DB/OL]<a href="https://www.zgbk.com/ecph/words?SiteID=1&ID=456822&Type=bkzyb&SubID=137849" target="_blank" target="_blank">https://www.zgbk.com/ecph/words?SiteID=1&ID=456822&Type=bkzyb&SubID=137849</a>    整个数学大厦的基础。数学追求严密性，被认为是知识的典范。数学命题需要被证明才能成立，而这又要求数学概念被严格定义。用于证明的最终前提，与无法再被定义的基本概念即构成数学的基础。   证明不能求诸权威、习俗、或实用性；人们必须研究和揭示数学的基础及其可靠性。通常，在数学产生危机时，数学基础研究尤为兴盛。在当代，公理集合论被普遍接受为经典数学的基础。大部分数学家不再关心基础问题，而将数学基础研究看作数理逻辑的一个分支；而今天的数学哲学的研究则侧重于探讨关于经典数学的本体论、认识论、与语义学等问题。   ZFC公理集合论是万有理论，能够推导出经典数学的所有理论。但是，公理集合论无法被证明是一致的，人们只是在事实上迄今为止未在其中发现悖论（矛盾）；并且，其中的选择公理的地位一直为人所质疑。虽然数学仍未建立在严格的基础之上，但20世纪30、40年代后，大部分数学家已不再关心数学基础的问题。[12] 2022-01-20，第三次数学危机/the third mathematical crisis/刘叶涛，中国大百科全书，第三版网络版[DB/OL]<a href="https://www.zgbk.com/ecph/words?SiteID=1&ID=204999&Type=bkzyb" target="_blank" target="_blank">https://www.zgbk.com/ecph/words?SiteID=1&ID=204999&Type=bkzyb</a>    第三次数学危机是在数理逻辑发展史的奠基时期产生的，是数理逻辑的一些重要分支得以创立和发展的助推器。   正是这些集合论悖论的出现，导致了“数学基础的危机”。然而，危机的出现有力地推动了数学家和逻辑学家对数学基础与数理逻辑问题进行研究，发展出很多崭新的理论。   围绕如何解决此次数学危机，当时数理逻辑和数学哲学领域出现了三大学派，以罗素为代表的逻辑主义学派、以L.E.J.布劳威尔为代表的直觉主义学派、以希尔伯特为代表的希尔伯特学派。公理集合论方案如策梅洛-弗兰克尔公理集合论、冯·诺伊曼-贝尔纳斯-哥德尔公理集合论，罗素的类型论以及W.V.O.蒯因的改进型数理逻辑系统，直觉主义、多值逻辑等非经典逻辑方案，希尔伯特发展出的元数学、证明论等，都是试图解决集合论悖论、化解危机的具体方案，都是由第三次数学危机所催生的重要成果。[13] 2023-07-15，逻辑主义/logicism/刘叶涛，中国大百科全书，第三版网络版[DB/OL]<a href="https://www.zgbk.com/ecph/words?SiteID=1&ID=205196&Type=bkzyb&SubID=101959" target="_blank" target="_blank">https://www.zgbk.com/ecph/words?SiteID=1&ID=205196&Type=bkzyb&SubID=101959</a>    逻辑主义分为强的和弱的两个版本，共同主张是：①构成数学各分支的所有对象都是逻辑的对象。②逻辑能够为所有数学分支的初始概念提供定义，数学上的初始原理也可以从逻辑推导出来，因而各数学分支可以最终还原为逻辑。   不过，1931年哥德尔提出了形式算术的不完全性定理，证明逻辑公理是不完全的，不可能无遗漏地包容全部数学直至推出数学，甚至都不能推出算术，从而说明逻辑主义夸大了逻辑在数学发展中的作用。后来又出现了很多新型的逻辑主义理论，如建构性逻辑主义、模态新逻辑主义。[14] 2022-01-20，直觉主义/intuitionism/刘叶涛，中国大百科全书，第三版网络版[DB/OL]<a href="https://www.zgbk.com/ecph/words?SiteID=1&ID=205198&Type=bkzyb&SubID=101959" target="_blank" target="_blank">https://www.zgbk.com/ecph/words?SiteID=1&ID=205198&Type=bkzyb&SubID=101959</a>    直觉主义有一个著名口号——“（数学）存在等于被构造”，基本观点是，数学是一种纯粹的心智活动，数学陈述的真假只能通过这种心智的构造加以确定，对数的认识不能依赖逻辑和经验，其唯一来源是数学所固有的带有构造性的直觉。   从不承认排中律出发，直觉主义也不承认反证律、双重否定消去律等经典逻辑基本规则，从而从总体上对整个经典逻辑提出了质疑。[15] 2022-01-20，形式主义/formalism/刘叶涛，中国大百科全书，第三版网络版[DB/OL]<a href="https://www.zgbk.com/ecph/words?SiteID=1&ID=205197&Type=bkzyb&SubID=101959" target="_blank" target="_blank">https://www.zgbk.com/ecph/words?SiteID=1&ID=205197&Type=bkzyb&SubID=101959</a>    1904年，希尔伯特在第3届国际数学家大会上发表讲演，首次提出要用公理化方法为算术规律和逻辑规律奠基，同时首次提出应该把证明本身也作为研究对象。这就是希尔伯特的证明论纲领。[16] 2022-01-20，希尔伯特现代公理化方法/modern axiomatic method of Hilbert/杜国平，中国大百科全书，第三版网络版[DB/OL]<a href="https://www.zgbk.com/ecph/words?SiteID=1&ID=229204&Type=bkzyb" target="_blank" target="_blank">https://www.zgbk.com/ecph/words?SiteID=1&ID=229204&Type=bkzyb</a>    该方法是对肇始于古希腊的公理化方法的发展，其具有几个非常重要的特征：①并不直接给出基本概念的具体定义，而是通过若干公理来间接地描述基本概念所指称的对象、对象的性质及其相互之间的关系。②区分对象语言和元语言、对象理论和元理论，不仅将几何学整理为一个演绎的公理化的理论系统（对象理论），而且在更高层次上探讨该理论系统的一般性质，如几何公理之间的相互独立性、公理系统的相容性和完备性等（元理论）。③采用严格的形式化方法，尽管该公理系统可以解释为一个几何理论系统，但是并不限于解释为几何理论系统，只要一个非几何对象符合该理论系统的公理描述，那么该系统也可以解释为一个非几何的理论系统。④遵循严格的公理化方法，定理的证明不依赖于直觉，而是完全根据公理和规则进行操作。[17] 胡作玄. 第三次数学危机. 成都：四川人民出版社，1985-04.中国科普博览，<a href="http://www.kepu.net.cn/gb/basic/szsx/4/44/4_44_1001.htm" target="_blank" target="_blank">http://www.kepu.net.cn/gb/basic/szsx/4/44/4_44_1001.htm</a> <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">          <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);"><p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; background-color: rgb(255, 255, 255);">相关链接：[1] 2024-11-17，[数学文化，客观派，讨论] 欧几里得对“素数有无穷多个”研究的有效性<a href="https://blog.sciencenet.cn/blog-107667-1460458.html" target="_blank" target="_blank">https://blog.sciencenet.cn/blog-107667-1460458.html</a> [2] 2024-11-15，[数学文化] 数学研究的“客观派”正式成立！<a href="https://blog.sciencenet.cn/blog-107667-1460209.html" target="_blank" target="_blank">https://blog.sciencenet.cn/blog-107667-1460209.html</a>           [3] 2024-11-10，[数学文化，笔记] 素数有无穷多个之九类证明<a href="https://blog.sciencenet.cn/blog-107667-1459433.html" target="_blank" target="_blank">https://blog.sciencenet.cn/blog-107667-1459433.html</a> [4] 2024-11-02，[笔记，科普，资料] 素数 prime number 入门<a href="https://blog.sciencenet.cn/blog-107667-1458252.html" target="_blank" target="_blank">https://blog.sciencenet.cn/blog-107667-1458252.html</a> [5] 2024-07-24，[笔记，科普，资料] The Millennium Prize Problems 千禧年大奖难题<a href="https://blog.sciencenet.cn/blog-107667-1443609.html" target="_blank" target="_blank">https://blog.sciencenet.cn/blog-107667-1443609.html</a> [6] 2024-04-01，[笔记，数学文化] “千禧年大奖难题”，“发现全新的研究方向或领域”，后者更难能可贵<a href="https://blog.sciencenet.cn/blog-107667-1427807.html" target="_blank" target="_blank">https://blog.sciencenet.cn/blog-107667-1427807.html</a> [7] 2023-08-17，[小资料] 阿贝尔奖 The Abel Prize，挪威<a href="https://blog.sciencenet.cn/blog-107667-1399417.html" target="_blank" target="_blank">https://blog.sciencenet.cn/blog-107667-1399417.html</a> <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-family: Tahoma, Helvetica, SimSun, sans-serif, Hei; font-size: 14px; white-space: normal; text-align: center; background-color: rgb(255, 255, 255);">                  <p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-size: 14px; white-space: normal; text-align: center; line-height: 25px; font-family: tahoma, helvetica, simsun, sans-serif, hei; background-color: rgb(255, 255, 255);"><p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-size: 14px; white-space: normal; text-align: center; line-height: 25px; font-family: tahoma, helvetica, simsun, sans-serif, hei; background-color: rgb(255, 255, 255);">感谢您的指教！<p style="overflow-wrap: break-word; margin-top: 1.12em; margin-bottom: 1.12em; padding: 0px; color: rgb(51, 51, 51); font-size: 14px; white-space: normal; text-align: center; 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