# 略谈国内高中数学所谓分析法和综合法

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1   内容提要

(1)    国内高中教材中所谓的的分析法综合法不等式证明方法，分别源自二千多年前的古希腊的analysissynthesis，但与之意义并不完全相同。

(2)    术语analytic proof在现代数学中已有完全不同的含义。synthetic proof

(3)analytic methodsynthetic method都是多义的词语，其古希腊含义

2   引子

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3   笔者的困惑

1: 分析法证明实例

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2: 综合法证明实例

4   西方数学史家的记录

The terms synthesis and analysis (术语综合分析) are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have (它们在古代数学中的含义与现代不同) .

The oldest definition of mathematical analysis as opposed to synthesis (数学上最古老的分析综合的定义) is that given in Euclid, XIII. 5 (见于欧几里得《几何原本》第 13 卷命题 5。在欧氏原文中应该没有这句话，应是

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The analytic method (分析法) is not conclusive, unless all operations involved in it are known to be reversible (分析法只有每一步都可逆时，证明才可靠) . To remove all doubt, the Greeks, as a rule, added to the analytic process a synthetic one (综合法) , consisting of a reversion of all operations occurring in the analysis (综合法与分析法的步骤相反) . Thus the aim of anal-ysis was to aid in the discovery of synthetic proofs or solutions (分析

Mathematics (希腊数学简史) (Cambridge Library Collection 版本，第 177 ) 则把综合

In other words, the synthetic proof(综合法证明) proceeds by shewing that certain admitted truths involve the proposed new truth (揭示已知真理中蕴含待证的新真理) : the analytic proof (分析法证明) proceeds by shewing that the proposed new truth involves certain admitted truths (揭示待证新的新真理中蕴含了已知真理) . An analytic proof begins by an assumption, upon which a synthetic reasoning is founded.

The Greeks distinguished theoretic from problematic analysis (古希腊人区分定理型的分析问题型的分析) . A theoretic analysis is of the following kind. To prove that A is B (为了证明命题 A 等价于命题 B) , assume first that A is B (先假设 A 等价于 B) . If so, then, since B is C and C is D and D is E, therefore A is E. If this be known a falsity, A is not B. But if this be a known truth and all the intermediate propositions be convertible (以上是分析过程，以下是相反的综合证明过程) , then the reverse process, A is E, E is D, D is C, C is B, therefore A is B, constitutes a synthetic proof of the original theorem.

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The Thirteen Books of the Elements Vol. 1 (《原本》第一卷》) 里的说法。

Heath 花了近 6 页的篇幅专门介绍Analysis and Synthesis这两个术语 (在其导言部分第 IX ) ，里面除了有类似上述引文的内容之外，还有这么一段引文 (原文无分段，据称它源自公元 3-4 世纪的 Pappus)

Now analysis is of two kinds, the one directed to searching for the truth and called theoretical (定理型的) , the other directed to finding what we are told to find and called problematical (问题型的) .

(1)  In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hyhypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis (情形 (a) 与前面引文相同) , but (b), if we come upon something admittedly false, that which is sought will also be false. (情形 (b) 却不同：如果得到错误

(2)  (以下引文解释了什么是问题型分析，与本文关系不大)  In the prob-

lematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtain-able, that is, what mathematicians call given (给定，比如给定三条线段的长度) , what was originally proposed will also be possible, and the proof will again correspond in reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible. (在几何命题中，问题型的分析是为了解决一个作图问题，定理型的分析则是为了证明一个定理。à)

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由上面if we come upon something admittedly false 这半句话可知，在古希腊人那里，现在所谓的反证法也是分析法的一种。事实上，Heath 爵士还有以下一段说明 (ibid)

Reductio ad absurdum a variety of analysis. (归谬法也是一种分析法)

In the process of analysis starting from the hypothesis that a proposition A is true and passing through B, C... as successive consequences we may arrive at a proposition K which, instead of being admittedly true, is either admittedly false or the contradictory of the original hypothesis A or of some one or more of the propositions B, C... intermediate between A and K. (分析过程得到一个错误的命题或与假设矛盾)

Now correct inference from a true proposition cannot lead to a false proposition; and in this case therefore we may at once conclude, without any inquiry whether the various steps in the argument are convertible or not, that the hypothesis A is false, for, if it were true, all the consequences correctly inferred from it would be true and no incompatibility could arise.

This method of proving that a given hypothesis is false furnishes an indi-rect method of proving (由此产生一种间接证明方法) that a given hypothesis

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à顺便说一下，最近上海三联书店出版了冯翰翘先生的《The Thirteen Books of the Elements》汉译全本

Again the deductions from the first principles are divided into problems and theorems, the former embracing the generation, division, subtraction or addition of figures, and generally the changes which are brought about in them, the latter exhibiting the essential attributes of each

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A is true, since we have only to take the contradictory of A and to prove that it is false. This is the method of reductio ad absurdum (归谬法) , which is therefore a variety of analysis.

The contradictory of A, or not-A will generally include more than one case and, in order to prove its falsity, each of the cases must be separately disposed of (如果反面有多种情况，就要一一驳倒) : e.g., if it is desired to prove that a certain part of a figure is equal to some other part, we take separately the hypotheses (1) that it is greater, (2) that it is less, and prove that each of these hypotheses leads to a conclusion either admittedly false or contradictory to the hypothesis itself or to some one of its consequences. (顺便说一下，按照这里的说法，蔡天新教授对古希腊归谬法 (reductio ad absurdum)的理解似乎也不正确，见脚注à)

The ancient Analysis has been made the subject of careful studies by several writers during the last half-century, the most complete being those of Hankel, Duhamel and Zeuthen; others by Ofterdinger and Cantor (Moritz Benedikt Cantor) should also be mentioned.

5   现代西方人眼中的分析法综合法

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à蔡天新教授对归功于柏拉图的归谬法有如下脚注：

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In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not predominantly make use of algebraic or geometrical methods. The term was first used by Bernard Bolzano (波尔查诺，1781-1848，捷克数学家) , who first provided a non-analytic proof of his intermediate value theorem and then, several years later provided a proof of the theorem which was free from intuitions con-cerning lines crossing each other at a point, and so he felt happy calling it analytic.

by Bernard Bolzano (这个术语最早是由波尔查诺使用的)这句话就很能说明问题，现

Klein. Elementary Mathematics from a Higher Standpoint: Volume II, Geometry, Springer (2016)

However, I should like to add to this account an explanation of the dif-ference between analytic and synthetic geometry (论述解析几何和综合几何之间的区别) , which always plays a part in such discussions. According to their original meaning, emphcsynthesis and analysis are different methods of presentation (它们的最初含义是两种不同的表现方式) . Synthesis begins with details, and builds up from them more general, and finally the most general notions. Analysis, on the contrary, starts with the most general, and separates out more and more the details (不论克莱因教授的说法是不是它们的最初含义

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istry (合成化学分析化学，正由上述含义的差别而来) Likewise, in school geometry, we speak of the analysis of geometric constructions: we assume there that the desired triangle has been found, and we then dissect the given problem into separate partial problems.à

In higher mathematics, however, these words have, curiously, taken on an entirely different meaning (很奇怪地，它们在高等数学中具有跟上面完全不同的含义) . Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates (综合几何通过图形来研究图形，不借助公式；解析几何

fau.edu/FG2016volume16/FG201608.pdf) 。不必细看该论文，只需观察其中的符号，

thetic proof呢？在数十本国外高等数学和中学数学教材或参考书中，笔者很少见

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à《高观点下的初等数学》第二卷中译本第 66 页对本段的翻译如下：

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Synthetic proofs. These are the standard proofs that you see in text-books. They build up to the conclusion one step at a time. You can easily follow a synthetic proof, but it’s hard to construct them, except the easiest ones. (我们很容易看懂综合法证明，但不易构造它，特简单的除外)

One kind of synthetic proof is a direct proof (综合法的一种类型是直接证明) . For a direct proof of an implication P = Q, assume the hypothesis P and derive the conclusion Q. There may intermediate steps. From P you derive R, from R you derive S, and from S you derive the final conclusion Q.

There are other kinds of synthetic proofs such as indirect proofs and proof by contrapositive (综合法的另一种类型是间接证明和利用逆否命题证明) . For an indirect proof, also called a proof by contradiction (反证法) , to prove P, instead assume P is false and derive any contradiction, that is, any statement of the form Q and not Q (这里又把反证法看作是一种综合证明) . For a proof by contrapositive, to prove P = Q, you can instead prove the contrapositive, which is logically equivalent, that is, assume that Q is false and derive that P is false……

Analytic proofs. Although it’s often easier to find an indirect proof than a direct proof, there’s a different kind of proof that’s closer to what we do when we’re looking for a proof. That is, we start at the end and work back. To prove an implication P = Q, start with the goal Q and break it down into simpler statements that imply it. You might find that Q follows from S, then S follows from R, and then R follows from P. You’ve succeeded in showing that P = Q. You can always turn an analytic proof into a synthetic proof by reversing the order of your statements. It makes it easier to follow the proof, but it hides the process you used to find the proof. (这个分析的表述，与前面引述的古希腊意义相近，但显然不包括归谬法)

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6   建议慎重使用综合法分析法

(1)    尽量厘清综合法分析法在不同历史阶段、不同语境下的含义，并让学生们对此有所了解。

(2)    要处理好与其他方法的逻辑层次关系。例如，笔者以为，将作差法”“

(3)    最好不要把综合法分析法二词局限在不等式证明这一狭窄的领域，毕竟它们源自古希腊几何学，因此起码在平面几何证明中介绍它们(哪怕是作为历史背景) 会是更自然、更符合历史。

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