数学史是由胜利者书写的吗？

注：英文引文，机翻未校。

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IS MATHEMATICAL HISTORY WRITTEN BY THE VICTORS?

数学史是由胜利者书写的吗？

ÉRIE HENRY, VLADIMIR KANOVEI, KARIN U. KATZ, MIKHAIL G. KATZ, SEMEN S. KUTATELADZE, THOMAS MCGAFFEY, DAVID M. SCHAPS, DAVID SHERRY, AND STEVEN SHNIDER

From: Mikhail G. Katz

It can be formalized as follows: 3 ^3 3
( ∀ a , b ) ( ∃ n ∈ N ) [ n a > b ] , w h e r e n a = a + . . . + a ⏟ n t i m e s . (2.1) (\forall a, b)(\exists n \in \mathbb{N})[n a>b], where n a=\underbrace{a+...+a}_{n\; times}. \tag{2.1} (∀a,b)(∃n∈N)[na>b],wherena=ntimes a+...+a.(2.1)

今日所称的阿基米德公理最早见于欧几里得《原本》第五卷定义 4（欧几里得 [35]，定义 V.4），并用于命题 V.8（欧几里得 [35]）。为清晰起见，附上括号内的符号表示：

两个量 [ a , b ] [a, b] [a,b] 称为可公度，若其一经若干倍乘 [ n a ] [n a] [na] 后能超过另一量 [ n a > b ] [n a>b] [na>b]。

其形式化表达如下： 3 ^3 3
( ∀ a , b ) ( ∃ n ∈ N ) [ n a > b ] , 其中 n a = a + . . . + a ⏟ n 次 . (2.1) (\forall a, b)(\exists n \in \mathbb{N})[n a>b], 其中 n a=\underbrace{a+...+a}_{n\; 次}. \tag{2.1} (∀a,b)(∃n∈N)[na>b],其中na=n次 a+...+a.(2.1)

Next, it appears in the papers of Archimedes as the following lemma (see Archimedes [2], I, Lamb. 5):

随后，该公理以引理形式出现在阿基米德著作中（阿基米德 [2]，第一卷，引理 5）：

Of unequal lines, unequal surfaces, and unequal solids [ a , b , c ] [a, b, c] [a,b,c], the greater exceeds the lesser [ a < b ] [a<b] [a<b] by such a magnitude [ b − a ] [b-a] [b−a] as, when added to itself [ n ( b − a ) ] [n(b-a)] [n(b−a)], can be made to exceed any assigned magnitude [ c ] [c] [c] among those which are comparable with one another (Heath 1897 [51, p. 4]).

对于不相等的线、面、体 [ a , b , c ] [a, b, c] [a,b,c]，较大量 [ a < b ] [a<b] [a<b] 超过较小量的部分 [ b − a ] [b-a] [b−a]，经自身累加 [ n ( b − a ) ] [n(b-a)] [n(b−a)] 后，可超过任一与之可公度的给定量 [ c ] [c] [c]（希思 1897 [51，第 4 页]）。

This can be formalized as follows:

其形式化表达如下：
( ∀ a , b , c ) ( ∃ n ∈ N ) [ a < b → n ( b − a ) > c ] . (2.2) (\forall a, b, c)(\exists n \in \mathbb{N})[a<b \to n(b-a)>c]. \tag{2.2} (∀a,b,c)(∃n∈N)[a<b→n(b−a)>c].(2.2)

Note that Euclid's definition V.4 and the lemma of Archimedes are not logically equivalent (see entry 3.3, footnote 11).

注意：欧几里得定义 V.4 与阿基米德引理逻辑不等价（见 3.3 节脚注 11）。

The Archimedean axiom plays no role in the plane geometry as developed in Books I--IV of The Elements. 4 ^4 4 Interpreting geometry in ordered fields, or in geometry over fields for short, one knows that F 2 \mathbb{F}^{2} F2 is a model of Euclid's plane, where ( F , + , ⋅ , 0 , 1 , < ) (\mathbb{F},+, \cdot, 0,1,<) (F,+,⋅,0,1,<) is a Euclidean field, i.e., an ordered field closed under the square root operation. Consequently, R ∗ × R ∗ \mathbb{R}^{*} \times \mathbb{R}^{*} R∗×R∗ (where R ∗ \mathbb{R}^{*} R∗ is a hyperreal field) is a model of Euclid's plane, as well (see entry 4.6 on modern implementations). Euclid's definition V.4 is discussed in more detail in entry 3.3.

阿基米德公理在《原本》第一至四卷的平面几何中并未发挥作用。 4 ^4 4 若在有序域上解释几何（简称域上几何），则 F 2 \mathbb{F}^{2} F2 是欧氏平面的模型，其中 ( F , + , ⋅ , 0 , 1 , < ) (\mathbb{F},+, \cdot, 0,1,<) (F,+,⋅,0,1,<) 为欧几里得域，即对开方运算封闭的有序域。因此， R ∗ × R ∗ \mathbb{R}^{*} \times \mathbb{R}^{*} R∗×R∗（ R ∗ \mathbb{R}^{*} R∗ 为超实数域）也是欧氏平面的模型（见 4.6 节现代实现）。3.3 节将更详细讨论欧几里得定义 V.4。

Otto Stolz rediscovered the Archimedean axiom for mathematicians, making it one of his axioms for magnitudes and giving it the following form: if a > b a>b a>b, then there is a multiple of b such that n b > a n b>a nb>a (Stolz 1885 [113, p. 69]). 5 ^5 5 At the same time, in his development of the integers Stolz implicitly used the Archimedean axiom. Stolz's visionary realisation of the importance of the Archimedean axiom, and his work on nonArchimedean systems, stand in sharp contrast with Cantor's remarks on infinitesimals (see entry 4.5 on mathematical rigor).

奥托·斯托尔兹重新向数学界揭示阿基米德公理，并将其作为量的公理之一，表述为：若 a > b a>b a>b，则存在 b b b 的倍量 n b n b nb 使得 n b > a n b>a nb>a（斯托尔兹 1885 [113，第 69 页]）。 5 ^5 5 同时，斯托尔兹在整数理论中隐含使用了该公理。斯托尔兹前瞻性地认识到阿基米德公理的重要性，并开展非阿基米德系统研究，与康托尔对无穷小的评论形成鲜明对比（见 4.5 节数学严格性）。

In modern mathematics, the theory of ordered fields employs the following form of the Archimedean axiom (see e.g., Hilbert 1899 [56, p. 27]):

现代数学中，有序域理论采用以下形式的阿基米德公理（例如希尔伯特 1899 [56，第 27 页]）：
( ∀ x > 0 ) ( ∀ ϵ > 0 ) ( ∃ n ∈ N ) [ n ϵ > x ] , (\forall x>0)(\forall \epsilon>0)(\exists n \in \mathbb{N})[n \epsilon>x], (∀x>0)(∀ϵ>0)(∃n∈N)[nϵ>x],

or equivalently

或等价形式
( ∀ ϵ > 0 ) ( ∃ n ∈ N ) [ n ϵ > 1 ] . (2.3) (\forall \epsilon>0)(\exists n\in \mathbb {N}) \quad[n \epsilon>1]. \tag{2.3} (∀ϵ>0)(∃n∈N)[nϵ>1].(2.3)

A number system satisfying (2.3) will be referred to as an Archimedean continuum. In the contrary case, there is an element ϵ > 0 \epsilon>0 ϵ>0 called an infinitesimal such that no finite sum ϵ + ϵ + . . . + ϵ \epsilon+\epsilon+...+\epsilon ϵ+ϵ+...+ϵ will ever reach 1; in other words,

A number system satisfying (2.4) is referred to as a Bernoullian continuum (i.e., a non-Archimedean continuum); see entry 2.5.

满足式 (2.3) 的数系称为阿基米德连续统 。反之，若存在 ϵ > 0 \epsilon>0 ϵ>0（称为无穷小 ），使得任意有限和 ϵ + ϵ + . . . + ϵ \epsilon+\epsilon+...+\epsilon ϵ+ϵ+...+ϵ 均无法达到 1，则该数系满足

称为伯努利连续统（即非阿基米德连续统）；见 2.5 节。

2.3. Berkeley, George

2.3 乔治·贝克莱

George Berkeley (1685--1753) was a cleric whose empiricist (i.e., based on sensations, or sensationalist) metaphysics tolerated no conceptual innovations, like infinitesimals, without an empirical counterpart or referent. Berkeley was similarly opposed, on metaphysical grounds, to infinite divisibility of the continuum (which he referred to as extension), an idea widely taken for granted today. In addition to his outdated metaphysical criticism of the infinitesimal calculus of Newton and Leibniz, Berkeley also formulated a logical criticism. 6 ^6 6 Berkeley claimed to have detected a logical fallacy at the basis of the method. In terms of Fermat's E E E occuring in his adequality (entry 2.1), Berkeley's objection can be formulated as follows:

The increment E E E is assumed to be nonzero at the beginning of the calculation, but zero at its conclusion, an apparent logical fallacy.

However, E E E is not assumed to be zero at the end of the calculation, but rather is discarded at the end of the calculation (see entry 2.4 for more details). Such a technique was the content of Fermat's adequality (see entry 2.1) and Leibniz's transcendental law of homogeneity (see entry 4.4), where the relation of equality has to be suitably interpreted (see entry 5.2 on relation ≈ \approx ≈). The technique is closely related to taking the limit (of a typical expression such as f ( A + E ) − f ( A ) E \frac{f(A+E)-f(A)}{E} Ef(A+E)−f(A) for example) in Weierstrass's approach, and to taking the standard part (see entry 5.3) in Robinson's approach.

乔治·贝克莱（1685--1753）是一位神职人员，其经验主义（基于感知）形而上学不接受无经验对应物的概念创新，例如无穷小。贝克莱同样基于形而上学立场，反对连续统的无限可分性（他称之为广延），而这一观念在今日被普遍接受。除了对牛顿与莱布尼茨无穷小微积分的过时形而上学批判外，贝克莱还提出了逻辑批判 。 6 ^6 6 贝克莱宣称发现了该方法基础中的逻辑谬误。用费马拟等式中的增量 E E E（见 2.1 节）表述，贝克莱的质疑可概括为：

增量 E E E 在计算开始时被假定非零，在计算结束时又被视为零，这是明显的逻辑矛盾。

但事实上， E E E 在计算末尾并未被设为零，而是被舍弃（详见 2.4 节）。这一操作正是费马拟等式（2.1 节）与莱布尼茨齐次性超越定律（4.4 节）的核心，其中等号需做恰当解释（见 5.2 节无限接近关系 ≈ \approx ≈）。该操作与魏尔斯特拉斯路径中取极限（如对 f ( A + E ) − f ( A ) E \frac{f(A+E)-f(A)}{E} Ef(A+E)−f(A) 取极限）、鲁宾逊路径中取标准部分（5.3 节）密切相关。

Meanwhile, Berkeley's own attempt to explain the calculation of the derivative of x 2 x^{2} x2 in The Analyst contains a logical circularity. Namely, Berkeley's argument relies on the determination of the tangents of a parabola by Apollonius (which is equivalent to the calculation of the derivative). This circularity in Berkeley's argument was analyzed in (Andersen 2011 [1]).

与此同时，贝克莱在《分析者》中自行解释 x 2 x^2 x2 导数的推导时，存在逻辑循环：其论证依赖于阿波罗尼奥斯对抛物线切线的求解（而这等价于导数计算）。贝克莱论证中的循环性见安德森（2011 [1]）。

2.4. Berkeley's logical criticism

2.4 贝克莱的逻辑批判

Berkeley's logical criticism of the calculus amounts to the contention that the evanescent increment is first assumed to be non-zero to set up an algebraic expression, and then treated as zero in discarding the terms that contained that increment when the increment is said to vanish. In modern terms, Berkeley was claiming that the calculus was based on an inconsistency of type

贝克莱对微积分的逻辑批判可概括为：消逝量在开头被假定非零以建立代数表达式，在末尾称其消逝并舍弃含该增量的项时又被视为零。用现代语言说，贝克莱认为微积分基于如下矛盾：
( d x ≠ 0 ) ∧ ( d x = 0 ) (d x \neq 0) \wedge (d x=0) (dx=0)∧(dx=0)

The criticism, however, involves a misunderstanding of Leibniz's method. The rebuttal of Berkeley's criticism is that the evanescent increment need not be "treated as zero", but, rather, is merely discarded through an application of the transcendental law of homogeneity by Leibniz, as illustrated in entry 5.1 in the case of the product rule.

但这一批判误解了莱布尼茨的方法。对贝克莱的反驳是：消逝增量并非"被视为零"，而是通过莱布尼茨齐次性超越定律被舍弃，如 5.1 节乘积法则的例子所示。

While consistent (in the sense of level (2) of entry 4.5), Leibniz's system unquestionably relied on heuristic principles such as the laws of continuity and homogeneity, and thus fell short of a standard of rigor if measured by today's criteria (see entry 4.5 on mathematical rigor). On the other hand, the consistency and resilience of Leibniz's system is confirmed through the development of modern implementations of Leibniz's heuristic principles (see entry 4.6).

尽管莱布尼茨体系是相容的（按 4.5 节第 2 层意义），但它无疑依赖连续性、齐次性等启发式原理，以今日标准衡量严格性不足（见 4.5 节数学严格性）。另一方面，莱布尼茨启发式原理的现代实现（见 4.6 节），证实了其体系的相容性与稳健性。

2.5. Bernoulli, Johann

2.5 约翰·伯努利

Johann Bernoulli (1667--1747) was a disciple of Leibniz's who, having learned an infinitesimal methodology for the calculus from the master, never wavered from it. This is in contrast to Leibniz himself, who, throughout his career, used both

约翰·伯努利（1667--1747）是莱布尼茨的弟子，他从莱布尼茨处习得无穷小微积分方法后终身坚守。这与莱布尼茨本人形成对比：莱布尼茨在整个学术生涯中并行使用两种方法：

(A) an Archimedean methodology (proof by exhaustion), and

阿基米德方法（穷竭法证明）；

(B) an infinitesimal methodology,

无穷小方法。

in a symbiotic fashion. Thus, Leibniz relied on the A-methodology to underwrite and justify the B-methodology, and he exploited the B-methodology to shorten the path to discovery (Ars Inveniendi). Historians often name Bernoulli as the first mathematician to have adhered systematically to the infinitesimal approach as the basis for the calculus. We refer to an infinitesimal-enriched number system as a B-continuum, as opposed to an Archimedean A-continuum, i.e., a continuum satisfying the Archimedean axiom (see entry 2.2).

莱布尼茨用方法 A 为方法 B 奠基与辩护，用方法 B 简化发现路径（发明术）。史家通常将伯努利视为首位系统以无穷小方法作为微积分基础 的数学家。我们将包含无穷小的数系称为B-连续统 ，以区别于满足阿基米德公理的A-连续统（见 2.2 节）。

2.6. Bishop, Errett

2.6 埃雷特·毕晓普

Errett Bishop (1928--1983) was a mathematical constructivist who, unlike his fellow intuitionist 7 ^7 7 Arend Heyting (see entry 3.8), held a dim view of classical mathematics in general and Robinson's infinitesimals in particular. Discouraged by the apparent non-constructivity of his early work in functional analysis (notably [10]), he believed to have found the culprit in the law of excluded middle (LEM), the key logical ingredient in every proof by contradiction. He spent the remaining 18 years of his life in an effort to expunge the reliance on LEM (which he dubbed "the principle of omniscience" in [12]) from analysis, and sought to define meaning itself in mathematics in terms of such LEM-extirpation.

埃雷特·毕晓普（1928--1983）是数学构造主义者。与同为直觉主义者 7 ^7 7的阿伦·海廷不同（见 3.8 节），他对经典数学整体、尤其对鲁宾逊的无穷小持否定态度。早期泛函分析工作的非构造性（尤其 [10]）令他不满，他认为排中律是根源------反证法的核心逻辑工具。他余生 18 年致力于从分析中剔除排中律（他在 [12] 中称之为"全知原理"），并试图以剔除排中律的方式定义数学的意义。

Accordingly, he described classical mathematics as both a debasement of meaning (Bishop 1973 [14, p. 1]) and sawdust (Bishop 1973 [14, p. 14]), and did not hesitate to speak of both crisis (Bishop 1975 [12]) and schizophrenia (Bishop 1973 [14]) in contemporary mathematics, predicting an imminent demise of classical mathematics in the following terms:

Very possibly classical mathematics will cease to exist as an independent discipline (Bishop 1968 [11, p. 54]).

据此，他将经典数学称为"意义的堕落"（毕晓普 1973 [14，第 1 页]）与"木屑"（毕晓普 1973 [14，第 14 页]），直言当代数学存在"危机"（毕晓普 1975 [12]）与"精神分裂"（毕晓普 1973 [14]），并预言经典数学即将消亡：

经典数学极有可能不再作为独立学科存在（毕晓普 1968 [11，第 54 页]）。

His attack in (Bishop 1977 [13]) on calculus pedagogy based on Robinson's infinitesimals was a natural outgrowth of his general opposition to the logical underpinnings of classical mathematics, as analyzed in (Katz & Katz 2011 [68]). Robinson formulated a brief but penetrating appraisal of Bishop's ventures into the history and philosophy of mathematics as follows:

The sections of [Bishop's] book that attempt to describe the philosophical and historical background of [the] remarkable endeavor [of Intuitionism] are more vigorous than accurate and tend to belittle or ignore the efforts of others who have worked in the same general direction (Robinson 1968 [102, p. 921]).

See entry 2.9 for a related criticism by Alain Connes.

他在（毕晓普 1977 [13]）中对基于鲁宾逊无穷小的微积分教学的攻击，是其反对经典数学逻辑基础的自然结果，分析见卡茨、卡茨（2011 [68]）。鲁宾逊对毕晓普涉足数学史与数学哲学的工作给出简短而犀利的评价：

毕晓普书中试图描述直觉主义哲学与历史背景的部分，言辞激烈而失准，贬低或忽视了同方向其他学者的工作（鲁宾逊 1968 [102，第 921 页]）。

阿兰·孔涅的相关批判见 2.9 节。

2.7. Cantor, Georg

2.7 格奥尔格·康托尔

Georg Cantor (1845--1918) is familiar to the modern reader as the underappreciated creator of the "Cantorian paradise" which David Hilbert would not be expelled out of, as well as the tragic hero, allegedly persecuted by Kronecker, who ended his days in a lunatic asylum. Cantor historian J. Dauben notes, however, an underappreciated aspect of Cantor's scientific activity, namely his principled persecution of infinitesimalists:

格奥尔格·康托尔（1845--1918）为现代读者熟知的形象是：被低估的"康托尔乐园"缔造者（戴维·希尔伯特称无人能将我们从此乐园驱逐），以及被克罗内克迫害、最终在精神病院离世的悲剧英雄。然而，康托尔研究者 J·道本指出其学术活动中易被忽视的一面：他有组织地打压无穷小研究者：

Cantor devoted some of his most vituperative correspondence, as well as a portion of the Beiträge, to attacking what he described at one point as the 'infinitesimal Cholera bacillus of mathematics', which had spread from Germany through the work of Thomae, du Bois Reymond and Stolz, to infect Italian mathematics . . . Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz and Veronese was to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible (Dauben 1980 [28, pp. 216--217]).

A discussion of Cantor's flawed investigation of the Archimedean axiom (see entry 2.2) may be found in entry 4.5 on mathematical rigor. 8 ^8 8

康托尔在其措辞最刻薄的通信与《基础文集》部分章节中，攻击无穷小为"数学中的霍乱杆菌"，称其经托马、杜·波依斯·雷蒙德、斯托尔兹的工作从德国蔓延，侵染意大利数学......承认无穷小的存在，必然意味着他的数论不完备。因此，接受托马、杜·波依斯·雷蒙德、斯托尔兹与韦罗内塞的工作，等于否定康托尔理论的完备性。不难理解，康托尔发起全面运动，极尽所能诋毁韦罗内塞的工作（道本 1980 [28，第 216--217 页]）。

康托尔对阿基米德公理的错误研究见 4.5 节数学严格性。 8 ^8 8

2.8. Cauchy, Augustin-Louis

2.8 奥古斯丁-路易·柯西

Augustin-Louis Cauchy (1789--1857) is often viewed in the history of mathematics literature as a precursor of Weierstrass. Note, however, that contrary to a common misconception, Cauchy never gave an ϵ , δ \epsilon, \delta ϵ,δ definition of either limit or continuity (see entry 5.4 on variable quantity for Cauchy's definition of limit). Rather, his approach to continuity was via what is known today as microcontinuity al. [15]; Borovik & Katz [17]; Bråting [21]; Katz & Katz [67], [69]; Katz & Tall [74]; Tall & Katz [114]), have argued that a proto-Weierstrassian reading of Cauchy is one-sided and obscures Cauchy's important contributions, including not only his infinitesimal definition of continuity but also such innovations as his infinitesimally defined ("Dirac") delta function, with applications in Fourier analysis and evaluation of singular integrals, and his study of orders of growth of infinitesimals that anticipated the work of Paul du Bois-Reymond, Borel, Hardy, and ultimately Skolem ([109], [110], [111]) and Robinson.

奥古斯丁-路易·柯西（1789--1857）在数学史文献中常被视为魏尔斯特拉斯的先驱。但需纠正普遍误解：柯西从未给出极限或连续性的 ϵ , δ \boldsymbol{\epsilon,\delta} ϵ,δ 定义 （柯西的极限定义见 5.4 节变量）。相反，他的连续性定义基于今日所称的微连续性。多项研究（布拉什奇克等 [15]；博罗维克、卡茨 [17]；布拉廷 [21]；卡茨、卡茨 [67]、[69]；卡茨、托尔 [74]；托尔、卡茨 [114]）指出，将柯西解读为前魏尔斯特拉斯主义是片面的，掩盖了其重要贡献：不仅包括基于无穷小的连续性定义，还包括无穷小定义的"狄拉克"δ函数（应用于傅里叶分析与奇异积分），以及对无穷小阶的研究------该工作预见了杜·波依斯·雷蒙德、博雷尔、哈代，最终斯科勒姆（[109]、[110]、[111]）与鲁宾逊的工作。

To elaborate on Cauchy's "Dirac" delta function, note the following formula from (Cauchy 1827 [24, p. 188]) in terms of an infinitesimal α \alpha α:

关于柯西"狄拉克"δ函数的具体形式，见柯西（1827 [24，第 188 页]）以无穷小 α \alpha α 表示的公式：
1 2 ∫ a − ϵ a + ϵ F ( μ ) α d μ α 2 + ( μ − a ) 2 = π 2 F ( a ) . \frac{1}{2} \int_{a-\epsilon}^{a+\epsilon} F(\mu) \frac{\alpha d \mu}{\alpha^{2}+(\mu-a)^{2}}=\frac{\pi}{2} F(a). 21∫a−ϵa+ϵF(μ)α2+(μ−a)2αdμ=2πF(a).

Replacing Cauchy's expression α α 2 + ( μ − a ) 2 \frac{\alpha}{\alpha^{2}+(\mu-a)^{2}} α2+(μ−a)2α by δ a ( μ ) \delta_{a}(\mu) δa(μ), one obtains Dirac's formula up to trivial modifications (see Dirac [31, p. 59]):

将柯西表达式 α α 2 + ( μ − a ) 2 \frac{\alpha}{\alpha^{2}+(\mu-a)^{2}} α2+(μ−a)2α 替换为 δ a ( μ ) \delta_{a}(\mu) δa(μ)，经平凡调整即得狄拉克公式（见狄拉克 [31，第 59 页]）：
∫ − ∞ ∞ f ( x ) δ ( x ) = f ( 0 ) . \int_{-\infty}^{\infty} f(x) \delta(x)=f(0). ∫−∞∞f(x)δ(x)=f(0).

Cauchy's 1853 paper on a notion closely related to uniform convergence was recently examined in (Katz & Katz 2011 (67)) and (Blaszczyk et al. 2012 [15]). Cauchy handles the said notion using infinitesimals, including one generated by the null sequence ( 1 n ) (\frac{1}{n}) (n1)

柯西 1853 年关于一致收敛相关概念的论文，近期由卡茨、卡茨（2011 [67]）与布拉什奇克等（2012 [15]）重新考察。柯西用无穷小处理该概念 ，包括由零序列 ( 1 n ) (\frac{1}{n}) (n1) 生成的无穷小。

Meanwhile, Núñez et al. (1999 [97, p. 54]) coined the term 'Cauchy--Weierstrass definition of continuity'. Since Cauchy gave an infinitesimal definition and Weierstrass, an δ \delta δ one, such a coinage is an oxymoron. J. Gray (2008a [46, p. 62]) lists continuity among concepts Cauchy allegedly defined using 'limiting arguments', but Gray unfortunately confuses the term 'limit' as bound with 'limit' as in variable tending to a quantity, since the term 'limits' appear in Cauchy's definition only in the sense of endpoints (bounds) of an interval. Not to be outdone, Kline (1980 [78, p. 273]) claims that "Cauchy's work not only banished [infinitesimals] but disposed of any need for them." Hawking (2007 [49, p. 639]) does reproduce Cauchy's infinitesimal definition, yet on the same page 639 claims that Cauchy "was particularly concerned to banish infinitesimals," apparently unaware of a comical non-sequitur he committed.

努涅斯等（1999 [97，第 54 页]）创造"柯西--魏尔斯特拉斯连续性定义"一词。然而柯西用无穷小定义，魏尔斯特拉斯用 ϵ , δ \epsilon,\delta ϵ,δ 定义，该术语本身矛盾。J·格雷（2008a [46，第 62 页]）将连续性列为柯西用"极限论证"定义的概念，但他混淆了作为"边界"的 limit 与作为"变量趋近"的极限：柯西定义中的 limits 仅指区间端点（边界）。克莱因（1980 [78，第 273 页]）声称"柯西的工作不仅摒弃了无穷小，还消除了对它的需求"。霍金（2007 [49，第 639 页]）虽转录了柯西的无穷小定义，却在同一页称柯西"致力于摒弃无穷小"，显然未意识到自己的荒谬矛盾。

2.9. Chimeras

2.9 幻造物

Alain Connes (1947--) formulated criticisms of Robinson's infinitesimals between the years 1995 and 2007, on at least seven separate occasions (see Kanovei et al. (2012) [62], Section 3.1, Table 1). These range from pejorative epithets such as "inadequate", "disappointing", "chimera", and "irremediable defect", to "the end of the rope for being 'explicit' ".

阿兰·孔涅（1947--）在 1995 至 2007 年间至少七次批判鲁宾逊的无穷小（见卡诺维奇等 2012 [62]，第 3.1 节，表 1）。批判言辞从贬损性称谓（"不恰当""令人失望""幻造物""无可救药的缺陷"）到"'显明性'走到尽头"。

Connes sought to exploit the Solovay model S S S (Solovay 1970 [112]) as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes' own earlier work in functional analysis. Connes described the hyperreals as both a "virtual theory" and a "chimera", yet acknowledged that his argument relies on the transfer principle (see entry 4.6). In S S S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being "virtual" if it is not definable in a suitable model of ZFC. If so, Connes' claim that a theory of the hyperreals is "virtual" is refuted by the existence of a definable model of the hyperreal field (Kanovei & Shelah [64]). Free ultrafilters aren't definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in his magnum opus Noncommutative Geometry (Connes 1994 [27, ch. V, sect. 6. δ \delta δ, Def. 11]), raising the question whether the latter may not be vulnerable to Connes' criticism of virtuality. The article [62] analyzed the philosophical underpinnings of Connes' argument based on Gödel's incompleteness theorem, and detected an apparent circularity in Connes' logic. The article [62] also documented the reliance on non-constructive foundational material, and specifically on the Dixmier trace f f f (featured on the front cover of Connes' magnum opus) and the Hahn--Banach theorem, in Connes' own framework; see also [70].

孔涅试图用索洛维模型 S S S（索洛维 1970 [112]）攻击非标准分析，但该模型反而反噬，削弱其早期泛函分析工作。孔涅将超实数称为"虚拟理论"与"幻造物"，却承认其论证依赖转移原理 （见 4.6 节）。在模型 S S S 中，所有可定义实数集均勒贝格可测，这表明孔涅认为：若一个理论不能在合适的 ZFC 模型中定义，则是"虚拟的"。若如此，超实数域可定义模型 的存在性（卡诺维奇、谢拉赫 [64]）已反驳孔涅的论断。自由超滤不可定义，但孔涅在 1970--80 年代因子分类工作与巨著《非交换几何》（孔涅 1994 [27，第五章，第 6.δ 节，定义 11]）中均使用此类超滤，这使其理论同样面临"虚拟性"指责。文献 [62] 分析了孔涅基于哥德尔不完备定理的论证哲学基础，发现其逻辑存在明显循环。该文还记录了孔涅自身框架对非构造性基础工具的依赖，尤其对迪西埃迹 （其著作封面标识）与哈恩--巴拿赫定理的依赖；另见 [70]。

3. Continuity to indivisibles

3. 从连续性到不可分量

3.1. Continuity

3.1 连续性

Of the two main definitions of continuity of a function, Definition A (see below) is operative in either a B-continuum or an A-continuum (satisfying the Archimedean axiom; see entry 2.2), while Definition B only works in a B-continuum (i.e., an infinitesimal-enriched or Bernoullian continuum; see entry 2.5).

Definition A ( ϵ , δ \epsilon, \delta ϵ,δ approach): A real function f f f is continuous at a real point x x x if and only if

函数连续性有两种主流定义：定义 A（见下文）在 A-连续统（满足阿基米德公理，见 2.2 节）与 B-连续统中均成立；定义 B 仅在 B-连续统（即含无穷小的伯努利连续统，见 2.5 节）中成立。

定义 A（ ϵ , δ \epsilon,\delta ϵ,δ 方法）：实函数 f f f 在实点 x x x 连续当且仅当
( ∀ ϵ > 0 ) ( ∃ δ > 0 ) ( ∀ x ′ ) [ ∣ x − x ′ ∣ < δ → ∣ f ( x ) − f ( x ′ ) ∣ < ϵ ] . (\forall \epsilon>0)(\exists \delta>0)\left(\forall x'\right)\left[\left|x-x'\right|<\delta \to\left|f(x)-f\left(x'\right)\right|<\epsilon\right]. (∀ϵ>0)(∃δ>0)(∀x′)[∣x−x′∣<δ→∣f(x)−f(x′)∣<ϵ].

Definition B (microcontinuity): A real function f f f is continuous at a real point x x x if and only if

定义 B（微连续性）：实函数 f f f 在实点 x x x 连续当且仅当
( ∀ x ′ ) [ x ′ ≈ x → f ( x ′ ) ≈ f ( x ) ] . (3.1) \left(\forall x'\right)\left[x' \approx x \to f\left(x'\right) \approx f(x)\right]. \tag{3.1} (∀x′)[x′≈x→f(x′)≈f(x)].(3.1)

In formula (3.1), the natural extension of f f f is still denoted f f f, and the symbol " ≈ \approx ≈" stands for the relation of being infinitely close; thus, x ′ ≈ x x' \approx x x′≈x if and only if x ′ − x x'-x x′−x is infinitesimal (see entry 5.2 on relation " ≈ \approx ≈").

式 (3.1) 中， f f f 的自然扩张仍记为 f f f；符号" ≈ \approx ≈"表示无限接近关系 ： x ′ ≈ x x' \approx x x′≈x 当且仅当 x ′ − x x'-x x′−x 为无穷小（见 5.2 节无限接近关系 ≈ \approx ≈）。

3.2. Diophantus

3.2 丢番图

Diophantus of Alexandria (who lived about 1800 years ago) contributed indirectly to the development of infinitesimal calculus through the technique called π α ρ ι σ ο ˊ τ η ς \pi\alpha\rho\iota\sigma\acute{\omicron}\tau\eta\varsigma παρισοˊτης, developed in his work Arithmetica, Book Five, problems 12, 14, and 17. The term π α ρ ι σ ο ˊ τ η ς \pi\alpha\rho\iota\sigma\acute{\omicron}\tau\eta\varsigma παρισοˊτης can be literally translated as "approximate equality". This was rendered as adaequalitas in Bachet's Latin translation [4], and adégalité in French (see entry 2.1 on adequality). The term was used by Fermat to describe the comparison of values of an algebraic expression, or what would today be called a function f f f, at nearby points A A A and A + E A+E A+E, and to seek extrema by a technique that would be re-formulated today in terms of the vanishing of f ( A + E ) − f ( A ) E \frac{f(A+E)-f(A)}{E} Ef(A+E)−f(A) after discarding the remaining E E E-terms; see (Katz, Schaps & Shnider 2013 [71]).

亚历山大的丢番图（约 1800 年前）通过其《算术》第五卷问题 12、14、17 中提出的 π α ρ ι σ ο ˊ τ η ς \pi\alpha\rho\iota\sigma\acute{\omicron}\tau\eta\varsigma παρισοˊτης 方法，间接推动了无穷小微积分的发展。该术语字面意为"近似相等"，在巴谢的拉丁文译本 [4] 中译为 adaequalitas，法文为 adégalité（见 2.1 节拟等式）。费马用该术语描述代数表达式（今日称为函数 f f f）在邻近点 A A A 与 A + E A+E A+E 处的取值比较，并通过舍弃剩余 E E E 项后令 f ( A + E ) − f ( A ) E \frac{f(A+E)-f(A)}{E} Ef(A+E)−f(A) 为零的方法求极值；见卡茨、沙普斯、施奈德（2013 [71]）。

3.3. Euclid's definition V.4

3.3 欧几里得《原本》第五卷定义 4

Euclid's definition V.4 was already mentioned in entry 2.2. In addition to Book V, it appears in Books X and XII and is used in the method of exhaustion (see Euclid [35], Propositions X.1, XII.2). The method of exhaustion was exploited intensively by both Archimedes and Leibniz (see entry 4.2 on Leibniz's work De Quadratura). It was revived in the 19th century in the theory of the Riemann integral.

欧几里得定义 V.4 已在 2.2 节提及。除第五卷外，该定义还出现在第十卷与第十二卷，并用于穷竭法（见欧几里得 [35]，命题 X.1、XII.2）。穷竭法被阿基米德与莱布尼茨广泛使用（见 4.2 节莱布尼茨《论算术求积》），并在 19 世纪重生于黎曼积分理论。

Euclid's Book V sets the basis for the theory of similar figures developed in Book VI. Great mathematicians of the 17th century like Descartes, Leibniz, and Newton exploited Euclid's theory of similar figures of Book VI while paying no attention to its axiomatic background. 9 ^9 9 Over time Euclid's Book V became a subject of interest for historians and editors alone.

欧几里得第五卷为第六卷相似形理论奠基。17 世纪笛卡尔、莱布尼茨、牛顿等大数学家使用第六卷相似形理论，却无视其公理基础。 9 ^9 9 久而久之，第五卷仅成为史家与校勘者的研究对象。

To formalize Definition V.4, one needed a formula for Euclid's notion of "multiple" and an idea of total order. Some progress in this direction was made by Robert Simson in 1762. 10 ^{10} 10 In 1876, Hermann Hankel provided a modern reconstruction of Book V. Combining his own historical studies with an idea of order compatible with addition developed by Hermann Grassmann (1861 [45]), he gave a formula that to this day is accepted as a formalisation of Euclid's definition of proportion in V.5 (Hankel 1876 [50, pp. 389--398]).

为形式化定义 V.4，需要严格表达欧几里得的"倍量"概念与全序思想。罗伯特·辛普森在 1762 年对此有所推进。 10 ^{10} 10 1876 年，赫尔曼·汉克尔对第五卷给出现代重构：结合自身史学研究与赫尔曼·格拉斯曼（1861 [45]）提出的加法相容序思想，给出至今仍被认可的欧几里得 V.5 比例定义形式化（汉克尔 1876 [50，第 389--398 页]）。

Euclid's proportion is a relation among four "magnitudes", such as

欧几里得的比例是四个"量"之间的关系，形如
Λ ( n A = m B → n C = m D ) Λ ( n A < 1 m B → n C < 2 m D ) ] , \left.\Lambda(n A=m B \to n C=m D) \Lambda\left(n A<{1} m B \to n C<{2} m D\right)\right], Λ(nA=mB→nC=mD)Λ(nA<1mB→nC<2mD)],

It was interpreted by Hankel as the relation where n n n, m m m are natural numbers. The indices on the inequalities emphasize the fact that the "magnitudes" A A A, B B B have to be of "the same kind", e.g., line segments, whereas C C C, D D D could be of another kind, e.g., triangles.

汉克尔将其解释为： n , m n,m n,m 为自然数；不等式下标强调量 A , B A,B A,B 必须"同类"（如线段），而 C , D C,D C,D 可为另一类（如三角形）。

In 1880, J. L. Heiberg in his edition of Archimedes' Opera omnia, in a comment on a lemma of Archimedes, cites Euclid's definition V.4, noting that these two are the same axioms (Heiberg 1880 [53, p. 11]). 11 ^{11} 11

1880 年，J·L·海伯格在其编订的《阿基米德全集》中，评注阿基米德引理时引用欧几里得定义 V.4，称二者为同一公理（海伯格 1880 [53，第 11 页]）。 11 ^{11} 11

This is the reason why Euclid's definition V.4 is commonly known as the Archimedean axiom. Today we formalize Euclid's definition V.4 as in (2.1), while the Archimedean lemma is rendered by formula (2.2).

这就是欧几里得定义 V.4 通常被称为阿基米德公理的原因。今日我们将欧几里得定义 V.4 形式化为式 (2.1)，阿基米德引理形式化为式 (2.2)。

3.4. Euler, Leonhard

3.4 莱昂哈德·欧拉

Euler's Introductio in Analysin Infinitorum (1748 [36]) contains remarkable calculations carried out in an extended number system in which the basic algebraic operations are applied to infinitely small and infinitely large quantities. Thus, in Chapter 7, "Exponentials and Logarithms Expressed through Series", we find a derivation of the power series for a z a^{z} az starting from the formula a ω = 1 + k ω a^{\omega}=1+k \omega aω=1+kω, for ω \omega ω infinitely small and then raising the equation to the infinitely great power 12 ^{12} 12 j = z ω j=\frac{z}{\omega} j=ωz for a finite (appreciable) z z z to give

欧拉《无穷分析引论》（1748 [36]）包含在扩张数系 中完成的精彩计算：基本代数运算直接应用于无穷小与无穷大量。例如在第 7 章"指数与对数的级数表示"中，他从无穷小 ω \omega ω 满足 a ω = 1 + k ω a^{\omega}=1+k \omega aω=1+kω 出发，对有限（可观）量 z z z，将等式取无穷大幂次 12 ^{12} 12 j = z ω j=\frac{z}{\omega} j=ωz，得
a z = a j ω = ( 1 + k ω ) j a^{z}=a^{j \omega}=(1+k \omega)^{j} az=ajω=(1+kω)j

and finally expanding the right hand side as a power series by means of the binomial formula.

最终用二项式公式将右端展开为幂级数。

In the chapters following Euler finds infinite product expansions factoring the power series expansion for transcendental functions (see entry 3.5 for his infinite product formula for sine). By Chapter 10, he has the tools to sum the series for ζ ( 2 ) \zeta(2) ζ(2) where ζ ( s ) = ∑ n n − s \zeta(s)=\sum_{n} n^{-s} ζ(s)=∑nn−s. He explicitly calculates ζ ( 2 k ) \zeta(2 k) ζ(2k) for k = 1 , . . . , 13 k=1, ..., 13 k=1,...,13 as well as many other related infinite series.

在后续章节中，欧拉得到超越函数幂级数的无穷乘积展开（正弦无穷乘积公式见 3.5 节）。到第 10 章，他已掌握求和 ζ ( 2 ) \zeta(2) ζ(2) 的工具，其中 ζ ( s ) = ∑ n n − s \zeta(s)=\sum_{n} n^{-s} ζ(s)=∑nn−s。他显式计算了 k = 1 , ... , 13 k=1,\dots,13 k=1,...,13 时的 ζ ( 2 k ) \zeta(2k) ζ(2k) 以及众多相关无穷级数。

In Chapter 3 of his Institutiones Calculi Differentialis (1755 [38]), Euler deals with the methodology of the calculus, such as the nature of infinitesimal and infinitely large quantities. We will cite the English translation [39] of the Latin original [38]. Here Euler writes that

e\]ven if someone denies that infinite numbers really exist in this world, still in mathematical speculations there arise questions to which answers cannot be given unless we admit an infinite number (ibid., § 82) \[emphasis added--the authors\]. 在《微分学原理》（1755 \[38\]）第 3 章中，欧拉阐述微积分方法论，包括无穷小与无穷大量的本性。此处引用拉丁文原著 \[38\] 的英译本 \[39\]： 即便有人否认无穷数在现实世界中真实存在，在数学研究中仍会出现唯有承认无穷数才能解答的问题（同上，第 82 节）【作者强调】。 Euler's approach, countenancing the possibility of denying that "infinite numbers really exist", is consonant with a Leibnizian view of infinitesimal and infinite quantities as "useful fictions" (see Katz \& Sherry \[73\]; Sherry \& Katz \[107\]). Euler then notes that "an infinitely small quantity is nothing but a vanishing quantity, and so it is really equal to 0" (ibid., § 83). The "equality" in question is an arithmetic one (see below). 欧拉的方法允许否认"无穷数真实存在"，这与莱布尼茨将无穷小与无穷大视为"**有用虚构**"的观点一致（见卡茨、谢里 \[73\]；谢里、卡茨 \[107\]）。欧拉随后指出："无穷小量不过是消逝量，因此它实则等于 0"（同上，第 83 节）。此处的"相等"是算术意义上的（见下文）。 Similarly, Leibniz combined a view of infinitesimals as "useful fictions" and inassignable quantities, with a generalized notion of "equality" which was an equality up to an incomparably negligible term. Leibniz sought to codify this idea in terms of his transcendental law of homogeneity (TLH); see entry 4.4. Thus, Euler's formulas like 同样，莱布尼茨将无穷小视为"有用虚构"与**不可 assignable 量** ，并结合广义"相等"概念------差为不可比拟的可忽略项。莱布尼茨试图用**齐次性超越定律（TLH）** 系统化这一思想；见 4.4 节。因此，欧拉的公式 a + d x = a (3.2) a+d x=a \\tag{3.2} a+dx=a(3.2) (where a a a "is any finite quantity"; ibid., 86, 87) are consonant with a Leibnizian tradition (cf. formula (4.1) in entry 4.4). （其中 a a a 为"任意有限量"；同上，第 86、87 节）与莱布尼茨传统一致（参见 4.4 节公式 (4.1)）。 To explain formulas like (3.2), Euler elaborated two distinct ways (arithmetic and geometric) of comparing quantities in the following terms: 为解释式 (3.2) 这类公式，欧拉详细区分了两种量的比较方式（算术比较与几何比较）： Since we are going to show that an infinitely small quantity is really zero, we must meet the objection of why we do not always use the same symbol 0 for infinitely small quantities, rather than some special ones. . . \[S\]ince we have two ways to compare them, either arithmetic or geometric, let us look at the quotients of quantities to be compared in order to see the difference. 既然要证明无穷小量实则为零，就必须回应质疑：为何不用统一符号 0 表示无穷小，而用特殊符号......因为我们有两种比较方式：算术比较与几何比较。通过考察被比较量的商，可看清二者区别。 If we accept the notation used in the analysis of the infinite, then d x d x dx indicates a quantity that is infinitely small, so that both d x = 0 d x=0 dx=0 and a d x = 0 a d x=0 adx=0, where a a a is any finite quantity. Despite this, the geometric ratio a d x : d x a d x: d x adx:dx is finite, namely a : 1 a: 1 a:1. For this reason, these two infinitely small quantities, d x d x dx and a d x a d x adx, both being equal to 0, cannot be confused when we consider their ratio. In a similar way, we will deal with infinitely small quantities d x d x dx and d y d y dy (ibid., 86, p. 51--52) \[emphasis added--the authors\]. 若接受无穷分析的记号，则 d x d x dx 表示无穷小量，故 d x = 0 d x=0 dx=0 且 a d x = 0 a d x=0 adx=0（ a a a 为任意有限量）。尽管如此，**几何比** a d x : d x a d x: d x adx:dx 是有限的，即 a : 1 a:1 a:1。因此，这两个同为零的无穷小量 d x d x dx 与 a d x a d x adx，在考虑比值时不可混淆。同理处理无穷小量 d x d x dx 与 d y d y dy（同上，第 86 节，第 51--52 页）【作者强调】。 Euler proceeds to clarify the difference between the arithmetic and geometric comparisons as follows: 欧拉进一步澄清算术比较与几何比较的区别： Let a a a be a finite quantity and let d x dx dx be infinitely small. The arithmetic ratio of equals is clear: Since n d x = 0 n dx=0 ndx=0, we have 设 a a a 为有限量， d x dx dx 为无穷小量。相等量的**算术比** 显而易见：由于 n d x = 0 n dx=0 ndx=0，故有 a ± n d x = a a \\pm n dx = a a±ndx=a On the other hand, the geometric ratio is clearly of equals, since 另一方面，相等量的**几何比** 同样显然成立，因为 a ± n d x a = 1. (3.3) \\frac{a \\pm n dx}{a}=1. \\tag{3.3} aa±ndx=1.(3.3) From this we obtain the well-known rule that the infinitely small vanishes in comparison with the finite and hence can be neglected (Euler 1755 \[39, §87\]) \[emphasis in the original--the authors\]. 由此我们得到熟知的法则：无穷小量与有限量相比可完全忽略（欧拉 1755 \[39, §87\]）【原文强调】。 Like Leibniz, Euler considers more than one way of comparing quantities. Euler's formula (3.3) indicates that his geometric comparison is procedurally identical with the Leibnizian TLH. Namely, Euler's geometric comparision of a pair of quantities amounts to their ratio being infinitely close to 1; the same is true for TLH. Thus, one has a + d x = a a+dx=a a+dx=a in this sense for an appreciable a ≠ 0 a \\neq 0 a=0, but not d x = 0 dx=0 dx=0 (which is true only arithmetically in Euler's sense). Euler's "geometric" comparison was dubbed "the principle of cancellation" in (Ferraro 2004 \[40, p. 47\]). 与莱布尼茨一样，欧拉采用多种方式比较量。欧拉公式 (3.3) 表明，他的**几何比较** 在操作上与莱布尼茨齐次性超越定律（TLH）完全一致。具体来说，欧拉对两个量的几何比较等价于它们的比值无限接近 1；TLH 亦是如此。因此，对非零有限量 a a a，在该意义下有 a + d x = a a+dx=a a+dx=a，但并非 d x = 0 dx=0 dx=0（后者仅在欧拉的算术意义下成立）。欧拉的"几何比较"在费拉罗（2004 \[40, p. 47\]）中被称为**消去原理**。 Euler proceeds to present the usual rules of infinitesimal calculus, which go back to Leibniz, L'Hôpital, and the Bernoullis, such as 欧拉继而给出无穷小微积分的常规运算法则，这些法则可追溯至莱布尼茨、洛必达与伯努利家族，例如 a   d x m + b   d x n = a   d x m (3.4) a\\, dx\^{m}+b\\, dx\^{n}=a\\, dx\^{m} \\tag{3.4} adxm+bdxn=adxm(3.4) provided m \< n m\ * calculus - Is mathematical history written by the victors? - 2013 * reference request - Good "history of mathematical ideas" book? - 2016