In his impressive Leibniz on Time, Space, and Relativity, Ric Arthur manages to juggle a daunting array of tasks: tracking the chronological development of Leibniz’s views over more than half a century; explicating Leibniz’s groundbreaking mathematics; assembling texts—primary and secondary—in at least five languages; and, as if in passing, offering original translations and assessments of countless source materials. All this erudition is put to the service of offering detailed interpretations of Leibniz’s challenging theories of time, space, and motion. Arthur’s performance is a lifetime in the making, and his Leibniz on Time, Space, and Relativity is certain to be essential reading for those interested in the topics it covers for many years to come.Leibniz’s subtle theory of time defies easy summary. According to Arthur, Leibniz’s theory of time is ultimately grounded in relations among states of substances. States of substances are representations of a world from a perspective. States that do not contradict one another occur at the same time. States that do contradict one another are successive. Among successive states, some provide reasons for others. A state that provides a reason for another state is temporally prior to that state. One thing exists before, after, or at the same time as another thing not because of the way both things are related to some special, independent third thing—namely, time—but rather because of the relations they bear to one another (and other similar things). A occurs before, after, or at the same time as B because of the relations between A and B. For Arthur’s Leibniz, time itself is an abstract ordering relation that structures not only all actual temporal relations but also all possible temporal relations. It guarantees not only that my fifth birthday must precede my fiftieth but also that my fiftieth must precede my merely possible five hundredth.Is Leibniz eliminating time? Many commentators have thought so. Leibniz was a nominalist, and nominalists typically deny that abstract objects exist. In holding that time is abstract, mustn’t Leibniz also hold that time doesn’t really exist? “No,” says Arthur. Abstract objects, for Leibniz, have a home in the “divine mind” (61). Thus, while the abstract structure that orders all possible existing things can’t itself exist in the concrete world, that doesn’t mean that it doesn’t exist at all. Furthermore, and perhaps even more importantly, the temporal relations holding between things in the world are not abstract. Even if time itself were not real, Arthur’s Leibniz would still insist that my fifth birthday occurred before my fiftieth birthday and at roughly the same time as my older brother’s seventh birthday. Even if time is abstract, temporal relations are not.But wait, doesn’t Leibniz also hold that relations are ideal? And in holding that temporal relations are ideal, isn’t Leibniz suggesting that temporal relations themselves are not real? Again, Arthur thinks not. He argues that while “Leibniz denies reality to relations as entities in themselves, he does not thereby deny the reality of relational facts, such as the fact of two states of a substance occurring one after the other” (61). Suppose that Plato is older than Aristotle. For various reasons, Leibniz thinks that any relation between Plato and Aristotle can exist neither solely in Plato nor solely in Aristotle, nor in both of them jointly. Any relation between Plato and Aristotle must therefore be grounded at least partially in a mind that considers them both. Relations are, for Leibniz, ideal in precisely that sense. But “Plato is older than Aristotle” is still true even if no one considers Plato and Aristotle. That Plato is older than Aristotle is grounded in Plato’s intrinsic nature and Aristotle’s intrinsic nature. The older-than relation is metaphysically ideal, but relational “facts” are not. Arthur’s Leibniz has a somewhat complex view concerning the foundations of time, but it would be inaccurate, Arthur thinks—and I agree—to say that Leibniz means to eliminate time.Arthur attributes to Leibniz a similar view of space. In his famous exchange of letters with Samuel Clarke, Leibniz contrasts his mature view of space (and time) with Newton’s views. In broad strokes, Clarke and Newton think of space as an immobile, unchanging backdrop. Bodies are in space by being collocated with regions of space. Bodies move through space by collocating with different regions of space at different times. In opposition to this picture, and in keeping with his own understanding of time, Leibniz suggests that existing things stand directly in spatial relations to one another. As with time, one thing is cospatial with another thing not because of the way that both are related to some special, independent third thing—namely, space—but rather because of the relations they bear to one another (and other similar things). And, again, as with time, space itself is an abstract ordering relation that structures not only all actual spatial relations at a time but also all possible spatial relations at a time. It is, Leibniz tells us, the “order of coexistence,” and, like time, it is an abstract, ideal structure that resides in the divine intellect.Leibniz wrote his letters to Clarke in the final years of his life. (One imagines him dying with pen in hand, one more argument against his great adversary still in his inkwell.) Leibniz’s thinking about the nature of space, however, goes back to some of his very earliest writings. In those early writings, according to Arthur, Leibniz developed the idea that “the divine attribute of immensity is the basis of space” and that divine immensity is divided into a network of regions by the motions of bodies contained in it (123). On Arthur’s telling, Leibniz was then led by his study of Hobbes to develop a new mathematics of situation, Leibniz’s famed analysis situs. Leibniz’s analysis situs, in turn, provided him with formal tools for characterizing space both as a changing network of regions and as a system of relative positions. It became possible to understand space as an abstract system of possible positions structuring the spatial relations between existing things. On Arthur’s reading, there is thus a progressive, mostly continuous line of development in Leibniz’s thinking about the nature of space from the late 1670s until his death in 1716.Arthur’s progressive, mostly continuous reading of Leibniz’s views on space stands in provocative contrast to past and recent work by Vincenzo de Risi. In forthcoming work, De Risi (forthcoming) argues that Leibniz’s views on space can be divided into three periods: an early substantivalist view, a middle Scholastic view rooted in the idea of imaginary space, and his famous late relational view. Most intriguingly, De Risi proposes that Leibniz’s relational view emerges for the first time only in the late 1680s and that it is prompted precisely by Leibniz’s clandestine reading of Newton’s Principia. On De Risi’s reading, Leibniz’s views on space are not an evolving refinement of a core theory but rather a series of essentially distinct theories. Leibniz’s relational theory is not the almost miraculously coincidental twin of Newton’s substantivalism, as it might appear to be from the Leibniz–Clark correspondence. Rather, it was crafted from the ground up by Leibniz as an alternative to the position of his great rival. Arthur and de Risi have opened up an exciting interpretative area of great consequence for early modern philosophy and science. Scholars looking for the next big thing in Leibniz studies may need to look no further.In taking up Leibniz’s views on motion, Arthur emphasizes a distinction Leibniz makes between motion conceived geometrically and motion with respect to cause. Motion conceived geometrically involves nothing more than a mere change of situation. Leibniz maintains that between any two bodies geometrical motion is relative and mutual. If our ships pass in the night, your ship moves relative to mine and mine moves relative to yours. There’s no saying—it doesn’t even make sense to ask—which is really moving. Motion with respect to cause “involves identifying the subject of motion by reference to the best explanation of the change” (321). Suppose there is a breeze and your sails are raised. The best explanation for our change of relative situation is that your boat is being driven by the wind while mine is resting in the water. In this case, the best hypothesis is that your boat is truly moving with respect to cause while mine is not. Motion with respect to cause is thus not—in contrast to geometrical motion—necessarily relative. In the case of motion with respect to cause, we can ask, and even hazard an educated guess, whether it is your boat or my boat that is really moving.Arthur shows how Leibniz’s distinction between geometrical and causal motion may be used to untangle some prima facie puzzles that have been raised for Leibniz’s theory of motion. Take, for example, the apparent tension between Leibniz’s commitment to the “equipollence of hypotheses” and his apparent endorsement of Copernicanism. In explicating the equipollence of hypotheses, Leibniz insists that “all hypotheses about the motions of any bodies whatsoever, however numerous, that are moved solely by corporeal impacts, it follows that not even an angel could discern, in mathematical rigor, which of the many bodies of this kind is at rest, and is the center of motion of the others” (248–49). Copernicanism, however, suggests that we—not to mention the angels—can say that the earth and planets move and that the sun is at rest. Faced with the obvious tension between these two commitments, it is tempting simply to favor one commitment at the expense of the other. It is tempting to suppose either that Leibniz’s equipollence of hypotheses is a disingenuous sop to the powers of Rome or that Leibniz’s commitment to Copernicanism is faint of heart—that Leibniz, for example, is only commending Copernicanism as merely one of many possible systems.On Arthur’s reading, Leibniz’s distinction between geometrical and causal motion suggests a more satisfying middle path. Leibniz does think that from a geometrical point of view, Copernicanism should be counted as merely one among many possible systems. And he really does believe that even an angel couldn’t say, from a geometrical point of view, whether the planets or the sun is in motion. From a geometrical point of view, such questions don’t even make sense. But they do make sense from a causal point of view, and from a causal point of view we can make better and worse hypotheses about which bodies are really moving. Leibniz can thus coherently insist that from a geometrical point of view Copernicanism is merely one among many possible systems, while still maintaining that from a causal point of view the Copernican system is our best hypothesis, that the planets really move, that the sun is really at rest. On Arthur’s reading, even if Leibniz’s language is conciliatory, there is nothing faint-hearted about his commitment to Copernicanism.Leibniz On Time, Space, and Relativity is a masterpiece of scholarship. It is historically, philosophically, and technically rich. (Did I mention that it includes a series of appendixes containing formal expositions, detailed explications of Arthur’s background views, and original translations? There is even a glossary of technical terms!) It is not, it should be conceded, an easy book. Although clearly written and organized, it is long and detailed. There is enough material between its covers for at least three important books, one on each of its principal subjects. Indeed, it is so chock-full of arguments, textual notes, and scholarly details that one might be tempted to suppose that it is dense in the mathematician’s sense, that between any two philosophical arguments, notes, or scholarly details, one will always be able to find another argument, note, or scholarly detail. For those with a taste for virtuosic scholarship and an interest in Leibniz, early modern philosophy, or the philosophy of space and time, Arthur’s masterpiece promises a dense but endless series of delights. Treat yourself.

中文翻译：

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莱布尼茨论时间、空间和相对论

在他令人印象深刻的《莱布尼茨关于时间、空间和相对论》一书中，里克·阿瑟成功地完成了一系列艰巨的任务：追踪莱布尼茨观点半个多世纪以来的时间发展；解释莱布尼茨开创性的数学；汇编至少五种语言的文本（主要和次要）；并且，仿佛顺便提供了无数原始材料的原始翻译和评估。所有这些博学都是为了对莱布尼茨具有挑战性的时间、空间和运动理论提供详细的解释。亚瑟的表演是一生的创作，他的《莱布尼茨论时间、空间和相对论》对于那些对未来许多年里所涵盖的主题感兴趣的人来说肯定是必读的。莱布尼茨微妙的时间理论很难简单概括。根据阿瑟的说法，莱布尼茨的时间理论最终是以物质状态之间的关系为基础的。物质状态是从某种角度来看的世界的表征。彼此不矛盾的状态同时出现。相互矛盾的国家是连续的。在连续的状态中，有些状态为其他状态提供了理由。为另一个状态提供原因的状态在时间上先于该状态。一件事在另一件事之前、之后或同时存在，并不是因为这两件事与某些特殊的、独立的第三件事（即时间）相关的方式，而是因为它们彼此之间的关系（以及其他事物）类似的事情）。由于 A 和 B 之间的关系，A 出现在 B 之前、之后或同时。对于亚瑟的莱布尼茨来说，时间本身是一种抽象的排序关系，它不仅构造了所有实际的时间关系，而且还构造了所有可能的时间关系。它不仅保证我的五岁生日必须先于我的五十岁，而且还保证我的五十岁生日必须先于我仅有的可能的五百岁。莱布尼茨正在消除时间吗？很多评论家都这么认为。莱布尼茨是一位唯名论者，而唯名论者通常否认抽象对象的存在。莱布尼茨在认为时间是抽象的同时，难道不也认为时间实际上并不存在吗？“不，”亚瑟说。对于莱布尼茨来说，抽象对象在“神圣心灵”中拥有一个家（61）。因此，虽然对所有可能存在的事物进行排序的抽象结构本身不能存在于具体世界中，但这并不意味着它根本不存在。此外，也许更重要的是，世界上事物之间的时间关系并不是抽象的。即使时间本身不是真实的，亚瑟的莱布尼茨仍然坚持认为我的五岁生日发生在我五十岁生日之前，并且与我哥哥的七岁生日大致在同一时间。即使时间是抽象的，时间关系也不是。但是等等，莱布尼茨不是也认为关系是理想的吗？莱布尼茨认为时间关系是理想的，这难道不是在暗示时间关系本身并不真实吗？再次，亚瑟不这么认为。他认为，虽然“莱布尼茨否认关系作为实体本身的现实性，但他并没有因此否认关系事实的现实性，例如物质的两种状态相继发生的事实”（61）。假设柏拉图比亚里士多德年长。由于种种原因，莱布尼茨认为柏拉图与亚里士多德之间的任何关系既不能单独存在于柏拉图，也不能单独存在于亚里士多德，更不能共同存在于他们两者之中。因此，柏拉图和亚里士多德之间的任何关系都必须至少部分地建立在考虑他们两人的头脑中。对于莱布尼茨来说，关系正是在这个意义上是理想的。但即使没有人考虑柏拉图和亚里士多德，“柏拉图比亚里士多德更古老”仍然是正确的。柏拉图比亚里士多德年长，这是基于柏拉图的内在本质和亚里士多德的内在本质。更古老的关系在形而上学上是理想的，但关系“事实”却不是。亚瑟的莱布尼茨对于时间的基础有一个有些复杂的观点，但亚瑟认为——我同意——说莱布尼茨意味着消除时间是不准确的。亚瑟认为莱布尼茨对空间有类似的看法。在与塞缪尔·克拉克的著名信件往来中，莱布尼茨将他成熟的空间（和时间）观点与牛顿的观点进行了对比。从广义上讲，克拉克和牛顿认为空间是一个固定的、不变的背景。身体通过与空间区域的搭配而处于空间中。身体通过在不同时间与空间的不同区域搭配来在空间中移动。与这一图景相反，并与他自己对时间的理解保持一致，莱布尼茨认为现有事物直接处于彼此的空间关系中。与时间一样，一件事与另一件事共存，不是因为两者与某些特殊的、独立的第三件事（即空间）相关，而是因为它们彼此（以及其他类似事物）之间的关系。而且，与时间一样，空间本身也是一种抽象的有序关系，它不仅构造一次所有实际的空间关系，而且构造一次所有可能的空间关系。莱布尼茨告诉我们，它是“共存的秩序”，就像时间一样，它是一种抽象的、理想的结构，存在于神圣的智力中。莱布尼茨在生命的最后几年写下了给克拉克的信。（人们想象他手里拿着笔死去，而他的伟大对手的另一场辩论仍在他的墨水池中。） 然而，莱布尼茨对空间本质的思考可以追溯到他最早的一些著作。根据亚瑟的说法，在这些早期著作中，莱布尼茨提出了“神圣的巨大属性是空间的基础”的观点，并且神圣的巨大被其中包含的物体的运动划分为一个区域网络（123）。据阿瑟讲述，莱布尼茨在对霍布斯的研究的引导下，发展了一种新的情境数学，即莱布尼茨著名的情境分析。莱布尼茨的分析位置反过来，为他提供了将空间描述为不断变化的区域网络和相对位置系统的正式工具。人们可以将空间理解为一个由可能位置组成的抽象系统，构成现有事物之间的空间关系。因此，在阿瑟的阅读中，从 1670 年代末到 1716 年他去世，莱布尼茨对空间本质的思考有一个渐进的、基本上连续的发展路线。阿瑟对莱布尼茨空间观的渐进的、基本上连续的阅读与过去形成了鲜明的对比。以及 Vincenzo de Risi 的最新作品。在即将出版的著作中，德里西认为莱布尼茨的空间观可以分为三个时期：早期的实体主义观点、中期根植于虚空间观念的经院主义观点以及他著名的晚期关系主义观点。最有趣的是，德里西提出，莱布尼茨的关系观点直到 1680 年代末才首次出现，而它正是由莱布尼茨秘密阅读牛顿原理所促成的。根据德里西的解读，莱布尼茨的空间观并不是核心理论的不断完善，而是一系列本质上不同的理论。莱布尼茨的关系理论并不是牛顿实体主义的近乎奇迹般巧合的孪生兄弟，因为它可能来自莱布尼茨与克拉克的对应关系。相反，它是莱布尼茨从头开始精心设计的，作为他伟大竞争对手地位的替代方案。阿瑟和德里西开辟了一个令人兴奋的解释领域，对早期现代哲学和科学产生了重大影响。在莱布尼茨研究中寻找下一个重大成果的学者们可能不需要再进一步寻找。 在讨论莱布尼茨关于运动的观点时，阿瑟强调了莱布尼茨在几何构想的运动和关于原因的运动之间的区别。几何上设想的运动只不过是情况的改变。莱布尼茨认为，任何两个物体之间的几何运动都是相对的、相互的。如果我们的船在夜间经过，你的船相对于我的船移动，我的船相对于你的船移动。没有什么说法——甚至连问都没有意义——是真正令人感动的。关于原因的动议“涉及通过参考变化的最佳解释来识别动议的主题”（321）。假设有一阵微风，你的帆扬起。对于我们相对状况的变化，最好的解释是，你的船在风中行驶，我的船在水中休息。在这种情况下，最好的假设是，你的船确实在按原因移动，而我的却没有。因此，与几何运动相反，相对于原因的运动不一定是相对的。在相对于原因的运动的情况下，我们可以询问，甚至冒险进行有根据的猜测，到底是你的船还是我的船在真正运动。阿瑟展示了莱布尼茨对几何运动和因果运动之间的区别如何被用来解开莱布尼茨运动理论提出的一些表面上的难题。例如，莱布尼茨对“假设的等价性”的承诺与他对哥白尼主义的明显支持之间存在着明显的紧张关系。在解释假设的等价性时，莱布尼茨坚持认为，“关于任何物体运动的所有假设，无论数量有多少，仅由物质撞击所移动，因此，即使是天使也无法以数学严谨的方式辨别出众多物体中的哪一个。这种物体处于静止状态，并且是其他物体的运动中心”（248-49）。然而，哥白尼主义表明，我们——更不用说天使了——可以说地球和行星在运动，而太阳是静止的。面对这两项承诺之间明显的紧张关系，人们很容易倾向于牺牲其中一项承诺而牺牲另一项承诺。人们很容易认为莱布尼茨假设的等价性是对罗马权力的不诚实的安抚，或者莱布尼茨对哥白尼主义的承诺是胆怯的——例如，莱布尼茨只是称赞哥白尼主义只是许多可能的体系之一。亚瑟的阅读，莱布尼茨对几何运动和因果运动的区分提出了一条更令人满意的中间道路。莱布尼茨确实认为，从几何的角度来看，哥白尼主义应该仅仅被视为许多可能体系中的一种。他确实相信，即使是天使也无法从几何角度判断行星或太阳是否在运动。从几何的角度来看，这样的问题根本没有意义。但从因果的角度来看，它们确实有意义，而且从因果的角度来看，我们可以对哪些物体真正在运动做出更好和更差的假设。因此，莱布尼茨可以一致地坚持认为，从几何的角度来看，哥白尼主义只是许多可能系统中的一个，同时仍然坚持认为，从因果的角度来看，哥白尼系统是我们最好的假设，行星确实在运动，太阳确实在运动。在休息。在阿瑟的阅读中，即使莱布尼茨的语言是和解性的，他对哥白尼主义的承诺也没有丝毫胆怯。《莱布尼茨论时间、空间和相对论》是一部学术杰作。它在历史、哲学和技术上都很丰富。（我有没有提到它包括一系列附录，其中包含正式的阐述、亚瑟背景观点的详细解释以及原始翻译？甚至还有一个技术术语表！） 应该承认，这不是一本简单的书。虽然写得清楚、组织得当，但它又长又详细。其封面之间有足够的材料，可供至少三本重要书籍使用，其中每个主要主题各一本。事实上，它充满了论点、文字注释，和学术细节，人们可能会忍不住认为它是数学家意义上的密集，在任何两个哲学论证、注释或学术细节之间，人们总是能够找到另一个论证、注释或学术细节。对于那些热衷于精湛学术并对莱布尼茨、早期现代哲学或时空哲学感兴趣的人来说，亚瑟的杰作必将带来一系列密集而无穷的乐趣。请你自便。