确实，在这些世界中也存在某些绝对的类时间关系，例如由线元 给出的类时间距离（当 时）；但这并不与康德的时间学说相矛盾⁶，因为这些绝对关系的结构缺乏上述提到的时间的基本特性；康德也并不否认某种与事物固有的关联的存在，这种关联具有不同于直观时间的特性。相反，他明确提到“那些感受”（显然是指“物自身”的感受），“我们将其表象为变化”，并且“在具有其他认知形式的存在者中，这些感受会引发一种感知，而这种感知中时间的概念……根本不会出现”。（B 54）。康德关于时间的观点与相对论性世界中存在的条件之间确实存在重要差异，⁷ 特别是在康德学说的积极部分中，尤其是在通常意义上的时间是否存在这一问题上。然而，即使在这一点上，也存在显著的原则上的相似性。满足上述要求的时间（即事件的一维时间排序）在康德看来是相对于感知主体或更确切地说是相对于其“感性”⁸而存在的；在 R-世界中，它是相对于某些更普遍和抽象的实体而存在的，例如物质点、世界线和坐标系。然而，这些实体同样可以最方便地被视为一个可能观察者的特性或归属于一个可能观察者。康德将任何客观现实赋予时间的意义实际上是指：时间的属性代表了事物与感知主体之间的某种关系，这可以从他许多著作中的段落看出。例如，在解释“假象（Schein）”与“现象（Erscheinung）”的区别时，他明确指出，现象的属性虽然不是事物本身的属性，但也不是单纯的幻觉，因为它们代表了事物与主体之间的关系。康德自己的话是：“现象的谓词可以归因于对象本身，但这是相对于我们的感官的关系”，以及“不能在对象本身中找到的，而总是以其与主体的关系存在的……是现象”（B 69 n.）。在同一讨论中，康德还说，我们赋予现象中的对象的属性“依赖于主体的直观方式，以及与所给对象的关系”（B 69 [重点为哥德尔所加]）。在其他段落中，他提到外感和内感“只能在其表象中包含对象与主体的关系”（B 67）；他还说，时间和空间是“事物本身所不具备的特性，而仅在于它们与我们感性的关系中”；¹⁰ 现象是某种“其可能性基于某些不可知事物……与我们感性之间关系的事物”。最后，康德还提到¹¹：“我们对空间的表象完全适合于我们的感性与对象的关系”（显然，这也适用于时间）。康德的时间相对性绝不是指仅仅涉及时间属性的定性特征的弱意义。如果是这样，至少时间所给出的顺序将是“固有于对象”的东西，这一点被康德明确否定了（参见本文开头引用的表述）。另一方面，康德的时间相对性也不能被解释为时间属性对于事物而言除了我们想象它们具有这些属性外没有其他含义。这显然与上述一些引用段落相矛盾，这些段落表明，我们对时间条件的观念一方面取决于某些客观的“修正”（参见第2页提到的 [B 54] 段落），另一方面取决于对象与我们之间的某些“关系”（参见第4页提到的 [B 69] 段落）。因此，康德的时间相对性在某种程度上类似于相对论的时间观。然而，有一个本质的区别，即康德几乎不可能认为事物的时间属性会因观察者不同而不同。相反，除了不同个体在时间中的位置不同外，时间属性对所有人类都是相同的，仅可能因完全不同性质的存在者而异（或者根本不存在）。现在让我们更详细地比较康德的时间理论与 R-世界中的条件，并从康德学说的消极部分开始。这里我们首先注意到，如果通过钟表的直接测量和爱因斯坦的同时性概念¹² 定义时间，康德在开头表述的论断是简单相对论事实的直接结果，即使不考虑世界的具体结构也是如此。通过这种方式获得的基本时间关系“事件 A 领先事件 B t 秒”，在相对论中对于适当地选择的不同观察者总是量化不同的，甚至在两个观察者之间可能是相反的方向。因此，它显然不是固有于事件本身的东西，也不允许时间作为一个“基质”来承载事件。然而，这并不排除某种物理结构的存在，这种结构独立于观察者或任何其他参考对象，具有牛顿时间的特性，或者至少具有第 [2] 页提到的最基本特性，但它与钟表读数（至少与任何任意观察者的钟表读数）没有简单和直接的联系。实际上，这种“绝对的世界时间”可以在所有已知的宇宙学解（即相对论中可能的世界）中被定义¹³。然而，在 R-世界中，这种定义显然是不可能的；也就是说，可以引入的各种“时间点”系统之间在客观上无法区分，但其中任何一个都只能通过参考具体的个体对象（例如我们的地球）¹⁴ 来单独确定。如上所述，在 R-世界中也存在某些绝对的类时间关系，但它们与通常意义上的“时间”完全不同。首先，它们仅定义了部分排序，或者更确切地说：无论人们如何引入一个绝对的“之前”，总会存在时间上不可比较的事件或循环排序的事件。这意味着任何可定义的绝对时间距离都缺乏加法性（即 ，特殊情况除外），因此不能由一条线表示，而只能由一个多于一维的空间表示。这些结构差异进一步表明，在 R-世界中，直观时间观念中所包含的客观时间流逝是不可能的。因为我们通过时间流逝想象现实由一个个连续存在的无限层次组成。但 R-世界中的这些层次只能相对于某个观察者或参考对象进行划分，这些层次将依赖于这些参考点。至少，在 R-世界中，直接经历的时间流逝没有客观意义，这也可以从以下事实得出：在这些世界中，可以任意远地穿越到未来或过去并返回，就像在其他世界中可以任意远地旅行到遥远的空间部分一样。¹⁵因此，综合上述考虑可以说，在 R-世界中作为客观现实的时间，无论是独立于观察者还是其他参考对象，都既不能定义事件的线性排序，也不由一维的点系统组成，也不能具有“流逝”的特征。然而，这种东西很难被称为时间，并且与康德理解的时间完全不同。因此，至少康德时间理论的消极部分在 R-世界中无疑是成立的。至于其积极部分，则不那么[明显]，但在原则上仍然存在不可否认的一致性。不仅在 R-世界中满足第 [2] 页提出的要求的时间仅作为某种关系存在，而且对于每个可能的观察者而言，还存在一种可以称之为“他的时间”的时间，因为它是通过将该观察者直接体验或测量到的时间最简单地推及到整个世界而获得的。这种时间（如同在康德那里）表达了事物与该观察者之间的关系。然而，在相对论中，这种关系只是观察者身体的纯粹物理关系（即，它相对于他作为一个最一般意义上的感性存在而存在）；而在康德那里，这是一种与其感性的关系（康德显然的意思是，它仅相对于其感官和想象器官的特殊结构而存在）。这无疑意味着，相较于相对论，即使在 R-世界中，康德对时间仅相对存在的主张具有更强烈的意义。两种理论之间还存在类似关系，尤其是在康德无疑认为直观时间与其客观对应物之间的差异远远大于相对论中这一差异，甚至在 R-世界中也是如此；事实上，这种差异大到无法用人类能够理解的概念加以描述。因此，一般可以说，就时间（以及空间，见下文）的本质而言，康德和相对论理论完全沿着相同的方向前进，但相对论理论仅仅沿着康德指明的方向迈出了一步。¹⁶尽管如此，就康德理论的积极部分而言，确实存在一种深层次的一致性，而不仅仅是人为制造的联系。对于一个给定观察者有效的时间确实表达了“事物与其感性之间的关系”，尽管比康德的意义更为广泛。因为这是通过将事件投射到其身体的世界线上实现的。即，通过他与外部现实之间直接感官接触的联系线。这里的“感性”一词确实与康德的含义并不完全相同。因为（1）康德在时间的语境中讲的是内感性，[并且]（2）康德首先指的是感性的第二部分，即表象能力（这是某种智力活动），而不是感知能力。然而，尽管如此，仅仅是这两个情况使用相同的术语来描述已经足以证明在原理上具有一致性，尽管在细节上并不完全相符。需要注意的是，上述讨论的内容并不限于前面提到的那些被称为 R-世界的特殊可能宇宙，它们在某种弱化的形式下也适用于那些能够引入绝对时间的世界。对于这种绝对时间，虽然它在事件排序类型上与直观时间相一致，但在其他方面却完全不同。最重要的是，它并非由于自然规律的原因而存在，而只是由于这些世界的特殊结构，因此几乎不能合理地归因于时间的流逝。这表明，时间的客观流逝是否存在取决于物质及其运动在世界中是如何排列的。因此，如果在时间概念中除了其排序类型外还包括直观时间的某些特性，那么在目前已知的宇宙学解中，时间最多只是相对存在。这在旋转和膨胀的宇宙中更是如此（参见脚注 [14]）。有人可能会想，为什么前面的页面只讨论了时间。由于相对论中时间和空间之间的对称性，人们理应期望这些结论同样适用于空间。然而，事实上，这种期望仅在某些方面得到了部分验证，在一些重要的点上，这种类比并不成立。这是因为（1）时间的流逝在空间中没有对应的类比，（2）在考虑到物质的存在时，相对论中时间和空间之间的对称性并不完整。因为物质意味着物理上区分的一维类时间子空间（三参量系统，具体来说是物质的世界线）的存在，而一般来说并不存在物理上区分的单参数三维类空间子空间。对于正交于物质世界线的空间元素，通常不能将它们组合成三维空间。这导致的结果是，没有可能的世界可以具有与直观空间相比在绝对类空间关系结构上如此不同的情况，而这在 R-世界中是时间的情况。特别是，如果物质的速度矢量是连续的，那么相邻物质点之间的绝对距离空间（由其世界线的正交距离定义）是一个随时间变化的普通三维黎曼空间。这与经典物理中的[这种空间]不同，其对时间的依赖可以通过多种方式表示（由于同时性的相对性），这是因为在相对论中时间（而不是空间）具有不同的特性，这一点从所有随时间变化的物理属性都适用这一事实中可以看出。由于物质的平均速度矢量（取自适当地选择的天文尺度区域的平均值），甚至可以以某种意义引入“绝对”的空间点，但它们在多个方面与牛顿和直观空间的绝对点有本质上的不同，特别是在它们的相互距离不需要在时间上保持恒定（即，这种空间通常会收缩、扩展或变形）。有一种情况往往掩盖了相对论与康德之间在某些方面存在的紧密关系，即康德除了强调时间和空间的主观性外，还主张我们所有关于它们的知识的先验性，并且他甚至将证明它们主观性的主要依据建立在它们的先验性之上。然而，它们的先验性似乎显然与相对论理论相矛盾，因为时间和空间的新发现性质（例如同时性的相对性、空间的非欧几里得性、洛伦兹收缩等），远非先验可识别，甚至完全违背我们所谓的先验直觉。然而，进一步的检查表明，这些不能先验识别的性质涉及的实际上是我们时间和空间观念的客观对应物（即，用康德的术语，那些我们表象为时间和空间的感受），因此它们与某种主观时空方案的先验性是兼容的。这种时空方案是以人类的智力装备为基础的，但同样适用于物理现实的描述，即一种相对的时间和空间（坐标系）。以这种意义理解的牛顿时间和欧几里得空间的先验性，与相对论理论根本不矛盾，而是构成了它自身的一个问题。以几何为例，例如，围绕我们的物体按照非欧几里得几何的规律运动，这一事实丝毫不排除我们拥有一种欧几里得“感知形式”的可能性，即，我们应该拥有一种先验的欧几里得空间表象，并且能够仅通过将我们的感官投射到这种空间表象上来形成外部物体的图像。因此，即使我们出生在某种强烈的非欧几里得世界中，我们仍然会始终不变地将空间想象成欧几里得的，但将物质物体的大小和形状在相对于我们或我们相对于它们运动时以某种规律方式发生变化。物理几何的非欧几里得性（由刚体行为定义）并不意味着这种欧几里得纯粹直观性（如果它存在）是完全错误的。因为满足欧几里得公理的几何概念也可以在非欧几里得世界中定义，尽管（符合康德的观点）这些几何概念不是物体的客观¹⁸属性，而只是物体与我们的感官（或者可以补充说，与某些任意选择的对象）之间的关系。例如，可以通过以下方式获得此类定义：我们选择某些物理对象（例如地球或我们的身体），选择其上的一个点 ，以及在此点上的三个（最好是正交的）方向 。然后，我们通过沿着 方向（或者通过适当平行位移获得的方向）应用测量杆的次数，来定义任何物质点 的“坐标” ，以到达从 到 的位置。¹⁹ 如果在这些坐标中通过线性方程定义直线，则这些直线对于欧几里得公理来说是成立的，而与世界的物理特性无关。当然，其他几何概念和公理也可以采用相同的方法。因此，如果我们确实拥有一种能够向我们表征这类几何实体的内在直觉，²⁰ 则这种直观几何的公理虽然涉及现实（即测量过程），但将是先验真实的。²¹然而，根据康德的分析性概念，它们不会是分析的，因为几何被以这种方式还原到算术，而康德并不认为算术是分析的。²² 然而，康德未曾考虑到的是，（无论我们是否具有这种先验直观性）空间及其属性也在我们的感官中表现出来，而这些感官我们仅能后验地认识到；也就是说，通过以某种方式将感官投射到三维欧几里得空间中，这些连接感官的规律可以用某种简单形式表达出来。我们的感官不具有这种必要属性，这可以从物理非欧几里得几何的可能性及类似的其他考虑中得出，这些表明我们的感官可能是这样的：即使可以将它们投射到三维欧几里得空间中，这样做也完全没有任何用处。这一见解并不意味着空间的客观性。因为感官本身也是某种主观的东西，因此它们的任何属性可能与事物本身的属性有关，也可能与事物与我们之间的相互作用方式有关²³（决定取决于为了令人满意地解释现象必须假设的现实类型）。然而，这确实需要（即使在我们拥有欧几里得空间的内在直觉的情况下）对康德关于空间先验性的理论进行某种程度上的弱化，至少要承认，“我们对空间的表象与物体与我们感性的关系的适切性”只能通过经验得知，前提是为这一短语赋予了合理的含义²⁴。在每个世界中，欧几里得几何适当地表示了事物与我们的某些特定关系（参见例如上文的定义），但这当然并不意味着“完全适切于事物与我们感性之间的关系”。然而，康德可能在上述意义上是正确的。我不认为关于我们是否具有欧几里得空间的内在直觉，或者在一个强烈的非欧几里得世界中我们会实际发展出什么样的空间表象，这些问题已经有定论；也不认为与此相关的问题，即我们是否能够（在我们的世界中）学会想象²⁵一个非欧几里得空间，已经得到解答。我们能够想象在非欧几里得世界中物理刚体的行为²⁶并不能证明这一点。因为我们当然不会将物理刚体的行为与每个可能世界中的空间属性等同起来，例如，在一个大部分是欧几里得的世界中，但有某些非欧几里得的区域时。确实，在一个均匀的非欧几里得世界中，我们似乎更有可能发展出非欧几里得几何的直觉；然而，对于此点仍有合理怀疑的空间。但无论这些问题的答案是什么，它们显然都不可能导致康德与相对论之间的任何不兼容性，而顶多是康德与感知心理学（或现象学）之间的不兼容性。相对论与康德哲学之间的真正矛盾，在我看来，仅存在于一点上，即关于康德的观点：自然科学在描述世界时，必须保留我们感知形式的框架，并且只能在这个框架内建立表象之间的关系。康德的这一观点无疑源于他对“物自身”不可知性的信念（至少在理论理性方面）。在这一点上，在我看来，如果人们希望在康德的理论与现代物理学之间建立一致性，就必须对康德进行修改；即，应假定科学知识至少在某种程度上可以逐步超越表象并接近事物本身。²⁷放弃康德所称的“表象世界”的那种“自然”世界图景，正是现代物理学与牛顿物理学的主要区别所在。牛顿物理学除了消除了次级性质（原则上，早在德谟克利特就已知晓），只是对这一世界图景的改进，而不是校正；而现代物理学则具有完全不同的特性。这一点最清楚地体现在“实验室语言”和理论之间的发展区分上，而牛顿物理学可以完全用一种精炼的实验室语言来表达。因此，现代物理学的趋势在某种程度上与康德哲学相对立。然而，另一方面，也不应忽视的是，康德关于理论科学无法超越我们自然世界观的界限的观点的反驳，在几个方面证实了他关于这种自然世界图景的主要学说之一，即其在很大程度上的主观性²⁸，即使是对于那些似乎构成现实基础的概念也是如此。此外，需要注意的是，对超越表象的事物的知识的可能性，并不像某些康德追随者的观点那样，与康德本人的观点有如此严格的对立。因为（1）康德认为“物自身”的概念是有意义的，并多次强调必须假定它们的存在，[并且]（2）关于它们的知识不可能性对康德来说绝不是所有知识的必然结果，甚至在人类知识的每一个方面都没有持续存在。可以参照这个观点，例如，他在 《纯粹理性批判》 的 B xxi、B xxi n. 和 B xxvi n. 中关于其积极使用的论述，这不仅适用于宗教信仰，也可能适用于知识（见 B xxvi n.）。在上述段落的第二个引用中，康德甚至进一步将其主观主义知识理论的可能积极用途，与哥白尼通过观察者运动来解释行星表面运动的解释进行了比较，并指出哥白尼这一新视角引导人们发现了“连接宇宙结构的隐藏力量，即牛顿引力”。确实，康德希望将这种知识建立在伦理学基础上，但即使是在理论理性的层面，他显然并不想说“物自身”完全无法被断言。因为康德自己断言，例如，这些事物存在，影响我们的感性，并且不在时间和空间中存在，但时间和空间的观念完全适用于它们与我们感性之间的关系。无疑，康德最不会认为可以通过实验得出的结论，超越表象世界。然而，关于这一点需要注意的是，完全符合康德的观点，仅仅是观测结果本身，并不足以迫使我们放弃牛顿时间和空间作为客观现实²⁹，而是观测结果与某些一般原则共同作用，例如，两个无法通过观测区分的情境在客观上也是相等的原则。一般来说，可以认为相对论（尤其是广义相对论）比任何其他物理理论更依赖于某些非常普遍的原则和观念的应用，其大部分内容仅在后来的经验中得到了验证。
注释
我想一开始就说明，我并不是康德哲学的一贯支持者。以下的论述只是试图表明，相对论理论与康德关于时间和空间的学说之间在某些方面存在惊人的相似性，并且它们之间的矛盾，至少在它们出现的地方，远不像经常声称的那样根本。参见《纯粹理性批判》B 49，第 a 段。参见《未来形而上学导论》，第 11 节，[第 284 页]。即，由相对论理论中所遵循的自然法则所允许的世界。“绝对的”，在独立于参考框架或物体的意义上，与康德所用的短语“固有于对象”相当接近。关于“相对”的概念，在康德那里是其对立面。参见脚注 [17] 所引用的段落。在手稿中，这个脚注以“参见脚注所提到的段落”这句话结束，并在参考编号处留有空白。这引发了一个问题，即是否有依据确定所提到的段落。哥德尔在第 4 页引用的康德段落都涉及“对象”（或“事物”）的性质与其与我们感知的关系之间的对比；因此，这些段落中的每一段都可能是候选者（这些来源的引用，如今在文本中有所标注，而在哥德尔的原始脚注中则为不同段落提供了独立的脚注）。然而，这并未提供合理的标准来选择其中一个段落作为所提段落。此外，这些段落似乎都不完全符合哥德尔在当前脚注中的描述。但是，在戈德尔有争议的脚注 63（此版本中的脚注 [17]——参见附录 f 对后者的讨论）中提到的段落（B42，第 a 段）包含以下陈述：……空间并不代表任何附着于物体本身的规定性，并且这种规定性在抽象掉所有主观直观条件后仍然存在。因为任何规定性，无论是绝对的还是相对的，都不能先于其所属事物的存在而被直观，也因此都不能先验地被直观。（重点强调）在这里，我们明确看到了“绝对”和“相对”的对比——这种对比可能正是哥德尔在写道“对于‘相对’这一概念，康德将其视为相对立的概念”时所指的内容。（与什么的对立？——与绝对的对立。）这确实与康德体系中的某些其他部分相矛盾，即康德认为“物自体”（因此显然也是时间观念的客观对应物）是不可知的，不仅通过感官表象无法知晓，通过抽象思维也无法知晓。当然，并不需要主张相对论理论能够完全了解时间观念的客观对应物，而只是它朝这一方向迈出了一步（参见脚注 [16]），但即便如此，这似乎也与康德的观点不符（参见第 [27] 页）。在后续讨论中，我将这些世界称为“R-世界”（“R”是“旋转、旋转对称、静止”的缩写）。关于其数学描述，参见[1949]。在这些世界中，惯性罗盘相对于物质无处不旋转，这在我们的世界中将意味着它相对于星系系统的整体旋转。如果当前对遥远星云红移的解释是正确的，我们的世界就不是一个 R-世界。然而，它可能是一个扩展旋转的世界。在这种情况下，有关时间的非客观性的后续讨论（特别是关于时间旅行到过去和未来的讨论， 参见下文第 12 页）可能仍然适用。关于这一点的简要参考，参见哥德尔 1949b；关于哥德尔对爱因斯坦方程中此类解存在性的证明草图，参见哥德尔 1952。“感性”是根据康德的定义，在事物和事件的影响下获得感知的能力，包括对外部物体和自身的表象以及由这些感知产生的影响。康德将感性分为外感和内感。参见《未来形而上学导论》，第11节，[第284页]。参见《未来形而上学导论》，第13节，[第286页]。在这个段落以及上文提到的 B 69n. 的第二个段落中，需注意，对于康德，“现象”并不是指某物的显现，而是指一个具有被归属于显现对象的所有属性的思想对象（参见其在 B 34 的定义）。此外，这两段的上下文清楚地表明，“现象”在这里是按照这个意义来使用的。参见《未来形而上学导论》，第13节，注释II结尾，[第289-290页]。在“means”后面的短语“by it”已被编辑者删除，他们假定哥德尔的意图是引用德语单词“dabei”。因此，编辑者认为它的含义已在句子的其余部分中暗示。在 C2 版本中，哥德尔将“means by it that temporal properties represent”替换为“is viewing temporal properties as representing”（即“将时间属性视为表征”）。这个概念最初仅为狭义相对论设计，但显然它可以扩展到广义相对论，至少在 R-世界中满足某些条件时可以成立。仅适用于物质严格均匀分布的世界，但这一困难是可以克服的；例如，可以通过考虑与给定解相比（在某种意义上）总偏差最小的均匀解来解决。基于这些解在解释天文学事实中的实际成功，J. Jeans 主张回归客观流逝的绝对时间这一旧观念是合理的（参见 Jeans 1936，第22-23页）。然而，很难确定我们的世界是否真的可以由这些解之一来描述。此外，参见下文第 [17] 页的相关讨论。这种情况似乎意味着一种荒谬。例如，它允许一个人前往他曾经生活过的地方的近过去。在那里，他将遇到一个与他自己在多年以前相同的人。现在他可以对这个人做一些他通过记忆知道从未发生过的事情。这种情况和类似的矛盾，然而，为了证明 R-世界的不可能性，至少需要假定实际上可以实现这种过去旅行，而这很可能因需要接近光速的速度或其他情况而受到限制。当前的物理学状态似乎表明，未来的发展将沿着这些方向继续。Th. Kaluza 在 1921 年提出的第五维度，例如，就指出了这一方向。量子力学更是如此，它迄今未能给出对客观现实的令人满意的描述，以使其计算规则的成功变得可以理解。（参见 Einstein, Podolsky 和 Rosen 1935，其中证明了波函数不能代表整个现实，除非假定具有非常奇特特性的远距离作用。）如果物理学的未来发展确实如上所述方向前进，相对论理论将不得不被解释为仅仅是一个“客体化层次”（参见脚注 [26]），并将被其他层次所取代。应注意，康德明确否认空间的客观性，这不仅是在将其构想为独立于事物之外的实体时，而且是在将其构想为事物之间关系的系统时也是如此。（参见《纯粹理性批判》B 42，第 a 段。）由于哥德尔未在其脚注列表中将该脚注标记为删除，并且因为文本中一个脚注的交叉引用（编号 5，我们的编号）留空，可能是指向此脚注，因此我们选择将此脚注放在这里。（有关我们的脚注编号与原始编号的一致性，请参见文本注释。）进一步的证据表明，他并未打算删除该脚注，而只是忘记标记一个位置，其证据存在于哥德尔自己制作的对照表中。在那里，该编号出现在我们编号为 16 和 18 的脚注之间。（另请注意，该区域有些页面被删除。）我们选择的这一位置与该证据一致。至少，如果关于“客观性”的某些公理，例如两个不同观察者的可互换性，被假定成立。在一个闭合的世界（例如，黎曼几何世界）中，这种定义不会为坐标提供唯一值，即，它可能导致一个物体可以同时存在于许多不同的位置，或者对于每个物体存在无限多个完全相同的对象；或者，如果定义稍有改变，空间可能仅由欧几里得空间的有限部分组成。即，表征这些命题为显而易见的命题，这些命题适用于这些概念。当然，我并不是想说，根据康德，纯粹直观性确切地对应于上面定义的那些几何概念，或者这种类型的直观性实际上存在。我只是想表明，一种与现实相关且先验有效的欧几里得几何直观性在逻辑上是可能的，并且与非欧几里得几何和相对论理论的存在相兼容。这样的直观性可能也仅由空间本身的表象和其与现实相关性的总体观念所构成，而我们将感官投射到其上的特定方式则取决于经验。这似乎与爱因斯坦关于几何与经验关系的著名观点（参见 Einstein 1921）相矛盾，但这种矛盾更多是词语上的，而不是意义上的，因为爱因斯坦的“sich auf die Wirklichkeit beziehen”显然是指“说某些关于现实的事情”，而上述解释的几何公理，尽管它们涉及物理现实，却并未谈及其内容。同样，根据康德，几何命题是综合的，这不是因为它们谈及外部给定的内容（即感官），或事物本身，而是因为它们谈及我们，或更确切地说，我们的智力能力，在心理同化感官过程中想象或建构出的内容。然而，这些主观添加并不构成现实的适当意义上的一部分。需要注意的是，今天“综合”一词通常以与康德不同的意义使用，即表示一个状态的命题，而该状态独立于我们智力的活动而存在。在这个意义上，“综合”与“后验”无疑在康德那里也会一致。但康德会否认我们的智力活动仅限于明确的定义及其应用。此处所涉及的主观性当然与康德的不同，因为它不意味着先验性。然而，即使在这种第二种意义上，空间也可以非常适当地被称为“一种感性的形式”。即，“物体与感性的关系”应意味着外部物体如何产生感官，而不是我们如何在空间中表象事物，因为在后一种情况下，所讨论的适切性将毫无意义。在这一脚注旁边的左侧页边，哥德尔标注了一个问号。通常人们没有意识到，假如空间曲率足够大，非欧几里得空间直觉与我们的直觉会有多大的不同。例如，在某种洛巴切夫斯基空间中（其像我们的空间一样延伸至无限），半径为 1 英寸的圆的周长将为 1 码。参见，例如，Einstein 1921。参见 Bollert 1921，其中可以找到关于这些客体化步骤或层次的更详细描述，每一步骤都是通过消除先前某些主观元素获得的。“自然”世界图景，即康德的表象世界本身，也必须被视为这样一个层次，其中“感官世界”的许多主观元素已经被消除。不幸的是，尽管这种区分主观和客观知识元素的观点非常有益，但…参见 Bollert 1921，其中可以找到对这些客体化步骤或层次的更详细描述，每一个层次都是通过从前一个层次中消除某些主观元素而获得的。“自然”世界图景，即康德的表象世界本身，也必须被视为这样的一个层次，在这一层次中，“感官世界”的许多主观元素已经被消除。不幸的是，每当这种区分我们知识中的主观和客观元素的富有成效的观点（康德通过与哥白尼体系的比较提出了这一观点，详见下文第 [[29]] 页）在科学史上出现时，人们就倾向于将其夸大为一种无限的主观主义，从而抵消了其效果。康德关于“物自身”不可知的论点是一个例子，另一个例子是关于量子力学的实证主义解释必然是该理论最终阶段的偏见。可以说，（如同本文内容所表明的那样）相对论仅证明了时间的相对性，而非其主观性（即相对于我们的想象能力）。但需要注意的是，（1）绝对性（或唯一性）本身构成了直观时间的一个重要特性；（2）时间的相对流逝（如果这个短语可以赋予任何意义的话）将与第 [11] 页描述的流逝截然不同；因为其存在的本质是某种绝对的东西。因此，一个与直观时间观念在所有要素上真正对应的实体，也存在于相对论中，仅存于我们的想象中。在狭义相对论中，例如，一些任意选择的惯性系可以被视为代表绝对空间和时间（具有牛顿赋予它们的所有属性），而不同速度运动的观察者所得到的不同观测结果，可以通过相对于这种绝对空间的运动对物理体和过程，特别是测量仪器的影响来解释（这种影响恰恰是通过麦克斯韦电磁方程和假设物理体由电磁力或具有类似数学性质的其他力构成这一非常自然的假设，从经验上验证的结果中得出的，而无需任何附加假设）。然而，究竟哪个坐标系被视为这种绝对时空体系仍然完全是任意的，这一原则表明，绝对时间和空间实际上并不存在。
Some observations about the relationship between theory of relativity and Kantian philosophy (1946/9-C1)
Kurt Gödel
It is an interesting fact, to which very little attention is being paid in current philosophical discussions, that at least in one point relativity theory has furnished a very striking illustration, in some sense even a verification, of Kantian doctrines.¹ It is a certain part of Kant’s doctrine about time for which this is the case, namely, for his assertion that time is neither “something existing in itself” (i.e., a separate entity besides the objects in it) nor “a characteristic or ordering inherent in the objects”² but only a characteristic inherent in the relation of the objects to something else.³
This view of Kant’s, which constitutes the negative part and a fraction of the positive part of his doctrine about time, is indeed literally true in certain relativistic worlds,⁴ provided [that] by “time” is understood what everybody understood by it before relativity theory existed. For this idea of time no doubt includes as its most essential characteristic that time consists of a one-dimensional system of points, isomorphic with a straight line, in which every event happening in the world has a definite place. But in the worlds under consideration an absolute⁵ time of this kind does not exist.
There exist, it is true, also in these worlds certain absolute time-like relations between events, e.g., the time-like distance given by the line-element , in case ; but this does not contradict Kant’s doctrine about time⁶ because the structure of these absolute relations lacks the essential characteristic of time just mentioned; and Kant by no means denied the existence of some correlate, inherent in the things, of our idea of time with properties different from intuitive time. On the contrary, he speaks explicitly of “those affections” (evidently of the things in themselves) “which we represent to ourselves as changes” and which “in beings with other forms of cognition would give rise to a perception in which the idea of time ... would not occur at all” (B 54).
Essential differences between Kant’s view about time and the conditions prevailing in the relativistic worlds mentioned⁷ exist, it is true, as to the positive part of Kant’s doctrine, in particular as to the question relative to what a time in the usual sense does exist. But even in this respect there is a distinct similarity of principle. A time satisfying the requirement stated above (i.e., a one-dimensional temporal ordering of the events) exists for Kant, relative to the perceiving subject or more precisely its “sensibility”;⁸ in the R-worlds it exists relative to certain more general and abstract entities such as material points, world lines, and coordinate systems, which however likewise can be conceived most conveniently as characteristics of, or as belonging to, a possible observer.
That Kant, insofar as he attributes any objective reality to time, really means⁹ that temporal properties represent certain relations of the things to the perceiving subject appears from many passages of his writings. E.g., in his explanation of the difference between “Schein” and “Erscheinung” he says quite clearly that the properties of the appearances, although they are not properties of the things in themselves, still are not mere illusions, because they represent relations of the things to the subject. Kant’s own words, in literal translation, are: “the predicates of the appearance can be attributed to the object itself in relation to our sense,” and: “what is not to be found in the object in itself but always in its relation to the subject ... is appearance” (B 69 n.). Again in the same discussion Kant says that the properties which we attribute to the objects in the appearance “depend ... on the mode of intuition of the subject in the relation of the given object to it” (B 69 [emphasis Gödel’s]). In other passages, he says that the outer as well as the inner sense “can contain in its representation only the relation of an object to the subject” (B 67), that⁹ time and space are “characteristics not inherent in the things in themselves, but only in their relation to our sensibility,” that¹⁰ the appearances are something “whose possibility is based on the relation of certain things unknown in themselves ... to our sensibility.” Finally Kant also says¹¹ that “our representation of space is completely adequate to the relation which our sensibility has to the objects” (and this evidently is meant to apply to time as well).
This Kantian relativity of time certainly was not meant in so weak a sense as to refer solely to the qualitative character of temporal properties. For in that case at least the ordering given by time would be something “inherent in the objects,” which is explicitly denied by Kant (see the formulation quoted in the beginning of this paper). On the other hand, Kant’s relativity of time cannot be interpreted in so strong a sense that temporal properties would not mean anything else for the things except that we imagine them to have these properties. For this would clearly contradict some of the passages quoted above, which imply that our ideas of temporal conditions depend on the one hand on certain objective “modifications” (cf. the passage [from B 54] referred to on page 2’ above), on the other hand on certain “relations” of the objects to us (cf. the passage [from B 69] referred to on page 4 above).
So Kant’s relativity of time is to some extent similar to that of relativity theory. One essential difference subsists, however, insofar as Kant hardly meant that temporal properties of things could be different for different observers, but rather that (except for the various positions in time of various individuals) they are the same for all human beings, and could be different (or non-existent) only for beings of an entirely different nature.
Let us now compare in more detail Kant’s theory of time with the conditions prevailing in the R-worlds and let us begin with the negative part of Kant’s doctrine. Here the first thing we note is that if time is defined by direct measuring with clocks and Einstein’s concept of simultaneousness,¹² Kant’s assertion as formulated in the beginning is an immediate consequence of a simple relativistic fact, even irrespective of the particular structure of the world. For the fundamental temporal relation “A before B by t seconds” obtained in this way is, in relativity theory, always quantitatively different for suitably chosen different observers and may even be inverse in direction for two different observers. Hence it is certainly not something inherent in the events, nor [does it] allow time to be a substratum in which the events are lying.
This, however, does not yet exclude the existence of some physical structure independent of the observer or any other object of reference which has the properties of Newtonian time, or at least the most essential one mentioned on page [2], but is not connected with the clock readings (or at least not with the clock readings of any arbitrary observer) in the simple and direct way indicated above. Actually such an “absolute world time” can be defined¹³ in all known cosmological solutions (i.e., relativistically possible worlds).¹⁴ In the R-worlds, however, such a definition is demonstrably impossible; i.e., none of the various systems of “points of time” which can be introduced is objectively distinguishable from the others, but any one of them can be singled out only by reference to individual objects, such as our earth.
As was noted above there exist also in the R-worlds certain absolute time-like relations, but they are quite different from “time” in the usual sense of the term. Above all, they define only a partial ordering, or to be more exact: in whatever way one may introduce an absolute “before,” there always exist either temporally incomparable events or cyclically ordered events. It follows that every definable absolute temporal distance lacks the property of additivity (i.e., , except for special cases) and therefore cannot be represented by a line, but only by a more than one-dimensional space. These structural differences further imply that an objective lapse of time, such as is contained in the intuitive idea of time, is impossible in the R-worlds. For by the lapse of time, we imagine that reality consists of an infinity of layers which come into existence successively. But the R-worlds cannot be split up into such layers except relative to an observer or other object of reference on which the layers then will depend.
That at least that passing of time which is directly experienced has no objective meaning in the R-worlds also follows from the fact that it is possible in these worlds to travel arbitrarily far into the future or the past and back again exactly as it is possible in other worlds to travel to distant parts of space.¹⁵
Hence summarizing the foregoing considerations one may say that what remains of time in the R-worlds as an objective reality, independent of an observer or other objects of reference, neither defines a linear ordering of the events, nor consists of a one-dimensional system of points, nor can have the character of “passing by.” Something of this kind, however, can hardly be called time and is certainly quite different from what Kant understood by time. So at least the negative part of Kant’s theory of time is doubtless true in the R-worlds.
As to its positive part, the agreement is less [pronounced], but still there exists an undeniable affinity in principle. Not only does a time satisfying the requirement stated on page [2] exist in the R-worlds only as a relation to something, but in particular there exists for each possible observer a certain time which may be called his time, because it is obtained by the simplest extrapolation of the time directly experienced or measured by him to the whole world. This time (as in Kant) expresses a relation of the things to this observer.
There is, however, this difference that in relativity theory it is a purely physical relation to the body of the observer (i.e., it exists relative to him insofar as he is a sensual being in the most general sense); for Kant it is a relation to his sensibility (whereby Kant evidently meant that it exists only relative to the special structure of his organs of sense and imagination). This doubtless means that the only-relative existence of time is asserted in a much stronger sense by Kant than by relativity theory, even for the R-worlds.
A similar relationship subsists between the two theories also in other respects, in particular insofar as Kant doubtless held the difference between intuitive time and its objective correlate to be far greater than it is in relativity theory, even in the R-worlds; in fact, so great that the latter cannot be described at all in concepts understandable for human beings. So it may be said in general that, as far as the nature of time (and also of space, see below) is concerned, Kant and relativity theory go in exactly the same direction, but relativity theory goes only one step in the direction indicated by Kant.¹⁶
But nevertheless there really exists, also as to the positive part of Kant’s doctrine, a deep-rooted affinity, not only a connection made up artificially. For the time valid for a given observer really expresses a “relation of the things to his sensibility,” although in a more general sense than for Kant. For it comes about by projecting the events on the world line of his body. i.e., on the line of immediate sensorial contact he has with the reality outside himself.
It is perfectly true that the term “sensibility” here does not have exactly the same meaning as in Kant. For (1) Kant speaks (in the case of time) of inner sensibility [and] (2) Kant means in the first place the second part of sensibility, the faculty of representation (which is something intellectual) and not the faculty of sensation. But nevertheless the mere fact that the same general terms are appropriate to describe the situation in both cases proves an affinity of principle although not an agreement in detail.
It is important to note that the foregoing considerations are not entirely confined to that special kind of possible universes which were called R-worlds above, but that in a somewhat weakened form they also apply to those worlds in which an absolute time can be introduced. For this absolute time, although it agrees with intuitive time as to the type of ordering it creates for the events, still is in other respects quite different from it. Above all it does not exist owing to the laws of nature, but only owing to the particular structure of these worlds, and therefore it hardly conforms to reason to ascribe a passing to it. For this would imply that whether or not an objective passing of time exists depends on the particular way in which matter and its motion are arranged in the world. Hence if one includes in the concept of time certain characteristics of intuitive time in addition to its order type, then time has at most a relative existence also in the cosmological solutions known at present. All the more this would be true of the rotating and expanding universes (cf. footnote [14]).
One may wonder why in the foregoing pages only time was spoken of. Owing to the symmetry which subsists in relativity theory between time and space one should expect that the same would apply to space as well. As a matter of fact, however, this expectation verifies itself only in part and there are some essential points where the analogy does not go through.
This is due to the fact that (1) the passing of time has no analogue for space and that (2) the symmetry between space and time in relativity theory is not complete, if the existence of matter is taken into account. For matter entails the existence of a three-parametric system of physically distinguished one-dimensional time-like subspaces (namely, the world lines of matter), whereas there exists in general no one-parametric system of physically distinguished three-dimensional space-like subspaces. For the space-elements orthogonal to the world lines of matter can in general not be fitted together into three-dimensional spaces. This has the consequence that there exist no possible worlds in which the structure of absolute space-like relations would be so greatly different from intuitive space as is the case for time in the R-worlds. In particular, if the velocity vector of matter is continuous, the space of absolute distances between neighboring material points (defined by the orthogonal distances of their world lines) is an ordinary three-dimensional Riemannian space changing in time.That unlike [such space] in classical physics, its dependence on time can be represented in various ways (owing to the relativity of simultaneousness) is due to the different character which time (not space) has in relativity theory, as is seen from the fact that the same applies to every physical property which changes in time. Owing to the vector of mean velocity of matter (the mean being taken over suitably chosen regions of astronomical dimensions), it is even possible to introduce in some sense “absolute” points of space, but they differ in several respects essentially from the absolute points of Newtonian and intuitive space, in particular insofar as their mutual distances need not remain constant in time (i.e., this space in general contracts, expands or is being deformed).
There exists one circumstance which tends to conceal the close relationship which subsists in some respects between relativity theory and Kant, namely, [17] the fact that Kant, in addition to the subjectivity of time and space, also asserted the apriority of all our knowledge concerning them, and that he even based his main proof for their subjectivity on their apriority. Their apriority, however, seems flagrantly to contradict relativity theory, since the newly discovered properties of time and space (such as the relativity of simultaneousness, the non-Euclidicity of space, the Lorentz contraction, etc.), far from being a priori recognizable, even flatly contradict our supposed a priori intuition.
Closer examination, however, shows that these properties not recognizable a priori concern rather the objective correlate of our ideas of time and space (i.e., in Kant’s terminology, those affections which we represent to ourselves as time and space) and therefore are compatible with the apriority of some subjective space-time scheme which is founded in the intellectual equipment of man, but is likewise applicable to the description of physical reality, namely, as a kind of relative time and space (coordinate system). The apriority of Newtonian time and Euclidean space, understood in this sense, does not contradict relativity theory at all, but constitutes a question of its own.
In the case of geometry, e.g., the fact that the physical bodies surrounding us move by the laws of a non-Euclidean geometry does not exclude in the least that we should have a Euclidean “form of sense perception,” i.e., that we should possess an a priori representation of Euclidean space and be able to form images of outer objects only by projecting our sensations on this representation of space, so that, even if we were born in some strongly non-Euclidean world, we would nevertheless invariably imagine space to be Euclidean, but material objects to change their size and shape in a certain regular manner when they move with respect to us or we with respect to them.
Nor does the non-Euclidicity of physical geometry (defined by the behavior of rigid bodies) mean that this Euclidean pure intuition, if it exists, is simply wrong. For geometrical concepts satisfying Euclid’s axioms can be defined also in a non-Euclidean world, although (in conformity with Kant’s views) not as objective¹⁸ properties of the things but only as relations of the things to our sense organs (or, so one may add, to some other arbitrarily chosen objects).
Such definitions can be obtained, e.g., in the following way: we select some physical object (e.g., the earth or our body), a point on it and three (preferably orthogonal) directions at this point. Then we define the “coordinates” of any material point by the number of times a measuring rod must be applied along the directions (respectively, the directions obtained by suitable parallel displacements) in order to reach from .¹⁹ If then straight lines are defined by linear equations in these coordinates, the Euclidean axioms for them are true irrespective of the physical properties of the world. The same, of course, can be done for other concepts and axioms of geometry.
Hence, if we should possess an innate intuition representing to us geometrical entities of this kind,²⁰ the axioms of this intuitive geometry although referring to reality (namely, to measuring processes) would be apriori true.²¹ Nevertheless, they would not be analytic by Kant’s conception of analyticity, since arithmetic, to which geometry is reduced in this way, was not considered to be analytic by Kant.²² What, however, Kant did not take into account is that (irrespective of whether we have such an a priori intuition) space and its properties express themselves also in the sensations, which we know only a posteriori; namely, in the fact that by projecting the sensations in a certain way on a three-dimensional Euclidean space the laws connecting them can be expressed in a certain simple form. That this is not a necessary property of our sensations follows from the possibility of a physical non-Euclidean geometry and from other considerations along these lines, which show that our sensations might be such that to project them on a three-dimensional Euclidean space, though possible, would be of no use at all.
This insight does not imply anything about the objectivity of space. For the sensations also are something subjective and therefore any of their properties may as well be due to properties of the things as to the manner of interaction between the things and ourselves²³ (the decision depending on the kind of reality which must be assumed in order to explain the appearances satisfactorily). It does, however, necessitate (even in case we have an innate intuition of Euclidean space) a weakening of Kant’s theory of the apriority of space, at least to this extent, that the “adequacy of our representation of space to the relation of the objects to our sensibility” can only be known by experience, provided a reasonable meaning²⁴ is given to this phrase. That some particular relations of the things to us are adequately represented by Euclidean geometry in every world (cf., e.g., the definitions given above) certainly does not mean yet “a complete adequacy to the relation of the things to our sensibility.”
Nevertheless, Kant may be right in the sense explained above. I do not think the question whether we have an innate intuition of Euclidean space, or what representation of space we would actually develop in a strongly non-Euclidean world, has yet been decided; nor the related question whether we are able (in our world) to learn to imagine²⁵ a non-Euclidean space. That we are able to imagine the behaviour of physically rigid bodies in a non-Euclidean world²⁶ does not prove it. For we would certainly not identify (in our intuitive world picture) the behaviour of physically rigid bodies with the properties of space in every possible world, e.g., not in a world which is Euclidean for the most part, but has certain non-Euclidean spots. In a homogeneous non-Euclidean world, it is true, it seems more likely that we would develop an intuition of non-Euclidean geometry; there is, however, still room for reasonable doubt. But whatever the answers to these questions may be, there can certainly never result from them any incompatibility between Kant and relativity theory, but at most between Kant and psychology (or phenomenology) of sense perception.
A real contradiction between relativity theory and Kantian philosophy seems to me to exist only in one point, namely, as to Kant’s opinion that natural science, in the description it gives of the world, must necessarily retain the forms of our sense perception and can do nothing else but set up relations between appearances within this frame.
This view of Kant has doubtless its source in his conviction of the unknowability (at least by theoretical reason) of the things in themselves, and at this point, it seems to me, Kant should be modified, if one wants to establish agreement between his doctrines and modern physics; i.e., it should be assumed that it is possible for scientific knowledge, at least partially and step by step, to go beyond the appearances and approach the things in themselves.²⁷
The abandoning of that “natural” picture of the world which Kant calls the world of “appearance” is exactly the main characteristic distinguishing modern physics from Newtonian physics. Newtonian physics, except for the elimination of secondary qualities (which, in principle, was known already to Democritus), is only a refinement, but not a correction, of this picture of the world; modern physics, however, has an entirely different character. This is seen most clearly from the distinction which has developed between “laboratory language” and the theory, whereas Newtonian physics can be completely expressed in a refined laboratory language.
So the trend of modern physics is in one respect opposed to Kantian philosophy. On the other hand, however, it should not be overlooked that the very refutation of Kant’s view concerning the impossibility for theoretical science of stepping outside the limits of our natural conception of the world has furnished in several points a verification of one of his main doctrines concerning this natural world picture, namely, its largely subjectivistic²⁸ character, even as to those concepts which seem to constitute the very backbone of reality.
Moreover, it is to be noted that the possibility of a knowledge of things beyond the appearances is by no means so strictly opposed to the views of Kant himself as it is to those of some of his followers. For (1) Kant held the concept of things in themselves to be meaningful and emphasized repeatedly that their existence must be assumed, [and] (2) the impossibility of a knowledge concerning them is for Kant by no means a necessary consequence of the nature of all knowledge, nor subsists even for human knowledge in every respect. One may compare [with respect] to this point, e.g., what he says at B xxi, B xxi n. and B xxvi n. about the positive use of the Critique, not only for religious belief, but possibly also for knowledge (B xxvi n.). In the second of the passages cited above, Kant even goes so far as to compare his subjectivistic theory of knowledge as to its possible positive use with Copernicus’s explanation of the apparent motions of the planets by the motion of the observer, pointing out that this new viewpoint of Copernicus led to the discovery of the “hidden power connecting the structure of the universe, the Newtonian attraction.”
Kant, it is true, wanted to base such knowledge on ethics, but, even as far as theoretical reason is concerned, he evidently did not want to say that nothing whatsoever can be asserted about the things in themselves. For Kant himself asserted, e.g., that they exist, affect our sensibility, and do not exist in time and space, but that the ideas of time and space are completely adequate to their relationship to our sensibility.
Doubtless Kant would least of all have held an overstepping of the world of appearances to be possible, on the basis of conclusions drawn from experiments. But [with respect] to this point it is to be noted that, in perfect conformity with Kant, the observational results by themselves really do not force us to abandon Newtonian time and space as objective realities,²⁹ but only the observational results together with certain general principles, e.g., the principle that two states of affairs which cannot be distinguished by observations are also objectively equal. Generally speaking, it can be said that relativity theory (especially general relativity theory) owes its origin perhaps more than any other physical theory to the application of certain very general principles and ideas, and, for the most part, was only subsequently verified by experience.
Footnotes:I wish to say right in the beginning that I am not an adherent of Kantian philosophy in general. The subsequent considerations only try to show that a surprising similarity subsists in some respects between relativity theory and the Kantian doctrine about time and space and that contradictions between them, as far as they occur, are by far not so fundamental as is frequently maintained.See B 49, paragraph a).Cf. Prolegomena, §11, [p. 284].I.e., worlds possible by the laws of nature which hold in relativity theory.“Absolute,” in the sense of being independent of the frame or an object of reference, corresponds pretty closely to Kant’s phrase “inherent in the object.” For the concept “relative” appears in Kant as the opposite. Cf. the passage referred to in fn. [17].In the manuscript, this footnote ended with the phrase “Cnf. the passage referred to in footnote,” with a blank space where the reference number should be. This raised the question whether there is a basis for identifying the passage meant. Each of the quotations from Kant given by Gödel on p. 4 involves the contrast between properties of the “objects” (or “the things”) and relations of the objects to our sensibility; therefore, each of these is a possible candidate here (the references to the source, now given in the text, were given by Gödel in distinct footnotes for the distinct passages). This, however, provides no reasonable criterion for choosing one of those passages as the one meant here. Furthermore, none of these passages seems quite to fit Gödel’s words in the present footnote. But the passage (B42, paragraph a) referred to in Gödel’s problematic footnote 63 (fn. [17] in the present version—see editorial note f for discussion of the latter) contains the following statement:...space does not represent any determination that attaches to the objects themselves and which remains even when abstraction has been made of all the subjective conditions of intuition. For no determinations, whether absolute or relative, can be intuited prior to the existence of the things to which they belong, and none, therefore, can be intuited a priori. [Emphasis added.]Here we have the explicit contrast of “absolute” and “relative”—a contrast that Gödel may have been referring to when he wrote “For the concept ‘relative’ appears in Kant as the opposite.” (As the opposite of what?—of absolute.)It does contradict certain other parts of Kant’s system, namely, Kant’s view that the things in themselves (and therefore evidently also the objective correlate of the idea of time) are unknowable, not only by sensual imagination, but also by abstract thinking. Of course, it need not be maintained that relativity theory gives a complete knowledge of the objective correlate of the idea of time, but only that it goes one step in this direction (cf. fn. [16]), but even this would seem to disagree with Kant (cf. p. [27]).In the sequel, I shall refer to these worlds as “R-worlds” (“R” being an abbreviation for: “rotating, rotation-symmetric, stationary”). For their mathematical description cf. [1949]. The compass of inertia in these worlds rotates everywhere relative to matter, which in our world would mean that it rotates relative to the totality of galactic systems. If the current explanation of the red-shift of distant nebulae is correct, our world is not an R-world. It may, however, be an expanding rotating world. In that case, some of the subsequent considerations about the non-objectivity of time (in particular those about travelling into the past and future, cf. below p. 12) might remain applicable.For a brief reference to these, see Gödel 1949b; for Gödel’s sketch of a proof of the existence of such solutions to Einstein’s equation, see Gödel 1952."Sensibility" is, according to Kant, the faculty of having sensations under the influence of things and happenings and of forming images of outer objects and of oneself and his affections out of these sensations. Sensibility is divided by Kant into outer and inner sense.Cf. Prolegomena, §11, [p. 284].See Prolegomena, §13, [p. 286]. To this passage, as well as the second one from B 69n. referred to above, it is to be noted that for Kant “appearance” does not mean the fact that something appears, but rather an object of thought which has all the properties which are attributed to the object appearing (cf. his definition at B 34). Also the context in which these two passages occur shows clearly that “appearance” is meant in this sense.See: Prolegomena, §13, end of note II, [pp. 289-290].The phrase “by it” has been deleted following “means” by the editors, under the assumption that Gödel had in mind the German “dabei”. As such it was judged to be implied by the rest of the sentence. In version C2, Gödel had replaced “means by it that temporal properties represent” by “is viewing temporal properties as representing”.This concept was originally designed for special relativity theory only, but it is clear that it can be extended to general relativity theory, at least under certain conditions which are satisfied in the R-worlds.Immediately only for worlds in which matter is strictly homogeneously distributed,but this difficulty can be overcome; e.g., by considering the homogeneous solution with (in some sense) the least total deviation from the given one.On the ground of the practical success of these solutions for the explanation of astronomical facts, J. Jeans claimed a return to the old idea of an absolute time passing objectively to be justified (cf. Jeans 1936, pp. 22–23). However, it can hardly be maintained with certainty that our world really is described by one of these solutions. Moreover, cf. what is said below on p. [17].This state of affairs seems to imply an absurdity. For it enables one, e.g., to travel into the near past of those places where he has himself lived. There he would encounter a person who would be himself so and so many years ago. Now he could do something to this person which he knows by his memory has not happened to him. This and similar contradictions, however, in order to prove the impossibility of the R-worlds, presuppose at least the practical feasibility of the trip into the past, which may very well be precluded by the velocities very close to that of light which would be necessary for it, or by other circumstances.The present state of physics, however, seems to indicate that the future development will continue along these lines. The fifth dimension introduced by Th. Kaluza (cf. his 1921), e.g., points in this direction. Still more so quantum mechanics, which has so far not been able to give a satisfactory description of an objective reality which would make the success of its rules of computation understandable. [Cf. Einstein, Podolsky, and Rosen 1935, where it is proved that the wave function cannot represent the whole reality, unless an action at distance with very strange properties is assumed.] If the future development of physics really goes in the direction just indicated, relativity theory would have to be interpreted as only one “level of objectivation” (cf. fn. [26]) to be followed by others.It is to be noted that Kant explicitly denied the objectivity of space not only if conceived as an entity besides the things, but also if conceived as a system of relations between the things. (Cf. B 42, paragraph a.)Because Gödel had not marked this footnote for deletion from his list of footnotes, and because one of the footnote cross-references left blank in the text—namely, fn. 5 (our numbering)—probably refers to this footnote, we have elected to place this footnote here. (For a concordance of our footnote numbers with the original numbers, see the textual notes.) Further evidence that he did not intend to delete this footnote, but simply forgot to mark a location, consists in the occurrence of this footnote’s (original) number in a concordance made by Gödel himself. There the number appears between those for the footnotes that became our fns. 16 and 18. (Note also that there were some pages deleted in that region.) The location we have chosen is in agreement with that evidence.At least if certain axioms about “objectivity,” such as the interchangeability of two different observers, are postulated.In a closed (e.g., a Riemannian) world this definition would not give unique values for the coordinates, i.e., it would have the consequence that either one object can be in many different places simultaneously, or that for every object there exist infinitely many exactly equal ones; or, if the definition is slightly changed, space would consist of a finite portion of Euclidean space only.i.e., representing those propositions as evident, which hold for these concepts. Of course I do not want to say that pure intuition, according to Kant, refers exactly to those geometrical concepts which are defined above, or that an intuition of this kind actually exists. I only wanted to show that an innate Euclidean geometrical intuition which refers to reality and is a priori valid is logically possible and compatible with the existence of non-Euclidean geometry and with relativity theory. Such an intuition might also be so constituted that only the representation of space itself and the general idea of its relatedness to reality are inborn, while the particular way in which we project our sensations on it depends on experience.This seems to contradict Einstein’s well-known aperçu about the relation between geometry and experience (cf. Einstein 1921), but the contradiction is rather one of words than of meaning, since the phrase “sich auf die Wirklichkeit beziehen” is evidently meant by Einstein in the sense of “say something about reality,” whereas the geometrical axioms as interpreted above, although they refer to physical reality, do not say anything about it. Also according to Kant, the geometrical propositions are synthetic, not because they say anything about that which is given from outside (i.e., the sensations), or about the things in themselves, but because they say something about that which we, or rather our intellectual faculties, imagine or construct in the process of mental assimilation of the sensations. These subjective additions, however, do not form a part of objective reality in the proper sense of the term.It is to be noted that the term “synthetic” today is frequently used in a sense different from Kant’s, namely, for a proposition which expresses a state of affairs subsisting independently of the activity of our intellect apprehending it. In this sense, “synthetic” and “a posteriori” would doubtless coincide also for Kant. But Kant would deny that the activity of our intellect consists solely in explicit definitions and their application.The subjectivity here in question is of course of a different kind from Kant’s, as is seen from the fact that it does not imply apriority. Nevertheless space could very appropriately be called “a form of sensibility” also if it were subjective in this second sense.i.e., by “relation of the things to sensibility” must be meant the way in which outer objects produce sensations, not the fact that we represent things to ourselves in space, for in the latter case, the adequacy in question would mean nothing at all.In the left-hand margin beside this footnote, Gödel had a question mark.Usually it is not realized how greatly different a non-Euclidean spatial intuition would be from ours in case of a sufficiently large curvature of space. E.g., in a certain Lobachevskian space (which stretches into infinity like ours) the circumference of a circle of radius 1 inch would be 1 yard.Cf., e.g., Einstein 1921.Cf. in this connection Bollert 1921, where one may find a description in more detail of these steps or levels of objectivation, each of which is obtained from the preceding one by the elimination of certain subjective elements. The “natural” world picture, i.e., Kant’s world of appearances itself, also must of course be considered as one such level, in which a great many subjective elements of the “world of sensations” have already been eliminated. Unfortunately, whenever this fruitful viewpoint of a distinction between subjective and objective elements in our knowledge (which is so impressively suggested by Kant's comparison with the Copernican system, see below, p. [[29]) appearsin the history of science, there is at once a tendency to exaggerate it into a boundlesssubjectivism, whereby its efect is annulled, Kant's thesis of the unknowability of thethings in themselves is one example, another one is the prejudice that the positivisticinterpretation of quantum mechanics must necessarily be the final stage of the theory.One may say that (as also seems to appear from the contents of this paper) relativity theory has proved only the relativity of time, not its subjectivity (i.e., relativity to our imaginative faculty). But it is to be noted that (1) the absoluteness (or uniqueness) forms itself an essential characteristic of intuitive time, and that (2) a relative passing of time (if any sense can at all be given to this phrase) would be something radically different from the passing as described above on p. [11]; for existence by its nature is something absolute. Hence, an entity corresponding really in all essentials to the intuitive idea of time, also in relativity theory, exists only in our imagination.In special relativity theory, e.g., some arbitrarily chosen inertial system can be considered as representing absolute space and time (with all properties Newton attributed to them) and the varying observational results of observers moving with different velocities can be explained by the effect which motion relative to this absolute space has on the physical bodies and processes, in particular on the measuring instruments (effects which incidentally follow without any ad hoc hypothesis from the empirically verifiable electromagnetic equations of Maxwell and the very natural assumption that the constitution of physical bodies is based on electromagnetic forces or other forces with similar mathematical properties). It remains, however, entirely arbitrary which coordinate system is in this way distinguished as the absolute space-time scheme and this, by the principle formulated above, entails that absolute time and space do not exist.
