I think I love ideas. I like a story that’s got some concrete you know structure but also holds abstractions. Life is filled with abstractions and the way we make heads tails of it is through intuition. And so people get used to film that pretty much explains itself a hundred percent. And they kind of turn off that you know beautiful thing of intuition when they’re looking at a film that has some abstractions. And some people on the other hand love these abstractions. And it gives them room to dream. A abstractions to me is a thing that cinema can say. And it’ so beautiful for me anyway to think about these pictures and some sounds flowing along together in the time in a sequence making a thing that can only really be said in cinema. It’s not words. It’s not just music. It’s the whole bunch of things coming together and making a thing that didn’t exist before. And that’s what i really love about it. And then to answer your question a little further. It’s up to the people you know to you know find their own your know interpretation. It’s doesn’t really matter what I think. It’s all every screening no matter what even if all the frames of the film are exactly the same. But there no two screenings that are exactly the same. It’s the viwer and the picture and the sound and it makes a circle and it just goes like that. And so you just feel it and think it. That’s kind of intuition emotion and thinking together and come up make it have a sense to you.

The preface to the 1957 edition of Jorge Luis Borges’ collection of essays The Book of Imaginary Beings (幻兽辞典) contains the comment that monsters will always stalk mythic stories because real animals are a deeply important part of human experience and because monstrous beings are combinations of the real and the imagined, the stuff of nightmares and dreams. The Classical mythic centaur, which melds the forms of man and horse, has its Celtic counterpart in the Welsh horse-woman, Rhiannon. The Cretan Minotaur (米诺陶洛斯), a hideous blend of bull and human, can perhaps be seen transmuted in Irish mythology to become the great fighting bulls of Ulster and Connacht, which had human speech and understanding, or, in Wales, the enchanted boar Twrch Trwyth (图鲁夫图鲁维斯). Borges even goes so far as to argue that monsters are ‘necessary’ for human society. In our own day, fascinated by space and the possibility of worlds beyond, we conjure up fantastic images of galactic monsters, nowhere more clearly presented than in the Star Wars cantina, in which Skywalker and Solo encounter a collection of weird and wonderful beings from all over the Universe. Such are our modern mythic creations. 26

A persistent feature of both Irish and Welsh mythology is the theme of the magical cauldron, a vessel capable of raising the dead and of providing ever-replenishing supplies of food. The Irish god Daghdha, (‘the Good God’), possessed a huge inexhaustible cauldron. The central focus of the Irish Otherworld feast was the cauldron, which never ran out of food. One Irish cauldron-myth was associated with sacral kingship, where the new king of Ulster had to bathe in one, while consuming the meat and broth of a white mare he had ritually ‘married’. 29

$\S$ Ceridwen’s Cauldron: A Welsh mythic tale, preserved in a 13th-century text, The Book of Taliesin (塔列辛之书), contains a rich story of an enchanted cauldron, whose contents endowed those who ate or drank from it with knowledge and inspiration. The cauldron’s keeper was Ceridwen. She bore two children, Crearwy (‘the light or beautiful one’) and Afagddu (‘black’ or ‘ugly’). Wanting to compensate her son for his ill-favoured appearance, his mother mixed a special brew in the cauldron, designed to give him absolute wisdom. Because the potion needed to boil for a year, Ceridwen appointed a young boy, Gwion, to watch over it. As he was tending the cauldron, three drops of scalding liquid splashed onto his hand and, without thinking, he licked his fingers, thus inadvertently acquiring the wisdom intended for Afagddu. Gwion’s flight and pursuit by the angry Ceridwen eventually caused Gwion’s rebirth as the great visionary poet Taliesin. 30

Words are powerful things, the more so if they are spoken aloud, so that sound and meaning blend into a single powerful message that can be shared simultaneously by many people. Keepers of oral tradition had to have prodigiously long and accurate memories and the ability to learn long tales by heart, while adding embellishments along the way. Listeners, too, would remember stories they had heard all their lives, and would not have hesitated to point out errors or inconsistencies. 43

我们怀着深深的悲痛通知您, 著名数学家威廉·劳维尔 (William Lawvere) 博士于 2023 年 1 月 23 日去世. 劳维尔教授是加拿大共产党 (马列主义) 的好友, 他总在文明大道上前进, 致力于对抗并改变社会上的陈旧道德观念, 消除进步启蒙道路上的障碍.

从 1970 年代初开始, 他与 CPC (M-L) [译注: 即 Communist Party of Canada (Marxist-Leninist), 加拿大共产党 (马列主义)] 的创始人和领导人哈迪尔·贝恩斯 (Hardial Bains) 一起从事了几个与变革的必要性和辩证法相关的哲学和其他项目, 哈迪尔·贝恩斯本身就是一位科学家和哲学家. 从 1969 年到 1971 年, 他是哈利法克斯达尔豪斯大学 [Halifax, Dalhousie University] 的 Killam 学者, 是一个著名的国际数学家团队的负责人, 他们来那里与他一起学习. 但在 1971 年, 当劳维尔教授抗议《战争措施法》[War Measures Act] 和皮埃尔·特鲁多政府因暂停公民自由而犯下的罪行时, 达尔豪斯拒绝续签合同, 尽管他自己的数学家团队提出了抗议. 1,000 多名学生在 Dal Student Union 大楼的大厅集会, 反对任意解雇劳维尔教授.

在许多场合, 劳维尔教授在加拿大与我们一起讨论重要话题, 并贡献了他对这些话题的全面而广泛的了解. 1996 年 2 月, 他在温莎大学举行的主题为 “意识起源和社会变革” [The Origin of Consciousness and Social Change] 的跨学科会议上发表了演讲. 在那个场合发表的论文提出了对独立于我们而存在的意识、冷战史学、现代民主人格的出现等的重要见解. 劳维尔教授发表了一篇题为《数学改革的历史和哲学》[The History and Philosophy of Mathematic Reform] 的论文, 该论文研究了限制数学科学知识的教育学趋势的各个方面, 展示了这如何为促进神秘主义和盲目接受权威提供了有效的基础.

1997 年, 他与来自世界各地的学者一起参加了贝恩斯同志的葬礼. 他还参加了 1998 年的 CPC (M-L) 第七次代表大会, 在那里他对哈迪尔·贝恩斯的工作表示赞赏, 并为正在讨论的论文做出了贡献. 2005 年, 他参加了党的 35 周年庆典, 再次表达了对党在理论和现代定义方面的工作的赞赏. 他后来写道, 35 周年纪念日 “是一个非常有意义的时刻. […] 特别是, 这一场合使我的工作取得了进一步的进展, 特别是为纪念列宁哲学著作一百周年的项目, 详细介绍了它们如何从自然科学的角度以及从为启蒙而进行的阶级斗争的角度, 成为研究 19 世纪和 20 世纪哲学发展的有用指南. 自从 35 年前哈迪尔指出这些书的重要性以来, 它们一直是我的伴侣. […]”

听说劳维尔教授去世的消息后, 他的长期伙伴埃里克·霍夫曼 (Eric Hoffman) 在感谢哈迪尔·贝恩斯和比尔·劳维尔 (Bill Lawvere) [译注: 即 William Lawvere] 对他自己的生活和工作产生的深远影响时说:

“从他们在 1960 年代独立完成的最早的开创性工作发展起来, 出现了一种新的、令人惊讶的联系的最有力证据. 这种联系构成了人类与自然之间关系的一部分, 并嵌入其中. 比尔指出, 这种复合体的客观性迫使人们坚持不懈地坚持空间和数量研究的需要.

“在他博客上的一次采访中, 比尔说, ‘数学理论的核心是空间中数量的变化以及其中质量的出现.’ 无论考虑的是几何学、范畴、逻辑、政治形式的过渡、亲属关系还是任何其他因素, 比尔都证实了对立面的统一和同一性, 即辩证法, 是存在的.

“比尔的工作将作为扩大启蒙空间的贡献而经久不衰.”

在 CPC (M-L) 中央第一书记发给劳维尔教授家人的哀悼信中, 她写道:

“比尔在他的一生中做出了如此多的贡献, 我们非常感激. 他对发现新事物的热情和奉献精神总是令人鼓舞. 他的方法总是充满活力、专注、在许多方面都很新鲜. 他作为教师的耐心、他的论点的智慧和连贯性、他对年轻一代和教化他们的热情, 以及他对文明崇高道路的勇敢坚持, 都非常出色.

“与比尔会面总是非常高兴. 我们将永远珍惜他的友谊、慷慨的精神和对我们共同事业的奉献精神, 他为此毫不畏惧地大声疾呼. 这表明他是由特殊的东西组成的.

为了说明他作为数学教育家的工作质量, 他解释手头问题并与那些对他的工作表现出兴趣的人讨论的直截了当的方法, 我们举了他对提供 “为什么范畴论可能如此有用的广泛理由” 的请求的回答的例子. 以下是比尔不得不说的话:

人类的日常活动, 如在溪流旁的山上建造房屋、铺设电话管道网络、在太阳系中航行, 都需要可行的计划. 规划任何此类工作都需要发展对空间的思考. 每个发展都涉及许多思考步骤和许多相关的空间几何结构. 由于思考空间的必要多步骤性质, 必须采取独特的数学措施来使其可靠. 只有明确的思维原则 (逻辑) 和明确的空间原则 (几何) 才能保证可靠性. 艾伦伯格和麦克莱恩 60 年前发明的理论所取得的巨大进步使逻辑和几何学的原则变得明确; 这是通过发现逻辑和几何学的常见形式来实现的, 因此两者之间关系的原则也是明确的. 他们解决了亚里士多德在 2300 年前提出的一个问题, 当时亚里士多德最初在明确概念范畴方面取得了进展. 在 21 世纪, 他们的解决方案不仅适用于平面几何和中世纪三段论, 还适用于无穷维变换空间、数据 “空间” 以及每天应用数千次的其他概念工具. 逻辑和几何原理的形式都是由分类论者发现的, 它建立在空间之间转换和内部转换的 “自然性” 之上.

以下是 Lawvere 教授去世时发布的讣告. 我们添加了 1971 年 1 月 22 日《达尔豪斯公报》[Dalhousie Gazette] 上关于他被达尔豪斯大学开除的文章的传真件. 我们还提供了 Lawvere 博士在 1996 年 2 月 9 日至 11 日在温莎大学举行的关于意识起源和社会变革的跨学科会议上发表的题为 “数学改革的历史和哲学” [The History and Philosophy of Mathematic Reform] 的论文. 最后, 我们提供了对 Lawvere 博士的采访, 他用自己的话解释了他的工作.

我们向比尔的终身伴侣和合作者、他的子女、孙子、兄弟姐妹以及许许多多的朋友和同事表示最深切的哀悼. 愿他们都在共同创造的回忆中找到慰藉.