温伯格《解释世界》第十三章 方法再思考

亚里士多德科学探索方法到十六世纪末已深受质疑。自然而然人们开始对认识世界的方法寻求新的途径。建立新科学方法最知名的两位人物是弗朗西斯·培根和勒内·笛卡尔。不过在我看来他们两位在科学革命中的重要性被大大高估了。 弗朗西斯·培根出生于1561年，他父亲是尼古拉斯·培根，英格兰掌玺大臣。从剑桥三一学院毕业后，他取得了律师资格，从事法律，外交工作，后来从政。1618年他被授封为维鲁兰男爵，晋升为英格兰大法官，后来又被授封为奥尔本斯子爵。但是1621年他被控贪污受贿，议会宣布他不适合担任公职。 培根在科学史上的名望主要来自于他1620年出版的著作《新工具》（或关于解释自然的心方向）。培根不是科学家或数学家，他在此书中表达了极端经验科学观，不只否定亚里士多德思想，他也拒绝托勒密和哥白尼的方法。科学发现应该从仔细，不带偏见的对自然观察中得出，而不是从第一原理中推导出。他对任何没有直接实用价值的研究都嗤之以鼻。在《新亚特兰蒂斯》中他设想一个合作研究院--“所罗门之家”，其成员投身于收集有用的自然界事实真相。这样人类可以重新赢得被从伊甸园驱逐出来以后对自然的支配。 培根和柏拉图处于两个极端。当然两个极端都不对。科学进展取决于观察或实验以及对从这些观察或实验得出的普适性原理的验证的结合。寻求具有实用价值的知识有助于更正天马行空的猜想，但是无论是否会带来任何实用性，对自然世界的解释存在自身价值。十七和十八世纪科学家会借助于培根来抗衡柏拉图和亚里士多德，就像美国政治家会借助于杰斐逊，虽然他们没有受到杰斐逊所言所行的任何影响。我看不出有人的科学研究工作真正受到培根的什么影响。伽利略不需要培根告诉他去做实验，波义耳和牛顿也一样不需要。伽利略前一个世纪另一位佛罗伦萨人莱昂纳多·达·芬奇就已经做了落体，流体以及其他一些实验。我们只是从他去世以后汇编的一些关于绘画和流体运动的两部专著以及后来发现的一些笔记中才知道这些，虽然即使莱昂纳多的实验对科学的发展没有影响，但至少证实在培根之前已经有了实验。 勒内·笛卡尔比培根更值得关注。他1596年出生于法国一个地方法院法官的贵族家庭，在拉弗莱什的耶稣会学院接受教育，进入普瓦捷大学学习法律，荷兰独立战争期间加入纳索的莫里斯的军队。1619年笛卡尔觉得投身于哲学和数学，1628年他在荷兰永久定居之后他开始认真地进行研究。 笛卡尔十六世纪三十年代创作的《世界》中记录了他的力学观点，该书在他去世后的1664年才出版。1637年他发表了哲学作品《科学中正确运用理性和追求真理的方法论》。在他最长的著作《哲学原理》中他进一步发展了他的观点，该书1644年以拉丁文出版，1647年译为法文。在这些作品中笛卡尔表达了对遵从权威或感官的怀疑。对笛卡尔来说唯一确定的是“我思故我在”。他继而推断世界存在，因为他不需要意志力就可以感受到。他不相信亚里士多德的目的论—事物如其所是，不是为了服务什么目的。他给出几个上帝存在的证据（都没有说服力），但是他不接受宗教组织的权威。同时也反对超距（物体间相互直接作用的推力和拉力）的神秘力。 笛卡尔是将数学应用到物理学的主导者，但是与柏拉图一样他过度看重了数学推理的确定性。在《哲学原理》第一部分--“关于人类知识的原理”，笛卡尔描述了如何从纯思考中明确地推导出基础科学原理。我们可以相信”上帝赋予我们的自然启示或认识能力”因为“他不会欺骗我们。”奇怪的是笛卡尔认为可以容许地震和瘟疫发生的上帝会不欺骗一个哲学家。 笛卡尔确实接受将基础物理原理应用到特定领域会有不确定性，如果人们不知道系统内部细节时，需要用实验来确定。在《哲学原理》第三部分讨论天文学时，他考虑了行星系统各种假想，引述了伽利略对金星相的观测来证明哥白尼和第谷的假想要优于托勒密。 上面简短的总结不能全面概括笛卡尔的观点。他的哲学过去和现在一直受到推崇，特别在法国以及哲学家之中。我不理解。对于一个声称找到了探求可靠知识真实途径的学者，令人惊奇的是笛卡尔对自然界各方面的认识错的有多离谱。他的地球细长说（即地球两极间的距离大于赤道面距离）是错误的。他与亚里士多德一样错误地认为真空不可能存在。他的光瞬时传输的观点也是错误的。（注：笛卡尔将光比作刚性杆，推动其中一端，另一端瞬时移动。他对刚性杆的认识也是错误的，不过其理由他当时不可能知道。当推动刚性杆的一端时，另一端只有在压缩波（本质上是种声波）从一头传到另一头以后才会移动。这种波的传播速度随刚性杆强度增强而增快。但是爱因斯坦狭义相对论决定了任何物体都不可能是完全刚性的，波的传播速度不会大于光速。彼特·格里森在《笛卡尔对比：从无形到有形》（ISIS75, 322 – 1984）中探讨了笛卡尔使用的这种对比。）他说空间充满了物质旋涡携带行星在其轨道运行也是不对的。他错误地认为松果体是灵魂的居所，负责人的意识。他对碰撞时哪种量守恒的认识是错误的。他错误地认为自由落体下落速度正比于下落距离。最后，基于我对几只可爱的宠物猫的观察，我确信笛卡尔所谓的动物是没有意识的机器的说法是错误的。伏尔泰对笛卡尔也有同样疑议： 他把灵魂特性，上帝存在的证据，物质，运动规律，光的特性都弄错了。他认可天赋观念，杜撰一些新元素，创立了一个不同世界，他照着自己创造他人—事实上人们说这些人只是笛卡尔创造的人，与真正的人完全不同。 如果只是去评价一位创作伦理，政治哲学，或甚至形而上学方面学者的成就，那笛卡尔科学错误并不重要。但笛卡尔创作了“正确运用理性以及寻求科学真相的方法”，那他多次重复犯错显然会给他哲学判断蒙上阴影。笛卡尔承担不起逻辑推理之重。 最伟大的科学家也会出错。我们前面看到伽利略对潮汐和彗星的认识错误，后面我们还会看到牛顿在衍射方面的错误。笛卡尔虽然犯了许多错误，但与培根不同，他确实对科学做出了卓越贡献。这些以三个标题—几何，光学，气象学--作为附录发表在他的《方法论》著作中。我认为这些代表了他对科学的正面贡献，而不是他的哲学作品。 笛卡尔最伟大的贡献是我们现在称为解析几何的新数学方法--曲线和平面用满足曲线和平面上点坐标的数学方程表示。广义“坐标”可以是表示一个点位置的任何数，比如经度，纬度，高度等，而所谓的“笛卡尔坐标”特指该点与沿一组相互垂直数轴原点间的距离。比如在解析几何中半径为R的圆可以表式为一条曲线，曲线上一点的坐标x和y为与沿任何两个垂直数轴坐标原点的距离，并且满足方程 x2+y2=R2。（技术说明18介绍了对椭圆类似的描述。） 最早应用字母来表示未知数或其他数来自于十六世纪法国数学家，侍臣，以及密码专家弗朗索瓦·韦达，但是韦达仍然用文字来表述方程。现代代数形式及其对解析几何的应用要归功于笛卡尔。 应用解析几何我们可以通过求解曲线或平面方程得出两条曲线交点坐标，或两个相交平面产生的曲线方程。现今多数物理学家应用解析几何解决几何问题，而不是用古典的欧几里德方法。 笛卡尔物理方面卓越的贡献是对光的研究。首先在《光学》中笛卡尔介绍了当光线从一种介质A到另一种介质B（比如从空气到水）入射角和折射角的关系：设入射光与垂直介质分界面夹角为i，折射光与此垂面夹角为r，那i的正弦（注：前面讲过角的正弦等于直角三角形该角所对一边除以斜边。当角度从0度增加到90度，正弦值增加，小角度时正比于角度，然后缓慢增加）正比于除以r的正弦为与角度无关否认常数n: sin⁡〖(i)/sin⁡〖(r)=n〗 〗 通常介质A为空气（严格地说是真空），n被称为介质B的“折射率”。比如A为空气，B为水，n为水的折射率，约等于1.33。像这种n大于1的情形，折射角小于入射角I,光线进入更高密度介质折向垂直界面方向。 笛卡尔那时还不知道丹麦人（译注：应该是荷兰人）威里布里德·斯涅耳已于1621年采用经验方法得到了此关系，英国人托马斯·哈里奥特甚至更早得出此关系。公元十世纪阿拉伯人伊本·扎尔的一份手稿证实他也对此有所知。但是笛卡尔第一位正式发表此关系。现在人们一般称之为斯涅耳定律，除了法国，那里人们将之归功于笛卡尔。 人们不太容易理解笛卡尔对折射定律的推导，其中部分原因是无论在他推导过程或他的结果陈述都没有用到三角函数中角的正弦概念。他用纯粹几何术语描述，我们前面讲过阿尔·巴塔尼在七个世纪之前就已经从印度引入了正弦，而且阿尔·巴塔尼的工作在中世纪欧洲广为人知。笛卡尔的推导是基于这样的一个类比--当击一个网球穿过薄织物会发生什么。球会减速，但织物对球沿织物方向的速度没有影响。这样他就可以得出前面说过的结论（间技术说明27）：网球在击到界面之前和之后网球与垂直界面夹角的正弦比是与角度无关的常数n。虽然在笛卡尔的讨论中难以看到这一结果，但他一定非常理解，因为在后面要讨论的他的彩虹原理中他用一个合适的n值可以得出大致正确的数值结果。 笛卡尔的推导有两点显然是错误的。光不是网球，空气与水的界面也不是薄织物，所以他的类比并不恰当，而且笛卡尔认为光速无限大，这显然不像网球。另外笛卡尔的类比也使他得出错误的n值。对于网球（见技术说明27）他的假设意味着n等于球穿过界面后在介质B中的速度VB与球在击到界面之前在介质A中的速度VA之比。当然球穿过界面会慢下来，VB比VA小，它们的比值n应该小于1。应用于光的话这意味着折射光与垂直于界面的夹角大于入射光与垂直于界面的夹角。笛卡尔知道这点，他甚至用一张简图示意网球从垂直方向折射开来的路径。笛卡尔也知道光不是这样，至少从托勒密时代人们就已经观测到光线从空气进入水后会朝向垂直于水面的方向弯曲，这样i的正弦大于r的正弦，n值大于1。笛卡尔完全胡乱地解释了一气，我不能理解，他辩称光在水中比在空气中容易行进，所以对于光n值大于1。笛卡尔为自身目的没能解释n值其实关系不大，因为他可以而且确实用的是从实验得来的n值（可能来自于托勒密《光学》中的数据），当然n值大于1。 数学家皮埃尔·德·费马（1601-1665）提出更加令人信服的折射定律推导，类似于亚历山大里亚的希罗控制反射的等角率推导，不过现在假定光行进路线取最短时间，而不是最短距离。这个假设（见技术说明28）引导出正确的方程，n值为介质A中光速与介质B中光速之比，如果A为空气，B为玻璃或水，那n值大于1。笛卡尔不可能推导出此计算n值的公式，因为他认为光是瞬时行进的。（在第十四章我们会看到，克里斯蒂安·惠更斯也推导出了正确的结果，他的推导是基于他的光线扰动原理，不需要依据费马的光行进路线取最短时间先验假设。） 笛卡尔巧妙地应用了折射定律：在《气象学》中他利用入射角和折射角间的关系来解释彩虹。这是笛卡尔作为科学家的最佳表现。亚里士多德曾经主张彩虹的颜色是由于光被悬浮于空气中的小水滴反射形成的。我们在第九章和第十章讲过在中世纪阿尔·法里斯和弗赖堡的迪特里希都认为彩虹是由于光线进入和离开悬浮于空气中的雨滴发生的折射形成的。但是在笛卡尔之前没有人定量描述其机理。 笛卡尔首先做了一个实验，他用充满水的薄壁玻璃球来模拟雨滴。他观察到当太阳光从不同角度进入玻璃球后，与入射方向成42度角的返回光线“呈完全红色，比其他的亮很多。”他推断彩虹（或至少其红端）在天空中呈弧形，视线到彩虹与彩虹到太阳间的夹角为42度。笛卡尔猜想光线进入雨滴后发生折射弯曲，然后被雨滴后部反射，在从雨滴出来时再次发生折射弯曲。但是如何解释雨滴的这个属性--从雨滴出来的光与入射方向呈42度角？ 为了回答此问题，笛卡尔考虑了沿10条不同平行线进入球形水滴的光线。他用现代称之为影响参数的b值来标识这些光线，影响参数是指如果光线从水滴穿过不发生折射的话光线与水滴中心间的最短距离。第一条光线选为如果不发生折射该光线以距离水滴中心为水滴半径R的百分之十穿过水滴。（即b=0.1R），第十条光线选为从水滴表面掠过（这样b=R），中间这些光线均匀分布在此两者之间。笛卡尔应用欧几里德和希罗的等角率以及自己的折射定律计算出每条光线进入水滴的折射，水滴后部反射，和离开水滴的再次折射路径，他设水的折射率为4/3.下表给出笛卡尔计算的每条光线入射方向与返回方向夹角𝞿，以及我用同样的折射率做出的计算结果： b/R 𝞿(笛卡尔) 𝞿（重新计算） 0.1 5o40’ 5o44’ 0.2 11o19’ 11o20’ 0.3 17o56’ 17o6’ 0.4 22o30’ 22o41’ 0.5 27o52’ 28o6’ 0.6 32o56’ 33o14’ 0.7 37o26’ 37o49’ 0.8 40o44’ 41o13’ 0.9 40o57’ 41o30’ 1.0 13o40’ 14o22’

笛卡尔结果中的一些误差可以归因于他那时有限的辅助数学计算工具，我不知道他是否有正弦表，但他一定没有像现代计算器这样的设备。不过笛卡尔应该有更好的判断力，他应该把结果精度给到10弧秒，而不是到弧秒。 笛卡尔注意到存在宽范围的影响参数b值其夹角𝞿接近40度。他接着又计算了b值在水滴半径百分之八十到百分之一百内的18条更接近的光线，这里的𝞿大约40度。他发现18条中有14条的𝞿值在40o与最大值41o30’之间。这些理论计算解释了他早期实验观测得到的42度夹角。 技术说明29介绍了笛卡尔计算方法的现代版本。我们不像笛卡尔那样计算一系列光线中每条光线入射与返回间的夹角数值，我们推导出一个简单公式，可以计算对任何影响参数b值，以及任何空气中光速与水中光速比值n的光线𝞿值。然后我们用这个公式去找到射出光线集中的𝞿值。（注：这是通过找当b发生极小变化不会导致𝞿变化的b/R值实现的，在此𝞿值𝞿与b/R的关系曲线呈水平。该b/R值对应的𝞿达到最大值。（所有像𝞿与b/R关系这样上升到最大值后又下降的光滑曲线在顶点都呈水平。曲线不呈水平的点不可能是最大值，因为如果在某点曲线向右或左上升，那在右边或左边一定有曲线上更高的点。）当我们改变b/R时，𝞿与b/R关系曲线近水平处𝞿值变化非常缓慢，这样在此𝞿值范围存在相对大量的光线。）当n为4/3，正如笛卡尔发现的一样，射出光线相对集中的𝞿值为42度。笛卡尔甚至计算了光线在从雨滴射出之前在雨滴内进行了两次反射的霓的相应角度。 笛卡尔意识到彩虹颜色分布与光线通过棱镜折射呈现出的颜色间的联系，但是他既不能定量计算--因为他不知道太阳白光是由所有颜色光合成，也不理解光线折射率略取决于光线颜色。笛卡尔把水的折射率取值为4/3 = 1.333…，实际上 这接近于典型红光波长的1.330以及蓝光的1.343。大家可以算出（应用技术说明29推导的通式） 红光入射与射出夹角𝞿的最大值为42.8度，蓝光为40.7度。这就是之所以笛卡尔以与太阳光方向成42度的方向在他的水球中看到明亮红光的原因。这个角度𝞿大于蓝光从水球射出的最大角度40.7度，所以笛卡尔看不到光谱蓝端；但是其刚好小于红光𝞿的最大值42.8度，这样（如上一段内的注解）红光看起来特别明亮。 笛卡尔光学研究采用的完全是现代物理模式。笛卡尔先做出一个大胆猜想--光线穿过两种介质界面类似于网球穿过薄物，由此他推导出满足观察结果的入射角与反射角关系（选择合理的反射率n）。接下来用盛满水的玻璃球模拟雨滴，他通过观测提出形成彩虹的可能原由，然后他用数学证明观测与他的折射理论相符。他不了解彩虹的色彩，所以他回避了该问题，只发表了他所理解的部分。这正是物理学家今天所做的，除了将数学应用到物理，这与笛卡尔的《方法论》有什么关系？我看不到任何迹象表明他遵循了他自己所说的“正确运用理性以及寻求科学真相”。 我应该补充笛卡尔在《哲学原理》中大大改进了布里丹的动力概念。他主张“所有运动都保持直线运动，”所以（与亚里士多德和伽利略相反）需要外力维持行星在其曲线轨道运行。但是笛卡尔没有计算此外力。我们在第十四章会看到惠更斯计算了维持物体以给定速度在给定半径圆上运行需要的外力大小，牛顿解释了该外力--这是重力。 1649年笛卡尔到斯德哥尔摩任女王克里斯汀娜的导师。也许是由于瑞典寒冷的天气以及不习惯于很早就需要起来拜见克里斯蒂娜，笛卡尔次年与培根一样死于肺炎。十四年后他的著作与哥白尼和伽利略的作品一样被列入罗马天主教禁书目录。 笛卡尔有关科学方法的著作受到哲学家的高度关注，但是我不认为它们对从事科学研究（或甚至如前面所说，对笛卡尔自己最成功的科学研究）有多少正面作用。他的著作确有一个负面效果：它们推迟了法国人对牛顿物理的接受。《方法论》阐述的应用纯粹理性得出科学原理的方法从来没有有效，也不会有效。惠更斯年轻时认为自己追随笛卡尔，但是他渐渐认识到科学原理只是假想，需要通过对比其结果与观测来测试。 另外笛卡尔光学方面的研究表明他也理解这种科学假设有时是必须的。劳伦斯·劳丹在笛卡尔《哲学原理》中讨论化学时也发现同样的理解。这不禁让人疑问：是否有科学家真正从笛卡尔那里学到用实验测试假想的方法，劳丹认为波义耳是其中之一。我自己认为在笛卡尔之前人们早就理解了应用假想。否则如何理解伽利略应用匀加速落体假设得出抛体沿抛物线运行的结论以及用实验来测试？ 笛卡尔传记作者理查德·沃森说“如果没有笛卡尔的方法来分析物质的最基本组成，我们不可能研发出原子弹。十七世纪现代科学兴起，十八世纪启蒙，十九世纪工业革命，你二十世纪的计算机，以及二十世纪对大脑的破译--都是笛卡尔式的。”笛卡尔确实对数学做出了重大贡献，但是把任何这些进展归功于笛卡尔科学方法方面的著作都十分荒唐。 笛卡尔和培根是几个世纪以来一直试图为科学研究制定规则的其中两位哲学家。这从来不会有效。我们不是从任何从事科学的规则中学到科学方法，而是从科学研究经验中获得方法，其驱动力是当我们的方法成功解释某些事物后得到的快乐。

By the end of the sixteenth century the Aristotelian model for scientific investigation had been severely challenged. It was natural then to seek a new approach to the method for gathering reliable knowledge about nature. The two figures who became best known for attempts to formulate a new method for science are Francis Bacon and René Descartes. They are, in my opinion, the two individuals whose importance in the scientific revolution is most overrated. Francis Bacon was born in 1561, the son of Nicholas Bacon, Lord Keeper of the Privy Seal of England. After an education at Trinity College, Cambridge, he was called to the bar, and followed a career in law, diplomacy, and politics. He rose to become Baron Verulam and lord chancellor of England in 1618, and later Viscount St. Albans, but in 1621 he was found guilty of corruption and declared by Parliament to be unfit for public office. Bacon’s reputation in the history of science is largely based on his book Novum Organum (New Instrument, or True Directions Concerning the Interpretation of Nature), published in 1620. In this book Bacon, neither a scientist nor a mathematician, expressed an extreme empiricist view of science, rejecting not only Aristotle but also Ptolemy and Copernicus. Discoveries were to emerge directly from careful, unprejudiced observation of nature, not by deduction from first principles. He also disparaged any research that did not serve an immediate practical purpose. In The New Atlantis, he imagined a cooperative research institute, “Solomon’s House,” whose members would devote themselves to collecting useful facts about nature. In this way, man would supposedly regain the dominance over nature that was lost after the expulsion from Eden. Bacon died in 1626. There is a story that, true to his empirical principles, he succumbed to pneumonia after an experimental study of the freezing of meat. Bacon and Plato stand at opposite extremes. Of course, both extremes were wrong. Progress depends on a blend of observation or experiment, which may suggest general principles, and of deductions from these principles that can be tested against new observations or experiments. The search for knowledge of practical value can serve as a corrective to uncontrolled speculation, but explaining the world has value in itself, whether or not it leads directly to anything useful. Scientists in the seventeenth and eighteenth centuries would invoke Bacon as a counterweight to Plato and Aristotle, somewhat as an American politician might invoke Jefferson without ever having been influenced by anything Jefferson said or did. It is not clear to me that anyone’s scientific work was actually changed for the better by Bacon’s writing. Galileo did not need Bacon to tell him to do experiments, and neither I think did Boyle or Newton. A century before Galileo, another Florentine, Leonardo da Vinci, was doing experiments on falling bodies, flowing liquids, and much else.1 We know about this work only from a pair of treatises on painting and on fluid motion that were compiled after his death, and from notebooks that have been discovered from time to time since then, but if Leonardo’s experiments had no influence on the advance of science, at least they show that experiment was in the air long before Bacon. René Descartes was an altogether more noteworthy figure than Bacon. Born in 1596 into the juridical nobility of France, the noblesse de robe, he was educated at the Jesuit college of La Flèche, studied law at the University of Poitiers, and served in the army of Maurice of Nassau in the Dutch war of independence. In 1619 Descartes decided to devote himself to philosophy and mathematics, work that began in earnest after 1628, when he settled permanently in Holland. Descartes put his views about mechanics into Le Monde, written in the early 1630s but not published until 1664, after his death. In 1637 he published a philosophical work, Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences (Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth in the Sciences). His ideas were further developed in his longest work, the Principles of Philosophy, published in Latin in 1644 and then in a French translation in 1647. In these works Descartes expresses skepticism about knowledge derived from authority or the senses. For Descartes the only certain fact is that he exists, deduced from the observation that he is thinking about it. He goes on to conclude that the world exists, because he perceives it without exerting an effort of will. He rejects Aristotelian teleology—things are as they are, not because of any purpose they might serve. He gives several arguments (all unconvincing) for the existence of God, but rejects the authority of organized religion. He also rejects occult forces at a distance—things interact with each other through direct pulling and pushing. Descartes was a leader in bringing mathematics into physics, but like Plato he was too much impressed by the certainty of mathematical reasoning. In Part I of the Principles of Philosophy, titled “On the Principles of Human Knowledge,” Descartes described how fundamental scientific principles could be deduced with certainty by pure thought. We can trust in the “natural enlightenment or the faculty of knowledge given to us by God” because “it would be completely contradictory for Him to deceive us.”2 It is odd that Descartes thought that a God who allowed earthquakes and plagues would not allow a philosopher to be deceived. Descartes did accept that the application of fundamental physical principles to specific systems might involve uncertainty and call for experimentation, if one did not know all the details of what the system contains. In his discussion of astronomy in Part III of Principles of Philosophy, he considers various hypotheses about the nature of the planetary system, and cites Galileo’s observations of the phases of Venus as reason for preferring the hypotheses of Copernicus and Tycho to that of Ptolemy. This brief summary barely touches on Descartes’ views. His philosophy was and is much admired, especially in France and among specialists in philosophy. I find this puzzling. For someone who claimed to have found the true method for seeking reliable knowledge, it is remarkable how wrong Descartes was about so many aspects of nature. He was wrong in saying that the Earth is prolate (that is, that the distance through the Earth is greater from pole to pole than through the equatorial plane). He, like Aristotle, was wrong in saying that a vacuum is impossible. He was wrong in saying that light is transmitted instantaneously.* He was wrong in saying that space is filled with material vortices that carry the planets around in their paths. He was wrong in saying that the pineal gland is the seat of a soul responsible for human consciousness. He was wrong about what quantity is conserved in collisions. He was wrong in saying that the speed of a freely falling body is proportional to the distance fallen. Finally, on the basis of observation of several lovable pet cats, I am convinced that Descartes was also wrong in saying that animals are machines without true consciousness. Voltaire had similar reservations about Descartes:3

He erred on the nature of the soul, on the proofs of the existence of God, on the subject of matter, on the laws of motion, on the nature of light. He admitted innate ideas, he invented new elements, he created a world, he made man according to his own fashion—in fact, it is rightly said that man according to Descartes is Descartes’ man, far removed from man as he actually is.

Descartes’ scientific misjudgments would not matter in assessing the work of someone who wrote about ethical or political philosophy, or even metaphysics; but because Descartes wrote about “the method of rightly conducting one’s reason and of seeking truth in the sciences,” his repeated failure to get things right must cast a shadow on his philosophical judgment. Deduction simply cannot carry the weight that Descartes placed on it. Even the greatest scientists make mistakes. We have seen how Galileo was wrong about the tides and comets, and we will see how Newton was wrong about diffraction. For all his mistakes, Descartes, unlike Bacon, did make significant contributions to science. These were published as a supplement to the Discourse on Method, under three headings: geometry, optics, and meteorology.4 These, rather than his philosophical writings, in my view represent his positive contributions to science. Descartes’ greatest contribution was the invention of a new mathematical method, now known as analytic geometry, in which curves or surfaces are represented by equations that are satisfied by the coordinates of points on the curve or surface. “Coordinates” in general can be any numbers that give the location of a point, such as longitude, latitude, and altitude, but the particular kind known as “Cartesian coordinates” are the distances of the point from a center along a set of fixed perpendicular directions. For instance, in analytic geometry a circle of radius R is a curve on which the coordinates x and y are distances measured from the center of the circle along any two perpendicular directions, and satisfy the equation x2 + y2 = R2. (Technical Note 18 gives a similar description of an ellipse.) This very important use of letters of the alphabet to represent unknown distances or other numbers originated in the sixteenth century with the French mathematician, courtier, and cryptanalyst François Viète, but Viète still wrote out equations in words. The modern formalism of algebra and its application to analytic geometry are due to Descartes. Using analytic geometry, we can find the coordinates of the point where two curves intersect, or the equation for the curve where two surfaces intersect, by solving the pair of equations that define the curves or the surfaces. Most physicists today solve geometric problems in this way, using analytic geometry, rather than the classic methods of Euclid. In physics Descartes’ significant contributions were in the study of light. First, in his Optics, Descartes presented the relation between the angles of incidence and refraction when light passes from medium A to medium B (for example, from air to water): if the angle between the incident ray and the perpendicular to the surface separating the media is i, and the angle between the refracted ray and this perpendicular is r, then the sine* of i divided by the sine of r is an angle-independent constant n:

sine of i / sine of r = n

In the common case where medium A is the air (or, strictly speaking, empty space), n is the constant known as the “index of refraction” of medium B. For instance, if A is air and B is water then n is the index of refraction of water, about 1.33. In any case like this, where n is larger than 1, the angle of refraction r is smaller than the angle of incidence i, and the ray of light entering the denser medium is bent toward the direction perpendicular to the surface. Unknown to Descartes, this relation had already been obtained empirically in 1621 by the Dane Willebrord Snell and even earlier by the Englishman Thomas Harriot; and a figure in a manuscript by the tenth-century Arab physicist Ibn Sahl suggests that it was also known to him. But Descartes was the first to publish it. Today the relation is usually known as Snell’s law, except in France, where it is more commonly attributed to Descartes. Descartes’ derivation of the law of refraction is difficult to follow, in part because neither in his account of the derivation nor in the statement of the result did Descartes make use of the trigonometric concept of the sine of an angle. Instead, he wrote in purely geometric terms, though as we have seen the sine had been introduced from India almost seven centuries earlier by al-Battani, whose work was well known in medieval Europe. Descartes’ derivation is based on an analogy with what Descartes imagined would happen when a tennis ball is hit through a thin fabric; the ball will lose some speed, but the fabric can have no effect on the component of the ball’s velocity along the fabric. This assumption leads (as shown in Technical Note 27) to the result cited above: the ratio of the sines of the angles that the tennis ball makes with the perpendicular to the screen before and after it hits the screen is an angle- independent constant n. Though it is hard to see this result in Descartes’ discussion, he must have understood this result, because with a suitable value for n he gets more or less the right numerical answers in his theory of the rainbow, discussed below. There are two things clearly wrong with Descartes’ derivation. Obviously, light is not a tennis ball, and the surface separating air and water or glass is not a thin fabric, so this is an analogy of dubious relevance, especially for Descartes, who thought that light, unlike tennis balls, always travels at infinite speed.5 In addition, Descartes’ analogy also leads to a wrong value for n. For tennis balls (as shown in Technical Note 27) his assumption implies that n equals the ratio of the speed of the ball vB in medium B after it passes through the screen to its speed vA in medium A before it hits the screen. Of course, the ball would be slowed by passing through the screen, so vB would be less than vA and their ratio n would be less than 1. If this applied to light, it would mean that the angle between the refracted ray and the perpendicular to the surface would be greater than the angle between the incident ray and this perpendicular. Descartes knew this, and even supplied a diagram showing the path of the tennis ball being bent away from the perpendicular. Descartes also knew that this is wrong for light, for as had been observed at least since the time of Ptolemy, a ray of light entering water from the air is bent toward the perpendicular to the water’s surface, so that the sine of i is greater than the sine of r, and hence n is greater than 1. In a thoroughly muddled discussion that I cannot understand, Descartes somehow argues that light travels more easily in water than in air, so that for light n is greater than 1. For Descartes’ purposes his failure to explain the value of n didn’t really matter, because he could and did take the value of n from experiment (perhaps from the data in Ptolemy’s Optics), which of course gives n greater than 1. A more convincing derivation of the law of refraction was given by the mathematician Pierre de Fermat (1601–1665), along the lines of the derivation by Hero of Alexandria of the equal-angles rule governing reflection, but now making the assumption that light rays take the path of least time, rather than of least distance. This assumption (as shown in Technical Note 28) leads to the correct formula, that n is the ratio of the speed of light in medium A to its speed in medium B, and is therefore greater than 1 when A is air and B is glass or water. Descartes could never have derived this formula for n, because for him light traveled instantaneously. (As we will see in Chapter 14, yet another derivation of the correct result was given by Christiaan Huygens, a derivation based on Huygens’ theory of light as a traveling disturbance, which did not rely on Fermat’s a priori assumption that the light ray travels the path of least time.) Descartes made a brilliant application of the law of refraction: in his Meteorology he used his relation between angles of incidence and refraction to explain the rainbow. This was Descartes at his best as a scientist. Aristotle had argued that the colors of the rainbow are produced when light is reflected by small particles of water suspended in the air.6 Also, as we have seen in Chapters 9 and 10, in the Middle Ages both al-Farisi and Dietrich of Freiburg had recognized that rainbows are due to the refraction of rays of light when they enter and leave drops of rain suspended in the air. But no one before Descartes had presented a detailed quantitative description of how this works. Descartes first performed an experiment, using a thin-walled spherical glass globe filled with water as a model of a raindrop. He observed that when rays of sunlight were allowed to enter the globe along various directions, the light that emerged at an angle of about 42° to the incident direction was “completely red, and incomparably more brilliant than the rest.” He concluded that a rainbow (or at least its red edge) traces the arc in the sky for which the angle between the line of sight to the rainbow and the direction from the rainbow to the sun is about 42°. Descartes assumed that the light rays are bent by refraction when entering a drop, are reflected from the back surface of the drop, and then are bent again by refraction when emerging from the drop back into the air. But what explains this property of raindrops, of preferentially sending light back at an angle of about 42° to the incident direction? To answer this, Descartes considered rays of light that enter a spherical drop of water along 10 different parallel lines. He labeled these rays by what is today called their impact parameter b, the closest distance to the center of the drop that the ray would reach if it went straight through the drop without being refracted. The first ray was chosen so that if not refracted it would pass the center of the drop at a distance equal to 10 percent of the drop’s radius R (that is, with b = 0.1 R), while the tenth ray was chosen to graze the drop’s surface (so that b = R), and the intermediate rays were taken to be equally spaced between these two. Descartes worked out the path of each ray as it was refracted entering the drop, reflected by the back surface of the drop, and then refracted again as it left the drop, using the equal-angles law of reflection of Euclid and Hero, and his own law of refraction, and taking the index of refraction n of water to be 4/3. The following table gives values found by Descartes for the angle φ (phi) between the emerging ray and its incident direction for each ray, along with the results of my own calculation using the same index of refraction: b/R φ (Descartes) φ (recalculated) 0.1 5° 40' 5° 44' 0.2 11° 19' 11° 20' 0.3 17° 56' 17° 6' 0.4 22° 30' 22° 41' 0.5 27° 52' 28° 6' 0.6 32° 56' 33° 14' 0.7 37° 26' 37° 49' 0.8 40° 44' 41° 13' 0.9 40° 57' 41° 30' 1.0 13° 40' 14° 22'

The inaccuracy of some of Descartes’ results can be set down to the limited mathematical aids available in his time. I don’t know if he had access to a table of sines, and he certainly had nothing like a modern pocket calculator. Still, Descartes would have shown better judgment if he had quoted results only to the nearest 10 minutes of arc, rather than to the nearest minute. As Descartes noticed, there is a relatively wide range of values of the impact parameter b for which the angle φ is close to 40°. He then repeated the calculation for 18 more closely spaced rays with values of b between 80 percent and 100 percent of the drop’s radius, where φ is around 40°. He found that the angle φ for 14 of these 18 rays was between 40° and a maximum of 41° 30'. So these theoretical calculations explained his experimental observation mentioned earlier, of a preferred angle of roughly 42°. Technical Note 29 gives a modern version of Descartes’ calculation. Instead of working out the numerical value of the angle φ between the incoming and outgoing rays for each ray in an ensemble of rays, as Descartes did, we derive a simple formula that gives φ for any ray, with any impact parameter b, and for any value of the ratio n of the speed of light in air to the speed of light in water. This formula is then used to find the value of φ where the emerging rays are concentrated.* For n equal to 4/3 the favored value of φ, where the emerging light is somewhat concentrated, turns out to be 42.0°, just as found by Descartes. Descartes even calculated the corresponding angle for the secondary rainbow, produced by light that is reflected twice within a raindrop before it emerges. Descartes saw a connection between the separation of colors that is characteristic of the rainbow and the colors exhibited by refraction of light in a prism, but he was unable to deal with either quantitatively, because he did not know that the white light of the sun is composed of light of all colors, or that the index of refraction of light depends slightly on its color. In fact, while Descartes had taken the index for water to be 4/3 = 1.333 . . . , it is actually closer to 1.330 for typical wavelengths of red light and closer to 1.343 for blue light. One finds (using the general formula derived in Technical Note 29) that the maximum value for the angle φ between the incident and emerging rays is 42.8° for red light and 40.7° for blue light. This is why Descartes saw bright red light when he looked at his globe of water at an angle of 42° to the direction of the Sun’s rays. That value of the angle φ is above the maximum value 40.7° of the angle that can emerge from the globe of water for blue light, so no light from the blue end of the spectrum could reach Descartes; but it is just below the maximum value 42.8° of φ for red light, so (as explained in the previous footnote) this would make the red light particularly bright. The work of Descartes on optics was very much in the mode of modern physics. Descartes made a wild guess that light crossing the boundary between two media behaves like a tennis ball penetrating a thin screen, and used it to derive a relation between the angles of incidence and refraction that (with a suitable choice of the index of refraction n) agreed with observation. Next, using a globe filled with water as a model of a raindrop, he made observations that suggested a possible origin of rainbows, and he then showed mathematically that these observations followed from his theory of refraction. He didn’t understand the colors of the rainbow, so he sidestepped the issue, and published what he did understand. This is just about what a physicist would do today, but aside from its application of mathematics to physics, what does it have to do with Descartes’ Discourse on Method? I can’t see any sign that he was following his own prescriptions for “Rightly Conducting One’s Reason and of Seeking Truth in the Sciences.” I should add that in his Principles of Philosophy Descartes offered a significant qualitative improvement to Buridan’s notion of impetus.7 He argued that “all movement is, of itself, along straight lines,” so that (contrary to both Aristotle and Galileo) a force is required to keep planetary bodies in their curved orbits. But Descartes made no attempt at a calculation of this force. As we will see in Chapter 14, it remained for Huygens to calculate the force required to keep a body moving at a given speed on a circle of given radius, and for Newton to explain this force, as the force of gravitation. In 1649 Descartes traveled to Stockholm to serve as a teacher of the reigning Queen Christina. Perhaps as a result of the cold Swedish weather, and having to get up to meet Christina at an unwontedly early hour, Descartes in the next year, like Bacon, died of pneumonia. Fourteen years later his works joined those of Copernicus and Galileo on the Index of books forbidden to Roman Catholics. The writings of Descartes on scientific method have attracted much attention among philosophers, but I don’t think they have had much positive influence on the practice of scientific research (or even, as argued above, on Descartes’ own most successful scientific work). His writings did have one negative effect: they delayed the reception of Newtonian physics in France. The program set out in the Discourse on Method, of deriving scientific principles by pure reason, never worked, and never could have worked. Huygens when young considered himself a follower of Descartes, but he came to understand that scientific principles were only hypotheses, to be tested by comparing their consequences with observation.8 On the other hand, Descartes’ work on optics shows that he too understood that this sort of scientific hypothesis is sometimes necessary. Laurens Laudan has found evidence for the same understanding in Descartes’ discussion of chemistry in the Principles of Philosophy.9 This raises the question whether any scientists actually learned from Descartes the practice of making hypotheses to be tested experimentally, as Laudan thought was true of Boyle. My own view is that this hypothetical practice was widely understood before Descartes. How else would one describe what Galileo did, in using the hypothesis that falling bodies are uniformly accelerated to derive the consequence that projectiles follow parabolic paths, and then testing it experimentally? According to the biography of Descartes by Richard Watson,10 “Without the Cartesian method of analyzing material things into their primary elements, we would never have developed the atom bomb. The seventeenth-century rise of Modern Science, the eighteenth-century Enlightenment, the nineteenth- century Industrial Revolution, your twentieth-century personal computer, and the twentieth-century deciphering of the brain—all Cartesian.” Descartes did make a great contribution to mathematics, but it is absurd to suppose that it is Descartes’ writing on scientific method that has brought about any of these happy advances. Descartes and Bacon are only two of the philosophers who over the centuries have tried to prescribe rules for scientific research. It never works. We learn how to do science, not by making rules about how to do science, but from the experience of doing science, driven by desire for the pleasure we get when our methods succeed in explaining something.