作为经验对象的几何学：康德与庞加莱与爱因斯坦之间错过的争论
Geometry as an object of experience: Kant and the missed debate between Poincaré and Einstein
摘要：
庞加莱认为，几何是一个公约，不能通过实验来测试。这一立场显然被广义相对论和它的预言的成功证实所反驳;不幸的是，庞加莱没有活着捍卫他的论文。在本文中，我认为：1）与许多作者所声称的相反，非欧几里德几何并不排除康德的命题，即空间是给定{ \it a priori}的直观形式;相反，欧几里德几何是任何更一般几何的可能性的条件。2)时空作为黎曼流形的概念是描述引力场的一种极其巧妙的方法，但是，正如Utiyama在1956年所表明的，广义相对论实际上是与洛伦兹群相关联的规范理论。Utiyama的方法并不依赖于时空弯曲的假设，尽管规范理论的方程与广义相对论的方程相同。因此，按照庞加莱的说法，我们可以说，把引力场描述成黎曼流形或欧几里得空间中的规范场只是一种惯例。
英文摘要：
Poincaré held the view that geometry is a convention and cannot be tested experimentally. This position was apparently refuted by the general theory of relativity and the successful confirmation of its predictions; unfortunately, Poincaré did not live to defend his thesis. In this paper, I argue that: 1) Contrary to what many authors have claimed, non-euclidean geometries do not rule out Kant's thesis that space is a form of intuition given {\it a priori}; on the contrary, Euclidean geometry is the condition for the possibility of any more general geometry. 2) The conception of space-time as a Riemannian manifold is an extremely ingenious way to describe the gravitational field, but, as shown by Utiyama in 1956, general relativity is actually the gauge theory associated to the Lorentz group. Utiyama's approach does not rely on the assumption that space-time is curved, though the equations of the gauge theory are identical to those of general relativity. Thus, following Poincaré, it can be claimed that it is only a matter of convention to describe the gravitational field as a Riemannian manifold or as a gauge field in Euclidean space.
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