到19世纪晚期,已经开发了几个一致的非欧几里德几何,在数学上与欧几里德几何不同。欧氏几何具有独特的平行公理,三角形的角度和等于180°,而球形几何具有零平行公理,三角形的角度和大于或等于180°。这些几何形状提高了物理空间可能是非欧几里德的可能性。经验主义者认为我们可以通过实验确定物理空间是否是欧几里德。例如,高斯据称试图测量三个山顶之间三角形的角度,以测试物理空间是否为欧几里得。现实主义者认为物理空间具有某种确定的几何特征,即使我们无法发现它具有什么特征。康德人认为物理空间必须是欧几里德,因为只有欧几里德几何与我们的感性形式一致。庞加莱(Poincaré,1913)认为经验主义者,现实主义者和康德主义者是错误的:物理空间的几何学在经验上不是可确定的,事实的或合成的先验。假设高斯的实验给出了三角形的角度和为180°。这将支持这样的假设:物理空间只是在关于光学与几何的协调的某些预设下的欧几里德:未受干扰的光线的最短路径是欧几里德直线。相反,例如,180°测量也可以通过预先假定光线穿过球形空间中的最短路径但受到力的干扰来适应,因此物理空间是“真正的”非欧几里德:三角形的真实角度和大于180°,但令人不安的力使它“出现”空间为欧几里德,三角形的角度和为180°。认为关于物理空间的几何形状没有事实。庞加莱提出了传统主义:我们常规地决定几何是欧几里得,力是牛顿,光是欧几里德直线,我们看看实验结果是否符合这些惯例。传统主义并不是一种“随时随地”的学说 – 并非所有规定都能容纳证据 – 它是声称测量和证据的物理意义是由传统采用的框架决定的。线和角度的测量通常依赖于光传播最短路径的假设。但除非我们决定最短路径是欧几里得还是非欧几里得,否则这缺乏物理意义。由于实验仅具有相对于它们的物理意义,因此无法通过实验反驳或确认这些惯例。我们采用哪一组约定取决于实用因素:在其他条件相同的情况下,我们选择使物理更简单,更易处理,更熟悉等的约定。例如,庞加莱认为,由于其简单性,我们永远不会放弃欧几里德几何。

新加坡理工学院Essay代写:庞加莱的传统主义

By the late 19th century several consistent non-Euclidean geometries, mathematically distinct from Euclidean geometry, had been developed. Euclidean geometry has a unique parallels axiom and angle sum of triangles equals 180º, whereas, for example, spherical geometry has a zero-parallel axiom and angle sum of triangles greater than or equal to 180º. These geometries raise the possibility that physical space could be non-Euclidean. Empiricists think we can determine whether physical space is Euclidean through experiments. For example, Gauss allegedly attempted to measure the angles of a triangle between three mountaintops to test whether physical space is Euclidean. Realists think physical space has some determinate geometrical character even if we cannot discover what character it has. Kantians think that physical space must be Euclidean because only Euclidean geometry is consistent with the form of our sensibility. Poincaré (1913) argued that empiricists, realists, and Kantians are wrong: the geometry of physical space is not empirically determinable, factual, or synthetic a priori. Suppose Gauss’s experiment gave the angle-sum of a triangle as 180º. This would support the hypothesis that physical space is Euclidean only under certain presuppositions about the coordination of optics with geometry: that the shortest path of an undisturbed light ray is a Euclidean straight line. Instead, for example, the 180º measurement could also be accommodated by presupposing that light rays traverse shortest paths in spherical space but are disturbed by a force, so that physical space is “really” non-Euclidean: the true angle-sum of the triangle is greater than 180º, but the disturbing force makes it “appear” that space is Euclidean and the angle-sum of the triangle is 180º. Arguing that there is no fact of the matter about the geometry of physical space. Poincaré proposed conventionalism: we decide conventionally that geometry is Euclidean, forces are Newtonian, light travels in Euclidean straight lines, and we see if experimental results will fit those conventions. Conventionalism is not an “anything-goes” doctrine—not all stipulations will accommodate the evidence—it is the claim that the physical meaning of measurements and evidence is determined by conventionally adopted frameworks. Measurements of lines and angles typically rely on the hypothesis that light travels shortest paths. But this lacks physical meaning unless we decide whether shortest paths are Euclidean or non-Euclidean. These conventions cannot be experimentally refuted or confirmed since experiments only have physical meaning relative to them. Which group of conventions we adopt depends on pragmatic factors: other things being equal, we choose conventions that make physics simpler, more tractable, more familiar, and so forth. Poincaré, for example, held that, because of its simplicity, we would never give up Euclidean geometry.

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