纯粹的思维和经验(pure thought and empirical)

The axiomatic geometry of Euclid was the model for correct reasoning from at least as early as 300 BC to the mid-1800s. Here was a system of thought that started with basic definitions and axioms and then proceeded to prove theorem after theorem about geometry, all done without any empirical input. It was believed that Euclidean geometry correctly described the space that we live in. Pure thought seemingly told us about the physical world, which is a heady idea for mathematicians. But by the early 1800s, non-Euclidean geometries had been discovered, culminating in the early 1900s in the special and general theory of relativity, by which time it became clear that, since there are various types of geometry, the type of geometry that describes our universe is an empirical question. Pure thought can tell us the possibilities but does not appear able to pick out the correct one.

(For a popular account of this development by a fine mathematician and mathematical gadfly, see Kline's Mathematics and the Search for Knowledge [73].)

Euclid started with basic definitions and attempted to give definitions for his terms. Today, this is viewed as a false start. An axiomatic system starts with a collection of undefined terms and a collection of relations (axioms) among these undefined terms. We can then prove theorems based on these axioms. An axiomatic system "works" if no contradictions occur. Hyperbolic and elliptic geometries were taken seriously when it was shown that any possible contradiction in them could be translated back into a contradiction in Euclidean geometry, which no one seriously believes contains a contradiction. This will be discussed in the appropriate sections of this chapter.

欧几里得的公理几何学从公元前300年左右开始，一直到19世纪中期，都是正确推理的典范。它是一个从基本定义和公理出发，逐步证明几何学定理的思想体系，且完全不依赖任何经验输入。人们相信欧几里得几何学准确地描述了我们生活的空间，似乎纯粹的思维就能告诉我们有关物理世界的真理。对于数学家来说，这是一个非常激动人心的想法。然而，到了19世纪初，非欧几里得几何学被发现，并在20世纪初通过狭义和广义相对论达到了顶峰。此时人们意识到，由于存在多种几何学，哪种几何学描述我们的宇宙成了一个经验问题。纯粹的思维可以告诉我们各种可能性，但似乎无法选择出正确的那一种。
欧几里得从基本定义开始，并试图为他的术语给出定义。如今，这被视为一个错误的开端。一个公理系统应该从一组未定义的术语和这些术语之间的一组关系（即公理）开始。然后，我们可以基于这些公理证明定理。如果没有出现矛盾，那么这个公理系统就被认为是"有效的"。当证明双曲几何和椭圆几何中的任何潜在矛盾都可以转化为欧几里得几何中的矛盾时，这些几何学才被认真对待，而人们普遍认为欧几里得几何学是没有矛盾的。该话题将在本章的相关部分中讨论。

Reference

All the mathematics You missed But Need to Know for Graduate School -Thomas A. Garrity

** 问题：What's the meaning of 'empirical'? And what's the antonym of this word? **

The word empirical refers to knowledge or information obtained through observation, experience, or experiments, rather than theory or pure logic. In science, an empirical approach is one based on practical evidence, such as data from experiments or real-world observations.

For example, in an empirical study, researchers collect actual data to support or refute a hypothesis, as opposed to relying on abstract reasoning alone.

The antonym of empirical is theoretical or speculative. These terms refer to knowledge based on theory, reasoning, or conjecture without direct observation or practical experimentation.