AXIOM, a postulate or assumption, accepted as true without proof. Every deductive system of reasoning must have a set of axioms as initial premises. The axioms, undefined terms, and defined terms enable one to prove theorems.

公理，一种无需证明就可接受为事实的假设或假定。每个推理的演绎系统必须有一组作为初始前提的公理。那些公理，能使人们证明定理的未定义项和已定义项。

A choice of axioms is arbitrary. One may choose many or few, but it is best to choose only a few simple ones to avoid misunderstandings and hidden contradictions. A set of axioms must be consistent—they must not be contradictory or lead to contradictory theorems. A set of axioms should also be independent; that is, no axiom should be deducible from the others.

公理的选择是任意的。人们可以选择多个或几个，但最好只选择几个简单的公理以避免误解和潜在的矛盾。一组公理必须一致—它们一定不能相互矛盾或导致矛盾的定理。一组公理也应该是独立的；就是说，任何公理不会从其它公理中推导出来。

Euclid structured his entire geometry on only 10 consistent and independent axioms. His concepts and axioms were well chosen. Many theorems in geometry could be deduced from them, and others could follow the same deductive steps.

欧几里德仅凭借10个一致和独立的公理就构建了他的整个几何学。他的概念和公理都选得恰到好处。许多几何学的定理都可以从它们中推导出来，而且其他人也可以遵循同样的演绎步骤。

In the 19^th century, mathematicians gave much thought to the significance of axioms, Euclid’s axiom on parallel lines was replaced by a different one, and logical non-Euclidean geometry was constructed. It was descriptive of space and did not conflict with man’s experience or his measurements. This choice of axioms started a revolution in thought concerning the nature of mathematics.

在19世纪，数学家们对公理的重要性给予了许多思考，欧几里德有关平行线的公理被另一个公理所取代，并建立了符合逻辑的非欧几里德几何学。它是对空间的描述，与人的经验或测量值并不冲突。这一公理的选择开启了一场有关数学本质的思想革命。

In his study of infinite classes (1871—1874), Georg Cantor abandoned the Euclidean axiom that “the whole is greater than any of its parts.” He showed that in the case of an infinite class, there may be as many members in part of the class as in the whole class. For example, there are as many even integers as there are odd and even integers.

在格奥尔格·康托尔的无穷类（1871年—1874年）研究中，他放弃了欧几里德“整体大于它的任何部分”的公理。他证明在无穷类的情况下，类的部分项与类的整体项可以一样多。例如，偶数整数与奇数和偶数整数的数量一样多。

The teaching of arithmetic now is oriented toward making structure explicit, and emphasis is placed on the conscious use of axioms. Children can easily see that bringing a set of two blocks to a set of THREE blocks (3 + 2) results in the same set of two blocks (2 + 3). This insight is generalized to the commutative axiom for addition. A similar explanation of other axioms helps give arithmetic a firm logical basis.

现在的演算教学重视使结构清楚明了，并强调有意识地使用公理。孩子们很容易明白，将一组两块积木变成一组三块积木（3 + 2），结果与两块积木（2 +3）是一样的。这一领悟被推广到加法的交换公理中。对其它公理的相同解释有助于为演算提供一种坚实的逻辑基础。

                                              LEE E. BOYER

                        Harrisburg (Pa.) Area Community College

                                              李·E. 博耶

                              哈里斯堡（宾州）地区社区学院

                                         2023年3月5日译

（译者注：该词条位列《大美百科全书》1985年版，第2卷，第381页）