（奎因《短文集：一本不拘泥于分类学标准的哲学词典》，译者：翟玉章）
【未被完全指定的数x2或x+5是一回事，函数λx(x2)或λx(x+5)又是另一回事。函数是一种运算，这种运算加之于一个数（或其他对象）时，就会产生一个数（或其他对象）。函数是一个关系，是函数值对自变值之间的关系。于是，λx(x2)是25对5、4对2，以及一般地，x2对x的关系。一位函数和多位函数。】
The word is awkward in its ambiguity. We often need it in its mathematical sense even in other than mathematical contexts, and we often need it in its other sense of 'use' or 'purpose' even in fairly mathematical contexts; and thus we are put now and again to groping for paraphrases. I shall be concerned with the mathematical sense.
“function”这个词的歧义性会让人犯难。即使在非数学的语境中，我们也经常需要在“函数”这一数学意义上使用这个词；即使在相当数学化的语境中，我们也经常需要在“作用”或“功能”的意义上使用这个词。因此我们时不时地需要找出它的准确含义。我这里只讨论这个词的数学意义。
The impatient mathematician tells us that x2, x+5, etc. are functions of x, and that this means that for each value of xthere is a unique value of x2, x+5, etc. Wrong; x2, x+5, etc. are not functions but numbers, if anything; they would be numbers if we were told what number xis supposed to be. Well, they are variable numbers, varying with x. Wrong, there are no variable numbers. Well, they are expressions. Wrong again; ‘x2’and ‘x+5’ are expressions. See USE VERSUS MENTION.
急性子的数学家告诉我们，x2, x+5等是x的函数，这意味着对于x的每个值，x2, x+5等都有一个唯一的值。错；x2, x+5等如果是什么东西的话，那也是数，而不是函数；如果我们被告知x是哪个数，那么x2, x+5等也成了数。哦，它们是随x而变的变数。错，变数是不存在的。哦，它们是表达式。又错了，‘x2’和‘x+5’才是表达式。见“使用与提及”。
Frege was clear on the matter, all unheeded, in 1879. The incompletely specified number x2or x+5, is one thing; the function λx(x2) or λx(x+5) is another. (The lambda notation is Alonzo Church's modification of Frege's.) The function is an operator or operation which, applied to a number (or other object), yields a number (or other object). Thus λx(x2)applied to 5, yields 25. Applied to any number x it yields x2. This much by way of clarification could have allayed the bewilderment of countless bright students at their introduction to the differential calculus down the decades.
弗雷格早在1879年就把这件事弄清楚了，只是没有引起注意。未被完全指定的数x2或x+5是一回事，函数λx(x2)或λx(x+5)又是另一回事。（λ-记法是阿朗佐·丘奇对弗雷格记法的改进。）函数是一种运算子或运算。这种运算加之于一个数（或其他对象）时，就会产生一个数（或其他对象）。因此，将λx(x2)加之于5就得到25，加之于任意数x就得到x2。几十年来，无数聪明的学生在刚学微分时，都会感到困惑，而上面的解释应该能起到解惑的作用。
So a function is an operator, or operation. Intent on further clarity, one may still reasonably ask what sort of thing that is. Giuseppe Peano recorded the inevitable answer only in 1911, but it is one to which I think Frege might have responded “Natürlich!” already in 1879. The function, Peano explained, is a relation. It is the relation of its values to its arguments, to revert to mathematical jargon. The "values" are the results of applying the function; the "arguments" are the things to which it is applied; and the function is the relation of the former to the latter. So λx(x2) is the relation that 25 bears to 5, and 4 to 2, and, in general, x2 tox. (But see MATHEMATOSIS.)
所以函数是一个运算子，或运算。为着进一步的澄清，人们可以合理地问：函数到底是怎样一类事物呢？直到1911年，朱塞佩·皮亚诺 (Giuseppe Peano)才正式地提出不可避免的答案；我想，如果弗雷格于1879年提前看到这个答案，一定会说“那是理所当然的”。皮亚诺解释说，函数是一个关系。用数学术语来说，它是函数值对自变值之间的关系。函数值是应用函数所得到的事物，自变值则是函数应用于其上的事物；函数就是前者对后者的关系。于是，λx(x2)是25对5、4对2，以及一般地，x2对x的关系。（但请参阅“数学病”。）
And what is a relation? It is a class of ordered pairs; see COMPLEX NUMBERS.
那么，关系又是什么？关系是有序对的类；请参阅“复数”。
In λx(x2) and λx(x+5) we have one-place functions, or, as mathematicians have long since put it, functions of one variable. Addition is a two-place function, λxy(x+y); multiplication is another, λxy(xy). Just as one-place functions are dyadic relations, or classes of ordered pairs, so two-place functions can be construed as triadic relations, or classes of ordered triples; and so on up to three-place functions and higher. An ordered triple <x, y, z>can be explained as a pair <x, <y, z>>, a quadruple as <x, y, <z, u>>,and so on.
λx(x2)或λx(x+5)是一位函数，也就是数学家长久以来所说的含有一个自变量的函数。加法是二位函数：λxy(x+y)；乘法则是另一个二位函数：λxy(xy)。正如一位函数是二元关系或有序对的类一样，二位函数可以被解释为三元关系或有序三元组的类；三位函数或更高位的函数的解释，可以依此类推。有序三元组<x, y, z>可以被解释为有序对<x, <y, z>>，有序四元组<x, y, z, u>可以解释为有序三元组<x, y, <z, u>>，以此类推。
But Frege had a neat alternative way of accommodating many-place functions, and it has fitted nicely into some offbeat logics (combinatory logic, term-functor logic). He construed a two-place function f as a one-place function whose values are one-place functions in turn. He explained λxy(x+y) as λx(λy(x+y)): the function which, when applied to a number x, yields as value the function which, when applied to a numbery, yields the number x+y. Thus f(x, y) is explained as (f(x))(y). Similarly for three places and more.
但弗雷格另有一种解释多位函数的方法，这种方法很好地适应了一些另类逻辑（组合逻辑，词项函子逻辑）。他把二位函数解释为一位函数，只是这个一位函数的值也是一位函数。他把加法函数λxy(x+y)解释为λx(λy(x+y))：这个函数应用于某个数x时，所产生的值是一个函数【λy(x+y)】，而当这后一个函数应用于某个值y时，就会产生数x+y。一般地，二位函数f(x, y)被解释为(f(x))(y)。对三位和多位函数也可做出类似的解释。
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