《一个有趣的悖论》A Curious Paradox/雷蒙•斯穆连(Raymond Smullyan)(美国)著/刘明星(马来西亚)译

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以下是我从马丁加德纳集会(G4G)看到的一篇短文,“有趣”与否,请君入瓮:

译文:
有两个正整数x和y,其一是另一的两倍。x和y何者较大没有交代。我现在将证明以下两个明显并不相容的命题。

命题一、x – y 的余数,若x比y大,是比y – x的余数大的,若y比x大。

命题二、两个数量其实是一样的(即x – y的余数,若x比y大,相等于 y – x的余数,若y比x大)。

命题一的证明:假设x比y大。那么x = 2y,x – y的余数则为y。因此,x -y的余数,若x比y大,则为y。现在,假设y比x大。那么x=1/2y,y – x的余数则是y – 1/2y = 1/2y。因此,y – x的余数,若y比x大,是1/2y。由于y比1/2y大,这就证明了x – y的余数,若x比y大,是比y – x的余数更大的,若y比x大。因此命题一成立。

命题二的证明:设d为x和y的差,或者以另一种相同的说法,两者中较小的。那么,明显地x – y的余数,若x比y大,是d,而y – x的余数,若y比x大,也是d。因为d = d,命题二成立!

然而,命题一、二不能两者都对!你到底相信哪一个命题?

大多数人似乎更趋向选命题二。但是,假设y是100,则x – y的余数,若x比y大,肯定是100,而y – x的余数,若y比x大,肯定是50(因为x是50)。而100难道不是肯定比50大的吗?

原文:
Consider two positive integers x and y, one of which is twice as great as the
other. We are not told whether it is x or y that is the greater of the two. I
will now prove the following two obviously incompatible propositions.

Proposition 1. The excess of x over y, if x is greater than y, is greater than the
excess of y over x, if y is greater than x.

Proposition 2. The two amounts are really the same (i.e., the excess of x over y,
if x is greater than y, is equal to the excess of y over x, if y is greater than x).

Proof of Proposition 1. Suppose x is greater than y. Then x = 2y, hence the
excess of x over y is then y. Thus the excess of x over y, if x is greater than
y, is y. Now, suppose y is greater than x. Then x = 1/2y, hence the excess
of y over x is then y – 1/2y = 1/2y. Thus the excess of y over x, if y is greater
than x, is 1/2y. Since y is greater than 1/2y, this proves that the excess of x
over y, if x is greater than y, is greater than the excess of y over x, if y is
greater than x. Thus Proposition 1 is established.

Proof of Proposition 2. Let d be the difference between x and y — or what is
the same thing, the lesser of the two. Then obviously the excess of x over y,
if x is greater than y, is d, and the excess of y over x, if y is greater than x,
is again d. Since d = d, Proposition 2 is established!

Now, Propositions 1 and 2 can’t both be true! Which of the two propositions
do you actually believe?

Most people seem to opt for Proposition 2. But look, suppose y, say, is
100. Then the excess of x over y, if x is greater than y, is certainly 100, and
the excess of y over x, if y is greater than x, is certainly 50 (since x is then
50). And isn’t 100 surely greater than 50?

编注:作者为当代美国数学家、演奏会钢琴家、逻辑学家、哲学家(道家)、魔术师。关于G4G,可参考以下链接:http://gathering4gardner.org/ABOUT.htm

摄影:周嘉惠(马来西亚)

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