Definition. 对偶数 [dual-number]
Definition. 对偶数 [dual-number]
$\gdef\R{\mathbf{R}}$ $\gdef\C{\mathbf{C}}$
一种通俗的讲法是, 认为对偶数 $R[x]/(x^2)$ 可以作为复数 $\R[x]/(x^2+1)$ 的类比. 如果有读者能轻率地暂时忽略 $R[x]/(x^2)$ 这种结构的存在性, 或者说接受了下面的 $d$ 的定义
$$ \exists ~ d \in R\smallsetminus\{0\}, ~ d^2 = 0 $$
不能接受也很正常, 因为这轻易就会导致某个经典逻辑中的 矛盾. 为此我们需要下面的准备工作.
Axiom is incompatible with the law of excluded middle. Either the one or the other has to leave the scene. In Part I of this book, the law of excluded middle has to leave, being incompatible with the natural synthetic reasoning on smooth geometry to be presented here. In the terms which the logicians use, this means that the logic employed is ‘constructive’ or ‘intuitionistic’. We prefer to think of it just as ‘that reasoning which can be carried out in all sufficiently good cartesian closed categories’.
— Anders Kock, Synthetic Differential Geometry
无论是单纯接受这个 公理 还是认可该公理存在的舞台 $\mathcal{E}$, 其实都会导致完全相同的结果, 那就是 $\mathcal{E}$ 当中一般函数的性质发生了变化, 让所有的函数都变得光滑. 与之相对的, 这样的好性质所要求的代价是, $\mathcal{E}$ 当中不能使用经典逻辑中的选择公理、排中律、反证法等命题.