Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped with the rich and original structure of a non-commutative ordered ring, and that Bolzano’s views on the mathematical infinite are, after all, consistent.

Bernard Bolzano（1781-1848）通常被认为试图发展一种无限集合的大小理论，该理论遵循所谓的部分-整体原则，根据该原则，整体总是大于其任何适当的部分。在本文中，我们对博尔扎诺成熟的无限理论进行了新颖的解释，并表明，与主流解释相反，最好将其理解为无穷和理论。我们的正式结果表明，Bolzano 的无穷和可以配备非交换有序环的丰富而原始的结构，并且 Bolzano 对数学无穷大的观点毕竟是一致的。