论文标题
一位供宣言：我们需要重新制定数学基础吗？
A Finitist's Manifesto: Do we need to Reformulate the Foundations of Mathematics?
论文作者
论文摘要
经典数学的基础存在问题，即使是计算机科学的基础，数学家也忽略了数学家。这篇文章是为了让他们在无限制的数学天堂中睡觉的数学家呼吁。大部分数学依赖于（i）包含无限数量元素的对象的“存在”，（ii）我们的能力“理论上”，以任意的精度计算，或者（iii）我们的能力，“理论上”，以计算任意大量时间步骤的数量。所有的微积分都取决于极限的概念。真实和复杂分析的巨大结果取决于真实数字的“连续性”的无缝概念，这些概念在飞机上扩展到复数，并为我们提供了“严格的”连续性定义，衍生性，不同的积分，各种不同的积分，以及caltere的基本和以前的衍生作用，并且是衍生的，并且是衍生的。说$ \ mathbb {c} $上的每个多项式都有一个复杂的根。这篇文章是对是否有任何方法可以将意义分配给上述（i）至（iii）的“存在”和“在理论上”的概念。
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been sleep-walking in their infinitary mathematical paradise to take heed. Much of mathematics relies upon either (i) the "existence'" of objects that contain an infinite number of elements, (ii) our ability, "in theory", to compute with an arbitrary level of precision, or (iii) our ability, "in theory", to compute for an arbitrarily large number of time steps. All of calculus relies on the notion of a limit. The monumental results of real and complex analysis rely on a seamless notion of the "continuum" of real numbers, which extends in the plane to the complex numbers and gives us, among other things, "rigorous" definitions of continuity, the derivative, various different integrals, as well as the fundamental theorems of calculus and of algebra -- the former of which says that the derivative and integral can be viewed as inverse operations, and the latter of which says that every polynomial over $\mathbb{C}$ has a complex root. This essay is an inquiry into whether there is any way to assign meaning to the notions of "existence" and "in theory'" in (i) to (iii) above.
