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.

中文翻译：

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一个有限论者的宣言：我们是否需要重新表述数学的基础？

经典数学的基础存在一个问题，甚至可能是计算机科学的基础，数学家们基本上都忽略了这个问题。这篇文章呼吁那些在他们无限的数学天堂中梦游的实践数学家要注意。许多数学依赖于（i）包含无限数量元素的对象的“存在”，（ii）我们“理论上”以任意精度水平计算的能力，或（iii）我们的能力，“理论上”，计算任意数量的时间步长。所有的微积分都依赖于极限的概念。实数和复数分析的巨大结果依赖于实数“连续统”的无缝概念，它在平面上扩展到复数，并为我们提供了连续性、导数、各种不同积分的“严格”定义，以及微积分和代数的基本定理——前者说导数和积分可以看作是逆运算，后者表示 $\mathbb{C}$ 上的每个多项式都有一个复根。本文旨在探讨是否有任何方法可以为上述（i）至（iii）中的“存在”和“理论上”的概念赋予意义。以及微积分和代数的基本定理——前者说导数和积分可以看作是逆运算，后者说 $\mathbb{C}$ 上的每个多项式都有一个复数根。本文旨在探讨是否有任何方法可以为上述（i）至（iii）中的“存在”和“理论上”的概念赋予意义。以及微积分和代数的基本定理——前者说导数和积分可以看作是逆运算，后者说 $\mathbb{C}$ 上的每个多项式都有一个复数根。本文旨在探讨是否有任何方法可以为上述（i）至（iii）中的“存在”和“理论上”的概念赋予意义。