哥德爾的不完備性定理與心靈是否為機器的論爭

Abstract

哥德爾的不完備性定理是現代邏輯發展過程中所發現的最重要的獨立性結果。在晚近的文獻中常可看到有些學者試圖利用不完備性定理來證明不可能有能夠完整地模擬人類心靈的機器存在，持此立場之較有代表性的學者有盧卡斯 (Lucas)、潘若斯 (Penrose) 以及麥扣 (McCall) 等。這些學者個別的主張雖不盡相同，但都認為: 對任何機器而言，根據不完備性定理，都會有一些數學命題是它所無法證明的，但人類心靈卻可得知其為真。而持反對立場的學者，如弗蘭森 (Franzén)、林德斯仲 (Lindström)、夏皮洛 (Shapiro) 及蓋夫曼 (Gaifman) 等人則分別指出：上述學者的論證不足以保證所宣稱的心靈相對於機器的優勢。在本文中，筆者將審視關於這個議題的主要的正反論證，並釐清牽涉於論爭的幾個重要概念，如「機器」、「證明」以及「一致」等。筆者將指出：限制在不完備性定理的脈絡下，心靈是否為機器這個問題不可能有數學上或邏輯上的明確答案，然而就哲學的觀點而言，機器論者必須承擔較大的舉證責任。

Gödel’s incompleteness theorems are the most important independent results ever discovered so far in the continuing development of modern logic. Recently in the literature several attempts have been made to use the incompleteness theorems to argue that no machine can fully capture what the human mind can do: for the incompleteness theorems seem to imply that, for any machine, there are always some mathematical propositions which the human mind can know to be true but which machine cannot prove. I shall look into the major arguments, both pro and con, that can be found in the literature and clarify some relevant concepts such as “machine”, “proof” and “consistency”. Then I shall argue that in the vein of the incompleteness theorems there is no answer with mathematical certitude to the mind-machine debate and that much of the philosophical burden of proof will lie with mechanists in this debate.