Wittgenstein's Philosophy of Mathematics
維特根斯坦的數學哲學

Ludwig Wittgenstein’s Philosophy of Mathematics is undoubtedly the most unknown and under-appreciated part of his philosophical opus. Indeed, more than half of Wittgenstein’s writings from 1929 through 1944 are devoted to mathematics, a fact that Wittgenstein himself emphasized in 1944 by writing that his “chief contribution has been in the philosophy of mathematics” (Monk 1990: 466).
路德維希·維根斯坦的數學哲學無疑是他哲學作品中最不為人知和被低估的部分。事實上，維根斯坦在 1929 年至 1944 年間的著作中，有超過一半是專注於數學，這一事實在 1944 年得到了維根斯坦自己的強調，他寫道他的「主要貢獻在於數學哲學」（Monk 1990: 466）。

The core of Wittgenstein’s conception of mathematics is very much set by the Tractatus Logico-Philosophicus (1922; hereafter Tractatus), where his main aim is to work out the language-reality connection by determining what is required for language, or language usage, to be about the world. Wittgenstein answers this question, in part, by asserting that the only genuine propositions that we can use to make assertions about reality are contingent (‘empirical’) propositions, which are true if they agree with reality and false otherwise (4.022, 4.25, 4.062, 2.222). From this it follows that all other apparent propositions are pseudo-propositions of various types and that all other uses of ‘true’ and ‘truth’ deviate markedly from the truth-by-correspondence (or agreement) that contingent propositions have in relation to reality. Thus, from the Tractatus to at least 1944, Wittgenstein maintains that “mathematical propositions” are not real propositions and that “mathematical truth” is essentially non-referential and purely syntactical in nature. On Wittgenstein’s view, we invent mathematical calculi and we expand mathematics by calculation and proof, and though we learn from a proof that a theorem can be derived from axioms by means of certain rules in a particular way, it is not the case that this proof-path pre-exists our construction of it.
維特根斯坦對數學的概念的核心在於《邏輯哲學論》（1922；以下簡稱《邏輯哲學論》），他的主要目標是通過確定語言或語言使用需要什麼來解決語言與現實之間的聯繫。維特根斯坦部分地回答了這個問題，聲稱我們可以用來對現實進行斷言的唯一真正命題是偶然的（“經驗的”）命題，這些命題如果與現實一致則為真，否則為假（4.022, 4.25, 4.062, 2.222）。由此可知，所有其他表面上的命題都是各種類型的偽命題，所有其他對“真”和“真理”的使用與偶然命題在與現實的關係中所具有的真實性（或一致性）顯著偏離。因此，從《邏輯哲學論》到至少 1944 年，維特根斯坦堅持認為“數學命題”並不是真正的命題，而“數學真理”本質上是非指稱的，並且純粹是語法性的。 在維根斯坦的觀點中，我們發明數學計算法，並通過計算和證明擴展數學，儘管我們從證明中學到一個定理可以通過某些規則以特定方式從公理推導出來，但這條證明路徑並不是在我們構建它之前就已經存在的。

As we shall see, Wittgenstein’s Philosophy of Mathematics begins in a rudimentary way in the Tractatus, develops into a finitistic constructivism in the middle period (Philosophical Remarks (1929–30) and Philosophical Grammar (1931–33), respectively; hereafter PR and PG, respectively), and is further developed in new and old directions in the manuscripts used for Remarks on the Foundations of Mathematics (1937–44; hereafter RFM). As Wittgenstein’s substantive views on mathematics evolve from 1918 through 1944, his writing and philosophical styles evolve from the assertoric, aphoristic style of the Tractatus to a clearer, argumentative style in the middle period, to a dialectical, interlocutory style in RFM and the Philosophical Investigations (hereafter PI).
正如我們將看到的，維根斯坦的數學哲學在《邏輯哲學論》中以一種初步的方式開始，並在中期（《哲學備忘錄》（1929–30）和《哲學文法》（1931–33），以下簡稱《備忘錄》和《文法》）發展為有限主義的建構主義，並在用於《數學基礎的備忘錄》（1937–44；以下簡稱《數學備忘錄》）的手稿中進一步向新舊方向發展。隨著維根斯坦對數學的實質觀點從 1918 年到 1944 年的演變，他的寫作和哲學風格也從《邏輯哲學論》的斷言性、格言式風格演變為中期的更清晰、論證性的風格，再到《數學備忘錄》和《哲學研究》（以下簡稱《研究》）中的辯證性、對話式風格。

• 1. Wittgenstein on Mathematics in the Tractatus
  1. 維特根斯坦在《邏輯哲學論》中的數學
• 2. The Middle Wittgenstein’s Finitistic Constructivism
  2. 中期維根斯坦的有限建構主義
  • 2.1 Wittgenstein’s Intermediate Constructive Formalism
    2.1 維根斯坦的中介建構形式主義
  • 2.2 Wittgenstein’s Intermediate Finitism
    2.2 維根斯坦的中介有限主義
  • 2.3 Wittgenstein’s Intermediate Finitism and Algorithmic Decidability
    2.3 維特根斯坦的中介有限主義與算法可決性
  • 2.4 Wittgenstein’s Intermediate Account of Mathematical Induction and Algorithmic Decidability
    2.4 維根斯坦的數學歸納法與算法可決性的中介說明
  • 2.5 Wittgenstein’s Intermediate Account of Irrational Numbers
    2.5 維根斯坦對非理性數的中介說明
    • 2.5.1 Wittgenstein’s Anti-Foundationalism and Genuine Irrational Numbers
      2.5.1 維特根斯坦的反基礎主義與真正的非理性數字
    • 2.5.2 Wittgenstein’s Real Number Essentialism and the Dangers of Set Theory
      2.5.2 維根斯坦的實數本質主義與集合論的危險
  • 2.6 Wittgenstein’s Intermediate Critique of Set Theory
    2.6 維特根斯坦對集合論的中介批評
    • 2.6.1 Intensions, Extensions, and the Fictitious Symbolism of Set Theory
      2.6.1 意圖、擴展與集合論的虛構符號主義
    • 2.6.2 Against Non-Denumerability
      2.6.2 反對非可數性
• 3. The Later Wittgenstein on Mathematics: Some Preliminaries
  3. 後期維根斯坦的數學：一些初步探討
  • 3.1 Mathematics as a Human Invention
    3.1 數學作為人類的發明
    • 3.1.1 Wittgenstein’s Later Anti-Platonism: The Natural History of Numbers and the Vacuity of Platonism
      3.1.1 維特根斯坦的後期反柏拉圖主義：數字的自然歷史與柏拉圖主義的空虛
  • 3.2 Wittgenstein’s Later Finitistic Constructivism
    3.2 維特根斯坦的後期有限主義建構論
  • 3.3 The Later Wittgenstein on Decidability and Algorithmic Decidability
    3.3 後期維根斯坦關於可決定性和算法可決定性
  • 3.4 Wittgenstein’s Later Critique of Set Theory: Non-Enumerability vs. Non-Denumerability
    3.4 維根斯坦對集合論的後期批評：不可列舉性與不可計數性
  • 3.5 Extra-Mathematical Application as a Necessary Condition of Mathematical Meaningfulness
    3.5 作為數學意義必要條件的額外數學應用
  • 3.6 Wittgenstein on Gödel and Undecidable Mathematical Propositions
    3.6 維特根斯坦對哥德爾和不可判定數學命題的看法
• 4. The Impact of Philosophy of Mathematics on Mathematics
  4. 數學哲學對數學的影響
• Bibliography 參考文獻
  • Wittgenstein’s Writings 維特根斯坦的著作
  • Notes on Wittgenstein’s Lectures and Recorded Conversations
    維特根斯坦的講座和錄音對話筆記
  • Secondary Sources and Relevant Primary Literature
    次級資料和相關的主要文獻
• Academic Tools 學術工具
• Other Internet Resources 其他網路資源
• Related Entries 相關條目

---

1. Wittgenstein on Mathematics in the Tractatus
1. 維特根斯坦在《邏輯哲學論》中的數學

Wittgenstein’s non-referential, formalist conception of mathematical propositions and terms begins in the Tractatus.^[1] Indeed, insofar as he sketches a rudimentary Philosophy of Mathematics in the Tractatus, he does so by contrasting mathematics and mathematical equations with genuine (contingent) propositions, sense, thought, propositional signs and their constituent names, and truth-by-correspondence.
維特根斯坦的非指稱性、形式主義數學命題和術語的概念始於邏輯哲學論。^[1] 事實上，因為他在邏輯哲學論中勾勒出一個基本的數學哲學，他是通過對比數學和數學方程與真正的（偶然的）命題、意義、思想、命題符號及其組成名稱，以及通過對應的真理來進行的。

In the Tractatus, Wittgenstein claims that a genuine proposition, which rests upon conventions, is used by us to assert that a state of affairs (i.e., an elementary or atomic fact; ‘Sachverhalt’) or fact (i.e., multiple states of affairs; ‘Tatsache’) obtain(s) in the one and only real world. An elementary proposition is isomorphic to the possible state of affairs it is used to represent: it must contain as many names as there are objects in the possible state of affairs. An elementary proposition is true iff its possible state of affairs (i.e., its ‘sense’; ‘Sinn’) obtains. Wittgenstein clearly states this Correspondence Theory of Truth at (4.25):
在《論述》中，維根斯坦聲稱，真正的命題依賴於約定，我們用它來斷言某種事態（即，基本或原子事實；‘事實狀態’）或事實（即，多個事態；‘事實’）在唯一的真實世界中存在。一個基本命題與它所用來表示的可能事態同構：它必須包含與可能事態中的物件數量相等的名稱。一個基本命題當且僅當其可能事態（即，它的‘意義’；‘意義’）存在時為真。維根斯坦在（4.25）明確闡述了這一真理的對應理論：

If an elementary proposition is true, the state of affairs exists; if an elementary proposition is false, the state of affairs does not exist.
如果一個基本命題是真，那麼狀態就存在；如果一個基本命題是假的，那麼狀態就不存在。

But propositions and their linguistic components are, in and of themselves, dead—a proposition only has sense because we human beings have endowed it with a conventional sense (5.473). Moreover, propositional signs may be used to do any number of things (e.g., insult, catch someone’s attention); in order to assert that a state of affairs obtains, a person must ‘project’ the proposition’s sense—its possible state of affairs—by ‘thinking’ of (e.g., picturing) its sense as one speaks, writes or thinks the proposition (3.11). Wittgenstein connects use, sense, correspondence, and truth by saying that “a proposition is true if we use it to say that things stand in a certain way, and they do” (4.062; italics added).
但是命題及其語言成分本身是死的——一個命題只有因為我們人類賦予它一個約定俗成的意義（5.473）。此外，命題符號可以被用來做許多事情（例如，侮辱、引起某人的注意）；為了斷言某種事態的存在，一個人必須通過“思考”（例如，想像）其意義來“投射”命題的意義——其可能的事態——在說、寫或思考命題時（3.11）。維根斯坦通過說“如果我們用它來說明事物以某種方式存在，那麼命題就是真的，而事實確實如此”（4.062；斜體字為補充）來連結使用、意義、對應和真理。

The Tractarian conceptions of genuine (contingent) propositions and the (original and) core concept of truth are used to construct theories of logical and mathematical ‘propositions’ by contrast. Stated boldly and bluntly, tautologies, contradictions and mathematical propositions (i.e., mathematical equations) are neither true nor false—we say that they are true or false, but in doing so we use the words ‘true’ and ‘false’ in very different senses from the sense in which a contingent proposition is true or false. Unlike genuine propositions, tautologies and contradictions “have no ‘subject-matter’” (6.124), “lack sense”, and “say nothing” about the world (4.461), and, analogously, mathematical equations are “pseudo-propositions” (6.2) which, when ‘true’ (‘correct’; ‘richtig’ (6.2321)), “merely mark … [the] equivalence of meaning [of ‘two expressions’]” (6.2323). Given that “[t]autology and contradiction are the limiting cases—indeed the disintegration—of the combination of signs” (4.466; italics added), where
特拉克特派對於真正（偶然）命題的概念以及（原始和）核心的真理概念被用來構建邏輯和數學“命題”的理論相對而言。直言不諱地說，重言式、矛盾和數學命題（即數學方程式）既不是真也不是假——我們說它們是真或假，但這樣做時，我們使用“真”和“假”這兩個詞的意義與偶然命題的真或假有著非常不同的含義。與真正的命題不同，重言式和矛盾“沒有‘主題’”（6.124）、“缺乏意義”，並且“對世界‘什麼也不說’”（4.461），類似地，數學方程式是“偽命題”（6.2），當它們‘真’（‘正確’；‘正確’（6.2321））時，“僅僅標記……[‘兩個表達式’]的意義等價”（6.2323）。考慮到“[t]重言式和矛盾是符號組合的極限情況——實際上是解體”（4.466；斜體字已添加），其中

the conditions of agreement with the world—the representational relations—cancel one another, so that [they] do[] not stand in any representational relation to reality,
與世界的協議條件——表徵關係——相互抵消，因此[它們]不與現實存在任何表徵關係，

tautologies and contradictions do not picture reality or possible states of affairs and possible facts (4.462). Stated differently, tautologies and contradictions do not have sense, which means we cannot use them to make assertions, which means, in turn, that they cannot be either true or false. Analogously, mathematical pseudo-propositions are equations, which indicate or show that two expressions are equivalent in meaning and therefore are intersubstitutable. Indeed, we arrive at mathematical equations by “the method of substitution”:
同義反覆和矛盾並不描繪現實或可能的情況和事實（4.462）。換句話說，同義反覆和矛盾沒有意義，這意味著我們不能用它們來做出斷言，這反過來意味著它們不能是真或假。類似地，數學偽命題是方程式，這表明或顯示兩個表達式在意義上是等價的，因此可以互相替代。事實上，我們通過“替代法”得出數學方程式：

starting from a number of equations, we advance to new equations by substituting different expressions in accordance with the equations. (6.24)
從一系列方程式開始，我們通過根據方程式替換不同的表達式來推進到新的方程式。(6.24)

We prove mathematical ‘propositions’ ‘true’ (‘correct’) by ‘seeing’ that two expressions have the same meaning, which “must be manifest in the two expressions themselves” (6.23), and by substituting one expression for another with the same meaning. Just as “one can recognize that [‘logical propositions’] are true from the symbol alone” (6.113), “the possibility of proving” mathematical propositions means that we can perceive their correctness without having to compare “what they express” with facts (6.2321; cf. RFM App. III, §4).
我們通過“看到”兩個表達式具有相同的意義來證明數學“命題”的“真” （“正確”），這“必須在這兩個表達式本身中顯現”（6.23），並通過用具有相同意義的另一個表達式替代一個表達式來進行證明。正如“人們可以僅從符號認識到[‘邏輯命題’]是真實的”（6.113），“證明”數學命題的可能性意味著我們可以在不必將“它們所表達的”與事實進行比較的情況下感知它們的正確性（6.2321；參見RFM 附錄 III, §4）。

The demarcation between contingent propositions, which can be used to correctly or incorrectly represent parts of the world, and mathematical propositions, which can be decided in a purely formal, syntactical manner, is maintained by Wittgenstein until his death in 1951 (Zettel §701, 1947; PI II, 2001 edition, pp. 192–193e, 1949). Given linguistic and symbolic conventions, the truth-value of a contingent proposition is entirely a function of how the world is, whereas the “truth-value” of a mathematical proposition is entirely a function of its constituent symbols and the formal system of which it is a part. Thus, a second, closely related way of stating this demarcation is to say that mathematical propositions are decidable by purely formal means (e.g., calculations), while contingent propositions, being about the ‘external’ world, can only be decided, if at all, by determining whether or not a particular fact obtains (i.e., something external to the proposition and the language in which it resides) (2.223; 4.05).
在 1951 年去世之前，維根斯坦維持了偶然命題與數學命題之間的區分，前者可以正確或不正確地表示世界的某些部分，而後者則可以以純粹形式、語法的方式來決定（Zettel §701, 1947; PI II, 2001 年版, 頁 192–193e, 1949）。根據語言和符號的約定，偶然命題的真值完全取決於世界的狀況，而數學命題的“真值”則完全取決於其組成符號及其所屬的形式系統。因此，表述這一區分的第二種密切相關的方式是，數學命題可以通過純粹的形式手段（例如計算）來決定，而偶然命題因為涉及“外部”世界，只有在確定某個特定事實是否存在的情況下才能被決定（即，某些外部於命題及其所處語言的事物）（2.223; 4.05）。

The Tractarian formal theory of mathematics is, specifically, a theory of formal operations. Over the past 20 years, Wittgenstein’s theory of operations has received considerable examination (Frascolla 1994, 1997; Marion 1998; Potter 2000; and Floyd 2002), which has interestingly connected it and the Tractarian equational theory of arithmetic with elements of Alonzo Church’s λ$\lambda$-calculus and with R. L. Goodstein’s equational calculus (Marion 1998: chapters 1, 2, and 4). Very briefly stated, Wittgenstein presents:
特拉克塔里安的數學形式理論，具體來說，是一種理論 of 正式運算。在過去的 20 年中，維根斯坦的運算理論受到了相當多的檢視（Frascolla 1994, 1997; Marion 1998; Potter 2000; 和 Floyd 2002），有趣的是，它將其與特拉克塔理論的算術方程理論以及阿隆佐·丘奇的 λ$\lambda$ -微積分和 R. L. 古德斯坦的方程式聯繫了起來。 微積分（Marion 1998：第 1、2 和 4 章）。簡而言之， 維特根斯坦呈現：

1. … the sign ‘[a,x,O'x$a,x,O\text{'}x$]’ for the general term of the series of forms a$a$, O'a$O\text{'}a$, O'O'a$O\text{'}O\text{'}a$…. (5.2522)
   … 符號 ‘[ a,x,O'x$a,x,O\text{'}x$ ]’ 用於系列形式 a$a$ , O'a$O\text{'}a$ , O'O'a$O\text{'}O\text{'}a$ … 的一般術語。(5.2522)
2. … the general form of an operation Ω'(η⎯⎯⎯)$\mathrm{\Omega }\text{'}(\overline{\eta })$ [as]
   … 操作的一般形式 Ω'(η⎯⎯⎯)$\mathrm{\Omega }\text{'}(\overline{\eta })$ [as]
   [ξ⎯⎯⎯,N(ξ⎯⎯⎯)]'(η⎯⎯⎯)(=[η⎯⎯⎯,ξ⎯⎯⎯,N(ξ⎯⎯⎯)]).(6.01)$$[\overline{\xi },N(\overline{\xi })]\text{'}(\overline{\eta })(=[\overline{\eta },\overline{\xi },N(\overline{\xi })]).(6.01)$$
3. … the general form of a proposition (“truth-function”) [as] [p⎯⎯⎯,ξ⎯⎯⎯,N(ξ⎯⎯⎯)]$[\overline{p},\overline{\xi },N(\overline{\xi })]$. (6)
   … 命題的一般形式 （“真值函數”）[作為] [p⎯⎯⎯,ξ⎯⎯⎯,N(ξ⎯⎯⎯)]$[\overline{p},\overline{\xi },N(\overline{\xi })]$ 。（6）
4. The general form of an integer [natural number] [as] [0,ξ,ξ+1]$[0,\xi ,\xi +1]$. (6.03)
   整數 [自然數] 的一般形式 [為] [0,ξ,ξ+1]$[0,\xi ,\xi +1]$ 。 (6.03)

adding that “[t]he concept of number is… the general form of a number” (6.022). As Frascolla (and Marion after him) have pointed out, “the general form of a proposition is a particular case of the general form of an ‘operation’” (Marion 1998: 21), and all three general forms (i.e., of operation, proposition, and natural number) are modeled on the variable presented at (5.2522) (Marion 1998: 22). Defining “[a]n operation [as] the expression of a relation between the structures of its result and of its bases” (5.22), Wittgenstein states that whereas “[a] function cannot be its own argument,… an operation can take one of its own results as its base” (5.251).
添加“[t]數的概念是……數的一般形式”（6.022）。正如 Frascolla（以及隨後的 Marion）所指出的，“命題的一般形式是‘運算’的一般形式的特例”（Marion 1998: 21），而這三種一般形式（即運算、命題和自然數）都是基於（5.2522）中呈現的變數建模的（Marion 1998: 22）。定義“[一]個運算[為]其結果與其基礎之間關係的表達”（5.22），維根斯坦指出，雖然“[一]個函數不能是它自己的參數，……一個運算可以將其自身的結果之一作為其基礎”（5.251）。

On Wittgenstein’s (5.2522) account of ‘[a,x,O'x$a,x,O\text{'}x$]’,
在維根斯坦（5.2522）對「[ a,x,O'x$a,x,O\text{'}x$ ]」的說法中，

the first term of the bracketed expression is the beginning of the series of forms, the second is the form of a term x$x$ arbitrarily selected from the series, and the third [O'x$O\text{'}x$] is the form of the term that immediately follows x$x$ in the series.
括號內表達式的第一項是的開始 一系列的形式，第二個是從該系列中任意選擇的術語 x$x$ 的形式，第三個[ O'x$O\text{'}x$ ]是該系列中緊接在 x$x$ 之後的術語形式。

Given that “[t]he concept of successive applications of an operation is equivalent to the concept ‘and so on’” (5.2523), one can see how the natural numbers can be generated by repeated iterations of the general form of a natural number, namely ‘[0,ξ,ξ+1$0,\xi ,\xi +1$]’. Similarly, truth-functional propositions can be generated, as Russell says in the Introduction to the Tractatus (p. xv), from the general form of a proposition ‘[p⎯⎯⎯$\overline{p}$, ξ⎯⎯⎯$\overline{\xi }$, N(ξ⎯⎯⎯)$N(\overline{\xi })$]’ by
考慮到「[t]連續應用的概念」 操作等同於「等等」的概念 (5.2523)，可以看到自然數是如何生成的 自然數的一般形式的重複迭代，即 ‘[ 0,ξ,ξ+1$0,\xi ,\xi +1$ ]’。同樣地，真值函數命題可以生成，正如羅素在《邏輯哲學論》的引言中所說（第 xv 頁），從命題的一般形式‘[ p⎯⎯⎯$\overline{p}$ , ξ⎯⎯⎯$\overline{\xi }$ , N(ξ⎯⎯⎯)$N(\overline{\xi })$ ]’通過

taking any selection of atomic propositions [where p$p$ “stands for all atomic propositions”; “the bar over the variable indicates that it is the representative of all its values” (5.501)], negating them all, then taking any selection of the set of propositions now obtained, together with any of the originals [where x$x$ “stands for any set of propositions”]—and so on indefinitely.
對任何原子命題的選擇進行取用[其中 p$p$ “代表所有原子命題”；“變數上方的橫線表示它是所有值的代表”（5.501）]，對它們全部取反，然後對現在獲得的命題集合進行任何選擇，連同任何原始命題[其中 x$x$ “代表任何命題集合”]——如此類推 持續無限期。

On Frascolla’s (1994: 3ff) account,
根據 Frascolla（1994：3ff）的說法，

a numerical identity “t=s$\mathbf{t}=\mathbf{s}$” is an arithmetical theorem if and only if the corresponding equation “Ωt'x=Ωs'x${\mathrm{\Omega }}^t\text{'}x={\mathrm{\Omega }}^s\text{'}x$”, which is framed in the language of the general theory of logical operations, can be proven.
數值身份 “ t=s$\mathbf{t}=\mathbf{s}$ ” 是一個算術定理當且僅當相應的方程 “ Ωt'x=Ωs'x${\mathrm{\Omega }}^t\text{'}x={\mathrm{\Omega }}^s\text{'}x$ ”, 這是框架 在邏輯運算的一般理論語言中，可以是 已證明。

By proving  通過證明

the equation “Ω2×2'x=Ω4'x${\mathrm{\Omega }}^{2\times 2}\text{'}x={\mathrm{\Omega }}^4\text{'}x$”, which translates the arithmetic identity “2×2=4$2\times 2=4$” into the operational language (6.241),
該方程式 “ Ω2×2'x=Ω4'x${\mathrm{\Omega }}^{2\times 2}\text{'}x={\mathrm{\Omega }}^4\text{'}x$ ”, 將算術恆等式 “ 2×2=4$2\times 2=4$ ” 轉換為操作語言 (6.241)，

Wittgenstein thereby outlines “a translation of numerical arithmetic into a sort of general theory of operations” (Frascolla 1998: 135).
維特根斯坦因此勾勒出「將數字算術翻譯成一種操作的一般理論」（Frascolla 1998: 135）。

Despite the fact that Wittgenstein clearly does not attempt to reduce mathematics to logic in either Russell’s manner or Frege’s manner, or to tautologies, and despite the fact that Wittgenstein criticizes Russell’s Logicism (e.g., the Theory of Types, 3.31–3.32; the Axiom of Reducibility, 6.1232, etc.) and Frege’s Logicism (6.031, 4.1272, etc.),^[2] quite a number of commentators, early and recent, have interpreted Wittgenstein’s Tractarian theory of mathematics as a variant of Logicism (Quine 1940 [1981: 55]; Benacerraf & Putnam 1964a: 14; Black 1964: 340; Savitt 1979 [1986: 34]; Frascolla 1994: 37; 1997: 354, 356–57, 361; 1998: 133; Marion 1998: 26 & 29; and Potter 2000: 164 and 182–183). There are at least four reasons proffered for this interpretation.
儘管維根斯坦顯然並不試圖像羅素或弗雷格那樣將數學簡化為邏輯，或簡化為同義反覆，並且儘管維根斯坦批評了羅素的邏輯主義（例如，類型理論，3.31–3.32；可約性公理，6.1232 等）和弗雷格的邏輯主義（6.031，4.1272 等），相當多的評論者，無論是早期還是近期，都將維根斯坦的《邏輯哲學論》中的數學理論解釋為邏輯主義的一種變體（奎因 1940 [1981: 55]；本納塞拉夫與普特南 1964a: 14；布萊克 1964: 340；薩維特 1979 [1986: 34]；弗拉斯科拉 1994: 37；1997: 354, 356–57, 361；1998: 133；馬里昂 1998: 26 & 29；以及波特 2000: 164 和 182–183）。對於這種解釋，至少有四個理由被提出。

1. Wittgenstein says that “[m]athematics is a method of logic” (6.234).
   維根斯坦說：“[數學是一種邏輯的方法]”（6.234）。
2. Wittgenstein says that “[t]he logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics” (6.22).
   維特根斯坦說：“世界的邏輯，通過邏輯的命題在同義反覆中顯示，通過數學的方程式顯示。”（6.22）
3. According to Wittgenstein, we ascertain the truth of both mathematical and logical propositions by the symbol alone (i.e., by purely formal operations), without making any (‘external’, non-symbolic) observations of states of affairs or facts in the world.
   根據維根斯坦的說法，我們僅通過符號（即純粹的形式操作）來確定數學和邏輯命題的真理，而不需要對世界中的狀態或事實進行任何（‘外部’，非符號的）觀察。
4. Wittgenstein’s iterative (inductive) “interpretation of numerals as exponents of an operation variable” is a “reduction of arithmetic to operation theory”, where “operation” is construed as a “logical operation” (italics added) (Frascolla 1994: 37), which shows that “the label ‘no-classes logicism’ tallies with the Tractatus view of arithmetic” (Frascolla 1998: 133; 1997: 354).
   維特根斯坦的迭代（歸納）“將數字解釋為運算變數的指數”是“將算術簡化為運算理論”，其中“運算”被理解為“邏輯運算”（斜體字添加）（Frascolla 1994: 37），這顯示出“標籤‘無類邏輯主義’與邏輯哲學論的算術觀相符”（Frascolla 1998: 133; 1997: 354）。

Though at least three Logicist interpretations of the Tractatus have appeared within the last 20 years, the following considerations (Rodych 1995; Wrigley 1998) indicate that none of these reasons is particularly cogent.
儘管在過去 20 年中至少出現了三種邏輯主義對《邏輯哲學論》的詮釋，但以下考量（Rodych 1995；Wrigley 1998）表明這些理由都不是特別有說服力的。

For example, in saying that “[m]athematics is a method of logic” perhaps Wittgenstein is only saying that since the general form of a natural number and the general form of a proposition are both instances of the general form of a (purely formal) operation, just as truth-functional propositions can be constructed using the general form of a proposition, (true) mathematical equations can be constructed using the general form of a natural number. Alternatively, Wittgenstein may mean that mathematical inferences (i.e., not substitutions) are in accord with, or make use of, logical inferences, and insofar as mathematical reasoning is logical reasoning, mathematics is a method of logic.
例如，當說「[m]數學是一種邏輯的方法」時，或許維根斯坦只是想說，自然數的一般形式和命題的一般形式都是（純形式）運算的一般形式的實例，就像可以使用命題的一般形式構建真值函數命題一樣，（真實的）數學方程式也可以使用自然數的一般形式構建。或者，維根斯坦可能是指數學推理（即，不是替代）與邏輯推理一致，或利用邏輯推理，並且在數學推理是邏輯推理的範疇內，數學就是一種邏輯的方法。

Similarly, in saying that “[t]he logic of the world” is shown by tautologies and true mathematical equations (i.e., #2), Wittgenstein may be saying that since mathematics was invented to help us count and measure, insofar as it enables us to infer contingent proposition(s) from contingent proposition(s) (see 6.211 below), it thereby reflects contingent facts and “[t]he logic of the world”. Though logic—which is inherent in natural (‘everyday’) language (4.002, 4.003, 6.124) and which has evolved to meet our communicative, exploratory, and survival needs—is not invented in the same way, a valid logical inference captures the relationship between possible facts and a sound logical inference captures the relationship between existent facts.
類似地，維根斯坦在說“[t]世界的邏輯”是由同義反覆和真實的數學方程式所顯示（即#2）時，可能是在說，由於數學是為了幫助我們計數和測量而發明的，因為它使我們能夠從偶然命題推導出偶然命題（見下文 6.211），因此它反映了偶然事實和“[t]世界的邏輯”。雖然邏輯——這是固有於自然（‘日常’）語言（4.002, 4.003, 6.124）並且已經演變以滿足我們的交流、探索和生存需求——並不是以相同的方式被發明的，但有效的邏輯推理捕捉了可能事實之間的關係，而合理的邏輯推理則捕捉了存在事實之間的關係。

As regards #3, Black, Savitt, and Frascolla have argued that, since we ascertain the truth of tautologies and mathematical equations without any appeal to “states of affairs” or “facts”, true mathematical equations and tautologies are so analogous that we can “aptly” describe “the philosophy of arithmetic of the Tractatus… as a kind of logicism” (Frascolla 1994: 37). The rejoinder to this is that the similarity that Frascolla, Black and Savitt recognize does not make Wittgenstein’s theory a “kind of logicism” in Frege’s or Russell’s sense, because Wittgenstein does not define numbers “logically” in either Frege’s way or Russell’s way, and the similarity (or analogy) between tautologies and true mathematical equations is neither an identity nor a relation of reducibility.
關於第 3 點，布萊克、薩維特和弗拉斯科拉主張，由於我們在不訴諸“事態”或“事實”的情況下確定同義命題和數學方程的真理，真正的數學方程和同義命題是如此類似，以至於我們可以“恰當地”描述《邏輯哲學論》的“算術哲學……作為一種邏輯主義”（弗拉斯科拉 1994: 37）。對此的反駁是，弗拉斯科拉、布萊克和薩維特所認識的相似性並不使維根斯坦的理論成為弗雷格或羅素意義上的“邏輯主義”，因為維根斯坦並未以弗雷格或羅素的方式“邏輯地”定義數字，而同義命題和真正的數學方程之間的相似性（或類比）既不是同一性也不是可約性關係。

Finally, critics argue that the problem with #4 is that there is no evidence for the claim that the relevant operation is logical in Wittgenstein’s or Russell’s or Frege’s sense of the term—it seems a purely formal, syntactical operation. “Logical operations are performed with propositions, arithmetical ones with numbers”, says Wittgenstein (WVC 218); “[t]he result of a logical operation is a proposition, the result of an arithmetical one is a number”. In sum, critics of the Logicist interpretation of the Tractatus argue that ##1–4 do not individually or collectively constitute cogent grounds for a Logicist interpretation of the Tractatus.
最後，批評者認為第 4 點的問題在於，沒有證據支持相關操作在維根斯坦、羅素或弗雷格的意義上是邏輯的——這似乎是一種純粹的形式、語法操作。維根斯坦說：“邏輯操作是用命題進行的，算術操作是用數字進行的”（WVC 218）；“邏輯操作的結果是一個命題，算術操作的結果是一個數字”。總之，對於邏輯主義詮釋《邏輯哲學論》的批評者認為，##1–4 無論是單獨還是集合起來都不構成對《邏輯哲學論》的邏輯主義詮釋的有力依據。

Another crucial aspect of the Tractarian theory of mathematics is captured in (6.211).
另一個關鍵的方面在於Tractarian數學理論中體現在(6.211)中。

Indeed in real life a mathematical proposition is never what we want. Rather, we make use of mathematical propositions only in inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question, ‘What do we actually use this word or this proposition for?’ repeatedly leads to valuable insights.)
事實上，在現實生活中，數學命題從來不是我們所想要的。相反，我們僅在從不屬於數學的命題推理到同樣不屬於數學的其他命題時，才利用數學命題。（在哲學中，問題「我們實際上用這個詞或這個命題做什麼？」反覆引導出有價值的見解。）

Though mathematics and mathematical activity are purely formal and syntactical, in the Tractatus Wittgenstein tacitly distinguishes between purely formal games with signs, which have no application in contingent propositions, and mathematical propositions, which are used to make inferences from contingent proposition(s) to contingent proposition(s). Wittgenstein does not explicitly say, however, how mathematical equations, which are not genuine propositions, are used in inferences from genuine proposition(s) to genuine proposition(s) (Floyd 2002: 309; Kremer 2002: 293–94). As we shall see in §3.5, the later Wittgenstein returns to the importance of extra-mathematical application and uses it to distinguish a mere “sign-game” from a genuine, mathematical language-game.
儘管數學和數學活動純粹是形式和語法上的，但在《邏輯哲學論》中，維根斯坦默默區分了純粹形式的符號遊戲，這些遊戲在偶然命題中沒有應用，與用於從偶然命題推理到偶然命題的數學命題。維根斯坦並沒有明確說明，然而，數學方程式，這些方程式並不是實際的命題，如何在從實際命題推理到實際命題中使用（Floyd 2002: 309; Kremer 2002: 293–94）。正如我們在§3.5 中將看到的，後期的維根斯坦回到了超數學應用的重要性，並利用它來區分僅僅是“符號遊戲”和真正的數學語言遊戲。

This, in brief, is Wittgenstein’s Tractarian theory of mathematics. In the Introduction to the Tractatus, Russell wrote that Wittgenstein’s “theory of number” “stands in need of greater technical development”, primarily because Wittgenstein had not shown how it could deal with transfinite numbers (Russell 1922[1974]: xx). Similarly, in his review of the Tractatus, Frank Ramsey wrote that Wittgenstein’s ‘account’ does not cover all of mathematics partly because Wittgenstein’s theory of equations cannot explain inequalities (Ramsey 1923: 475). Though it is doubtful that, in 1923, Wittgenstein would have thought these issues problematic, it certainly is true that the Tractarian theory of mathematics is essentially a sketch, especially in comparison with what Wittgenstein begins to develop six years later.
簡而言之，這是維根斯坦的《邏輯哲學論》數學理論。在《邏輯哲學論》的引言中，羅素寫道維根斯坦的“數字理論” “需要更大的技術發展”，主要是因為維根斯坦沒有展示它如何處理超限數（羅素 1922[1974]: xx）。同樣，在他對《邏輯哲學論》的評論中，法蘭克·拉姆齊寫道維根斯坦的“說明”並未涵蓋所有數學，部分原因是維根斯坦的方程理論無法解釋不等式（拉姆齊 1923: 475）。雖然在 1923 年，維根斯坦是否會認為這些問題有爭議是值得懷疑的，但毫無疑問的是，《邏輯哲學論》的數學理論本質上是一個草圖，尤其是與維根斯坦六年後開始發展的內容相比。

After the completion of the Tractatus in 1918, Wittgenstein did virtually no philosophical work until February 2, 1929, eleven months after attending a lecture by the Dutch mathematician L.E.J. Brouwer.
在 1918 年完成《邏輯哲學論》後，維根斯坦幾乎沒有進行任何哲學工作，直到 1929 年 2 月 2 日，這是在聽完荷蘭數學家 L.E.J.布勞威的講座後的十一個月。

2. The Middle Wittgenstein’s Finitistic Constructivism
2. 中期維根斯坦的有限主義建構論

There is little doubt that Wittgenstein was invigorated by L.E.J. Brouwer’s March 10, 1928 Vienna lecture “Science, Mathematics, and Language” (Brouwer 1929), which he attended with F. Waismann and H. Feigl, but it is a gross overstatement to say that he returned to Philosophy because of this lecture or that his intermediate interest in the Philosophy of Mathematics issued primarily from Brouwer’s influence. In fact, Wittgenstein’s return to Philosophy and his intermediate work on mathematics is also due to conversations with Ramsey and members of the Vienna Circle, to Wittgenstein’s disagreement with Ramsey over identity, and several other factors.
毫無疑問，維根斯坦受到 L.E.J.布勞威於 1928 年 3 月 10 日在維也納的講座“科學、數學與語言”（布勞威 1929）的激勵，他與 F. Waismann 和 H. Feigl 一起參加了這場講座，但說他因為這場講座而回到哲學，或者他的數學哲學的中間興趣主要源於布勞威的影響，這是一種過度的誇張。事實上，維根斯坦回到哲學以及他在數學上的中間工作也與他與蘭姆齊及維也納學派成員的對話、他與蘭姆齊在身份問題上的分歧以及其他幾個因素有關。

Though Wittgenstein seems not to have read any Hilbert or Brouwer prior to the completion of the Tractatus, by early 1929 Wittgenstein had certainly read work by Brouwer, Weyl, Skolem, Ramsey (and possibly Hilbert) and, apparently, he had had one or more private discussions with Brouwer in 1928 (Finch 1977: 260; Van Dalen 2005: 566–567). Thus, the rudimentary treatment of mathematics in the Tractatus, whose principal influences were Russell and Frege, was succeeded by detailed work on mathematics in the middle period (1929–1933), which was strongly influenced by the 1920s work of Brouwer, Weyl, Hilbert, and Skolem.
雖然維特根斯坦似乎在完成《邏輯哲學論》之前並未閱讀任何希爾伯特或布勞威爾的著作，但到 1929 年初，維特根斯坦肯定已經閱讀了布勞威爾、維爾、斯科倫、拉姆齊（可能還有希爾伯特）的作品，顯然他在 1928 年與布勞威爾進行過一個或多個私人討論（芬奇 1977: 260；范達倫 2005: 566–567）。因此，《邏輯哲學論》中對數學的初步處理，其主要影響來自羅素和弗雷格，隨後在中期（1929–1933）進行了詳細的數學研究，這受到 1920 年代布勞威爾、維爾、希爾伯特和斯科倫工作的強烈影響。

2.1 Wittgenstein’s Intermediate Constructive Formalism
2.1 維根斯坦的中介建構形式主義

To best understand Wittgenstein’s intermediate Philosophy of Mathematics, one must fully appreciate his strong variant of formalism, according to which “[w]e make mathematics” (WVC 34, note 1; PR §159) by inventing purely formal mathematical calculi, with ‘stipulated’ axioms (PR §202), syntactical rules of transformation, and decision procedures that enable us to invent “mathematical truth” and “mathematical falsity” by algorithmically deciding so-called mathematical ‘propositions’ (PR §§122, 162).
要最佳理解維根斯坦的中介數學哲學，必須充分欣賞他強烈的形式主義變體，根據該變體，“[我們] 創造 數學” (WVC 34, 註 1; PR §159)，通過發明純粹形式的數學計算，具有‘規定的’公理 (PR §202)、轉換的語法規則，以及使我們能夠通過算法決定所謂的數學‘命題’來發明“數學真理”和“數學虛假”的決策程序 (PR §§122, 162)。

The core idea of Wittgenstein’s formalism from 1929 (if not 1918) through 1944 is that mathematics is essentially syntactical, devoid of reference and semantics. The most obvious aspect of this view, which has been noted by numerous commentators who do not refer to Wittgenstein as a ‘formalist’ (Kielkopf 1970: 360–38; Klenk 1976: 5, 8, 9; Fogelin 1968: 267; Frascolla 1994: 40; Marion 1998: 13–14), is that, contra Platonism, the signs and propositions of a mathematical calculus do not refer to anything. As Wittgenstein says at (WVC 34, note 1), “[n]umbers are not represented by proxies; numbers are there”. This means not only that numbers are there in the use, it means that the numerals are the numbers, for “[a]rithmetic doesn’t talk about numbers, it works with numbers” (PR §109).
維根斯坦在 1929 年（如果不是 1918 年）到 1944 年的形式主義的核心思想是，數學本質上是語法性的，缺乏參照和語義。這一觀點最明顯的方面是，許多不將維根斯坦稱為“形式主義者”的評論者已經注意到（Kielkopf 1970: 360–38; Klenk 1976: 5, 8, 9; Fogelin 1968: 267; Frascolla 1994: 40; Marion 1998: 13–14），即，反對柏拉圖主義，數學演算的符號和命題並不指涉任何事物。正如維根斯坦在（WVC 34, note 1）中所說，“[n]數字不是由代理表示的；數字就在那裡”。這不僅意味著數字在使用中存在，還意味著數字本身就是數字，因為“[a]算術並不談論數字，它與數字一起工作”（PR §109）。

What arithmetic is concerned with is the schema ||||$||||$.—But does arithmetic talk about the lines I draw with pencil on paper?—Arithmetic doesn’t talk about the lines, it operates with them. (PG 333)
算術所關心的是結構 ||||$||||$ 。——但算術是否談論我用鉛筆在紙上畫的線？——算術不談論這些線，它是操作這些線的。 (PG 333)

In a similar vein, Wittgenstein says that (WVC 106) “mathematics is always a machine, a calculus” and “[a] calculus is an abacus, a calculator, a calculating machine”, which “works by means of strokes, numerals, etc”. The “justified side of formalism”, according to Wittgenstein (WVC 105), is that mathematical symbols “lack a meaning” (i.e., Bedeutung)—they do not “go proxy for” things which are “their meaning[s]”.
在類似的情況下，維根斯坦說（WVC 106）“數學始終是一台機器，一種計算法”，並且“[一]種計算法是一個算盤，一個計算器，一台計算機”，它“通過筆劃、數字等方式運作”。根據維根斯坦（WVC 105），“形式主義的正當一面”是數學符號“缺乏意義”（即，Bedeutung）——它們並不“代理”事物，這些“是它們的意義”。

You could say arithmetic is a kind of geometry; i.e. what in geometry are constructions on paper, in arithmetic are calculations (on paper).—You could say it is a more general kind of geometry. (PR §109; PR §111)
你可以說算術是一種幾何；即在幾何中是紙上的構造，在算術中是（紙上的）計算。——你可以說這是一種更一般的幾何。

This is the core of Wittgenstein’s life-long formalism. When we prove a theorem or decide a proposition, we operate in a purely formal, syntactical manner. In doing mathematics, we do not discover pre-existing truths that were “already there without one knowing” (PG 481)—we invent mathematics, bit-by-little-bit. “If you want to know what 2+2=4$2+2=4$ means”, says Wittgenstein, “you have to ask how we work it out”, because “we consider the process of calculation as the essential thing” (PG 333). Hence, the only meaning (i.e., sense) that a mathematical proposition has is intra-systemic meaning, which is wholly determined by its syntactical relations to other propositions of the calculus.
這是維根斯坦終生形式主義的核心。當我們 為了證明一個定理或決定一個命題，我們以一種純形式、語法的方式進行。在做數學時，我們並不是發現那些“已經存在但未被人知曉”的真理（PG 481）——我們是逐步發明數學的。“如果你想知道 2+2=4$2+2=4$ 的意思，”維根斯坦說，“你必須問我們是如何推導出來的”，因為“我們認為計算的過程是最重要的”（PG 333）。因此，一個數學命題唯一的意義（即，意義）是系統內部的意義，這完全由它與微積分中其他命題的語法關係所決定。

A second important aspect of the intermediate Wittgenstein’s strong formalism is his view that extra-mathematical application (and/or reference) is not a necessary condition of a mathematical calculus. Mathematical calculi do not require extra-mathematical applications, Wittgenstein argues, since we “can develop arithmetic completely autonomously and its application takes care of itself since wherever it’s applicable we may also apply it” (PR §109; cf. PG 308, WVC 104).
中期維根斯坦強形式主義的第二個重要方面是他認為超數學的應用（和/或參考）並不是數學演算的必要條件。維根斯坦主張，數學演算並不需要超數學的應用，因為我們“可以完全自主地發展算術，其應用自會照顧自己，因為無論在哪裡適用，我們也可以應用它”（PR §109；參見 PG 308, WVC 104）。

As we shall shortly see, the middle Wittgenstein is also drawn to strong formalism by a new concern with questions of decidability. Undoubtedly influenced by the writings of Brouwer and David Hilbert, Wittgenstein uses strong formalism to forge a new connection between mathematical meaningfulness and algorithmic decidability.
正如我們將很快看到的，中期的維根斯坦也因對可決性問題的新關注而被強形式主義所吸引。無疑受到布勞威爾和大衛·希爾伯特著作的影響，維根斯坦利用強形式主義在數學的意義和算法的可決性之間建立了一種新的聯繫。

An equation is a rule of syntax. Doesn’t that explain why we cannot have questions in mathematics that are in principle unanswerable? For if the rules of syntax cannot be grasped, they’re of no use at all…. [This] makes intelligible the attempts of the formalist to see mathematics as a game with signs. (PR §121)
方程式是一種語法規則。這難道不解釋了為什麼我們在數學中不能有原則上無法回答的問題嗎？因為如果語法規則無法被理解，那它們根本沒有用處……[這]使形式主義者試圖將數學視為一種符號遊戲的努力變得可理解。

In Section 2.3, we shall see how Wittgenstein goes beyond both Hilbert and Brouwer by maintaining the Law of the Excluded Middle in a way that restricts mathematical propositions to expressions that are algorithmically decidable.
在第 2.3 節中，我們將看到維根斯坦如何超越希爾伯特和布勞威爾，通過維持排中律，以限制數學命題為算法可決的表達式。

2.2 Wittgenstein’s Intermediate Finitism
2.2 維根斯坦的中介有限主義

The single most important difference between the Early and Middle Wittgenstein is that, in the middle period, Wittgenstein rejects quantification over an infinite mathematical domain, stating that, contra his Tractarian view, such ‘propositions’ are not infinite conjunctions and infinite disjunctions simply because there are no such things.
早期和中期維根斯坦之間最重要的區別在於，在中期，維根斯坦拒絕對無限數學範疇進行量化，並指出，與他的《論述》觀點相反，這樣的「命題」並不是無限的合取和無限的析取，因為根本不存在這樣的事物。

Wittgenstein’s principal reasons for developing a finitistic Philosophy of Mathematics are as follows.
維特根斯坦發展有限主義數學哲學的主要原因如下。

1. Mathematics as Human Invention: According to the middle Wittgenstein, we invent mathematics, from which it follows that mathematics and so-called mathematical objects do not exist independently of our inventions. Whatever is mathematical is fundamentally a product of human activity.
   數學作為人類的發明：根據中期維根斯坦的說法，我們發明了數學，因此數學和所謂的數學對象並不獨立於我們的發明而存在。任何數學的事物根本上都是人類活動的產物。
2. Mathematical Calculi Consist Exclusively of Intensions and Extensions: Given that we have invented only mathematical extensions (e.g., symbols, finite sets, finite sequences, propositions, axioms) and mathematical intensions (e.g., rules of inference and transformation, irrational numbers as rules), these extensions and intensions, and the calculi in which they reside, constitute the entirety of mathematics. (It should be noted that Wittgenstein’s usage of ‘extension’ and ‘intension’ as regards mathematics differs markedly from standard contemporary usage, wherein the extension of a predicate is the set of entities that satisfy the predicate and the intension of a predicate is the meaning of, or expressed by, the predicate. Put succinctly, Wittgenstein thinks that the extension of this notion of concept-and-extension from the domain of existent (i.e., physical) objects to the so-called domain of “mathematical objects” is based on a faulty analogy and engenders conceptual confusion. See #1 just below.)
   數學演算僅由意向和外延組成：鑑於我們僅發明了數學外延（例如，符號、有限集合、有限序列、命題、公理）和數學意向（例如，推理和轉換規則、無理數作為規則），這些外延和意向，以及它們所存在的演算，構成了數學的全部。（應注意，維根斯坦對於數學中「外延」和「意向」的用法與當代標準用法有顯著不同，後者中，謂詞的外延是滿足該謂詞的實體集合，而謂詞的意向是該謂詞所表達的意義。簡而言之，維根斯坦認為，這一概念與外延的概念從存在（即物理）對象的範疇延伸到所謂的「數學對象」範疇是基於錯誤的類比，並產生了概念上的混淆。請參見下面的#1。）

These two reasons have at least five immediate consequences for Wittgenstein’s Philosophy of Mathematics.
這兩個原因對維根斯坦的數學哲學至少有五個直接的後果。

1. Rejection of Infinite Mathematical Extensions: Given that a mathematical extension is a symbol (‘sign’) or a finite concatenation of symbols extended in space, there is a categorical difference between mathematical intensions and (finite) mathematical extensions, from which it follows that “the mathematical infinite” resides only in recursive rules (i.e., intensions). An infinite mathematical extension (i.e., a completed, infinite mathematical extension) is a contradiction-in-terms
   拒絕無限數學擴展：考慮到數學擴展是一個符號（‘符號’）或符號的有限串聯擴展於空間中，數學意圖和（有限）數學擴展之間存在類別上的差異，因此“數學無限”僅存在於遞歸規則中（即，意圖）。無限數學擴展（即，一個完成的、無限數學擴展）是一種自相矛盾的說法。
2. Rejection of Unbounded Quantification in Mathematics: Given that the mathematical infinite can only be a recursive rule, and given that a mathematical proposition must have sense, it follows that there cannot be an infinite mathematical proposition (i.e., an infinite logical product or an infinite logical sum).
   數學中對無界量化的拒絕：鑑於數學無限只能是一個遞歸規則，並且鑑於數學命題必須有意義，因此不可能存在無限的數學命題（即無限的邏輯乘積或無限的邏輯和）。
3. Algorithmic Decidability vs. Undecidability: If mathematical extensions of all kinds are necessarily finite, then, in principle, all mathematical propositions are algorithmically decidable, from which it follows that an “undecidable mathematical proposition” is a contradiction-in-terms. Moreover, since mathematics is essentially what we have and what we know, Wittgenstein restricts algorithmic decidability to knowing how to decide a proposition with a known decision procedure.
   算法可判定性與不可判定性：如果所有類型的數學擴展必然是有限的，那麼，原則上，所有數學命題都是算法可判定的，由此可得“不可判定的數學命題”是一種自相矛盾。此外，由於數學本質上是我們所擁有和所知道的，維根斯坦將算法可判定性限制在知道如何用已知的決策程序來判定一個命題。
4. Anti-Foundationalist Account of Real Numbers: Since there are no infinite mathematical extensions, irrational numbers are rules, not extensions. Given that an infinite set is a recursive rule (or an induction) and no such rule can generate all of the things mathematicians call (or want to call) “real numbers”, it follows that there is no set of ‘all’ the real numbers and no such thing as the mathematical continuum.
   反基礎主義的實數觀：由於不存在無限的數學擴展，無理數是規則，而不是擴展。考慮到無限集合是一個遞歸規則（或歸納法），而且沒有任何這樣的規則可以生成數學家所稱（或想要稱）的“實數”，因此可以得出結論，並不存在“所有”實數的集合，也不存在數學上的連續體。
5. Rejection of Different Infinite Cardinalities: Given the non-existence of infinite mathematical extensions, Wittgenstein rejects the standard interpretation of Cantor’s diagonal proof as a proof of infinite sets of greater and lesser cardinalities.
   拒絕不同的無窮基數：鑑於無限數學擴展的不存在，維根斯坦拒絕康托爾對角證明的標準解釋，認為這是對無限集合的更大和更小基數的證明。

Since we invent mathematics in its entirety, we do not discover pre-existing mathematical objects or facts or that mathematical objects have certain properties, for “one cannot discover any connection between parts of mathematics or logic that was already there without one knowing” (PG 481). In examining mathematics as a purely human invention, Wittgenstein tries to determine what exactly we have invented and why exactly, in his opinion, we erroneously think that there are infinite mathematical extensions.
由於我們完全發明了數學，我們並不是發現已存在的數學對象或事實，或者數學對象具有某些屬性，因為「人們無法發現數學或邏輯的部分之間已經存在的任何聯繫，除非他們知道」（PG 481）。在將數學視為純粹的人類發明時，維根斯坦試圖確定我們究竟發明了什麼，以及為什麼在他看來，我們錯誤地認為存在無限的數學擴展。

If, first, we examine what we have invented, we see that we have invented formal calculi consisting of finite extensions and intensional rules. If, more importantly, we endeavour to determine why we believe that infinite mathematical extensions exist (e.g., why we believe that the actual infinite is intrinsic to mathematics), we find that we conflate mathematical intensions and mathematical extensions, erroneously thinking that there is “a dualism” of “the law and the infinite series obeying it” (PR §180). For instance, we think that because a real number “endlessly yields the places of a decimal fraction” (PR §186), it is “a totality” (WVC 81–82, note 1), when, in reality, “[a]n irrational number isn’t the extension of an infinite decimal fraction,… it’s a law” (PR §181) which “yields extensions” (PR §186). A law and a list are fundamentally different; neither can ‘give’ what the other gives (WVC 102–103). Indeed, “the mistake in the set-theoretical approach consists time and again in treating laws and enumerations (lists) as essentially the same kind of thing” (PG 461). 

Closely related with this conflation of intensions and extensions is the fact that we mistakenly act as if the word ‘infinite’ is a “number word”, because in ordinary discourse we answer the question “how many?” with both (PG 463; cf. PR §142). But “‘[i]nfinite’ is not a quantity”, Wittgenstein insists (WVC 228); the word ‘infinite’ and a number word like ‘five’ do not have the same syntax. The words ‘finite’ and ‘infinite’ do not function as adjectives on the words ‘class’ or ‘set’, (WVC 102), for the terms “finite class” and “infinite class” use ‘class’ in completely different ways (WVC 228). An infinite class is a recursive rule or “an induction”, whereas the symbol for a finite class is a list or extension (PG 461). It is because an induction has much in common with the multiplicity of a finite class that we erroneously call it an infinite class (PR §158). 

In sum, because a mathematical extension is necessarily a finite sequence of symbols, an infinite mathematical extension is a contradiction-in-terms. This is the foundation of Wittgenstein’s finitism. Thus, when we say, e.g., that “there are infinitely many even numbers”, we are not saying “there are an infinite number of even numbers” in the same sense as we can say “there are 27 people in this house”; the infinite series of natural numbers is nothing but “the infinite possibility of finite series of numbers”—“[i]t is senseless to speak of the whole infinite number series, as if it, too, were an extension” (PR §144). The infinite is understood rightly when it is understood, not as a quantity, but as an “infinite possibility” (PR §138).
總之，因為數學擴展必然是一個有限的符號序列，無限的數學擴展是一種自相矛盾的說法。這是維根斯坦有限主義的基礎。因此，當我們說，例如，“有無限多的偶數”時，我們並不是在說“有無限多的偶數”，就像我們可以說“這個房子裡有 27 個人”一樣；自然數的無限序列不過是“有限數列的無限可能性”——“談論整個無限數列是毫無意義的，就好像它也是一個擴展” （PR §144）。當無限被理解為“無限可能性”時，它才是正確的理解（PR §138）。

Given Wittgenstein’s rejection of infinite mathematical extensions, he adopts finitistic, constructive views on mathematical quantification, mathematical decidability, the nature of real numbers, and Cantor’s diagonal proof of the existence of infinite sets of greater cardinalities.
鑑於維特根斯坦對無限數學擴展的拒絕，他採取了有限主義的建構性觀點，涉及數學量化、數學可決性、實數的本質以及康托爾對存在更大基數的無限集合的對角線證明。

Since a mathematical set is a finite extension, we cannot meaningfully quantify over an infinite mathematical domain, simply because there is no such thing as an infinite mathematical domain (i.e., totality, set), and, derivatively, no such things as infinite conjunctions or disjunctions (G.E. Moore 1955: 2–3; cf. AWL 6; and PG 281). 

[I]t still looks now as if the quantifiers make no sense for numbers. I mean: you can’t say ‘(n)ϕn$(n)\varphi n$’, precisely because ‘all natural numbers’ isn’t a bounded concept. But then neither should one say a general proposition follows from a proposition about the nature of number. 

But in that case it seems to me that we can’t use generality—all, etc.—in mathematics at all. There’s no such thing as ‘all numbers’, simply because there are infinitely many. (PR §126; PR §129) 

‘Extensionalists’ who assert that “ε(0).ε(1).ε(2)$\epsilon (0).\epsilon (1).\epsilon (2)$ and so on” is an infinite logical product (PG 452) assume or assert that finite and infinite conjunctions are close cousins—that the fact that we cannot write down or enumerate all of the conjuncts ‘contained’ in an infinite conjunction is only a “human weakness”, for God could surely do so and God could surely survey such a conjunction in a single glance and determine its truth-value. According to Wittgenstein, however, this is not a matter of human limitation. Because we mistakenly think that “an infinite conjunction” is similar to “an enormous conjunction”, we erroneously reason that just as we cannot determine the truth-value of an enormous conjunction because we don’t have enough time, we similarly cannot, due to human limitations, determine the truth-value of an infinite conjunction (or disjunction). But the difference here is not one of degree but of kind: “in the sense in which it is impossible to check an infinite number of propositions it is also impossible to try to do so” (PG 452). This applies, according to Wittgenstein, to human beings, but more importantly, it applies also to God (i.e., an omniscient being), for even God cannot write down or survey infinitely many propositions because for him too the series is never-ending or limitless and hence the ‘task’ is not a genuine task because it cannot, in principle, be done (i.e., “infinitely many” is not a number word). As Wittgenstein says at (PR 128; cf. PG 479): “‘Can God know all the places of the expansion of π$\pi$?’ would have been a good question for the schoolmen to ask”, for the question is strictly ‘senseless’. As we shall shortly see, on Wittgenstein’s account, “[a] statement about all numbers is not represented by means of a proposition, but by means of induction” (WVC 82). 

Similarly, there is no such thing as a mathematical proposition about some number—no such thing as a mathematical proposition that existentially quantifies over an infinite domain (PR §173). 

What is the meaning of such a mathematical proposition as ‘(∃n)4+n=7$(\mathrm{\exists }n)4+n=7$’? It might be a disjunction—(4+0=7)∨$(4+0=7)\vee$ (4+1=7)∨$(4+1=7)\vee$ etc. ad inf. But what does that mean? I can understand a proposition with a beginning and an end. But can one also understand a proposition with no end? (PR §127) 

We are particularly seduced by the feeling or belief that an infinite mathematical disjunction makes good sense in the case where we can provide a recursive rule for generating each next member of an infinite sequence. For example, when we say “There exists an odd perfect number” we are asserting that, in the infinite sequence of odd numbers, there is (at least) one odd number that is perfect—we are asserting ‘ϕ(1)∨ϕ(3)∨ϕ(5)∨$\varphi (1)\vee \varphi (3)\vee \varphi (5)\vee$ and so on’ and we know what would make it true and what would make it false (PG 451). The mistake here made, according to Wittgenstein (PG 451), is that we are implicitly “comparing the proposition ‘(∃n)$(\mathrm{\exists }n)$…’ with the proposition… ‘There are two foreign words on this page’”, which doesn’t provide the grammar of the former ‘proposition’, but only indicates an analogy in their respective rules. 

On Wittgenstein’s intermediate finitism, an expression quantifying over an infinite domain is never a meaningful proposition, not even when we have proved, for instance, that a particular number n$n$ has a particular property. 

The important point is that, even in the case where I am given that 32+42=52$3^2+4^2=5^2$, I ought not to say ‘(∃x,y,z,n)(xn+yn=zn)$(\mathrm{\exists }x,y,z,n)(x^n+y^n=z^n)$’, since taken extensionally that’s meaningless, and taken intensionally this doesn’t provide a proof of it. No, in this case I ought to express only the first equation. (PR §150) 

Thus, Wittgenstein adopts the radical position that all expressions that quantify over an infinite domain, whether ‘conjectures’ (e.g., Goldbach’s Conjecture, the Twin Prime Conjecture) or “proved general theorems” (e.g., “Euclid’s Prime Number Theorem”, the Fundamental Theorem of Algebra), are meaningless (i.e., ‘senseless’; ‘sinnlos’) expressions as opposed to “genuine mathematical proposition[s]” (PR §168). These expressions are not (meaningful) mathematical propositions, according to Wittgenstein, because the Law of the Excluded Middle does not apply, which means that “we aren’t dealing with propositions of mathematics” (PR §151). The crucial question why and in exactly what sense the Law of the Excluded Middle does not apply to such expressions will be answered in the next section. 

2.3 Wittgenstein’s Intermediate Finitism and Algorithmic Decidability 

The middle Wittgenstein has other grounds for rejecting unrestricted quantification in mathematics, for on his idiosyncratic account, we must distinguish between four categories of concatenations of mathematical symbols. 

1. Proved mathematical propositions in a particular mathematical calculus (no need for “mathematical truth”). 
2. Refuted mathematical propositions in (or of) a particular mathematical calculus (no need for “mathematical falsity”). 
3. Mathematical propositions for which we know we have in hand an applicable and effective decision procedure (i.e., we know how to decide them). 
4. Concatenations of symbols that are not part of any mathematical calculus and which, for that reason, are not mathematical propositions (i.e., are non-propositions). 

In his 2004 (p. 18), Mark van Atten says that 

… [i]ntuitionistically, there are four [“possibilities for a proposition with respect to truth”]: 

1. p$p$ has been experienced as true 
2. p$p$ has been experienced as false 
3. Neither 1 nor 2 has occurred yet, but we know a procedure to decide p$p$ (i.e., a procedure that will prove p$p$ or prove ¬p)$\mathrm{\neg }p)$ 
4. Neither 1 nor 2 has occurred yet, and we do not know a procedure to decide p$p$. 

What is immediately striking about Wittgenstein’s ##1–3 and Brouwer’s ##1–3 (Brouwer 1955: 114; 1981: 92) is the enormous similarity. And yet, for all of the agreement, the disagreement in #4 is absolutely crucial. 

As radical as the respective #3s are, Brouwer and Wittgenstein agree that an undecided ϕ$\varphi$ is a mathematical proposition (for Wittgenstein, of a particular mathematical calculus) if we know of an applicable decision procedure. They also agree that until ϕ$\varphi$ is decided, it is neither true nor false (though, for Wittgenstein, ‘true’ means no more than “proved in calculus Γ$\mathrm{\Gamma }$”). What they disagree about is the status of an ordinary mathematical conjecture, such as Goldbach’s Conjecture. Brouwer admits it as a mathematical proposition, while Wittgenstein rejects it because we do not know how to algorithmically decide it. Like Brouwer (1948 [1983: 90]), Wittgenstein holds that there are no “unknown truth[s]” in mathematics, but unlike Brouwer he denies the existence of “undecidable propositions” on the grounds that such a ‘proposition’ would have no ‘sense’, “and the consequence of this is precisely that the propositions of logic lose their validity for it” (PR §173). In particular, if there are undecidable mathematical propositions (as Brouwer maintains), then at least some mathematical propositions are not propositions of any existent mathematical calculus. For Wittgenstein, however, it is a defining feature of a mathematical proposition that it is either decided or decidable by a known decision procedure in a mathematical calculus. As Wittgenstein says at (PR §151),  

where the law of the excluded middle doesn’t apply, no other law of logic applies either, because in that case we aren’t dealing with propositions of mathematics. (Against Weyl and Brouwer).  

The point here is not that we need truth and falsity in mathematics—we don’t—but rather that every mathematical proposition (including ones for which an applicable decision procedure is known) is known to be part of a mathematical calculus. 

To maintain this position, Wittgenstein distinguishes between (meaningful, genuine) mathematical propositions, which have mathematical sense, and meaningless, senseless (‘sinnlos’) expressions by stipulating that an expression is a meaningful (genuine) proposition of a mathematical calculus iff we know of a proof, a refutation, or an applicable decision procedure (PR §151; PG 452; PG 366; AWL 199–200). “Only where there’s a method of solution [a ‘logical method for finding a solution’] is there a [mathematical] problem”, he tells us (PR §§149, 152; PG 393). “We may only put a question in mathematics (or make a conjecture)”, he adds (PR §151), “where the answer runs: ‘I must work it out’”. 

At (PG 468), Wittgenstein emphasizes the importance of algorithmic decidability clearly and emphatically:  

In mathematics everything is algorithm and nothing is meaning [Bedeutung]; even when it doesn’t look like that because we seem to be using words to talk about mathematical things. Even these words are used to construct an algorithm.  

When, therefore, Wittgenstein says (PG 368) that if “[the Law of the Excluded Middle] is supposed not to hold, we have altered the concept of proposition”, he means that an expression is only a meaningful mathematical proposition if we know of an applicable decision procedure for deciding it (PG 400). If a genuine mathematical proposition is undecided, the Law of the Excluded Middle holds in the sense that we know that we will prove or refute the proposition by applying an applicable decision procedure (PG 379, 387). 

For Wittgenstein, there simply is no distinction between syntax and semantics in mathematics: everything is syntax. If we wish to demarcate between “mathematical propositions” versus “mathematical pseudo-propositions”, as we do, then the only way to ensure that there is no such thing as a meaningful, but undecidable (e.g., independent), proposition of a given calculus is to stipulate that an expression is only a meaningful proposition in a given calculus (PR §153) if either it has been decided or we know of an applicable decision procedure. In this manner, Wittgenstein defines both a mathematical calculus and a mathematical proposition in epistemic terms. A calculus is defined in terms of stipulations (PR §202; PG 369), known rules of operation, and known decision procedures, and an expression is only a mathematical proposition in a given calculus (PR §155), and only if that calculus contains (PG 379) a known (and applicable) decision procedure, for “you cannot have a logical plan of search for a sense you don’t know” (PR §148). 

Thus, the middle Wittgenstein rejects undecidable mathematical propositions on two grounds. First, number-theoretic expressions that quantify over an infinite domain are not algorithmically decidable, and hence are not meaningful mathematical propositions. 

If someone says (as Brouwer does) that for (x)f1x=f2x$(x)f_{1}x=f_{2}x$, there is, as well as yes and no, also the case of undecidability, this implies that ‘(x)$(x)$…’ is meant extensionally and we may talk of the case in which all x$x$ happen to have a property. In truth, however, it’s impossible to talk of such a case at all and the ‘(x)$(x)$…’ in arithmetic cannot be taken extensionally. (PR §174) 

“Undecidability”, says Wittgenstein (PR §174) “presupposes… that the bridge cannot be made with symbols”, when, in fact, “[a] connection between symbols which exists but cannot be represented by symbolic transformations is a thought that cannot be thought”, for “[i]f the connection is there,… it must be possible to see it”. Alluding to algorithmic decidability, Wittgenstein stresses (PR §174) that “[w]e can assert anything which can be checked in practice”, because “it’s a question of the possibility of checking” (italics added). 

Wittgenstein’s second reason for rejecting an undecidable mathematical proposition is that it is a contradiction-in-terms. There cannot be “undecidable propositions”, Wittgenstein argues (PR §173), because an expression that is not decidable in some actual calculus is simply not a mathematical proposition, since “every proposition in mathematics must belong to a calculus of mathematics” (PG 376).
維特根斯坦拒絕不可決定數學命題的第二個理由是它是一種自相矛盾。維特根斯坦主張，不能有“不可決定的命題”(PR §173)，因為在某些實際計算中不可決定的表達式根本就不是一個數學命題，因為“數學中的每個命題必須屬於數學的計算”(PG 376)。

This radical position on decidability results in various radical and counter-intuitive statements about unrestricted mathematical quantification, mathematical induction, and, especially, the sense of a newly proved mathematical proposition. In particular, Wittgenstein asserts that uncontroversial mathematical conjectures, such as Goldbach’s Conjecture (hereafter ‘GC’) and the erstwhile conjecture “Fermat’s Last Theorem” (hereafter ‘FLT’), have no sense (or, perhaps, no determinate sense) and that the unsystematic proof of such a conjecture gives it a sense that it didn’t previously have (PG 374) because
這種對可判定性的激進立場導致了關於不受限制的數學量化、數學歸納法，尤其是新證明的數學命題的意義的各種激進和反直覺的陳述。特別是，維根斯坦主張，無爭議的數學猜想，如哥德巴赫猜想（以下簡稱「GC」）和曾經的猜想「費馬最後定理」（以下簡稱「FLT」），沒有意義（或者，也許，沒有確定的意義），而這樣的猜想的非系統性證明賦予了它之前所沒有的意義（PG 374），因為

it’s unintelligible that I should admit, when I’ve got the proof, that it’s a proof of precisely this proposition, or of the induction meant by this proposition. (PR §155)
我應該承認，當我有證據時，這證據恰恰是這個命題的證明，或是這個命題所指的歸納的證明，這是無法理解的。 (PR §155)

Thus Fermat’s [Last Theorem] makes no sense until I can search for a solution to the equation in cardinal numbers. And ‘search’ must always mean: search systematically. Meandering about in infinite space on the look-out for a gold ring is no kind of search. (PR §150)
因此，費馬的[最後定理]在我能夠在基數中尋找方程的解之前毫無意義。而「尋找」必須始終意味著：系統地尋找。在無限空間中漫遊，尋找一個金環並不是一種尋找。(PR §150)

I say: the so-called ‘Fermat’s Last Theorem’ isn’t a proposition. (Not even in the sense of a proposition of arithmetic.) Rather, it corresponds to an induction. (PR §189)
我說：所謂的「費馬最後定理」並不是一個命題。（甚至在算術命題的意義上也不是。）相反，它對應於一個歸納。（PR §189）

To see how Fermat’s Last Theorem isn’t a proposition and how it might correspond to an induction, we need to examine Wittgenstein’s account of mathematical induction.
要了解費馬最後定理為何不是一個命題，以及它如何可能與歸納法相對應，我們需要檢視維根斯坦對數學歸納法的闡述。

2.4 Wittgenstein’s Intermediate Account of Mathematical Induction and Algorithmic Decidability
2.4 維根斯坦的數學歸納法與算法可決性的中介說明

Given that one cannot quantify over an infinite mathematical domain, the question arises: What, if anything, does any number-theoretic proof by mathematical induction actually prove?
考慮到無法在無限的數學範疇中進行量化，問題隨之而來：任何數論的數學歸納證明究竟證明了什麼？

On the standard view, a proof by mathematical induction has the following paradigmatic form.
在標準視圖中，數學歸納法的證明具有以下典範形式。

If, however, “∀nϕ(n)$\mathrm{\forall }n\varphi (n)$” is not a meaningful (genuine) mathematical proposition, what are we to make of this proof?
如果，然而，“ ∀nϕ(n)$\mathrm{\forall }n\varphi (n)$ ” 不是一個有意義的（真實的）數學命題，我們該如何看待這個證明？

Wittgenstein’s initial answer to this question is decidedly enigmatic. “An induction is the expression for arithmetical generality”, but “induction isn’t itself a proposition” (PR §129).
維根斯坦對這個問題的初步回答無疑是神秘的。“歸納是算術一般性的表達”，但“歸納本身不是一個命題”（PR §129）。

We are not saying that when f(1)$f(1)$ holds and when f(c+1)$f(c+1)$ follows from f(c)$f(c)$, the proposition f(x)$f(x)$ is therefore true of all cardinal numbers: but: “the proposition f(x)$f(x)$ holds for all cardinal numbers” means “it holds for x=1$x=1$, and f(c+1)$f(c+1)$ follows from f(c)$f(c)$”. (PG 406)
我們並不是在說當 f(1)$f(1)$ 成立且當 f(c+1)$f(c+1)$ 從 f(c)$f(c)$ 推導出來時，命題 f(x)$f(x)$ 因此 對所有基數都是正確的：而是：“命題 f(x)$f(x)$ 對所有基數成立” 意味著 “它對 x=1$x=1$ 成立，並且 f(c+1)$f(c+1)$ 從 f(c)$f(c)$ 推導出來”。 (PG 406)

In a proof by mathematical induction, we do no actually prove the ‘proposition’ [e.g., ∀nϕ(n)$\mathrm{\forall }n\varphi (n)$] that is customarily construed as the conclusion of the proof (PG 406, 374; PR §164), rather this pseudo-proposition or ‘statement’ stands ‘proxy’ for the “infinite possibility” (i.e., “the induction”) that we come to ‘see’ by means of the proof (WVC 135). “I want to say”, Wittgenstein concludes, that “once you’ve got the induction, it’s all over” (PG 407). Thus, on Wittgenstein’s account, a particular proof by mathematical induction should be understood in the following way.
在數學歸納法的證明中，我們並不真正證明 「命題」[例如， ∀nϕ(n)$\mathrm{\forall }n\varphi (n)$ ] 通常被解釋為證明的結論（PG 406, 374；PR §164），而這個偽命題或「陳述」則代表著「無限的可能性」（即「歸納法」），我們通過證明來「看見」它（WVC 135）。“我想說”，維根斯坦總結道，“一旦你掌握了歸納法，一切就結束了”（PG 407）。因此，根據維根斯坦的說法，特定的數學歸納證明應該以以下方式理解。

Here the ‘conclusion’ of an inductive proof [i.e., “what is to be proved” (PR §164)] uses ‘m$m$’ rather than ‘n$n$’ to indicate that ‘m$m$’ stands for any particular number, while ‘n$n$’ stands for any arbitrary number. For Wittgenstein, the proxy statement “ϕ(m)$\varphi (m)$” is not a mathematical proposition that “assert[s] its generality” (PR §168), it is an eliminable pseudo-proposition standing proxy for the proved inductive base and inductive step. Though an inductive proof cannot prove “the infinite possibility of application” (PR §163), it enables us “to perceive” that a direct proof of any particular proposition can be constructed (PR §165). For example, once we have proved “ϕ(1)$\varphi (1)$” and “ϕ(n)→ϕ(n+1)$\varphi (n)\to \varphi (n+1)$”, we need not reiterate modus ponens m−1$m-1$ times to prove the particular proposition “ϕ(m)$\varphi (m)$” (PR §164). The direct proof of, say, “ϕ$\varphi$(714)” (i.e., without 713 iterations of modus ponens) “cannot have a still better proof, say, by my carrying out the derivation as far as this proposition itself” (PR §165).
這裡是歸納證明的「結論」[即， “要證明的內容” (PR §164)] 使用 ‘ m$m$ ’ 而不是 ‘ n$n$ ’ 來表示 ‘ m$m$ ’ 代表任何 特定 數字，而 ‘ n$n$ ’ 代表任何 任意 數字。對於維根斯坦來說，代理陳述“ ϕ(m)$\varphi (m)$ ” 不是一個“聲明其普遍性”的數學命題 (PR §168)，它是一個 可消除的 假命題，代表已證明的歸納基礎和歸納步驟。雖然歸納證明 無法證明 “應用的無限可能性” (PR §163)，但它使我們“能夠 感知”到任何 特定 命題的直接證明可以被構造 (PR §165)。 例如，一旦我們證明了“ ϕ(1)$\varphi (1)$ ”和“ ϕ(n)→ϕ(n+1)$\varphi (n)\to \varphi (n+1)$ ”, 我們就不需要重複模態肯定 m−1$m-1$ 次來證明特定命題“ ϕ(m)$\varphi (m)$ ”（PR §164）。直接證明，例如“ ϕ$\varphi$ (714)”（即，無需 713 次模態肯定）“不能有更好的證明，例如，我將推導進行到這個命題本身” （PR §165）。

A second, very important impetus for Wittgenstein’s radically constructivist position on mathematical induction is his rejection of an undecidable mathematical proposition.
對維根斯坦的激進建構主義立場而言，第二個非常重要的推動力是他對一個不可判定數學命題的拒絕。

In discussions of the provability of mathematical propositions it is sometimes said that there are substantial propositions of mathematics whose truth or falsehood must remain undecided. What the people who say that don’t realize is that such propositions, if we can use them and want to call them “propositions”, are not at all the same as what are called “propositions” in other cases; because a proof alters the grammar of a proposition. (PG 367)
在數學命題的可證性討論中，有時會說存在一些數學的實質命題，其真偽必須保持未決。說這種話的人沒有意識到，如果我們可以使用這些命題並想稱之為“命題”，那麼它們與其他情況下所謂的“命題”完全不同；因為證明改變了命題的語法。

In this passage, Wittgenstein is alluding to Brouwer, who, as early as 1907 and 1908, states, first, that “the question of the validity of the principium tertii exclusi is equivalent to the question whether unsolvable mathematical problems exist”, second, that “[t]here is not a shred of a proof for the conviction… that there exist no unsolvable mathematical problems”, and, third, that there are meaningful propositions/‘questions’, such as “Do there occur in the decimal expansion of π$\pi$ infinitely many pairs of consecutive equal digits?”, to which the Law of the Excluded Middle does not apply because “it must be considered as uncertain whether problems like [this] are solvable” (Brouwer, 1908 [1975: 109–110]). “A fortiori it is not certain that any mathematical problem can either be solved or proved to be unsolvable”, Brouwer says (1907 [1975: 79]), “though HILBERT, in ‘Mathematische Probleme’, believes that every mathematician is deeply convinced of it”.
在這段文字中，維根斯坦提到了布勞威爾，他早在 1907 年和 1908 年，國家首先指出“有效性問題 排除第三者原則等同於問題 是否存在無法解決的數學問題”，其次，“對於這種信念…不存在任何證據表明不存在無法解決的數學問題”，第三，存在有意義的命題/‘問題’，例如“是否存在 在 π$\pi$ 的小數展開中無限多對出現 連續相等的數字?”，對於此，排中律不適用因為“必須認為像[這樣的]問題是否可解是未知的”（布勞威爾，1908 [1975: 109–110]）。“更進一步說，任何數學問題是否可以被解決或證明為不可解都是不確定的，”布勞威爾說（1907 [1975: 79]），“儘管希爾伯特在《數學問題》中相信每位數學家都深信不疑。”

Wittgenstein takes the same data and, in a way, draws the opposite conclusion. If, as Brouwer says, we are uncertain whether all or some “mathematical problems” are solvable, then we know that we do not have in hand an applicable decision procedure, which means that the alleged mathematical propositions are not decidable, here and now. “What ‘mathematical questions’ share with genuine questions”, Wittgenstein says (PR §151), “is simply that they can be answered”. This means that if we do not know how to decide an expression, then we do not know how to make it either proved (true) or refuted (false), which means that the Law of the Excluded Middle “doesn’t apply” and, therefore, that our expression is not a mathematical proposition.
維特根斯坦採取相同的數據，並以某種方式得出相反的結論。如果，正如布勞威所說，我們對所有或某些“數學問題”是否可解感到不確定，那麼我們知道我們手中沒有一個適用的決策程序，這意味著所謂的數學命題在此時此地是不可判定的。“‘數學問題’與真正問題的共同點”，維特根斯坦說（PR §151），“僅僅在於它們可以被回答”。這意味著如果我們不知道如何決定一個表達式，那麼我們也不知道如何使它被證明（真）或駁斥（假），這意味著排中律“並不適用”，因此，我們的表達式不是一個數學命題。

Together, Wittgenstein’s finitism and his criterion of algorithmic decidability shed considerable light on his highly controversial remarks about putatively meaningful conjectures such as FLT and GC. GC is not a mathematical proposition because we do not know how to decide it, and if someone like G. H. Hardy says that he ‘believes’ GC is true (PG 381; LFM 123; PI §578), we must answer that s/he only “has a hunch about the possibilities of extension of the present system” (LFM 139)—that one can only believe such an expression is ‘correct’ if one knows how to prove it. The only sense in which GC (or FLT) can be proved is that it can “correspond to a proof by induction”, which means that the unproved inductive step (e.g., “G(n)→G(n+1)$G(n)\to G(n+1)$”) and the expression “∀nG(n)$\mathrm{\forall }nG(n)$” are not mathematical propositions because we have no algorithmic means of looking for an induction (PG 367). A “general proposition” is senseless prior to an inductive proof “because the question would only have made sense if a general method of decision had been known before the particular proof was discovered” (PG 402). Unproved ‘inductions’ or inductive steps are not meaningful propositions because the Law of the Excluded Middle does not hold in the sense that we do not know of a decision procedure by means of which we can prove or refute the expression (PG 400; WVC 82).
一起，維根斯坦的有限主義和他的標準 算法可判定性對他的高度提供了相當大的啟示 關於假定的有意義猜想（如 FLT 和 GC）的爭議性言論。GC 不是一個數學命題，因為我們不知道如何決定它，如果像 G. H. Hardy 這樣的人說他“相信”GC 是真的（PG 381；LFM 123；PI §578），我們必須回答說他/她只是“對當前系統的擴展可能性有一種直覺”（LFM 139）——只有當一個人知道如何證明它時，才能認為這樣的表達是‘正確的’。GC（或 FLT）能被證明的唯一意義在於它可以“對應於一個歸納證明”，這意味著未被證明的歸納步驟（例如，“ G(n)→G(n+1)$G(n)\to G(n+1)$ ”和表達“ ∀nG(n)$\mathrm{\forall }nG(n)$ ”不是數學命題，因為我們沒有算法手段來尋找歸納（PG 367）。 「一般命題」在歸納證明之前是毫無意義的，「因為這個問題只有在已知一種一般的決策方法的情況下才有意義，在特定證明被發現之前」（PG 402）。未經證明的「歸納」或歸納步驟不是有意義的命題，因為排中律在這種意義上不成立，因為我們不知道有哪種決策程序可以用來證明或反駁該表達式（PG 400；WVC 82）。

This position, however, seems to rob us of any reason to search for a ‘decision’ of a meaningless ‘expression’ such as GC. The intermediate Wittgenstein says only that “[a] mathematician is… guided by… certain analogies with the previous system” and that there is nothing “wrong or illegitimate if anyone concerns himself with Fermat’s Last Theorem” (WVC 144).
然而，這個立場似乎剝奪了我們尋找像 GC 這樣無意義的“表達”的“決定”的任何理由。中介的維根斯坦僅僅說“[一]個數學家是……受到……與先前系統的某些類比的指導”，並且如果有人關心費馬最後定理，則沒有什麼“錯誤或不合法”的地方（WVC 144）。

If e.g. I have a method for looking at integers that satisfy the equation x2+y2=z2$x^2+y^2=z^2$, then the formula xn+yn=zn$x^n+y^n=z^n$ may stimulate me. I may let a formula stimulate me. Thus I shall say, Here there is a stimulus—but not a question. Mathematical problems are always such stimuli. (WVC 144, Jan. 1, 1931)
如果例如我有一種方法來查看滿足的整數 方程式 x2+y2=z2$x^2+y^2=z^2$ ，那麼公式 xn+yn=zn$x^n+y^n=z^n$ 可能會刺激我。我可能會讓一個公式刺激我。因此我會說，這裡有一個 刺激—但不是一個 問題。數學問題總是這樣的刺激。 (WVC 144, 1931 年 1 月 1 日)

More specifically, a mathematician may let a senseless conjecture such as FLT stimulate her/him if s/he wishes to know whether a calculus can be extended without altering its axioms or rules (LFM 139).
更具體地說，一位數學家可能會讓像 FLT 這樣無意義的猜想激發她/他，如果她/他希望知道是否可以在不改變其公理或規則的情況下擴展微積分（LFM 139）。

What is here going [o]n [in an attempt to decide GC] is an unsystematic attempt at constructing a calculus. If the attempt is successful, I shall again have a calculus in front of me, only a different one from the calculus I have been using so far. (WVC 174–75; Sept. 21, 1931; italics added)
這裡正在進行的[為了決定 GC 的嘗試]是一種不系統的構建微積分的嘗試。如果這個嘗試成功，我將再次面對一個微積分，只是與我迄今為止使用的微積分不同。（WVC 174–75；1931 年 9 月 21 日；斜體字為新增）

If, e.g., we succeed in proving GC by mathematical induction (i.e., we prove “G(1)$G(1)$” and “G(n)→G(n+1)$G(n)\to G(n+1)$”), we will then have a proof of the inductive step, but since the inductive step was not algorithmically decidable beforehand (PR §§148, 155, 157; PG 380), in constructing the proof we have constructed a new calculus, a new calculating machine (WVC 106) in which we now know how to use this new “machine-part” (RFM VI, §13) (i.e., the unsystematically proved inductive step). Before the proof, the inductive step is not a mathematical proposition with sense (in a particular calculus), whereas after the proof the inductive step is a mathematical proposition, with a new, determinate sense, in a newly created calculus. This demarcation of expressions without mathematical sense and proved or refuted propositions, each with a determinate sense in a particular calculus, is a view that Wittgenstein articulates in myriad different ways from 1929 through 1944.
如果，例如，我們成功地通過數學歸納法證明 GC（即，我們 證明 “ G(1)$G(1)$ ” 和 “ G(n)→G(n+1)$G(n)\to G(n+1)$ ”), 我們將擁有歸納步驟的證明，但由於歸納步驟在此之前並不是算法可決的 (PR §§148, 155, 157; PG 380)，在構建證明的過程中，我們構建了一個 新的 演算，一個新的 計算機 (WVC 106)，在這個計算機中，我們 現在知道如何 使用這個新的“機器部件” (RFM VI, §13)（即，未經系統證明的歸納步驟）。在證明之前，歸納步驟並不是一個具有意義的數學命題（在特定的演算中），而在證明之後，歸納步驟 是 一個數學命題，具有新的、確定的意義，在新創建的演算中。這種對於沒有數學意義的表達和已證明或被駁斥的命題的劃分，每個命題在特定的演算中都有確定的意義，是維根斯坦在 1929 年至 1944 年間以無數不同方式表達的觀點。

Whether or not it is ultimately defensible—and this is an absolutely crucial question for Wittgenstein’s Philosophy of Mathematics—this strongly counter-intuitive aspect of Wittgenstein’s account of algorithmic decidability, proof, and the sense of a mathematical proposition is a piece with his rejection of predeterminacy in mathematics. Even in the case where we algorithmically decide a mathematical proposition, the connections thereby made do not pre-exist the algorithmic decision, which means that even when we have a “mathematical question” that we decide by decision procedure, the expression only has a determinate sense qua proposition when it is decided. On Wittgenstein’s account, both middle and later, “[a] new proof gives the proposition a place in a new system” (RFM VI, §13), it “locates it in the whole system of calculations”, though it “does not mention, certainly does not describe, the whole system of calculation that stands behind the proposition and gives it sense” (RFM VI, §11).
無論這是否最終是可辯護的——這對維根斯坦的數學哲學來說是一個絕對關鍵的問題——維根斯坦對算法可決性、證明和數學命題的意義的這一強烈反直覺的方面與他對數學中預定性的拒絕是一致的。即使在我們算法決定一個數學命題的情況下，由此產生的聯繫並不存在於算法決定之前，這意味著即使我們有一個通過決策程序決定的“數學問題”，該表達只有在被決定時才具有確定的意義作為命題。根據維根斯坦的說法，無論是中期還是後期，“[一]個新的證明給予命題在一個新系統中的位置” (RFM VI, §13)，它“將其定位於整個計算系統中”，儘管它“並不提及，當然也不描述，支撐該命題並賦予其意義的整個計算系統” (RFM VI, §11)。

Wittgenstein’s unorthodox position here is a type of structuralism that partially results from his rejection of mathematical semantics. We erroneously think, e.g., that GC has a fully determinate sense because, given “the misleading way in which the mode of expression of word-language represents the sense of mathematical propositions” (PG 375), we call to mind false pictures and mistaken, referential conceptions of mathematical propositions whereby GC is about a mathematical reality and so has just a determinate sense as “There exist intelligent beings elsewhere in the universe” (i.e., a proposition that is determinately true or false, whether or not we ever know its truth-value). Wittgenstein breaks with this tradition, in all of its forms, stressing that, in mathematics, unlike the realm of contingent (or empirical) propositions, “if I am to know what a proposition like Fermat’s last theorem says”, I must know its criterion of truth. Unlike the criterion of truth for an empirical proposition, which can be known before the proposition is decided, we cannot know the criterion of truth for an undecided mathematical proposition, though we are “acquainted with criteria for the truth of similar propositions” (RFM VI, §13).
維特根斯坦在這裡的非正統立場是一種結構主義，部分源於他對數學語義的拒絕。我們錯誤地認為，例如，GC 具有完全確定的意義，因為考慮到“詞語語言的表達方式如何誤導性地代表數學命題的意義”（PG 375），我們想起了錯誤的圖像和錯誤的指稱概念，認為 GC 是關於數學現實的，因此它的意義就像“宇宙中存在智慧生物”一樣是確定的（即，一個是確定真或假的命題，無論我們是否知道它的真值）。維特根斯坦與這一傳統在所有形式上決裂，強調在數學中，與偶然（或經驗）命題的領域不同，“如果我想知道像費馬最後定理這樣的命題說了什麼”，我必須知道它的真理標準。 與經驗命題的真理標準不同，該標準可以在命題被決定之前知道，我們無法知道未決定的數學命題的真理標準，儘管我們“熟悉類似命題的真理標準”（RFM VI, §13）。

2.5 Wittgenstein’s Intermediate Account of Irrational Numbers
2.5 維根斯坦對非理性數的中介說明

The intermediate Wittgenstein spends a great deal of time wrestling with real and irrational numbers. There are two distinct reasons for this.
中間的維根斯坦花了大量時間與實數和虛數搏鬥。這有兩個明確的原因。

First, the real reason many of us are unwilling to abandon the notion of the actual infinite in mathematics is the prevalent conception of an irrational number as a necessarily infinite extension. “The confusion in the concept of the ‘actual infinite’ arises” (italics added), says Wittgenstein (PG 471),
首先，我們許多人不願意放棄數學中實際無限的概念的真正原因，是對無理數的普遍看法，即它必然是無限延伸的。“‘實際無限’的概念中的混淆出現了”（斜體字為補充），維根斯坦說（PG 471），

from the unclear concept of irrational number, that is, from the fact that logically very different things are called ‘irrational numbers’ without any clear limit being given to the concept.
從不清晰的無理數概念出發，即從邏輯上非常不同的事物被稱為「無理數」的事實出發，並且沒有對該概念給出任何明確的界限。

Second, and more fundamentally, the intermediate Wittgenstein wrestles with irrationals in such detail because he opposes foundationalism and especially its concept of a “gapless mathematical continuum”, its concept of a comprehensive theory of the real numbers (Han 2010), and set theoretical conceptions and ‘proofs’ as a foundation for arithmetic, real number theory, and mathematics as a whole. Indeed, Wittgenstein’s discussion of irrationals is one with his critique of set theory, for, as he says, “[m]athematics is ridden through and through with the pernicious idioms of set theory”, such as “the way people speak of a line as composed of points”, when, in fact, “[a] line is a law and isn’t composed of anything at all” (PR §173; PR §§181, 183, & 191; PG 373, 460, 461, & 473).
其次，更根本的是，中期的維根斯坦詳細探討無理數，因為他反對基礎主義，特別是其“無間隙的數學連續體”概念、對實數的“全面”理論（Han 2010）以及將集合論的概念和“證明”作為算術、實數理論和整個數學的基礎。事實上，維根斯坦對無理數的討論與他對集合論的批評是一體的，正如他所說，“數學充斥著集合論的有害語言”，例如“人們談論一條線是由點組成的”，而事實上，“一條線是一條法則，根本不由任何東西組成”（PR §173；PR §§181, 183, & 191；PG 373, 460, 461, & 473）。

2.5.1 Wittgenstein’s Anti-Foundationalism and Genuine Irrational Numbers
2.5.1 維特根斯坦的反基礎主義與真正的非理性數字

Since, on Wittgenstein’s terms, mathematics consists exclusively of extensions and intensions (i.e., ‘rules’ or ‘laws’), an irrational is only an extension insofar as it is a sign (i.e., a ‘numeral’, such as ‘2‾√$\sqrt{2}$’ or ‘π$\pi$’). Given that there is no such thing as an infinite mathematical extension, it follows that an irrational number is not a unique infinite expansion, but rather a unique recursive rule or law (PR §181) that yields rational numbers (PR §186; PR §180).
由於在維根斯坦的術語中，數學完全由 擴展和意圖（即「規則」或 「法律」），一個不理性的東西僅僅是一個延伸，因為它 是一个符号（即“数字”，例如 ‘ 2‾√$\sqrt{2}$ ’ 或 ‘ π$\pi$ ’。 由於不存在無限的數學擴展，因此不理性數字並不是唯一的無限展開，而是一個獨特的遞歸規則或法則（PR §181），產生有理數（PR §186；PR §180）。

The rule for working out places of 2‾√$\sqrt{2}$ is itself the numeral for the irrational number; and the reason I here speak of a ‘number’ is that I can calculate with these signs (certain rules for the construction of rational numbers) just as I can with rational numbers themselves. (PG 484)
計算 2‾√$\sqrt{2}$ 的位置的規則本身就是這個無理數的數字；我在這裡提到「數字」的原因是，我可以用這些符號（某些有理數的構造規則）進行計算，就像我可以用有理數本身進行計算一樣。

Due, however, to his anti-foundationalism, Wittgenstein takes the radical position that not all recursive real numbers (i.e., computable numbers) are genuine real numbers—a position that distinguishes his view from even Brouwer’s.
然而，由於他的反基礎主義，維根斯坦採取了激進的立場，認為並非所有的遞歸實數（即可計算數字）都是真正的實數——這一立場使他的觀點與布勞威爾的觀點有所區別。

The problem, as Wittgenstein sees it, is that mathematicians, especially foundationalists (e.g., set theorists), have sought to accommodate physical continuity by a theory that ‘describes’ the mathematical continuum (PR §171). When, for example, we think of continuous motion and the (mere) density of the rationals, we reason that if an object moves continuously from A to B, and it travels only the distances marked by “rational points”, then it must skip some distances (intervals, or points) not marked by rational numbers. But if an object in continuous motion travels distances that cannot be commensurately measured by rationals alone, there must be ‘gaps’ between the rationals (PG 460), and so we must fill them, first, with recursive irrationals, and then, because “the set of all recursive irrationals” still leaves gaps, with “lawless irrationals”.
問題在於，正如維根斯坦所見，數學家，特別是基礎主義者（例如集合論者），試圖通過一種“描述”數學連續體的理論來適應物理連續性（PR §171）。例如，當我們想到連續運動和（僅僅）有理數的稠密性時，我們推理如果一個物體從 A 點連續移動到 B 點，並且它僅僅走過標記為“有理點”的距離，那麼它必須“跳過”一些未被有理數標記的距離（區間或點）。但是如果一個在連續運動中的物體走過的距離“無法”僅用有理數來度量，那麼在有理數之間必須存在“空隙”（PG 460），因此我們必須首先用遞歸無理數來填補這些空隙，然後，因為“所有遞歸無理數的集合”仍然留下空隙，所以用“無法律的無理數”來填補。

[T]he enigma of the continuum arises because language misleads us into applying to it a picture that doesn’t fit. Set theory preserves the inappropriate picture of something discontinuous, but makes statements about it that contradict the picture, under the impression that it is breaking with prejudices; whereas what should really have been done is to point out that the picture just doesn’t fit… (PG 471)
連續體的謎團出現是因為語言誤導我們將一個不合適的圖像應用於它。集合論保留了不適當的斷續性圖像，但對其做出的陳述卻與這個圖像相矛盾，給人一種打破偏見的印象；而實際上應該做的是指出這個圖像根本不合適……（PG 471）

We add nothing that is needed to the differential and integral calculi by ‘completing’ a theory of real numbers with pseudo-irrationals and lawless irrationals, first because there are no gaps on the number line (PR §§181, 183, & 191; PG 373, 460, 461, & 473; WVC 35) and, second, because these alleged irrational numbers are not needed for a theory of the ‘continuum’ simply because there is no mathematical continuum. As the later Wittgenstein says (RFM V, §32), “[t]he picture of the number line is an absolutely natural one up to a certain point; that is to say so long as it is not used for a general theory of real numbers”. We have gone awry by misconstruing the nature of the geometrical line as a continuous collection of points, each with an associated real number, which has taken us well beyond the ‘natural’ picture of the number line in search of a “general theory of real numbers” (Han 2010).
我們通過用偽無理數和無法無天的無理數“完善”實數理論，並沒有為微積分添加任何必要的內容，首先是因為數線上沒有空隙（PR §§181, 183, & 191; PG 373, 460, 461, & 473; WVC 35），其次是因為這些所謂的無理數對於“連續體”的理論並不需要，因為根本不存在數學上的連續體。正如後期維根斯坦所說（RFM V, §32），“數線的圖像在某一點之前是絕對自然的；也就是說，只要它不被用於實數的一般理論”。我們因為誤解了幾何線的本質，將其視為一個連續的點的集合，每個點都有一個相關的實數，這使我們在尋找“實數的一般理論”時，遠遠超出了數線的“自然”圖像（Han 2010）。

Thus, the principal reason Wittgenstein rejects certain constructive (computable) numbers is that they are unnecessary creations which engender conceptual confusions in mathematics (especially set theory). One of Wittgenstein’s main aims in his lengthy discussions of rational numbers and pseudo-irrationals is to show that pseudo-irrationals, which are allegedly needed for the mathematical continuum, are not needed at all.
因此，維根斯坦拒絕某些建構性（可計算）數字的主要原因是它們是多餘的創造，會在數學（特別是集合論）中產生概念上的混淆。維根斯坦在他對有理數和偽無理數的冗長討論中的主要目標之一是表明，所謂的數學連續體所需的偽無理數根本不需要。

To this end, Wittgenstein demands (a) that a real number must be “compar[able] with any rational number taken at random” (i.e., “it can be established whether it is greater than, less than, or equal to a rational number” (PR §191)) and (b) that “[a] number must measure in and of itself” and if a ‘number’ “leaves it to the rationals, we have no need of it” (PR §191) (Frascolla 1980: 242–243; Shanker 1987: 186–192; Da Silva 1993: 93–94; Marion 1995a: 162, 164; Rodych 1999b, 281–291; Lampert 2009).
為此，維根斯坦要求 (a) 一個實數必須能夠“與任何隨機取出的有理數進行比較”（即“可以確定它是否大於、小於或等於某個有理數” (PR §191)）以及 (b) “[一]個數字必須本身就能進行測量”，如果一個‘數字’“將其留給有理數，我們就不需要它” (PR §191) (Frascolla 1980: 242–243; Shanker 1987: 186–192; Da Silva 1993: 93–94; Marion 1995a: 162, 164; Rodych 1999b, 281–291; Lampert 2009)。

To demonstrate that some recursive (computable) reals are not genuine real numbers because they fail to satisfy (a) and (b), Wittgenstein defines the putative recursive real number
為了證明某些遞歸（可計算）實數並不是真正的實數，因為它們未能滿足（a）和（b），維根斯坦定義了假定的遞歸實數

as the rule “Construct the decimal expansion for 2‾√$\sqrt{2}$, replacing every occurrence of a ‘5’ with a ‘3’” (PR §182); he similarly defines π′${\pi }^{\prime }$ as
根據規則“構造 2‾√$\sqrt{2}$ 的十進制展開，將每個‘5’替換為‘3’”（PR §182）；他同樣定義 π′${\pi }^{\prime }$ 為

(PR §186) and, in a later work, redefines π′${\pi }^{\prime }$ as
(PR §186) 並且在後來的作品中，重新定義 π′${\pi }^{\prime }$ 為

(PG 475). (PG 475)。

Although a pseudo-irrational such as π′${\pi }^{\prime }$ (on either definition) is “as unambiguous as … π$\pi$ or 2‾√$\sqrt{2}$” (PG 476), it is ‘homeless’ according to Wittgenstein because, instead of using “the idioms of arithmetic” (PR §186), it is dependent upon the particular ‘incidental’ notation of a particular system (i.e., in some particular base) (PR §188; PR §182; and PG 475). If we speak of various base-notational systems, we might say that π$\pi$ belongs to all systems, while π′${\pi }^{\prime }$ belongs only to one, which shows that π′${\pi }^{\prime }$ is not a genuine irrational because “there can’t be irrational numbers of different types” (PR §180). Furthermore, pseudo-irrationals do not measure because they are homeless, artificial constructions parasitic upon numbers which have a natural place in a calculus that can be used to measure. We simply do not need these aberrations, because they are not sufficiently comparable to rationals and genuine irrationals. They are not irrational numbers according to Wittgenstein’s criteria, which define, Wittgenstein interestingly asserts, “precisely what has been meant or looked for under the name ‘irrational number’” (PR §191).
儘管像 π′${\pi }^{\prime }$ 這樣的偽無理數（無論哪種定義）是“與… π$\pi$ 或 2‾√$\sqrt{2}$ 一樣明確”（PG 476），但根據維根斯坦的說法，它是“無家可歸”的，因為它不是使用“算術的習語”（PR §186），而是依賴於特定系統的特定“偶然”符號（即，在某個特定的基數中）（PR §188；PR §182；和PG 475）。如果我們談論各種基數符號系統，我們可以說 π$\pi$ 屬於所有系統，而 π′${\pi }^{\prime }$ 僅屬於一個，這顯示 π′${\pi }^{\prime }$ 不是一個真正的無理數，因為“不同類型的無理數是不存在的”（PR §180）。此外，偽無理數不能測量，因為它們是無家可歸的，依賴於在可以用來測量的微積分中具有自然位置的數字的人工構造。我們根本不需要這些偏差，因為它們與有理數和真正的無理數相比不夠可比。 根據維根斯坦的標準，他們不是不理性的數字，維根斯坦有趣地聲稱，這些標準“準確地定義了在‘不理性數字’這個名稱下所意味或尋求的東西”（PR §191）。

For exactly the same reason, if we define a “lawless irrational” as either (a) a non-rule-governed, non-periodic, infinite expansion in some base, or (b) a “free-choice sequence”, Wittgenstein rejects “lawless irrationals” because, insofar as they are not rule-governed, they are not comparable to rationals (or irrationals) and they are not needed.
出於完全相同的原因，如果我們將“無法則的無理數”定義為 (a) 在某個基數下的非規則、非周期性、無限擴展，或 (b) “自由選擇序列”，維根斯坦拒絕“無法則的無理數”，因為在它們不受規則約束的情況下，它們無法與有理數（或無理數）進行比較，並且不需要它們。

[W]e cannot say that the decimal fractions developed in accordance with a law still need supplementing by an infinite set of irregular infinite decimal fractions that would be ‘brushed under the carpet’ if we were to restrict ourselves to those generated by a law,
我們不能說根據某個法則發展出來的小數分數仍然需要由一組無限的不規則無限小數分數來補充，如果我們僅限於那些由法則產生的話，這些將會被“掩蓋”

Wittgenstein argues, for “[w]here is there such an infinite decimal that is generated by no law” “[a]nd how would we notice that it was missing?” (PR §181; cf. PG 473, 483–84). Similarly, a free-choice sequence, like a recipe for “endless bisection” or “endless dicing”, is not an infinitely complicated mathematical law (or rule), but rather no law at all, for after each individual throw of a coin, the point remains “infinitely indeterminate” (PR §186). For closely related reasons, Wittgenstein ridicules the Multiplicative Axiom (Axiom of Choice) both in the middle period (PR §146) and in the latter period (RFM V, §25; VII, §33).
維特根斯坦主張，“[w]哪裡有這樣一個不受任何法則生成的無限小數” “[a]我們怎麼會注意到它的缺失？” (PR §181; cf. PG 473, 483–84)。同樣，自由選擇序列，如“無盡二分”或“無盡切割”的食譜，並不是一個無限複雜的數學法則（或規則），而是根本沒有法則，因為在每次擲硬幣之後，點仍然是“無限不確定的” (PR §186)。出於密切相關的原因，維特根斯坦在中期(PR §146)和後期(RFM V, §25; VII, §33)都嘲諷了乘法公理（選擇公理）。

2.5.2 Wittgenstein’s Real Number Essentialism and the Dangers of Set Theory
2.5.2 維特根斯坦的實數本質主義與集合論的危險

Superficially, at least, it seems as if Wittgenstein is offering an essentialist argument for the conclusion that real number arithmetic should not be extended in such-and-such a way. Such an essentialist account of real and irrational numbers seems to conflict with the actual freedom mathematicians have to extend and invent, with Wittgenstein’s intermediate claim (PG 334) that “[f]or [him] one calculus is as good as another”, and with Wittgenstein’s acceptance of complex and imaginary numbers. Wittgenstein’s foundationalist critic (e.g., set theorist) will undoubtedly say that we have extended the term “irrational number” to lawless and pseudo-irrationals because they are needed for the mathematical continuum and because such “conceivable numbers” are much more like rule-governed irrationals than rationals.
表面上看，維根斯坦似乎在提供一個本質主義的論證，得出結論認為實數算術不應該以這樣那樣的方式擴展。這種本質主義對實數和無理數的解釋似乎與數學家實際上擁有的擴展和創造的自由相衝突，與維根斯坦的中間主張（PG 334）“對[他]來說，一種微積分和另一種一樣好”，以及維根斯坦對複數和虛數的接受相矛盾。維根斯坦的基礎主義批評者（例如集合論者）無疑會說，我們已經將“無理數”這個術語擴展到無法無天和偽無理數，因為它們對數學連續體是必要的，並且這些“可想像的數字”與受規則約束的無理數相比，更像是無理數而不是有理數。

Though Wittgenstein stresses differences where others see similarities (LFM 15), in his intermediate attacks on pseudo-irrationals and foundationalism, he is not just emphasizing differences, he is attacking set theory’s “pernicious idioms” (PR §173) and its “crudest imaginable misinterpretation of its own calculus” (PG 469–70) in an attempt to dissolve “misunderstandings without which [set theory] would never have been invented”, since it is “of no other use” (LFM 16–17). Complex and imaginary numbers have grown organically within mathematics, and they have proved their mettle in scientific applications, but pseudo-irrationals are inorganic creations invented solely for the sake of mistaken foundationalist aims. Wittgenstein’s main point is not that we cannot create further recursive real numbers—indeed, we can create as many as we want—his point is that we can only really speak of different systems (sets) of real numbers (RFM II, §33) that are enumerable by a rule, and any attempt to speak of “the set of all real numbers” or any piecemeal attempt to add or consider new recursive reals (e.g., diagonal numbers) is a useless and/or futile endeavour based on foundational misconceptions. Indeed, in 1930 manuscript and typescript (hereafter MS and TS, respectively) passages on irrationals and Cantor’s diagonal, which were not included in PR or PG, Wittgenstein says: “The concept ‘irrational number’ is a dangerous pseudo-concept” (MS 108, 176; 1930; TS 210, 29; 1930). As we shall see in the next section, on Wittgenstein’s account, if we do not understand irrationals rightly, we cannot but engender the mistakes that constitute set theory.
儘管維根斯坦強調差異，而其他人則看到相似之處（LFM 15），在他對偽非理性和基礎主義的中間攻擊中，他不僅僅是在強調差異，他還在攻擊集合論的“有害成語”（PR §173）及其“對自身計算的最粗糙的可想像的誤解”（PG 469–70），試圖解決“沒有這些誤解，集合論就不會被發明”的問題，因為它“沒有其他用途”（LFM 16–17）。複數和虛數在數學中有機地發展，並在科學應用中證明了它們的價值，但偽非理性則是為了錯誤的基礎主義目標而發明的無機創造物。 維特根斯坦的主要觀點並不是我們無法創造更多的遞歸實數——事實上，我們可以創造任意多的——他的觀點是我們只能真正談論不同的實數系統（集合）（RFM II, §33），這些系統是可以通過某種規則來枚舉的，任何試圖談論“所有實數的集合”或任何零散的嘗試去添加或考慮新的遞歸實數（例如，對角數）都是基於根本誤解的無用和/或徒勞的努力。事實上，在 1930 年的手稿和打字稿（以下分別稱為 MS 和 TS）中，關於無理數和康托爾的對角線的段落，這些段落並未包含在 PR 或 PG 中，維特根斯坦說：“‘無理數’這個概念是一個危險的偽概念”（MS 108, 176; 1930; TS 210, 29; 1930）。正如我們在下一節中將看到的，根據維特根斯坦的說法，如果我們不正確理解無理數，我們就無法不產生構成集合論的錯誤。

2.6 Wittgenstein’s Intermediate Critique of Set Theory
維特根斯坦對集合論的中介批評 2.6

Wittgenstein’s critique of set theory begins somewhat benignly in the Tractatus, where he denounces Logicism and says (6.031) that “[t]he theory of classes is completely superfluous in mathematics” because, at least in part, “the generality required in mathematics is not accidental generality”. In his middle period, Wittgenstein begins a full-out assault on set theory that never abates. Set theory, he says, is “utter nonsense” (PR §§145, 174; WVC 102; PG 464, 470), ‘wrong’ (PR §174), and ‘laughable’ (PG 464); its “pernicious idioms” (PR §173) mislead us and the crudest possible misinterpretation is the very impetus of its invention (Hintikka 1993: 24, 27).
維特根斯坦對集合論的批評在《邏輯哲學論》中開始時顯得有些無害，他譴責邏輯主義並說（6.031）“在數學中，類的理論是完全多餘的”，因為至少在某種程度上，“數學所需的普遍性並不是偶然的普遍性”。在他的中期，維特根斯坦對集合論展開了全面的攻擊，這種攻擊從未減弱。他說，集合論是“完全的胡說”（《哲學研究》§§145, 174；WVC 102；《藍皮書》464, 470），是“錯誤的”（《哲學研究》§174），是“可笑的”（《藍皮書》464）；它的“有害的習語”（《哲學研究》§173）誤導了我們，而最粗糙的誤解正是其發明的動力（Hintikka 1993: 24, 27）。

Wittgenstein’s intermediate critique of transfinite set theory (hereafter “set theory”) has two main components: (1) his discussion of the intension-extension distinction, and (2) his criticism of non-denumerability as cardinality. Late in the middle period, Wittgenstein seems to become more aware of the unbearable conflict between his strong formalism (PG 334) and his denigration of set theory as a purely formal, non-mathematical calculus (Rodych 1997: 217–219), which, as we shall see in Section 3.5, leads to the use of an extra-mathematical application criterion to demarcate transfinite set theory (and other purely formal sign-games) from mathematical calculi.
維特根斯坦對超限集合論的中介批評（以下簡稱“集合論”）有兩個主要組成部分：（1）他對意向-延伸區別的討論，以及（2）他對不可數性作為基數的批評。在中期的晚期，維特根斯坦似乎越來越意識到他強烈形式主義（PG 334）與他對集合論的貶低之間無法忍受的衝突，後者被視為一種純粹的形式、非數學的計算（Rodych 1997: 217–219），正如我們在第 3.5 節中將看到的，這導致使用一個超數學的應用標準來區分超限集合論（以及其他純粹的形式符號遊戲）與數學計算。

2.6.1 Intensions, Extensions, and the Fictitious Symbolism of Set Theory
2.6.1 意圖、擴展與集合論的虛構符號學

The search for a comprehensive theory of the real numbers and mathematical continuity has led to a “fictitious symbolism” (PR §174).
對實數和數學連續性的全面理論的探索導致了一種「虛構符號」(PR §174)。

Set theory attempts to grasp the infinite at a more general level than the investigation of the laws of the real numbers. It says that you can’t grasp the actual infinite by means of mathematical symbolism at all and therefore it can only be described and not represented. … One might say of this theory that it buys a pig in a poke. Let the infinite accommodate itself in this box as best it can. (PG 468; cf. PR §170)
集合論試圖在比實數法則的研究更一般的層面上理解無限。它說你根本無法通過數學符號來把握實際的無限，因此它只能被描述而不能被表示。……人們可以說這個理論是在買一隻不知情的豬。讓無限在這個盒子裡盡可能地適應。

As Wittgenstein puts it at (PG 461),
正如維根斯坦在（PG 461）中所說，

the mistake in the set-theoretical approach consists time and again in treating laws and enumerations (lists) as essentially the same kind of thing and arranging them in parallel series so that one fills in gaps left by the other.
集合論方法中的錯誤一再表現為將法律和列舉（清單）視為本質上相同的事物，並將它們排列成平行系列，以便一個填補另一個留下的空白。

This is a mistake because it is ‘nonsense’ to say “we cannot enumerate all the numbers of a set, but we can give a description”, for “[t]he one is not a substitute for the other” (WVC 102; June 19, 1930); “there isn’t a dualism [of] the law and the infinite series obeying it” (PR §180).
這是一個錯誤，因為說“我們無法列舉一個集合的所有數字，但我們可以給出一個描述”是“無稽之談”，因為“[t]一者不能替代另一者”（WVC 102；1930 年 6 月 19 日）；“法律和遵循它的無限系列之間並不存在二元論”（PR §180）。

“Set theory is wrong” and nonsensical (PR §174), says Wittgenstein, because it presupposes a fictitious symbolism of infinite signs (PG 469) instead of an actual symbolism with finite signs. The grand intimation of set theory, which begins with “Dirichlet’s concept of a function” (WVC 102–03), is that we can in principle represent an infinite set by an enumeration, but because of human or physical limitations, we will instead describe it intensionally. But, says Wittgenstein, “[t]here can’t be possibility and actuality in mathematics”, for mathematics is an actual calculus, which “is concerned only with the signs with which it actually operates” (PG 469). As Wittgenstein puts it at (PR §159), the fact that “we can’t describe mathematics, we can only do it” in and “of itself abolishes every ‘set theory’”.
「集合論是錯誤的」和無意義的（PR §174），維根斯坦說，因為它假設了一種虛構的無限符號系統（PG 469），而不是一種具有有限符號的實際符號系統。集合論的偉大暗示，始於「狄利克雷的函數概念」（WVC 102–03），是我們可以原則上通過列舉來表示一個無限集合，但由於人類或物理的限制，我們將改為描述它的內涵。但維根斯坦說，「[t]數學中不可能有可能性和現實」，因為數學是一種實際的計算，這「僅關心它實際操作的符號」（PG 469）。正如維根斯坦在（PR §159）所說的，事實上「我們無法描述數學，我們只能做它」，而「這本身就廢除了每一種‘集合論’」。

Perhaps the best example of this phenomenon is Dedekind, who in giving his ‘definition’ of an “infinite class” as “a class which is similar to a proper subclass of itself” (PG 464), “tried to describe an infinite class” (PG 463). If, however, we try to apply this ‘definition’ to a particular class in order to ascertain whether it is finite or infinite, the attempt is ‘laughable’ if we apply it to a finite class, such as “a certain row of trees”, and it is ‘nonsense’ if we apply it to “an infinite class”, for we cannot even attempt “to co-ordinate it” (PG 464), because “the relation m=2n$m=2n$ [does not] correlate the class of all numbers with one of its subclasses” (PR §141), it is an “infinite process” which “correlates any arbitrary number with another”. So, although we can use m=2n$m=2n$ on the rule for generating the naturals (i.e., our domain) and thereby construct the pairs (2,1), (4,2), (6,3), (8,4), etc., in doing so we do not correlate two infinite sets or extensions (WVC 103). If we try to apply Dedekind’s definition as a criterion for determining whether a given set is infinite by establishing a 1–1 correspondence between two inductive rules for generating “infinite extensions”, one of which is an “extensional subset” of the other, we can’t possibly learn anything we didn’t already know when we applied the ‘criterion’ to two inductive rules. If Dedekind or anyone else insists on calling an inductive rule an “infinite set”, he and we must still mark the categorical difference between such a set and a finite set with a determinate, finite cardinality.
或許這一現象最好的例子是德德金，他在給予 他對「無限類別」的「定義」為 “一個類別，其類似於自身的適當子類別” （PG 464），“試圖描述一個無限類別”（PG 463）。然而，如果我們試圖將這個“定義”應用於特定的類別，以確定它是有限還是無限，則如果我們將其應用於一個有限類別，例如“某一排樹”，這種嘗試是“可笑的”；如果我們將其應用於“無限類別”，則這是“無意義的”，因為我們甚至無法嘗試“去協調它”（PG 464），因為“關係 m=2n$m=2n$ [並不]將所有數字的類別與其子類別之一相關聯”（PR §141），這是一個“無限過程”，它“將任何任意數字與另一個相關聯”。因此，儘管我們可以在生成自然數的規則上使用 m=2n$m=2n$ ，從而構建對（2,1），（4,2），（6,3），（8,4）等，但這樣做並不會將兩個無限集合或擴展相關聯（WVC 103）。 如果我們嘗試將德德金的定義作為一個標準，以確定給定的集合是否是無限的，通過在兩個生成“無限擴展”的歸納規則之間建立一對一的對應關係，其中一個是另一個的“外延子集”，那麼當我們將這個‘標準’應用於兩個歸納規則時，我們不可能學到任何我們之前不知道的東西。如果德德金或其他人堅持稱一個歸納規則為“無限集合”，他和我們仍然必須標記這樣的集合與具有確定的有限基數的有限集合之間的類別差異。

Indeed, on Wittgenstein’s account, the failure to properly distinguish mathematical extensions and intensions is the root cause of the mistaken interpretation of Cantor’s diagonal proof as a proof of the existence of infinite sets of lesser and greater cardinality.
確實，在維根斯坦的觀點中，未能正確區分數學的外延和內涵是錯誤解釋康托爾對角證明為存在較小和較大基數的無限集合的根本原因。

2.6.2 Against Non-Denumerability
2.6.2 反對非可數性

Wittgenstein’s criticism of non-denumerability is primarily implicit during the middle period. Only after 1937 does he provide concrete arguments purporting to show, e.g., that Cantor’s diagonal cannot prove that some infinite sets have greater ‘multiplicity’ than others.
維特根斯坦對不可數性的批評在中期主要是隱含的。只有在 1937 年之後，他才提供具體的論據，聲稱例如，康托爾的對角線無法證明某些無限集合的“多重性”大於其他集合。

Nonetheless, the intermediate Wittgenstein clearly rejects the notion that a non-denumerably infinite set is greater in cardinality than a denumerably infinite set.
儘管如此，中期的維根斯坦明確拒絕了非可數無窮集合在基數上大於可數無窮集合的觀念。

When people say ‘The set of all transcendental numbers is greater than that of algebraic numbers’, that’s nonsense. The set is of a different kind. It isn’t ‘no longer’ denumerable, it’s simply not denumerable! (PR §174)
當人們說「所有超越數的集合比代數數的集合更大」時，那是胡說八道。這個集合是不同類型的。它並不是「不再」可數的，它根本就是不可數的！

As with his intermediate views on genuine irrationals and the Multiplicative Axiom, Wittgenstein here looks at the diagonal proof of the non-denumerability of “the set of transcendental numbers” as one that shows only that transcendental numbers cannot be recursively enumerated. It is nonsense, he says, to go from the warranted conclusion that these numbers are not, in principle, enumerable to the conclusion that the set of transcendental numbers is greater in cardinality than the set of algebraic numbers, which is recursively enumerable. What we have here are two very different conceptions of a number-type. In the case of algebraic numbers, we have a decision procedure for determining of any given number whether or not it is algebraic, and we have a method of enumerating the algebraic numbers such that we can see that ‘each’ algebraic number “will be” enumerated. In the case of transcendental numbers, on the other hand, we have proofs that some numbers are transcendental (i.e., non-algebraic), and we have a proof that we cannot recursively enumerate each and every thing we would call a “transcendental number”.
與他對真正的非理性數和乘法公理的中間觀點一樣，維根斯坦在這裡將“超越數的集合”的非可數性對角證明視為僅顯示超越數無法被遞歸列舉的證明。他說，從這些數在原則上不可列舉的合理結論推導出超越數的集合在基數上大於代數數的集合（後者是可遞歸列舉的）是毫無意義的。我們在這裡有兩種非常不同的數字類型概念。在代數數的情況下，我們有一個決策程序來確定任何給定的數是否為代數數，並且我們有一種列舉代數數的方法，使我們可以看到“每一個”代數數“將會”被列舉。另一方面，在超越數的情況下，我們有證明某些數是超越的（即，非代數的），並且我們有證明我們無法遞歸列舉我們所稱的“超越數”的每一個事物。

At (PG 461), Wittgenstein similarly speaks of set theory’s “mathematical pseudo-concepts” leading to a fundamental difficulty, which begins when we unconsciously presuppose that there is sense to the idea of ordering the rationals by size—“that the attempt is thinkable”—and culminates in similarly thinking that it is possible to enumerate the real numbers, which we then discover is impossible.
在（PG 461）中，維根斯坦同樣提到集合論的“數學偽概念”導致了一個根本的困難，這個困難始於我們無意識地假設按大小對有理數進行排序的想法是有意義的——“這個嘗試是可以思考的”——並最終以同樣的方式認為可以列舉實數，而我們隨後發現這是不可能的。

Though the intermediate Wittgenstein certainly seems highly critical of the alleged proof that some infinite sets (e.g., the reals) are greater in cardinality than other infinite sets, and though he discusses the “diagonal procedure” in February 1929 and in June 1930 (MS 106, 266; MS 108, 180), along with a diagonal diagram, these and other early-middle ruminations did not make it into the typescripts for either PR or PG. As we shall see in Section 3.4, the later Wittgenstein analyzes Cantor’s diagonal and claims of non-denumerability in some detail.
雖然中期的維根斯坦似乎對某些無窮集合（例如實數）在基數上大於其他無窮集合的所謂證明持高度批評態度，並且他在 1929 年 2 月和 1930 年 6 月討論了“對角程序”（MS 106, 266; MS 108, 180），以及一個對角圖，但這些和其他早期中期的思考並未進入PR或PG的打字稿中。正如我們在第 3.4 節中將看到的，後期的維根斯坦對康托爾的對角線和不可數性的主張進行了詳細分析。

3. The Later Wittgenstein on Mathematics: Some Preliminaries
3. 後期維根斯坦的數學：一些初步探討

The first and most important thing to note about Wittgenstein’s later Philosophy of Mathematics is that RFM, first published in 1956, consists of selections taken from a number of manuscripts (1937–1944), most of one large typescript (1938), and three short typescripts (1938), each of which constitutes an Appendix to (RFM I). For this reason and because some manuscripts containing much material on mathematics (e.g., MS 123) were not used at all for RFM, philosophers have not been able to read Wittgenstein’s later remarks on mathematics as they were written in the manscripts used for RFM and they have not had access (until the 2000–2001 release of the Nachlass on CD-ROM) to much of Wittgenstein’s later work on mathematics. It must be emphasized, therefore, that this Encyclopedia article is being written during a transitional period. Until philosophers have used the Nachlass to build a comprehensive picture of Wittgenstein’s complete and evolving Philosophy of Mathematics, we will not be able to say definitively which views the later Wittgenstein retained, which he changed, and which he dropped. In the interim, this article will outline Wittgenstein’s later Philosophy of Mathematics, drawing primarily on RFM, to a much lesser extent LFM (1939 Cambridge lectures), and, where possible, previously unpublished material in Wittgenstein’s Nachlass.
關於維根斯坦後期的數學哲學，首先也是最重要的一點是，RFM首次於 1956 年出版，包含了從多份手稿（1937–1944 年）中選取的選段，其中大部分來自一份大型打字稿（1938 年）和三份短打字稿（1938 年），每一份都構成了(RFM I)的附錄。因此，由於一些包含大量數學材料的手稿（例如，MS 123）根本沒有被用於RFM，哲學家們無法閱讀維根斯坦在用於RFM的手稿中寫下的後期數學評論，並且在 2000–2001 年Nachlass的 CD-ROM 發行之前，他們也無法接觸到維根斯坦後期的數學作品。因此，必須強調的是，這篇百科全書文章是在一個過渡時期撰寫的。 直到哲學家們利用Nachlass來建立對維根斯坦完整且不斷演變的數學哲學的全面理解，我們將無法明確地說出後期維根斯坦保留了哪些觀點，改變了哪些觀點，以及放棄了哪些觀點。在此期間，本文將概述維根斯坦的後期數學哲學，主要依據RFM，在較小程度上依據LFM（1939 劍橋講座），並在可能的情況下，使用維根斯坦的Nachlass中未發表的材料。

It should also be noted at the outset that commentators disagree about the continuity of Wittgenstein’s middle and later Philosophies of Mathematics. Some argue that the later views are significantly different from the intermediate views (Frascolla 1994; Gerrard 1991: 127, 131–32; Floyd 2005: 105–106), while others argue that, for the most part, Wittgenstein’s Philosophy of Mathematics evolves from the middle to the later period without significant changes or renunciations (Wrigley 1993; Marion 1998). The remainder of this article adopts the second interpretation, explicating Wittgenstein’s later Philosophy of Mathematics as largely continuous with his intermediate views, except for the important introduction of an extra-mathematical application criterion.
值得注意的是，評論家對維根斯坦中期和後期數學哲學的連續性存在分歧。一些人認為後期觀點與中期觀點有顯著不同（Frascolla 1994；Gerrard 1991：127，131–32；Floyd 2005：105–106），而另一些人則認為，維根斯坦的數學哲學在很大程度上是從中期演變到後期，沒有重大變化或放棄（Wrigley 1993；Marion 1998）。本文的其餘部分採取第二種解釋，闡明維根斯坦的後期數學哲學在很大程度上與他的中期觀點保持連續，除了引入了一個重要的超數學應用標準。

3.1 Mathematics as a Human Invention
3.1 數學作為人類的發明

Perhaps the most important constant in Wittgenstein’s Philosophy of Mathematics, middle and late, is that he consistently maintains that mathematics is our, human invention, and that, indeed, everything in mathematics is invented. Just as the middle Wittgenstein says that “[w]e make mathematics”, the later Wittgenstein says that we ‘invent’ mathematics (RFM I, §168; II, §38; V, §§5, 9 and 11; PG 469–70) and that “the mathematician is not a discoverer: he is an inventor” (RFM, Appendix II, §2; (LFM 22, 82). Nothing exists mathematically unless and until we have invented it.
或許在維根斯坦的數學哲學中，中期和晚期最重要的常數是他始終堅持數學是我們人類的發明，事實上，數學中的一切都是被發明的。正如中期的維根斯坦所說的“我們創造數學”，晚期的維根斯坦則說我們‘發明’數學（RFM I, §168; II, §38; V, §§5, 9 和 11; PG 469–70），並且“數學家不是發現者：他是發明者”（RFM，附錄 II, §2; (LFM 22, 82)。除非我們已經發明了它，否則沒有任何東西在數學上存在。

In arguing against mathematical discovery, Wittgenstein is not just rejecting Platonism, he is also rejecting a rather standard philosophical view according to which human beings invent mathematical calculi, but once a calculus has been invented, we thereafter discover finitely many of its infinitely many provable and true theorems. As Wittgenstein himself asks (RFM IV, §48), “might it not be said that the rules lead this way, even if no one went it?” If “someone produced a proof [of ‘Goldbach’s theorem’]”, “[c]ouldn’t one say”, Wittgenstein asks (LFM 144), “that the possibility of this proof was a fact in the realms of mathematical reality”—that “[i]n order [to] find it, it must in some sense be there”—“[i]t must be a possible structure”?
在反對數學發現的論證中，維根斯坦不僅僅是在拒絕柏拉圖主義，他還在拒絕一種相當標準的哲學觀點，根據這種觀點，人類發明數學計算，而一旦計算被發明，我們隨後發現其無限多可證明和真實的定理中的有限多個。正如維根斯坦自己所問（RFM IV, §48），“難道不能說這些規則引導著這個方向，即使沒有人走過它？”如果“有人提供了一個[‘哥德巴赫猜想’的]證明”，維根斯坦問道（LFM 144），“難道不能說”，這個證明的可能性在數學現實的領域中是一個事實——“為了找到它，它必須在某種意義上存在”——“它必須是一個可能的結構”？

Unlike many or most philosophers of mathematics, Wittgenstein resists the ‘Yes’ answer that we discover truths about a mathematical calculus that come into existence the moment we invent the calculus (PR §141; PG 283, 466; LFM 139). Wittgenstein rejects the modal reification of possibility as actuality—that provability and constructibility are (actual) facts—by arguing that it is at the very least wrong-headed to say with the Platonist that because “a straight line can be drawn between any two points,… the line already exists even if no one has drawn it”—to say “[w]hat in the ordinary world we call a possibility is in the geometrical world a reality” (LFM 144; RFM I, §21). One might as well say, Wittgenstein suggests (PG 374), that “chess only had to be discovered, it was always there!”
與許多或大多數數學哲學家不同，維根斯坦拒絕了「是」的回答，即我們發現關於數學演算的真理，這些真理在我們發明演算的那一刻就「產生」了。維根斯坦反對將可能性具體化為現實的模式化觀點——即可證明性和可構造性是（實際的）事實——他認為，至少可以說，與柏拉圖主義者的觀點相反，說「一條直線可以在任何兩點之間畫出……這條線已經存在，即使沒有人畫過它」是錯誤的——說「在我們所稱的普通世界中的可能性，在幾何世界中是一種現實」。維根斯坦暗示，或許可以這樣說，「棋局只需被發現，它一直都在！」

At MS 122 (3v; Oct. 18, 1939), Wittgenstein once again emphasizes the difference between illusory mathematical discovery and genuine mathematical invention.
在 MS 122（3v；1939 年 10 月 18 日），維根斯坦再次強調了虛幻的數學發現與真正的數學創造之間的區別。

I want to get away from the formulation: “I now know more about the calculus”, and replace it with “I now have a different calculus”. The sense of this is always to keep before one’s eyes the full scale of the gulf between a mathematical knowing and non-mathematical knowing.^[3]
我想擺脫這種表述：“我現在對微積分有了更多的了解”，並將其替換為“我現在有了一種不同的微積分”。這樣的意義始終是要讓人時刻記住數學知識和非數學知識之間的巨大鴻溝。

And as with the middle period, the later Wittgenstein similarly says (MS 121, 27r; May 27, 1938) that “[i]t helps if one says: the proof of the Fermat proposition is not to be discovered, but to be invented”.
而與中期相似，晚期的維根斯坦同樣說道（MS 121, 27r；1938 年 5 月 27 日）：“如果說：費馬命題的證明不是要被發現，而是要被發明，這是有幫助的。”

The difference between the ‘anthropological’ and the mathematical account is that in the first we are not tempted to speak of ‘mathematical facts’, but rather that in this account the facts are never mathematical ones, never make mathematical propositions true or false. (MS 117, 263; March 15, 1940)
“人類學”的說法和數學的說法之間的區別在於，在前者中，我們不會被誘惑去談論“數學事實”，而是這種說法中的事實從來不是數學的，從來不會使數學命題為真或為假。（MS 117, 263；1940 年 3 月 15 日）

There are no mathematical facts just as there are no (genuine) mathematical propositions. Repeating his intermediate view, the later Wittgenstein says (MS 121, 71v; 27 Dec., 1938): “Mathematics consists of [calculi | calculations], not of propositions”. This radical constructivist conception of mathematics prompts Wittgenstein to make notorious remarks—remarks that virtually no one else would make—such as the infamous (RFM V, §9): “However queer it sounds, the further expansion of an irrational number is a further expansion of mathematics”.
數學事實並不存在，就像（真正的）數學命題不存在一樣。重申他的中間觀點，後期的維根斯坦說（MS 121, 71v; 1938 年 12 月 27 日）：“數學由[計算 | 計算]組成，而不是由命題組成”。這種激進的建構主義數學觀促使維根斯坦發表了臭名昭著的言論——幾乎沒有人會這樣說——例如那句臭名昭著的（RFM V, §9）：“無論聽起來多麼奇怪，無理數的進一步擴展就是數學的進一步擴展”。

3.1.1 Wittgenstein’s Later Anti-Platonism: The Natural History of Numbers and the Vacuity of Platonism
3.1.1 維特根斯坦的後期反柏拉圖主義：數字的自然歷史與柏拉圖主義的空虛

As in the middle period, the later Wittgenstein maintains that mathematics is essentially syntactical and non-referential, which, in and of itself, makes Wittgenstein’s philosophy of mathematics anti-Platonist insofar as Platonism is the view that mathematical terms and propositions refer to objects and/or facts and that mathematical propositions are true by virtue of agreeing with mathematical facts.
正如中期的維根斯坦所言，後期的維根斯坦認為數學本質上是語法性的和非指稱性的，這本身使得維根斯坦的數學哲學在某種程度上是反柏拉圖主義，因為柏拉圖主義的觀點認為數學術語和命題指稱對象和/或事實，並且數學命題因與數學事實一致而為真。

The later Wittgenstein, however, wishes to ‘warn’ us that our thinking is saturated with the idea of “[a]rithmetic as the natural history (mineralogy) of numbers” (RFM IV, §11). When, for instance, Wittgenstein discusses the claim that fractions cannot be ordered by magnitude, he says that this sounds ‘remarkable’ in a way that a mundane proposition of the differential calculus does not, for the latter proposition is associated with an application in physics,
然而，晚期的維根斯坦希望“警告”我們，我們的思維充滿了“[數字的]算術作為自然歷史（礦物學）”的觀念（RFM IV, §11）。例如，當維根斯坦討論分數不能按大小排序的說法時，他表示這聽起來“引人注目”，而這種引人注目的方式是微分計算中的平凡命題所不具備的，因為後者的命題與物理學中的應用相關。

whereas this proposition … seems to [solely] concern… the natural history of mathematical objects themselves. (RFM II, §40)
然而 這個命題 … 似乎僅僅關乎… 數學對象本身的自然歷史。 (RFM II, §40)

Wittgenstein stresses that he is trying to ‘warn’ us against this ‘aspect’—the idea that the foregoing proposition about fractions “introduces us to the mysteries of the mathematical world”, which exists somewhere as a completed totality, awaiting our prodding and our discoveries. The fact that we regard mathematical propositions as being about mathematical objects and mathematical investigation “as the exploration of these objects” is “already mathematical alchemy”, claims Wittgenstein (RFM V, §16), since
維特根斯坦強調他試圖“警告”我們注意這個“方面”——即前述關於分數的命題“將我們引入數學世界的神秘”，這個世界在某處作為一個完整的總體，等待著我們的推動和發現。我們將數學命題視為關於數學對象的，並將數學研究視為“對這些對象的探索”，這“已經是數學煉金術”，維特根斯坦聲稱（RFM V, §16），因為

it is not possible to appeal to the meaning [Bedeutung] of the signs in mathematics,… because it is only mathematics that gives them their meaning [Bedeutung].
在數學中，無法訴諸於符號的意義 [Bedeutung]，……因為只有數學賦予它們意義 [Bedeutung]。

Platonism is dangerously misleading, according to Wittgenstein, because it suggests a picture of pre-existence, predetermination and discovery that is completely at odds with what we find if we actually examine and describe mathematics and mathematical activity. “I should like to be able to describe”, says Wittgenstein (RFM IV, §13), “how it comes about that mathematics appears to us now as the natural history of the domain of numbers, now again as a collection of rules”.
根據維根斯坦，柏拉圖主義是危險地誤導，因為它暗示了一種先存、先決定和發現的圖景，這與我們如果實際檢查和描述數學及數學活動所發現的完全相悖。“我希望能夠描述”，維根斯坦說（RFM IV, §13），“數學是如何出現在我們面前的，現在作為數字領域的自然歷史，現在又作為一組規則。”

Wittgenstein, however, does not endeavour to refute Platonism. His aim, instead, is to clarify what Platonism is and what it says, implicitly and explicitly (including variants of Platonism that claim, e.g., that if a proposition is provable in an axiom system, then there already exists a path [i.e., a proof] from the axioms to that proposition (RFM I, §21; Marion 1998: 13–14, 226; Steiner 2000: 334). Platonism is either “a mere truism” (LFM 239), Wittgenstein says, or it is a ‘picture’ consisting of “an infinity of shadowy worlds” (LFM 145), which, as such, lacks ‘utility’ (cf. PI §254) because it explains nothing and it misleads at every turn.
然而，維根斯坦並不試圖反駁柏拉圖主義。他的目標是澄清柏拉圖主義是什麼以及它所說的內容，無論是隱含還是明示（包括聲稱例如如果一個命題在公理系統中是可證明的，那麼從公理到該命題之間已經存在一條路徑[即證明]的柏拉圖主義變體（《RFM》I, §21；Marion 1998: 13–14, 226；Steiner 2000: 334）。維根斯坦說，柏拉圖主義要麼是“僅僅是個真理”（《LFM》239），要麼是一種由“無限的陰影世界”組成的‘圖像’（《LFM》145），因此缺乏‘實用性’（參見《PI》§254），因為它什麼都解釋不了，並且在每一個轉折處都會誤導。

3.2 Wittgenstein’s Later Finitistic Constructivism
3.2 維特根斯坦的後期有限主義建構論

Though commentators and critics do not agree as to whether the later Wittgenstein is still a finitist and whether, if he is, his finitism is as radical as his intermediate rejection of unbounded mathematical quantification (Maddy 1986: 300–301, 310), the overwhelming evidence indicates that the later Wittgenstein still rejects the actual infinite (RFM V, §21; Zettel §274, 1947) and infinite mathematical extensions.
儘管評論家和批評家對於後期維根斯坦是否仍然是一位有限主義者以及如果是，他的有限主義是否與他對無界數學量化的中間拒絕一樣激進（Maddy 1986: 300–301, 310）並不一致，但壓倒性的證據表明，後期維根斯坦仍然拒絕實際的無限（RFM V, §21; Zettel §274, 1947）和無限的數學擴展。

The first, and perhaps most definitive, indication that the later Wittgenstein maintains his finitism is his continued and consistent insistence that irrational numbers are rules for constructing finite expansions, not infinite mathematical extensions. “The concepts of infinite decimals in mathematical propositions are not concepts of series”, says Wittgenstein (RFM V, §19), “but of the unlimited technique of expansion of series”. We are misled by “[t]he extensional definitions of functions, of real numbers etc”. (RFM V, §35), but once we recognize the Dedekind cut as “an extensional image”, we see that we are not “led to 2‾√$\sqrt{2}$ by way of the concept of a cut” (RFM V, §34). On the later Wittgenstein’s account, there simply is no property, no rule, no systematic means of defining each and every irrational number intensionally, which means there is no criterion “for the irrational numbers being complete” (PR §181).
後來的最初，或許是最明確的，跡象是 維特根斯坦堅持他的有限主義是他持續且一致的 堅持無理數是構造有限的規則 擴展，而不是無限的數學延伸。“數學命題中無限小數的概念並不是系列的概念，”維根斯坦說（《RFM》V, §19），“而是系列擴展的無限技術”。我們被“函數、實數等的外延定義”所誤導（《RFM》V, §35），但一旦我們認識到德德金切割作為“一種外延的圖像”，我們就會看到我們並不是“通過切割的概念導向 2‾√$\sqrt{2}$ ”（《RFM》V, §34）。在後期維根斯坦的說法中，根本不存在任何“性質”、任何“規則”、任何“系統化的方法”來定義每一個無理數的內涵，這意味著沒有“無理數是完整的”的標準（《PR》§181）。

As in his intermediate position, the later Wittgenstein claims that ‘ℵ0${\mathrm{\aleph }}_0$’ and “infinite series” get their mathematical uses from the use of ‘infinity’ in ordinary language (RFM II, §60). Although, in ordinary language, we often use ‘infinite’ and “infinitely many” as answers to the question “how many?”, and though we associate infinity with the enormously large, the principal use we make of ‘infinite’ and ‘infinity’ is to speak of the unlimited (RFM V, §14) and unlimited techniques (RFM II, §45; PI §218). This fact is brought out by the fact “that the technique of learning ℵ0${\mathrm{\aleph }}_0$ numerals is different from the technique of learning 100,000 numerals” (LFM 31). When we say, e.g., that “there are an infinite number of even numbers” we mean that we have a mathematical technique or rule for generating even numbers which is limitless, which is markedly different from a limited technique or rule for generating a finite number of numbers, such as 1–100,000,000. “We learn an endless technique”, says Wittgenstein (RFM V, §19), “but what is in question here is not some gigantic extension”.
在他中間的立場上，後期維根斯坦聲稱 ‘ ℵ0${\mathrm{\aleph }}_0$ ’ 和 “無限級數” 的數學用法源自於日常語言中對 ‘無限’ 的使用 (RFM II, §60)。雖然在日常語言中，我們經常用 ‘無限’ 和 “無限多” 來回答 “多少？” 的問題，並且我們將無限與極大的數量聯繫在一起，但我們對 ‘無限’ 和 ‘無限’ 的主要 用法 是談論 無限制的 (RFM V, §14) 和無限制的 技術 (RFM II, §45; PI §218)。這一事實通過 “學習 ℵ0${\mathrm{\aleph }}_0$ 數字的技術與學習 100,000 數字的技術是不同的” (LFM 31) 得以顯示出來。當我們說，例如，“偶數的數量是無限的” 時，我們的意思是我們有一種生成偶數的數學技術或規則，這是 無限制的，這與生成有限數量的數字（如 1–100,000,000）的有限技術或規則顯著不同。 「我們學習一種無盡的技術，」維根斯坦說（RFM V, §19），「但這裡所涉及的並不是某種巨大的擴展。」

An infinite sequence, for example, is not a gigantic extension because it is not an extension, and ‘ℵ0${\mathrm{\aleph }}_0$’ is not a cardinal number, for “how is this picture connected with the calculus”, given that “its connexion is not that of the picture | | | | with 4” (i.e., given that ‘ℵ0${\mathrm{\aleph }}_0$’ is not connected to a (finite) extension)? This shows, says Wittgenstein (RFM II, §58), that we ought to avoid the word ‘infinite’ in mathematics wherever it seems to give a meaning to the calculus, rather than acquiring its meaning from the calculus and its use in the calculus. Once we see that the calculus contains nothing infinite, we should not be ‘disappointed’ (RFM II, §60), but simply note (RFM II, §59) that it is not “really necessary… to conjure up the picture of the infinite (of the enormously big)”.
無限序列，例如，並不是一個巨大的擴展，因為 這不是一個擴展，而‘ ℵ0${\mathrm{\aleph }}_0$ ’也不是一個基數，因為“這幅圖與微積分有什麼關聯”，鑑於“它的聯繫並不是這幅圖 | | | | 與 4 的聯繫”（即，‘ ℵ0${\mathrm{\aleph }}_0$ ’與一個（有限）擴展沒有關聯）？這顯示了，維根斯坦說（RFM II, §58），我們應該在數學中避免使用“無限”這個詞，無論它似乎給微積分帶來了什麼意義，而不是從微積分及其在微積分中的使用中獲得意義。一旦我們看到微積分中沒有任何無限的東西，我們就不應該感到“失望”（RFM II, §60），而應該簡單地注意到（RFM II, §59）它並不是“真的必要……去想像無限（極其巨大的）圖像”。

A second strong indication that the later Wittgenstein maintains his finitism is his continued and consistent treatment of ‘propositions’ of the type “There are three consecutive 7s in the decimal expansion of π$\pi$” (hereafter ‘PIC’).^[4] In the middle period, PIC (and its putative negation, ¬$\mathrm{\neg }$PIC, namely, “It is not the case that there are three consecutive 7s in the decimal expansion of π$\pi$”) is not a meaningful mathematical “statement at all” (WVC 81–82: note 1). On Wittgenstein’s intermediate view, PIC—like FLT, GC, and the Fundamental Theorem of Algebra—is not a mathematical proposition because we do not have in hand an applicable decision procedure by which we can decide it in a particular calculus. For this reason, we can only meaningfully state finitistic propositions regarding the expansion of π$\pi$, such as “There exist three consecutive 7s in the first 10,000 places of the expansion of π$\pi$” (WVC 71; 81–82, note 1).
後期維根斯坦保持著他的第二個強烈指示 有限主義是他持續且一致的處理方式 “三個”的“命題” 連續的 7 在 π$\pi$ 的十進制展開中” (以下稱 ‘PIC’).^[4] 在中期，PIC（及其假定的否定， ¬$\mathrm{\neg }$ PIC，即“在 π$\pi$ 的十進制展開中不存在三個連續的 7”）並不是一個有意義的數學“陳述”（WVC 81–82：註 1）。根據維根斯坦的中間觀點，PIC—像 FLT、GC 和代數基本定理一樣—並不是一個數學命題，因為我們手中沒有適用的決策程序來決定它在特定的計算中。因此，我們只能有意義地陳述關於 π$\pi$ 展開的有限性命題，例如“在 π$\pi$ 的展開的前 10,000 位中存在三個連續的 7”（WVC 71；81–82，註 1）。

The later Wittgenstein maintains this position in various passages in RFM (Bernays 1959: 11–12). For example, to someone who says that since “the rule of expansion determine[s$s$] the series completely”, “it must implicitly determine all questions about the structure of the series”, Wittgenstein replies: “Here you are thinking of finite series” (RFM V, §11). If PIC were a mathematical question (or problem)—if it were finitistically restricted—it would be algorithmically decidable, which it is not (RFM V, §21; LFM 31–32, 111, 170; WVC 102–03). As Wittgenstein says at (RFM V, §9): “The question… changes its status, when it becomes decidable”, “[f]or a connexion is made then, which formerly was not there”. And if, moreover, one invokes the Law of the Excluded Middle to establish that PIC is a mathematical proposition—i.e., by saying that one of these “two pictures… must correspond to the fact” (RFM V, §10)—one simply begs the question (RFM V, §12), for if we have doubts about the mathematical status of PIC, we will not be swayed by a person who asserts “PIC ∨¬$\vee \mathrm{\neg }$PIC” (RFM VII, §41; V, §13).
晚期的維根斯坦在多個段落中保持這一立場 RFM（伯恩斯 1959：11–12）。例如，對於某人所說的“擴展規則決定[ s$s$ ] 完全系列”，因此“它必然隱含地決定所有關於系列結構的問題”，維根斯坦回答：“在這裡你是在思考有限系列”（RFM V，§11）。如果 PIC 是一個數學問題（或難題）——如果它是有限制的——那麼它將是算法可決定的，但事實上並非如此（RFM V，§21；LFM 31–32，111，170；WVC 102–03）。正如維根斯坦在（RFM V，§9）所說：“當問題變得可決定時……其地位會改變”，“[f]因為那時建立了一個以前不存在的聯繫”。而且，如果進一步引用排中律來確立 PIC 是一個數學命題——即，通過說這“兩幅圖像……必須與事實相符”（RFM V, §10）——人們只是懷疑問題（RFM V, §12），因為如果我們對 PIC 的數學地位有疑問，我們不會被一個聲稱“PIC ∨¬$\vee \mathrm{\neg }$ PIC”的人所說服（RFM VII, §41; V, §13）。

Wittgenstein’s finitism, constructivism, and conception of mathematical decidability are interestingly connected at (RFM VII, §41, par. 2–5).
維特根斯坦的有限主義、建構主義以及數學可決定性的概念在（RFM VII, §41, par. 2–5）中有趣地相互聯繫。

What harm is done e.g. by saying that God knows all irrational numbers? Or: that they are already there, even though we only know certain of them? Why are these pictures not harmless?
說上帝知道所有不理性的數字會造成什麼傷害？或者：它們已經存在，儘管我們只知道其中某些數字？為什麼這些畫面不是無害的？

For one thing, they hide certain problems.— (MS 124: 139; March 16, 1944)
一方面，他們隱藏了某些問題。— (MS 124: 139; 1944 年 3 月 16 日)

Suppose that people go on and on calculating the expansion of π$\pi$. So God, who knows everything, knows whether they will have reached ‘777’ by the end of the world. But can his omniscience decide whether they would have reached it after the end of the world? It cannot. I want to say: Even God can determine something mathematical only by mathematics. Even for him the mere rule of expansion cannot decide anything that it does not decide for us.
假設人們不斷計算 π$\pi$ 的擴展。因此，知道一切的上帝知道他們是否會在世界末日之前達到「777」。但他的全知能否決定他們在世界末日之後是否會達到它？這是不可能的。我想說：即使是上帝也只能通過數學來確定某些數學問題。即使對他來說，單純的擴展規則也無法決定它對我們無法決定的任何事情。

We might put it like this: if the rule for the expansion has been given us, a calculation can tell us that there is a ‘2’ at the fifth place. Could God have known this, without the calculation, purely from the rule of expansion? I want to say: No. (MS 124, pp. 175–176; March 23–24, 1944)
我們可以這樣說：如果擴展的規則已經給定，那麼一個計算可以告訴我們在第五個位置有一個‘2’。上帝能否在沒有計算的情況下，僅從擴展的規則中知道這一點？我想說：不能。(MS 124, pp. 175–176; 1944 年 3 月 23–24 日)

What Wittgenstein means here is that God’s omniscience might, by calculation, find that ‘777’ occurs at the interval [n,n+2$n,n+2$], but, on the other hand, God might go on calculating forever without ‘777’ ever turning up. Since π$\pi$ is not a completed infinite extension that can be completely surveyed by an omniscient being (i.e., it is not a fact that can be known by an omniscient mind), even God has only the rule, and so God’s omniscience is no advantage in this case (LFM 103–04; cf. Weyl 1921 [1998: 97]). Like us, with our modest minds, an omniscient mind (i.e., God) can only calculate the expansion of π$\pi$ to some n$n$^th decimal place—where our n$n$ is minute and God’s n$n$ is (relatively) enormous—and at no n$n$^th decimal place could any mind rightly conclude that because ‘777’ has not turned up, it, therefore, will never turn up.
維根斯坦在這裡的意思是上帝的全知性 可能，通過計算，發現‘777’出現在區間[ n,n+2$n,n+2$ ]，但另一方面，上帝可能會無限期地計算而‘777’永遠不會出現。由於 π$\pi$ 不是一個完整的無限延伸，無法被全知的存在完全觀察（即，這不是一個全知的心靈可以知道的事實），即使是上帝也只有規則，因此在這種情況下，上帝的全知並沒有優勢（LFM 103–04；參見 Weyl 1921 [1998: 97]）。像我們一樣，擁有謙遜心智的全知心靈（即，上帝）只能計算 π$\pi$ 的擴展到某個 n$n$ ^位小數點——我們的 n$n$ 是微小的，而上帝的 n$n$ 是（相對）巨大的——在任何 n$n$ ^位小數點上，任何心靈都無法正確地得出結論，因為‘777’尚未出現，因此它將永遠不會出現。

3.3 The Later Wittgenstein on Decidability and Algorithmic Decidability
3.3 後期維根斯坦關於可決定性和算法可決定性

On one fairly standard interpretation, the later Wittgenstein says that “true in calculus Γ$\mathrm{\Gamma }$” is identical to “provable in calculus Γ$\mathrm{\Gamma }$” and, therefore, that a mathematical proposition of calculus Γ$\mathrm{\Gamma }$ is a concatenation of signs that is either provable (in principle) or refutable (in principle) in calculus Γ$\mathrm{\Gamma }$ (Goodstein 1972: 279, 282; Anderson 1958: 487; Klenk 1976: 13; Frascolla 1994: 59). On this interpretation, the later Wittgenstein precludes undecidable mathematical propositions, but he allows that some undecided expressions are propositions of a calculus because they are decidable in principle (i.e., in the absence of a known, applicable decision procedure).
在一個相當標準的詮釋中，後期的維根斯坦說 那“在微積分中為真 Γ$\mathrm{\Gamma }$ ”與“在微積分中可證明 Γ$\mathrm{\Gamma }$ ”是相同的，因此，微積分中的數學命題 Γ$\mathrm{\Gamma }$ 是一串符號，這些符號在微積分中要麼是可證明的（原則上），要麼是可反駁的（原則上） Γ$\mathrm{\Gamma }$ （Goodstein 1972: 279, 282; Anderson 1958: 487; Klenk 1976: 13; Frascolla 1994: 59）。根據這種解釋，後期的維根斯坦排除了不可判定的數學命題，但他允許某些未決定的表達式是微積分的命題，因為它們在原則上是可判定的（即，在缺乏已知的、適用的決策程序的情況下）。

There is considerable evidence, however, that the later Wittgenstein maintains his intermediate position that an expression is a meaningful mathematical proposition only within a given calculus and iff we knowingly have in hand an applicable and effective decision procedure by means of which we can decide it. For example, though Wittgenstein vacillates between “provable in PM” and “proved in PM” at (RFM App. III, §6, §8), he does so in order to use the former to consider the alleged conclusion of Gödel’s proof (i.e., that there exist true but unprovable mathematical propositions), which he then rebuts with his own identification of “true in calculus Γ$\mathrm{\Gamma }$” with “proved in calculus Γ$\mathrm{\Gamma }$” (i.e., not with “provable in calculus Γ$\mathrm{\Gamma }$”) (Wang 1991: 253; Rodych 1999a: 177). This construal is corroborated by numerous passages in which Wittgenstein rejects the received view that a provable but unproved proposition is true, as he does when he asserts that (RFM III, §31, 1939) a proof “makes new connexions”, “[i]t does not establish that they are there” because “they do not exist until it makes them”, and when he says (RFM VII, §10, 1941) that “[a] new proof gives the proposition a place in a new system”. Furthermore, as we have just seen, Wittgenstein rejects PIC as a non-proposition on the grounds that it is not algorithmically decidable, while admitting finitistic versions of PIC because they are algorithmically decidable.
然而，有相當多的證據表明，後期的維根斯坦 保持他中立的立場，即表達是有意義的 數學命題僅在給定的微積分內，當且僅當我們明知手中有一個適用且有效的決策程序，藉以決定它。例如，儘管維根斯坦在“在 PM 中可證明”和“在 PM 中已證明”之間搖擺不定（RFM 附錄 III, §6, §8），他這樣做是為了使用前者來考慮哥德爾證明的所謂結論（即，存在真但不可證明的數學命題），他隨後用他自己對“在微積分 Γ$\mathrm{\Gamma }$ 中真”與“在微積分 Γ$\mathrm{\Gamma }$ 中已證明”的識別來反駁這一點（即，不是與“在微積分 Γ$\mathrm{\Gamma }$ 中可證明”）(Wang 1991: 253; Rodych 1999a: 177)。這種解釋得到了許多段落的證實，維根斯坦在這些段落中拒絕了普遍接受的觀點，即一個可證明但未證明的命題是真實的，正如他在（RFM III, §31, 1939）中所主張的，證明“創造了新的聯繫”，“[它]並不確立它們的存在”，因為“它們在證明創造之前並不存在”，以及當他在（RFM VII, §10, 1941）中說“[一]個新的證明給予命題在新系統中的位置”。 此外，正如我們剛才所見，維根斯坦拒絕將 PIC 視為非命題，理由是它不是算法可決的，同時承認 PIC 的有限版本，因為它們是算法可決的。

Perhaps the most compelling evidence that the later Wittgenstein maintains algorithmic decidability as his criterion for a mathematical proposition lies in the fact that, at (RFM V, §9, 1942), he says in two distinct ways that a mathematical ‘question’ can become decidable and that when this happens, a new connexion is ‘made’ which previously did not exist. Indeed, Wittgenstein cautions us against appearances by saying that “it looks as if a ground for the decision were already there”, when, in fact, “it has yet to be invented”. These passages strongly militate against the claim that the later Wittgenstein grants that proposition ϕ$\varphi$ is decidable in calculus Γ$\mathrm{\Gamma }$ iff it is provable or refutable in principle. Moreover, if Wittgenstein held this position, he would claim, contra (RFM V, §9), that a question or proposition does not become decidable since it simply (always) is decidable. If it is provable, and we simply don’t yet know this to be the case, there already is a connection between, say, our axioms and rules and the proposition in question. What Wittgenstein says, however, is that the modalities provable and refutable are shadowy forms of reality—that possibility is not actuality in mathematics (PR §§141, 144, 172; PG 281, 283, 299, 371, 466, 469; LFM 139). Thus, the later Wittgenstein agrees with the intermediate Wittgenstein that the only sense in which an undecided mathematical proposition (RFM VII, §40, 1944) can be decidable is in the sense that we know how to decide it by means of an applicable decision procedure.
或許是後期維根斯坦最有說服力的證據 以算法可判定性作為他對數學的標準 命題在於，根據（RFM V, §9, 1942），他以兩種不同的方式表示數學“問題”可以變得可決定，並且當這發生時，會產生一個之前不存在的新聯繫“建立”。事實上，維根斯坦警告我們要小心表象，他說“看起來似乎已經有一個決定的依據”，但實際上“這還需要被發明”。這些段落強烈反對後期維根斯坦承認命題 ϕ$\varphi$ 在演算 Γ$\mathrm{\Gamma }$ 中是可決定的，當且僅當它是可證明或可反駁的原則上。此外，如果維根斯坦持有這一立場，他會聲稱，與（RFM V, §9）相反，問題或命題並不變得可決定，因為它只是（總是）是可決定的。如果它是可證明的，而我們只是尚未知道這一點，那麼，舉例來說，我們的公理和規則與所討論的命題之間已經存在一個聯繫。 然而，維根斯坦所說的是，可證明和可駁斥的模式是現實的陰影形式——在數學中，可能性並不是現實（PR §§141, 144, 172; PG 281, 283, 299, 371, 466, 469; LFM 139）。因此，後期維根斯坦同意中期維根斯坦的觀點，即一個未決定的數學命題（RFM VII, §40, 1944）能夠可決定的唯一意義在於我們知道如何通過適用的決策程序來決定它。

3.4 Wittgenstein’s Later Critique of Set Theory: Non-Enumerability vs. Non-Denumerability
3.4 維根斯坦對集合論的後期批評：不可列舉性與非可列舉性

Largely a product of his anti-foundationalism and his criticism of the extension-intension conflation, Wittgenstein’s later critique of set theory is highly consonant with his intermediate critique (PR §§109, 168; PG 334, 369, 469; LFM 172, 224, 229; and RFM III, §43, 46, 85, 90; VII, §16). Given that mathematics is a “MOTLEY of techniques of proof” (RFM III, §46), it does not require a foundation (RFM VII, §16) and it cannot be given a self-evident foundation (PR §160; WVC 34 & 62; RFM IV, §3). Since set theory was invented to provide mathematics with a foundation, it is, minimally, unnecessary.
在很大程度上，這是他反基礎主義和對延伸-內涵混淆的批評的產物，維根斯坦後期對集合論的批評與他中期的批評高度一致（《哲學研究》§§109, 168；《邏輯哲學論》334, 369, 469；《邏輯形式的意義》172, 224, 229；以及《反思的邏輯》III, §43, 46, 85, 90；VII, §16）。考慮到數學是一種“證明技術的混合”（《反思的邏輯》III, §46），它不需要基礎（《反思的邏輯》VII, §16），也不能給予一個自明的基礎（《哲學研究》§160；《維根斯坦的語言》34 & 62；《反思的邏輯》IV, §3）。由於集合論是為了給數學提供基礎而發明的，因此它至少是不必要的。

Even if set theory is unnecessary, it still might constitute a solid foundation for mathematics. In his core criticism of set theory, however, the later Wittgenstein denies this, saying that the diagonal proof does not prove non-denumerability, for “[i]t means nothing to say: ‘Therefore the X numbers are not denumerable’” (RFM II, §10). When the diagonal is construed as a proof of greater and lesser infinite sets it is a “puffed-up proof”, which, as Poincaré argued (1913: 61–62), purports to prove or show more than “its means allow it” (RFM II, §21).
即使集合論是不必要的，它仍然可能構成數學的堅實基礎。然而，在他對集合論的核心批評中，後期維根斯坦否認了這一點，說對角線證明並不能證明不可數性，因為「說‘因此 X 數字是不可數的’毫無意義」（RFM II, §10）。當對角線被解釋為更大和更小的無窮集合的證明時，它是一個「膨脹的證明」，正如龐加萊所辯稱（1913: 61–62），它聲稱證明或顯示的內容超出了「其手段所允許的」（RFM II, §21）。

If it were said: Consideration of the diagonal procedure shews you that the concept ‘real number’ has much less analogy with the concept ‘cardinal number’ than we, being misled by certain analogies, are inclined to believe, that would have a good and honest sense. But just the opposite happens: one pretends to compare the ‘set’ of real numbers in magnitude with that of cardinal numbers. The difference in kind between the two conceptions is represented, by a skew form of expression, as difference of extension. I believe, and hope, that a future generation will laugh at this hocus pocus. (RFM II, §22)
如果說：對角程序的考量顯示出「實數」這個概念與「基數」這個概念的類比遠不如我們因某些類比而誤導的程度那麼高，這樣的說法會有良好而誠實的意義。但恰恰相反的情況發生了：人們假裝將實數的「集合」在大小上與基數的集合進行比較。這兩種概念之間的本質差異被以一種歪曲的表達形式表示為擴展的差異。我相信並希望未來的一代人會對這種把戲感到好笑。（RFM II, §22）

The sickness of a time is cured by an alteration in the mode of life of human beings… (RFM II, §23)
一個時代的病痛是通過人類生活方式的改變來治癒的……（RFM II, §23）

The “hocus pocus” of the diagonal proof rests, as always for Wittgenstein, on a conflation of extension and intension, on the failure to properly distinguish sets as rules for generating extensions and (finite) extensions. By way of this confusion “a difference in kind” (i.e., unlimited rule vs. finite extension) “is represented by a skew form of expression”, namely as a difference in the cardinality of two infinite extensions. Not only can the diagonal not prove that one infinite set is greater in cardinality than another infinite set, according to Wittgenstein, nothing could prove this, simply because “infinite sets” are not extensions, and hence not infinite extensions. But instead of interpreting Cantor’s diagonal proof honestly, we take the proof to “show there are numbers bigger than the infinite”, which “sets the whole mind in a whirl, and gives the pleasant feeling of paradox” (LFM 16–17)—a “giddiness attacks us when we think of certain theorems in set theory”—“when we are performing a piece of logical sleight-of-hand” (PI §412; §426; 1945). This giddiness and pleasant feeling of paradox, says Wittgenstein (LFM 16), “may be the chief reason [set theory] was invented”.
對於維根斯坦來說，對角證明的“魔法”始終基於對擴展和內涵的混淆，未能正確區分作為生成擴展的規則和（有限）擴展的集合。由於這種混淆，“種類上的差異”（即，無限規則與有限擴展）“以一種歪曲的表達形式呈現”，即作為兩個無限擴展的基數的差異。根據維根斯坦的說法，對角線無法證明一個無限集合的基數大於另一個無限集合，沒有什麼能證明這一點，因為“無限集合”不是擴展，因此也不是無限擴展。但我們並沒有誠實地解釋康托爾的對角證明，而是認為該證明“顯示有比無限更大的數字”，這“使整個心智陷入混亂，並帶來悖論的愉悅感”（LFM 16–17）——“當我們思考集合論中的某些定理時，會感到一陣眩暈”——“當我們在進行一種邏輯的戲法時”（PI §412; §426; 1945）。 這種眩暈和悖論的愉悅感，維根斯坦說（LFM 16），“可能是[集合論]被發明的主要原因”。

Though Cantor’s diagonal is not a proof of non-denumerability, when it is expressed in a constructive manner, as Wittgenstein himself expresses it at (RFM II, §1), “it gives sense to the mathematical proposition that the number so-and-so is different from all those of the system” (RFM II, §29). That is, the proof proves non-enumerability: it proves that for any given definite real number concept (e.g., recursive real), one cannot enumerate ‘all’ such numbers because one can always construct a diagonal number, which falls under the same concept and is not in the enumeration. “One might say”, Wittgenstein says,
雖然康托的對角線並不是不可數性的證明，但當它以建構性方式表達時，正如維根斯坦自己在(RFM II, §1)中所表達的，“它使數學命題有意義，即某個數字與系統中的所有數字不同” (RFM II, §29)。也就是說，這個證明證明了不可數性：它證明了對於任何給定的確定實數概念（例如，遞歸實數），人們無法列舉‘所有’這樣的數字，因為人們總是可以構造一個對角數，這個數屬於同一概念且不在列舉之中。“可以說”，維根斯坦說，

I call number-concept X non-denumerable if it has been stipulated that, whatever numbers falling under this concept you arrange in a series, the diagonal number of this series is also to fall under that concept. (RFM II, §10; cf. II, §§30, 31, 13)
如果已經規定，無論你將這個概念下的數字以何種方式排列成系列，這個系列的對角數字也必須屬於這個概念，我稱這個數字概念 X 為不可數的。 (RFM II, §10; cf. II, §§30, 31, 13)

One lesson to be learned from this, according to Wittgenstein (RFM II, §33), is that “there are diverse systems of irrational points to be found in the number line”, each of which can be given by a recursive rule, but “no system of irrational numbers”, and “also no super-system, no ‘set of irrational numbers’ of higher-order infinity”. Cantor has shown that we can construct “infinitely many” diverse systems of irrational numbers, but we cannot construct an exhaustive system of all the irrational numbers (RFM II, §29). As Wittgenstein says at (MS 121, 71r; Dec. 27, 1938), three pages after the passage used for (RFM II, §57):
根據維根斯坦（RFM II, §33），從中可以學到的一課是“在數線上存在著多樣的系統的無理點”，每一個都可以通過遞歸規則來給出，但“沒有無理數的系統”，而且“也沒有超系統，沒有‘更高階無窮的無理數集合’”。康托爾已經顯示我們可以構造“無限多”的多樣無理數系統，但我們無法構造一個全面的系統來包含所有的無理數（RFM II, §29）。正如維根斯坦在（MS 121, 71r; 1938 年 12 月 27 日）所說，在用於（RFM II, §57）的段落後的三頁中：

If you now call the Cantorian procedure one for producing a new real number, you will now no longer be inclined to speak of a system of all real numbers. (italics added)
如果你現在調用康托爾程序來產生一個新的實數，你將不再傾向於談論所有實數的系統。（斜體字已添加）

From Cantor’s proof, however, set theorists erroneously conclude that “the set of irrational numbers” is greater in multiplicity than any enumeration of irrationals (or the set of rationals), when the only conclusion to draw is that there is no such thing as the set of all the irrational numbers. The truly dangerous aspect to ‘propositions’ such as “The real numbers cannot be arranged in a series” and “The set… is not denumerable” is that they make concept formation [i.e., our invention] “look like a fact of nature” (i.e., something we discover) (RFM II §§16, 37). At best, we have a vague idea of the concept of “real number”, but only if we restrict this idea to “recursive real number” and only if we recognize that having the concept does not mean having a set of all recursive real numbers.
然而，從康托的證明來看，集合論者錯誤地得出結論，認為「無理數的集合」在數量上大於任何無理數的列舉（或有理數的集合），而唯一可以得出的結論是不存在所有 無理數的集合。像「實數無法排列成一個序列」和「該集合……不可數」這樣的「命題」真正危險的方面在於，它們使概念形成[即我們的發明]「看起來像自然的事實」（即我們發現的東西）(RFM II §§16, 37)。充其量，我們對「實數」的概念只有模糊的理解，但只有在我們將這個概念限制為「遞歸實數」並且認識到擁有這個概念並不意味著擁有所有遞歸實數的集合。

3.5 Extra-Mathematical Application as a Necessary Condition of Mathematical Meaningfulness
3.5 作為數學意義必要條件的額外數學應用

The principal and most significant change from the middle to later writings on mathematics is Wittgenstein’s (re-)introduction of an extra-mathematical application criterion, which is used to distinguish mere “sign-games” from mathematical language-games. “[I]t is essential to mathematics that its signs are also employed in mufti”, Wittgenstein states, for
從中期到後期的數學著作中，最主要和最重要的變化是維根斯坦重新引入了一個超數學的應用標準，用以區分單純的“符號遊戲”和數學語言遊戲。維根斯坦指出：“對於數學來說，其符號在便服中也被使用是至關重要的。”

[i]t is the use outside mathematics, and so the meaning [Bedeutung] of the signs, that makes the sign-game into mathematics. (i.e., a mathematical “language-game”; RFM V, §2, 1942; LFM 140–141, 169–70)
[i]它在數學之外的使用，以及符號的意義 [Bedeutung]，使得符號遊戲成為數學。（即數學的“語言遊戲”；RFM V, §2, 1942；LFM 140–141, 169–70）

As Wittgenstein says at (RFM V, §41, 1943),
正如維根斯坦在（RFM V, §41, 1943）中所說，

[c]oncepts which occur in ‘necessary’ propositions must also occur and have a meaning [Bedeutung] in non-necessary ones. (italics added)
[c]oncepts which occur in ‘necessary’ propositions 必須也 occur and have a meaning [意義] in non-necessary ones. (italics added)

If two proofs prove the same proposition, says Wittgenstein, this means that “both demonstrate it as a suitable instrument for the same purpose”, which “is an allusion to something outside mathematics” (RFM VII, §10, 1941; italics added).
如果兩個證明證明了相同的命題，維根斯坦說，這意味著「兩者都將其展示為適合同一目的的工具」，這「暗示著數學之外的某些東西」（RFM VII, §10, 1941；斜體字為補充）。

As we have seen, this criterion was present in the Tractatus (6.211), but noticeably absent in the middle period. The reason for this absence is probably that the intermediate Wittgenstein wanted to stress that in mathematics everything is syntax and nothing is meaning. Hence, in his criticisms of Hilbert’s ‘contentual’ mathematics (Hilbert 1925) and Brouwer’s reliance upon intuition to determine the meaningful content of (especially undecidable) mathematical propositions, Wittgenstein couched his finitistic constructivism in strong formalism, emphasizing that a mathematical calculus does not need an extra-mathematical application (PR §109; WVC 105).
正如我們所見，這一標準在《邏輯哲學論》（6.211）中存在，但在中期卻明顯缺失。這一缺失的原因可能是中期的維根斯坦想強調在數學中一切都是語法，沒有任何意義。因此，在他對希爾伯特的“內容數學”（希爾伯特 1925）和布勞威對直覺的依賴以確定（特別是不可判定的）數學命題的有意義內容的批評中，維根斯坦將他的有限主義建構主義包裝在強形式主義中，強調數學演算不需要額外的數學應用（《PR》§109；《WVC》105）。

There seem to be two reasons why the later Wittgenstein reintroduces extra-mathematical application as a necessary condition of a mathematical language-game. First, the later Wittgenstein has an even greater interest in the use of natural and formal languages in diverse “forms of life” (PI §23), which prompts him to emphasize that, in many cases, a mathematical ‘proposition’ functions as if it were an empirical proposition “hardened into a rule” (RFM VI, §23) and that mathematics plays diverse applied roles in many forms of human activity (e.g., science, technology, predictions). Second, the extra-mathematical application criterion relieves the tension between Wittgenstein’s intermediate critique of set theory and his strong formalism according to which “one calculus is as good as another” (PG 334). By demarcating mathematical language-games from non-mathematical sign-games, Wittgenstein can now claim that, “for the time being”, set theory is merely a formal sign-game.
後期維根斯坦重新引入超數學應用作為數學語言遊戲的必要條件似乎有兩個原因。首先，後期維根斯坦對自然語言和形式語言在多樣“生活形式”中的使用（《哲學研究》§23）有著更大的興趣，這促使他強調，在許多情況下，數學“命題”就像是一個“硬化為規則”的經驗命題（《數學的哲學》VI, §23），而數學在許多人類活動（例如科學、技術、預測）中扮演著多樣的應用角色。其次，超數學應用標準緩解了維根斯坦對集合論的中間批評與他強烈形式主義之間的緊張關係，根據該形式主義，“一種演算與另一種演算一樣好”（《邏輯哲學論》334）。通過將數學語言遊戲與非數學符號遊戲區分開來，維根斯坦現在可以聲稱，“暫時來看”，集合論僅僅是一種形式符號遊戲。

These considerations may lead us to say that 2ℵ0>ℵ0$2^{{\mathrm{\aleph }}_0}>{\mathrm{\aleph }}_0$.
這些考量可能使我們說 2ℵ0>ℵ0$2^{{\mathrm{\aleph }}_0}>{\mathrm{\aleph }}_0$ 。

That is to say: we can make the considerations lead us to that.
也就是說：我們可以讓考量引導我們到那裡。

Or: we can say this and give this as our reason.
或者：我們可以說這個，並給出這個作為我們的理由。

But if we do say it—what are we to do next? In what practice is this proposition anchored? It is for the time being a piece of mathematical architecture which hangs in the air, and looks as if it were, let us say, an architrave, but not supported by anything and supporting nothing. (RFM II, §35)
但如果我們這麼說——接下來該怎麼做？這個命題是基於什麼實踐？目前它是一個懸空的數學架構，看起來就像是一個橫樑，但沒有任何支撐，也不支撐任何東西。 (RFM II, §35)

It is not that Wittgenstein’s later criticisms of set theory change, it is, rather, that once we see that set theory has no extra-mathematical application, we will focus on its calculations, proofs, and prose and “subject the interest of the calculations to a test” (RFM II, §62). By means of Wittgenstein’s “immensely important” ‘investigation’ (LFM 103), we will find, Wittgenstein expects, that set theory is uninteresting (e.g., that the non-enumerability of “the reals” is uninteresting and useless) and that our entire interest in it lies in the ‘charm’ of the mistaken prose interpretation of its proofs (LFM 16). More importantly, though there is “a solid core to all [its] glistening concept-formations” (RFM V, §16), once we see it as “as a mistake of ideas”, we will see that propositions such as “2ℵ0>ℵ0$2^{{\mathrm{\aleph }}_0}>{\mathrm{\aleph }}_0$” are not anchored in an extra-mathematical practice, that “Cantor’s paradise” “is not a paradise”, and we will then leave “of [our] own accord” (LFM 103).
並不是維根斯坦對集合論的後期批評 改變，這是，相當於，當我們看到集合論沒有 額外的數學應用，我們將專注於其計算， 證明，以及散文並“將計算的興趣置於測試之下”（RFM II, §62）。通過維根斯坦“極其重要”的‘調查’（LFM 103），我們將發現，維根斯坦預期集合論是無趣的（例如，“實數”的不可數性是無趣且無用的），而我們對它的整個興趣在於其證明的錯誤散文詮釋的‘魅力’（LFM 16）。更重要的是，儘管“所有[其]閃亮的概念形成都有一個堅實的核心”（RFM V, §16），一旦我們將其視為“思想的錯誤”，我們將看到像“ 2ℵ0>ℵ0$2^{{\mathrm{\aleph }}_0}>{\mathrm{\aleph }}_0$ ”這樣的命題並未根植於額外的數學實踐中，“康托爾的樂園” “並不是一個樂園”，然後我們將自願離開（LFM 103）。

It must be emphasized, however, that the later Wittgenstein still maintains that the operations within a mathematical calculus are purely formal, syntactical operations governed by rules of syntax (i.e., the solid core of formalism).
然而，必須強調的是，後期的維根斯坦仍然認為數學演算中的運算純粹是形式上的、由語法規則所支配的語法運算（即形式主義的堅實核心）。

It is of course clear that the mathematician, in so far as he really is ‘playing a game’…[is] acting in accordance with certain rules. (RFM V, §1)
當然很清楚，數學家在某種程度上如果真的在「玩遊戲」……[是] 行動 根據某些規則。 (RFM V, §1)

To say mathematics is a game is supposed to mean: in proving, we need never appeal to the meaning [Bedeutung] of the signs, that is to their extra-mathematical application. (RFM V, §4)
說數學是一種遊戲應該意味著：在證明中，我們不需要訴諸符號的意義，即它們的超數學應用。（RFM V, §4）

Where, during the middle period, Wittgenstein speaks of “arithmetic [as] a kind of geometry” at (PR §109 & §111), the later Wittgenstein similarly speaks of “the geometry of proofs” (RFM I, App. III, §14), the “geometrical cogency” of proofs (RFM III, §43), and a “geometrical application” according to which the “transformation of signs” in accordance with “transformation-rules” (RFM VI, §2, 1941) shows that “when mathematics is divested of all content, it would remain that certain signs can be constructed from others according to certain rules” (RFM III, §38). Hence, the question whether a concatenation of signs is a proposition of a given mathematical calculus (i.e., a calculus with an extra-mathematical application) is still an internal, syntactical question, which we can answer with knowledge of the proofs and decision procedures of the calculus.
在中期，維根斯坦提到「算術[作為]一種幾何」(PR §109 & §111)，而後期的維根斯坦同樣提到「證明的幾何」(RFM I, App. III, §14)、證明的「幾何說服力」(RFM III, §43)，以及根據「變換規則」(RFM VI, §2, 1941)的「幾何應用」，顯示「當數學被剝奪所有內容時，仍然會有某些符號可以根據某些規則從其他符號構造出來」(RFM III, §38)。因此，符號的串聯是否為給定的數學演算（即具有額外數學應用的演算）的一個命題，仍然是一個內部的語法問題，我們可以通過對該演算的證明和決策程序的了解來回答。

3.6 Wittgenstein on Gödel and Undecidable Mathematical Propositions
3.6 維特根斯坦對哥德爾和不可判定數學命題的看法

RFM is perhaps most (in)famous for Wittgenstein’s (RFM App. III) treatment of “true but unprovable” mathematical propositions. Early reviewers said that “[t]he arguments are wild” (Kreisel 1958: 153), that the passages “on Gödel’s theorem… are of poor quality or contain definite errors” (Dummett 1959: 324), and that (RFM App. III) “throws no light on Gödel’s work” (Goodstein 1957: 551). “Wittgenstein seems to want to legislate [[q]uestions about completeness] out of existence”, Anderson said, (1958: 486–87) when, in fact, he certainly cannot dispose of Gödel’s demonstrations “by confusing truth with provability”. Additionally, Bernays, Anderson (1958: 486), and Kreisel (1958: 153–54) claimed that Wittgenstein failed to appreciate “Gödel’s quite explicit premiss of the consistency of the considered formal system” (Bernays 1959: 15), thereby failing to appreciate the conditional nature of Gödel’s First Incompleteness Theorem. On the reading of these four early expert reviewers, Wittgenstein failed to understand Gödel’s Theorem because he failed to understand the mechanics of Gödel’s proof and he erroneously thought he could refute or undermine Gödel’s proof simply by identifying “true in PM” (i.e., Principia Mathematica) with “proved/provable in PM”.
RFM 可能因維根斯坦的 (RFM 附錄 III) 對「真但不可證明」的數學命題的處理而最為人所知。早期的評論者表示「[這些]論點非常荒謬」（Kreisel 1958: 153），這些段落「關於哥德爾定理……質量很差或包含明確的錯誤」（Dummett 1959: 324），並且 (RFM 附錄 III)「對哥德爾的工作沒有任何啟發」（Goodstein 1957: 551）。安德森說：「維根斯坦似乎想要立法[[q]uestions about completeness]使其不存在」（1958: 486–87），而事實上，他確實無法通過「將真理與可證明性混淆」來處理哥德爾的證明。此外，伯奈斯、安德森（1958: 486）和克雷塞爾（1958: 153–54）聲稱維根斯坦未能理解「哥德爾所考慮的形式系統的一致性相當明確的前提」（伯奈斯 1959: 15），因此未能理解哥德爾第一不完備性定理的條件性質。 根據這四位早期專家的評審，維根斯坦未能理解哥德爾定理，因為他未能理解哥德爾證明的機制，並且他錯誤地認為僅僅通過將“在PM中為真”（即數學原理）與“在PM中被證明/可證明”相等，就可以駁斥或削弱哥德爾的證明。

Interestingly, we now have two pieces of evidence (Kreisel 1998: 119; Rodych 2003: 282, 307) that Wittgenstein wrote (RFM App. III) in 1937–38 after reading only the informal, ‘casual’ (MS 126, 126–127; Dec. 13, 1942) introduction of (Gödel 1931) and that, therefore, his use of a self-referential proposition as the “true but unprovable proposition” may be based on Gödel’s introductory, informal statements, namely that “the undecidable proposition [R(q);q$R(q);q$] states… that [R(q);q$R(q);q$] is not provable” (1931: 598) and that “[R(q);q$R(q);q$] says about itself that it is not provable” (1931: 599). Perplexingly, only two of the four famous reviewers even mentioned Wittgenstein’s (RFM VII, §§19, 21–22, 1941)) explicit remarks on ‘Gödel’s’ First Incompleteness Theorem (Bernays 1959: 2; Anderson 1958: 487), which, though flawed, capture the number-theoretic nature of the Gödelian proposition and the functioning of Gödel-numbering, probably because Wittgenstein had by then read or skimmed the body of Gödel’s 1931 paper.
有趣的是，我們現在有兩個證據 (Kreisel 1998: 119; Rodych 2003: 282, 307) 威特根斯坦在 1937–38 年寫的 (RFM 附錄 III)，是在閱讀了 (哥德爾 1931) 的非正式、‘隨意’ (MS 126, 126–127; 1942 年 12 月 13 日) 引言之後，因此，他使用自指命題作為“真但不可證明的命題”可能基於哥德爾的引言性、非正式的陳述，即“不可判定的命題 [ R(q);q$R(q);q$ ] 表示…… [ R(q);q$R(q);q$ ] 是不可證明的” (1931: 598) 和“[ R(q);q$R(q);q$ ] 自己說它是不可證明的” (1931: 599)。令人困惑的是，四位著名的評論者中只有兩位提到了威特根斯坦對‘哥德爾’第一不完全性定理的明確評論 (RFM VII, §§19, 21–22, 1941)) (伯奈斯 1959: 2; 安德森 1958: 487)，雖然有缺陷，但捕捉了哥德爾命題的數論性質 和 哥德爾編號的運作，可能是因為威特根斯坦那時已經閱讀或瀏覽了哥德爾 1931 年論文的內容。

The first thing to note, therefore, about (RFM App. III) is that Wittgenstein mistakenly thinks—again, perhaps because Wittgenstein had read only Gödel’s Introduction—(a) that Gödel proves that there are true but unprovable propositions of PM (when, in fact, Gödel syntactically proves that if PM is ω$\omega$-consistent, the Gödelian proposition is undecidable in PM) and (b) that Gödel’s proof uses a self-referential proposition to semantically show that there are true but unprovable propositions of PM.
因此，首先要注意的是（RFM 附錄 III）維根斯坦錯誤地認為——也許是因為維根斯坦只讀過哥德爾的《引言》——（a）哥德爾證明了存在真但不可證明的PM命題（實際上，哥德爾在語法上證明了如果PM是 ω$\omega$ -一致的，則哥德爾命題在PM中是不可判定的）以及（b）哥德爾的證明使用了一個自指命題來語義上顯示存在真但不可證明的PM命題。

For this reason, Wittgenstein has two main aims in (RFM App. III): (1) to refute or undermine, on its own terms, the alleged Gödel proof of true but unprovable propositions of PM, and (2) to show that, on his own terms, where “true in calculus Γ$\mathrm{\Gamma }$” is identified with “proved in calculus Γ$\mathrm{\Gamma }$”, the very idea of a true but unprovable proposition of calculus Γ$\mathrm{\Gamma }$ is meaningless.
因此，維根斯坦在（RFM 附錄 III）中有兩個主要目標：（1）在其自身的條件下，駁斥或削弱所謂的哥德爾對於PM的真但不可證明命題的證明，和（2）展示在他自己的條件下，當“在微積分中為真 Γ$\mathrm{\Gamma }$ ”被認同為“在微積分中被證明 Γ$\mathrm{\Gamma }$ ”時，微積分 Γ$\mathrm{\Gamma }$ 中真但不可證明命題的概念是毫無意義的。

Thus, at (RFM App. III, §8) (hereafter simply ‘§8’), Wittgenstein begins his presentation of what he takes to be Gödel’s proof by having someone say:
因此，在 (RFM 附錄 III, §8)（以下簡稱「§8」）中，維根斯坦開始他對他所認為的哥德爾證明的介紹，讓某人說：

I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: ‘P is not provable in Russell’s system’.
我已經構造了一個命題（我將用‘P’來指定它）在羅素的符號中，通過某些定義和變換，它可以被解釋為：‘P 在羅素的系統中是不可證明的’。

That is, Wittgenstein’s Gödelian constructs a proposition that is semantically self-referential and which specifically says of itself that it is not provable in PM. With this erroneous, self-referential proposition P [used also at (§10), (§11), (§17), (§18)], Wittgenstein presents a proof-sketch very similar to Gödel’s own informal semantic proof ‘sketch’ in the Introduction of his famous paper (1931: 598).
也就是說，維根斯坦的哥德爾構造了一個語義上自指的命題，並且特別聲明它在PM中是不可證明的。通過這個錯誤的自指命題P [在（§10）、（§11）、（§17）、（§18）中也有使用]，維根斯坦呈現了一個與哥德爾自己在其著名論文的引言中（1931: 598）所提供的非正式語義證明“草圖”非常相似的證明草圖。

Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable. (§8)
我難道不應該說這個命題一方面是真的，另一方面卻是無法證明的嗎？假設它是假的；那麼它就是真的可以被證明。而這肯定是不可能的！如果它被證明了，那麼就證明了它是無法證明的。因此它只能是真的，但無法證明。 (§8)

The reasoning here is a double reductio. Assume (a) that P must either be true or false in Russell’s system, and (b) that P must either be provable or unprovable in Russell’s system. If (a), P must be true, for if we suppose that P is false, since P says of itself that it is unprovable, “it is true that it is provable”, and if it is provable, it must be true (which is a contradiction), and hence, given what P means or says, it is true that P is unprovable (which is a contradiction). Second, if (b), P must be unprovable, for if P “is proved, then it is proved that it is not provable”, which is a contradiction (i.e., P is provable and not provable in PM). It follows that P “can only be true, but unprovable”.
這裡的推理是一個雙重還原。假設(a)在羅素的系統中，P必須是真或假，(b)在羅素的系統中，P必須是可證明或不可證明的。如果(a)，P必須是真，因為如果我們假設P是假的，既然P自稱是不可證明的，“它是可證明的是真的”，如果它是可證明的，那麼它必須是真的（這是一個矛盾），因此，根據P的意義或說法，P不可證明是真的（這是一個矛盾）。其次，如果(b)，P必須是不可證明的，因為如果P“被證明，那麼它被證明為不可證明”，這是一個矛盾（即，P在PM中是可證明且不可證明的）。因此，P“只能是真，但不可證明”。

To refute or undermine this ‘proof’, Wittgenstein says that if you have proved ¬P$\mathrm{\neg }P$, you have proved that P is provable (i.e., since you have proved that it is not the case that P is not provable in Russell’s system), and “you will now presumably give up the interpretation that it is unprovable” (i.e., ‘P is not provable in Russell’s system’), since the contradiction is only proved if we use or retain this self-referential interpretation (§8). On the other hand, Wittgenstein argues (§8), ‘[i]f you assume that the proposition is provable in Russell’s system, that means it is true in the Russell sense, and the interpretation “P is not provable” again has to be given up’, because, once again, it is only the self-referential interpretation that engenders a contradiction. Thus, Wittgenstein’s ‘refutation’ of “Gödel’s proof” consists in showing that no contradiction arises if we do not interpret ‘P’ as ‘P is not provable in Russell’s system’—indeed, without this interpretation, a proof of P does not yield a proof of ¬P$\mathrm{\neg }P$ and a proof of ¬P$\mathrm{\neg }P$ does not yield a proof of P. In other words, the mistake in the proof is the mistaken assumption that a mathematical proposition ‘P’ “can be so interpreted that it says: ‘P is not provable in Russell’s system’”. As Wittgenstein says at (§11), “[t]hat is what comes of making up such sentences”.
為了駁斥或削弱這個「證明」，維根斯坦說 如果你已經證明了 ¬P$\mathrm{\neg }P$ ，那麼你已經證明了 P 是可證明的（即，因為你已經證明了在羅素的系統中，並不是 P 是不可證明的），而“你現在大概會放棄它是不可證明的這一解釋”（即，‘P 在羅素的系統中是不可證明的’），因為只有在我們使用或保留這種自指的解釋時，矛盾才會被證明 (§8)。另一方面，維根斯坦辯稱 (§8)，“[如果]你假設該命題在羅素的系統中是可證明的，那麼這意味著它在羅素的意義上是正確的 ，而解釋‘P 是不可證明的’又必須放棄”，因為，再次強調，只有自指的解釋才會產生矛盾。因此，維根斯坦對“哥德爾的證明”的‘反駁’在於顯示，如果我們不將‘P’解釋為‘P 在羅素的系統中是不可證明的’，那麼就不會產生矛盾——事實上，沒有這種解釋，對 P 的證明不會產生對 ¬P$\mathrm{\neg }P$ 的證明，而對 ¬P$\mathrm{\neg }P$ 的證明也不會產生對 P 的證明。 換句話說，證明中的錯誤是錯誤地假設數學命題「P」可以被解釋為：「P 在羅素的系統中是不可證明的」。正如維根斯坦在（§11）所說，「這就是編造這種句子的結果」。

This ‘refutation’ of “Gödel’s proof” is perfectly consistent with Wittgenstein’s syntactical conception of mathematics (i.e., wherein mathematical propositions have no meaning and hence cannot have the ‘requisite’ self-referential meaning) and with what he says before and after (§8), where his main aim is to show (2) that, on his own terms, since “true in calculus Γ$\mathrm{\Gamma }$” is identical with “proved in calculus Γ$\mathrm{\Gamma }$”, the very idea of a true but unprovable proposition of calculus Γ$\mathrm{\Gamma }$ is a contradiction-in-terms.
這個對「哥德爾的」的「駁斥」 “證明”與維根斯坦的完全一致 數學的句法概念（即，數學中） 命題沒有意義，因此無法擁有 「必要的」自指意義) 以及他所 在第 8 條之前和之後所說的，他的主要目的是顯示(2) 那，按照他自己的條件，因為“在微積分中為真 Γ$\mathrm{\Gamma }$ ”與“在微積分中被證明 Γ$\mathrm{\Gamma }$ ”是相同的，微積分中真正但無法證明的命題的想法本身就是一種自相矛盾。

To show (2), Wittgenstein begins by asking (§5), what he takes to be, the central question, namely, “Are there true propositions in Russell’s system, which cannot be proved in his system?”. To address this question, he asks “What is called a true proposition in Russell’s system…?”, which he succinctly answers (§6): “‘p’ is true = p”. Wittgenstein then clarifies this answer by reformulating the second question of (§5) as “Under what circumstances is a proposition asserted in Russell’s game [i.e., system]?”, which he then answers by saying: “the answer is: at the end of one of his proofs, or as a ‘fundamental law’ (Pp.)” (§6). This, in a nutshell, is Wittgenstein’s conception of “mathematical truth”: a true proposition of PM is an axiom or a proved proposition, which means that “true in PM” is identical with, and therefore can be supplanted by, “proved in PM”. 

Having explicated, to his satisfaction at least, the only real, non-illusory notion of “true in PM”, Wittgenstein answers the (§8) question “Must I not say that this proposition… is true, and… unprovable?” negatively by (re)stating his own (§§5–6) conception of “true in PM” as “proved/provable in PM”:
在他至少滿意地闡明了“在PM中真實”的唯一真正的、非虛幻的概念後，維根斯坦以（重新）陳述他自己對“在PM中真實”的理解為“在PM中被證明/可證明”來對第（§8）個問題“我難道不應該說這個命題……是真實的，並且……無法證明？”作出否定的回答：

‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘false in Russell’s system’ means: the opposite has been proved in Russell’s system.
「在羅素的系統中為真」的意思是，如前所述：在羅素的系統中被證明；而「在羅素的系統中為假」的意思是：在羅素的系統中已證明相反的情況。

This answer is given in a slightly different way at (§7) where Wittgenstein asks “may there not be true propositions which are written in this [Russell’s] symbolism, but are not provable in Russell’s system?”, and then answers “‘True propositions’, hence propositions which are true in another system, i.e. can rightly be asserted in another game”. In light of what he says in (§§5, 6, and 8), Wittgenstein’s (§7) point is that if a proposition is ‘written’ in “Russell’s symbolism” and it is true, it must be proved/provable in another system, since that is what “mathematical truth” is. Analogously (§8), “if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell’s sense”, for “[w]hat is called ‘losing’ in chess may constitute winning in another game”. This textual evidence certainly suggests, as Anderson almost said, that Wittgenstein rejects a true but unprovable mathematical proposition as a contradiction-in-terms on the grounds that “true in calculus Γ$\mathrm{\Gamma }$” means nothing more (and nothing less) than “proved in calculus Γ$\mathrm{\Gamma }$”. 

On this (natural) interpretation of (RFM App. III), the early reviewers’ conclusion that Wittgenstein fails to understand the mechanics of Gödel’s argument seems reasonable. First, Wittgenstein erroneously thinks that Gödel’s proof is essentially semantical and that it uses and requires a self-referential proposition. Second, Wittgenstein says (§14) that “[a] contradiction is unusable” for “a prediction” that “that such-and-such construction is impossible” (i.e., that P is unprovable in PM), which, superficially at least, seems to indicate that Wittgenstein fails to appreciate the “consistency assumption” of Gödel’s proof (Kreisel, Bernays, Anderson). 

If, in fact, Wittgenstein did not read and/or failed to understand Gödel’s proof through at least 1941, how would he have responded if and when he understood it as (at least) a proof of the undecidability of P in PM on the assumption of PM’s consistency? Given his syntactical conception of mathematics, even with the extra-mathematical application criterion, he would simply say that P, qua expression syntactically independent of PM, is not a proposition of PM, and if it is syntactically independent of all existent mathematical language-games, it is not a mathematical proposition. Moreover, there seem to be no compelling non-semantical reasons—either intra-systemic or extra-mathematical—for Wittgenstein to accommodate P by including it in PM or by adopting a non-syntactical conception of mathematical truth (such as Tarski-truth (Steiner 2000)). Indeed, Wittgenstein questions the intra-systemic and extra-mathematical usability of P in various discussions of Gödel in the Nachlass and, at (§19), he emphatically says that one cannot “make the truth of the assertion [‘P’ or ‘Therefore P’] plausible to me, since you can make no use of it except to do these bits of legerdemain”. 

After the initial, scathing reviews of RFM, very little attention was paid to Wittgenstein’s (RFM App. III and RFM VII, §§21–22) discussions of Gödel’s First Incompleteness Theorem (Klenk 1976: 13) until Shanker’s sympathetic (1988b). In the last 22 years, however, commentators and critics have offered various interpretations of Wittgenstein’s remarks on Gödel, some being largely sympathetic (Floyd 1995, 2001) and others offering a more mixed appraisal (Rodych 1999a, 2002, 2003; Steiner 2001; Priest 2004; Berto 2009a). Recently, and perhaps most interestingly, Floyd & Putnam (2000) and Steiner (2001) have evoked new and interesting discussions of Wittgenstein’s ruminations on undecidability, mathematical truth, and Gödel’s First Incompleteness Theorem (Rodych 2003, 2006; Bays 2004; Sayward 2005; and Floyd & Putnam 2006). 

4. The Impact of Philosophy of Mathematics on Mathematics 

Though it is doubtful that all commentators will agree (Wrigley 1977: 51; Baker & Hacker 1985: 345; Floyd 1991: 145, 143; 1995: 376; 2005: 80; Maddy 1993: 55; Steiner 1996: 202–204), the following passage seems to capture Wittgenstein’s attitude to the Philosophy of Mathematics and, in large part, the way in which he viewed his own work on mathematics. 

What will distinguish the mathematicians of the future from those of today will really be a greater sensitivity, and that will—as it were—prune mathematics; since people will then be more intent on absolute clarity than on the discovery of new games. 

Philosophical clarity will have the same effect on the growth of mathematics as sunlight has on the growth of potato shoots. (In a dark cellar they grow yards long.) 

A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop. He has learned to regard them as something contemptible and… he has acquired a revulsion from them as infantile. That is to say, I trot out all the problems that a child learning arithmetic, etc., finds difficult, the problems that education represses without solving. I say to those repressed doubts: you are quite correct, go on asking, demand clarification! (PG 381, 1932) 

In his middle and later periods, Wittgenstein believes he is providing philosophical clarity on aspects and parts of mathematics, on mathematical conceptions, and on philosophical conceptions of mathematics. Lacking such clarity and not aiming for absolute clarity, mathematicians construct new games, sometimes because of a misconception of the meaning of their mathematical propositions and mathematical terms. Education and especially advanced education in mathematics does not encourage clarity but rather represses it—questions that deserve answers are either not asked or are dismissed. Mathematicians of the future, however, will be more sensitive and this will (repeatedly) prune mathematical extensions and inventions, since mathematicians will come to recognize that new extensions and creations (e.g., propositions of transfinite cardinal arithmetic) are not well-connected with the solid core of mathematics or with real-world applications. Philosophical clarity will, eventually, enable mathematicians and philosophers to “get down to brass tacks” (PG 467).