潘文全

(嶺南師范學(xué)院 馬克思主義學(xué)院，廣東 湛江 524048)

在維特根斯坦的所有著作中,《RemarksontheFoundationsofMathematics》基本上不被人接受。克萊塞(Kreisel G)把這本著作看成是天才心靈的無(wú)意義成果。[1]達(dá)米特(Dummett M)認(rèn)為此著作存在很多問(wèn)題,要么思想表達(dá)模糊;要么某些段落之間相互矛盾;要么某些段落的主題不明確。[2]491-509安德森(Anderson A)也認(rèn)為維特根斯坦沒(méi)有清楚地理解數(shù)學(xué)基礎(chǔ)中的本質(zhì)問(wèn)題。[3]481-490特別地,在維特根斯坦對(duì)哥德?tīng)柖ɡ淼脑u(píng)價(jià)上尤為突出?？巳R塞認(rèn)為維特根斯坦的論證非常野蠻。達(dá)米特認(rèn)為評(píng)論的某些章節(jié),特別是關(guān)于一致性和哥德?tīng)柖ɡ淼牟糠?要么沒(méi)有價(jià)值要么含有錯(cuò)誤。安德森認(rèn)為這些評(píng)論表明維特根斯坦誤解了哥德?tīng)柖ɡ淼膬?nèi)容和原因。古德斯坦(Goodstein R)在他的《Wittgenstein’sPhilosophyofMathematics》中認(rèn)為維特根斯坦評(píng)論不完備性定理的核心是“一個(gè)數(shù)學(xué)命題被說(shuō)成是真的,它的唯一意義是在某些系統(tǒng)中(不一定是完全形式化的)可被證明,在數(shù)學(xué)中,真意味著可被證明”,因?yàn)榫S特根斯坦認(rèn)為,這里的“真”意味著可以在其它系統(tǒng)中證明,因此哥德?tīng)栒Z(yǔ)句被認(rèn)為可以在某個(gè)系統(tǒng)中被證明,但在另一個(gè)系統(tǒng)中不能被證明。[4]271-286

但是也有研究者辯護(hù)了維特根斯坦。尚卡爾(Shanker S)[5]155-256和羅奇(Rodych V)[6]首先進(jìn)行了這個(gè)工作;另外值得注意的是弗洛伊德(Floyd J),她發(fā)表了一系列論文辯護(hù)維特根斯坦,[7]指出維特根斯坦不認(rèn)為在數(shù)理邏輯中構(gòu)造出的數(shù)學(xué)刻畫(huà)出了數(shù)學(xué)的本質(zhì),那么數(shù)理邏輯中的“證明”“真”就不能完全刻畫(huà)了數(shù)學(xué)中的“證明”“真”的本質(zhì),維特根斯坦是在利用不完備性定理去論證這個(gè)觀點(diǎn),[8]所以她認(rèn)為維特根斯坦理解了哥德?tīng)柖ɡ?而且把哥德?tīng)柖ɡ砼c數(shù)論中的不可能性證明相類比。[9] 407

所以這個(gè)問(wèn)題遠(yuǎn)遠(yuǎn)沒(méi)有達(dá)到完全澄清的地步,值得繼續(xù)研究,一方面有助于挖掘維特根斯坦這本書(shū)中有價(jià)值的地方,以及厘清他的錯(cuò)誤之處;另一方面,從維特根斯坦的角度來(lái)看不完備性定理,對(duì)近年來(lái)的哥德?tīng)査枷胙芯烤哂幸欢ㄖ妗?/p>

一、維特根斯坦對(duì)不完備性定理的評(píng)論

維特根斯坦對(duì)不完備性定理的理解是從RFM Appendix III開(kāi)始的。

首先,維特根斯坦討論了“質(zhì)疑”“命令”等語(yǔ)句形式,他認(rèn)為即使采用陳述句來(lái)表達(dá)這些語(yǔ)句形式,也沒(méi)有人會(huì)認(rèn)為它們具有真假值,因?yàn)樗鼈兣c “陳述”和“說(shuō)明”等語(yǔ)句形式非常不同,僅僅是一種語(yǔ)言規(guī)則,無(wú)所謂真假,但是當(dāng)陳述句被說(shuō)出時(shí),真值游戲參與其中,它們或真或假,為真當(dāng)且僅當(dāng)對(duì)應(yīng)一個(gè)事實(shí),為假當(dāng)且僅當(dāng)不對(duì)應(yīng)一個(gè)事實(shí)。(1)原文的引文都放在了腳注，后面不再特別說(shuō)明。It is easy to think of a language in which there is not a form for questions, or commands, but question and command are expressed in the form of statements, e.g. in forms corresponding to our: “I should like to know if…” and “My wish is that…”No one would say of a question (e.g. whether it is raining outside) that it was true or false. Of course it is English to say so of such a sentence as “I want to know whether…” But suppose this form were always used instead of the question?[10]116-123

但是在討論數(shù)學(xué)命題的時(shí)候情況發(fā)生了變化,維特根斯坦用下象棋來(lái)類比證明定理,它們都有固定的出發(fā)點(diǎn)和規(guī)則。他認(rèn)為既然在象棋中“輸”“贏”只是相對(duì)于象棋本身而言的,那么定理的“真”“假”也是相對(duì)于數(shù)學(xué)系統(tǒng)本身而言的,不存在系統(tǒng)之外的事實(shí)對(duì)應(yīng)數(shù)學(xué)命題,這就表明在數(shù)學(xué)語(yǔ)境中討論真假時(shí),或者討論排中律在數(shù)學(xué)命題中的運(yùn)用時(shí),就不再是依據(jù)“對(duì)應(yīng)”來(lái)判定真假,而是無(wú)所謂真假,類似于“質(zhì)疑”“命令”等語(yǔ)句形式。那么在此意義上討論數(shù)學(xué)命題時(shí),“真”“假”這些語(yǔ)詞是可以消除的,這就是維特根斯坦的真理冗余論。如果在討論數(shù)學(xué)命題時(shí)引入了語(yǔ)詞“真”,那么“真”只能用“可證”來(lái)解釋,“真”的語(yǔ)句只能是數(shù)學(xué)系統(tǒng)中的公理或者定理(2)And-you say -these sentences are true or false. Or, as I might also say, the game of truth-functions is played with them. For assertion is not something that gets added to the proposition, but an essential feature of the game we play with it. Comparable, say, to that characteristic of chess by which there is winning and losing in it, the winner being the one who takes the other’s king. Of course, there could be a game in a certain sense very near akin to chess, consisting in making the chess moves, but without there being any winning and losing in it; or with different conditions for winning.。

接著維特根斯坦舉例說(shuō)明:我們習(xí)慣說(shuō)“2乘以2等于4”,而動(dòng)詞“等于”使這變成了一個(gè)命題,并且顯然與我們稱之為命題的一切事物建立了密切的親緣關(guān)系。然而,這只是一種表面的關(guān)系。[10]116-123有意義的數(shù)學(xué)命題與有意義的偶然命題表面上很相似,然而它們是不同的,區(qū)別在于后者依據(jù)對(duì)應(yīng)必須馬上是或真或假的,但前者不是或真或假的,所以它們之間的相似只是一種表面關(guān)系,但這誤導(dǎo)了數(shù)學(xué)家用判定偶然命題真假的方法去判定數(shù)學(xué)命題的真假,甚至在判定真假前就誤導(dǎo)數(shù)學(xué)家去思考數(shù)學(xué)命題的真假,比如命題“偶數(shù)是能被2整除的數(shù)”中的“是”誤導(dǎo)數(shù)學(xué)家以為對(duì)應(yīng)使得此命題為真,所以這種表面關(guān)系錯(cuò)誤地使得真值游戲同數(shù)學(xué)命題聯(lián)系起來(lái)了。然而,維特根斯坦要求把這兩者嚴(yán)格區(qū)分開(kāi),他認(rèn)為不需要 “數(shù)學(xué)真”的概念,進(jìn)而認(rèn)為討論數(shù)學(xué)原理中存在不能被證明為真的數(shù)學(xué)陳述時(shí),應(yīng)該避免使用語(yǔ)詞“真”“假”,“P是真的”就是“斷定了P”,“P是假的”就是“否定P”或者“斷定了P”。[6]

在這種基本思想之下,維特根斯坦用他自己的術(shù)語(yǔ)綜述了不完備性定理。如果在羅素系統(tǒng)中構(gòu)造出了一個(gè)命題P,且它被解釋為不可證的,如何證明P呢?維特根斯坦復(fù)述了證明過(guò)程:如果它是假的,但又被證明了,這當(dāng)然不可能了,因?yàn)榱_素系統(tǒng)只能證明真的語(yǔ)句;如果它被證明了,那么就證明了不可證明的語(yǔ)句,羅素系統(tǒng)不可能做得到。所以P只能是真的但不可證的(3)I imagine someone asking my advice; he says: “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’. 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.。

接下維特根斯坦就反思了這個(gè)論證。羅素系統(tǒng)中存在真但不可證的命題嗎?這個(gè)系統(tǒng)中的真命題在其它系統(tǒng)中能被斷定嗎?它類似于在歐式幾何中能夠存在真但不可證的命題嗎?在歐式幾何中存在可證但在其它系統(tǒng)中假的命題嗎?維特根斯坦認(rèn)為這些都是不可能,因?yàn)樗鼈儾皇窃谕粋€(gè)意義上來(lái)討論的,那么如何把“真”和“可證”放在同一意義上來(lái)討論呢?(4)“But may there not be true propositions which are written in this symbolism, but are not provable in Russell’s system?”—‘True propositions’, hence propositions which are true in another system, i.e. can rightly be asserted in another game. Certainly; why should there not be such propositions; or rather: why should not propositions—of physics, e.g. —be written in Russell’s symbolism? The question is quite analogous to: Can there be true propositions in the language of Euclid, which are not provable in his system, but are true?—Why, there are even propositions which are provable in Euclid’s system, but are false in another system. May not triangles be—in another system—similar (very similar) which do not have equal angles?—“But that’s just a joke! For in that case they are not ‘similar’ to one another in the same sense!”—Of course not; and a proposition which cannot be proved in Russell’s system is “true” or “false” in a different sense from a proposition of Principia Mathematica.他認(rèn)為羅素系統(tǒng)中的真等價(jià)于羅素系統(tǒng)中的可證,羅素系統(tǒng)中的假等價(jià)于在羅素系統(tǒng)中證明了語(yǔ)句的否定。假設(shè)P在羅素系統(tǒng)中是可證的,這就意味著在羅素的意義上它是真的,那么必須放棄“P是不可證的”這種解釋;假設(shè)P在羅素系統(tǒng)中是真的,這就意味著在羅素的意義上它是可證的,那么也要放棄這種解釋。如果P在其它系統(tǒng)中是假的,這與它在羅素意義上是真的并不矛盾,因?yàn)樗诹_素的系統(tǒng)中被證明了(5)Just as we ask: “‘provable’ in what system?”, so we must also ask: “ ‘true’ in what system?” ‘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.—Now what does your “suppose it is false” mean? In the Russell sense it means ‘suppose the opposite is proved in Russell’s system’; if that is your assumption, you will now presumably give up the interpretation that it is unprovable. And by ‘this interpretation’ I understand the translation into this English sentence.—If 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. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: 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. (What is called “l(fā)osing” in chess may constitute winning another game.)。

因此,維特根斯坦對(duì)不完備性定理回答是明確的,用他自己的術(shù)語(yǔ)來(lái)說(shuō),一個(gè)“真但不可證”的數(shù)學(xué)命題是一種術(shù)語(yǔ)上的矛盾,因?yàn)橐粋€(gè)真的數(shù)學(xué)命題必須在某個(gè)演算中被證明,至少是可以證明的。如果一個(gè)命題在羅素系統(tǒng)中是可證的,這就意味著它在羅素的意義上是真的,所以維特根斯坦認(rèn)為“P是不可證的”這個(gè)解釋必須被放棄。

這里維特根斯坦犯了一個(gè)錯(cuò)誤,這也是招致克萊塞、達(dá)米特等人批評(píng)的最大原因。這個(gè)錯(cuò)誤是顯然的,因?yàn)椴煌陚湫远ɡ淼母绲聽(tīng)栕C明并不需要哥德?tīng)栒Z(yǔ)句P被理解為“P在羅素系統(tǒng)中不可證”這個(gè)語(yǔ)義解釋,也就是說(shuō)不需要假設(shè)P的自然語(yǔ)言意義以獲得矛盾,但是維特根斯坦在用這種解釋進(jìn)行論證。

基于這個(gè)錯(cuò)誤的理解,維特根斯坦甚至滑入到了認(rèn)為不完備性定理毫無(wú)價(jià)值的地步。他認(rèn)為P本身屬于羅素系統(tǒng)的定理,就屬于此系統(tǒng);P又被解釋為不可證的,依據(jù)他的觀點(diǎn),不可證的一定不是系統(tǒng)的公理或者定理,就不屬于此系統(tǒng),那么P既屬于又不屬于羅素系統(tǒng),矛盾就出現(xiàn)了。他認(rèn)為這種矛盾類似于說(shuō)謊者悖論,使得語(yǔ)言變得沒(méi)有使用價(jià)值,關(guān)注這些矛盾僅僅是因?yàn)樗苷勰ト祟?使人類對(duì)這種悖論產(chǎn)生興趣,除此之外沒(méi)有價(jià)值,所以他認(rèn)為不完備性定理的價(jià)值有限(6)Let us suppose I prove the unprovability (in Russell’s system) of P; then by this proof I have proved P. Now if this proof were one in Russell’s system—I should in that case have proved at once that it belonged and did not belong to Russell’s system. —That is what comes of making up such sentences. —But there is a contradiction here! —Well, then there is a contradiction here. Does it do any harm here?Is there any harm in the contradiction that arises when someone says: ‘I am lying. —So I am not lying. —So I am lying. —etc.’? I mean: does it make our language less usable if in this case, according to the ordinary rules, a proposition yields its contradictory, and vice versa?— the proposition itself is unusable, and these inferences equally; but why should they not be made?— It is a profitless performance!— It is a language-game with some similarity to the game of thumb-catching.Such a contradiction is of interest only because it has tormented people, and because this shews both how tormenting problems can grow out of language, and what kind of things can torment us.。

這些評(píng)論可能比其它評(píng)論更招人批評(píng),它們通常被籠統(tǒng)地看成是維特根斯坦對(duì)不完備性定理的攻擊,然而維特根斯坦沒(méi)有攻擊哥德?tīng)柖ɡ砗退淖C明,他攻擊的只是哥德?tīng)柖ɡ淼恼Z(yǔ)義證明的一個(gè)非形式化版本。[11]如何來(lái)解讀維特根斯坦的這種理解呢?

二、維特根斯坦的原因

為什么維特根斯坦對(duì)不完備性定理會(huì)做出這些評(píng)論?原因在于他堅(jiān)持一種數(shù)學(xué)立場(chǎng)——反柏拉圖主義,反對(duì)數(shù)學(xué)實(shí)在論。在它看來(lái),哲學(xué)渴望去理解那種終極的東西,而且使用日常的術(shù)語(yǔ)去解釋它們,為了反對(duì)這種做法,維特根斯坦的一個(gè)主要策略是去表明這些哲學(xué)概念中沒(méi)有一個(gè)具有穩(wěn)定的內(nèi)涵,它們是一種家族類似的概念,比如在PI中他明確地把“數(shù)”解釋為一個(gè)家族類似的概念。[12]37甚至“真”也是一樣,[7]一個(gè)命題是真的到底意味著什么?他有一個(gè)真理冗余論的回答——“P是真的=P”,這意味著判定“P是真的”的標(biāo)準(zhǔn)就是判定P的標(biāo)準(zhǔn),那么很自然地“P是真的”當(dāng)且僅當(dāng)“P是可證的”。

什么又是可證呢?他認(rèn)為只有當(dāng)數(shù)學(xué)命題位于證明的結(jié)尾或者作為一個(gè)公理時(shí),才能算是證明了此命題,也就是說(shuō)在羅素系統(tǒng)中證明了一個(gè)命題當(dāng)且僅當(dāng)它要么作為公理要么作為被推導(dǎo)出來(lái)的命題,不存在其它方式去證明一個(gè)命題。因此,“可證”是相對(duì)于特定系統(tǒng)而言的,系統(tǒng)不同“可證”也不同,所以“可證”僅僅是一個(gè)家族相似的概念,“真”也是一個(gè)家族相似的概念,“真”在哲學(xué)中沒(méi)有解釋的作用。另外,在后期維特根斯坦激進(jìn)構(gòu)造主義的立場(chǎng)上,他認(rèn)為數(shù)學(xué)是由運(yùn)用概念的規(guī)則所構(gòu)成的,采納了規(guī)則就創(chuàng)造了數(shù)學(xué)真,但不是發(fā)現(xiàn)數(shù)學(xué)真,“我將一次又一次地試圖表明所謂的數(shù)學(xué)發(fā)現(xiàn)最好被稱為數(shù)學(xué)發(fā)明”。[13]361-363所以“真”只能在數(shù)學(xué)規(guī)則的創(chuàng)造中被定義,也就是說(shuō)數(shù)學(xué)命題是在證明的語(yǔ)境中被斷定的,這對(duì)于數(shù)學(xué)原理的形式語(yǔ)境也是一樣,如果在數(shù)學(xué)原理的證明語(yǔ)境之外去判定一個(gè)語(yǔ)句的真假,這就是一種柏拉圖主義。

維特根斯坦用哥德?tīng)柖ɡ韥?lái)反駁柏拉圖主義。對(duì)于把“數(shù)學(xué)真”和“數(shù)學(xué)證明”聯(lián)系起來(lái)的哲學(xué)家而言,哥德?tīng)柖ɡ肀砻鳌皵?shù)學(xué)真”的概念不是絕對(duì)正確的,這也是維特根斯坦的立場(chǎng),然而他卻不是在數(shù)學(xué)實(shí)在論的立場(chǎng)上來(lái)討論不完備性定理,而是用哥德?tīng)柖ɡ韥?lái)論證自己的論點(diǎn)——“數(shù)”“證明”“游戲”等概念沒(méi)有意義,有窮多個(gè)甚至遞歸可數(shù)多個(gè)公理不能刻畫(huà)數(shù)的概念,也就是說(shuō)沒(méi)有形式過(guò)程能夠刻畫(huà)數(shù)的所有意義,[11]因?yàn)槿魏瓮陚涞男问较到y(tǒng)都用有限個(gè)遞歸公理來(lái)刻畫(huà)算術(shù),不完備性定理表明總是存在此系統(tǒng)不能表達(dá)的某個(gè)定理。進(jìn)而他利用不完備性定理去論證家族相似的觀點(diǎn),以此得出“真”“數(shù)”只具有相對(duì)的意義,反駁數(shù)學(xué)實(shí)在論。

具體而言,維特根斯坦如何進(jìn)行反駁?實(shí)在論意義上的真具有三個(gè)特點(diǎn):一、在羅素系統(tǒng)中真與可證是不一樣的,假設(shè)P是不可證的并不蘊(yùn)涵P是不真的;二、羅素系統(tǒng)的所有定理都是真的;三、沒(méi)有語(yǔ)句既真又假。這也是塔斯基(Tarski A)的真概念,[14]3-34羅素系統(tǒng)完全符合此概念,而塔斯基的真概念中含有形而上學(xué)的成分,維特根斯坦打算剔除這種成分,保留“真”剔除“塔斯基真”,那么就必須檢查由哥德?tīng)柖ɡ硭砻鞯倪@二者的關(guān)系,也就是說(shuō)對(duì)于已證明了哥德?tīng)柖ɡ淼臄?shù)學(xué)家而言,他們想讓羅素系統(tǒng)包含不可判定語(yǔ)句P,就必須依據(jù)哥德?tīng)柖ɡ聿捎盟够孀鳛橐话阏娴臄U(kuò)張,避免采用塔斯基真的唯一方法是攻擊哥德?tīng)柖ɡ?所以維特根斯坦為了反對(duì)數(shù)學(xué)實(shí)在論必須攻擊哥德?tīng)柖ɡ??？梢钥闯鼍S特根斯坦的目標(biāo)是反對(duì)形而上學(xué),形而上學(xué)在數(shù)學(xué)哲學(xué)中表現(xiàn)為脫離了證明的“真”,在理論上表現(xiàn)為塔斯基真,而攻擊塔斯基真的橋梁就是反對(duì)不完備性定理。

另外在維特根斯坦看來(lái),哥德?tīng)柖ɡ硪呀?jīng)成為數(shù)學(xué)實(shí)在論的一個(gè)標(biāo)志——數(shù)學(xué)真不同于定理的證明活動(dòng),而且哥德?tīng)柖ɡ硎沟迷谌魏蜗到y(tǒng)中數(shù)學(xué)真與可證性是不相同的,這就鼓勵(lì)了“數(shù)學(xué)真是多樣的”這種觀點(diǎn),而且鼓勵(lì)了“在任意系統(tǒng)中數(shù)學(xué)真都是不可證的”,那么柏拉圖主義就是不可避免的結(jié)果。為了反對(duì)數(shù)學(xué)實(shí)在論,維特根斯坦曾經(jīng)攻擊康托的集論,他認(rèn)為集論不是數(shù)學(xué)而是形而上學(xué),在維特根斯坦看來(lái)哥德?tīng)柖ɡ眍愃朴诩?而且被一種悖論的氣氛所環(huán)繞,不幸的是,哥德?tīng)栐谒攵ɡ淼姆切问交u(píng)論中,塑造了這種印象——他的論證類似于語(yǔ)義悖論,所以在維特根斯坦看來(lái)哥德?tīng)柖ɡ砭褪桥鴶?shù)學(xué)外衣的哲學(xué),這也成為他反對(duì)此定理的另一個(gè)原因。

所以“維特根斯坦反對(duì)不完備性定理”是一種誤解,他沒(méi)有反對(duì)不完備性定理的哥德?tīng)栕C明,只是反對(duì)不完備性定理的通俗解釋,甚至是柏拉圖式的解釋。

三、維特根斯坦對(duì)不完備性定理評(píng)論的價(jià)值

在維特根斯坦對(duì)不完備性定理的評(píng)論中存在哪些有益的啟發(fā)呢?歷史給出了答案:一個(gè)是弗協(xié)調(diào)邏輯的思想;另一個(gè)是如何確定數(shù)學(xué)命題的意義。

(一)弗協(xié)調(diào)邏輯的思想

首先關(guān)注維特根斯坦的一段評(píng)論:假設(shè)在羅素系統(tǒng)中證明了P的不可證性,不可證表明它不屬于系統(tǒng)的定理,又通過(guò)“P當(dāng)且僅當(dāng)P在數(shù)學(xué)原理中是不可證的”證明了P,P可證表明它屬于系統(tǒng)的定理,這就出現(xiàn)了一個(gè)矛盾——P既屬于又不屬于羅素系統(tǒng),這有什么危害嗎?當(dāng)某人說(shuō)“我正在撒謊,所以我沒(méi)有撒謊,所以我正在撒謊……”,這其中的矛盾有什么危害嗎?在這些情況下語(yǔ)言減少了可使用性,命題本身變得無(wú)用,它的用處僅僅是困擾人們(7)Let us suppose I prove the unprovability (in Russell’s system) of P; then by this proof I have proved P. Now if this proof were one in Russell’s system—I should in that case have proved at once that it belonged and did not belong to Russell’s system. —That is what comes of making up such sentences. —But there is a contradiction here!— Well, then there is a contradiction here. Does it do any harm here? Is there any harm in the contradiction that arises when someone says: ‘I am lying. —So I am not lying. —So I am lying. —etc.’? I mean: does it make our language less usable if in this case, according to the ordinary rules, a proposition yields its contradictory, and vice versa?— the proposition itself is unusable, and these inferences equally; but why should they not be made?— It is a profitless performance!— It is a language-game with some similarity to the game of thumb-catching.Such a contradiction is of interest only because it has tormented people, and because this shews both how tormenting problems can grow out of language, and what kind of things can torment us.。

這些評(píng)論表明維特根斯坦接受矛盾,支持在數(shù)學(xué)原理中P既可證又不可證,或者支持它們都是可證的,而且他喜歡這種觀點(diǎn)。此外維特根斯坦接受矛盾并不是一個(gè)孤立的觀點(diǎn),在評(píng)論的其它地方也存在類似表達(dá)。這就觸碰到了邏輯學(xué)對(duì)矛盾的恐懼之處,所以這些評(píng)論極其招人厭惡,也是他的數(shù)學(xué)哲學(xué)不被認(rèn)可的一個(gè)重要原因。

怎么去評(píng)論維特根斯坦接受矛盾呢?首先,維特根斯坦接受矛盾的論據(jù)是錯(cuò)誤的,他假設(shè)在羅素系統(tǒng)中證明了P的不可證性,顯然這是不成立的。其次,“矛盾是真的”這種觀點(diǎn)是不正確的,毫無(wú)疑問(wèn),維特根斯坦不為這種反駁所動(dòng),而且也給出了他反柏拉圖主義的辯護(hù),在語(yǔ)言游戲中為真的東西就是正確執(zhí)行游戲規(guī)則所得到的東西,如果一個(gè)語(yǔ)言游戲能正確生成形如P和P的語(yǔ)句,那么它們都是真的,所以矛盾在這種游戲中也是真的。最后,即使矛盾是真的,也很難像維特根斯坦所說(shuō)的那樣無(wú)害,因?yàn)樵诮?jīng)典邏輯系統(tǒng)中矛盾可以推出任意語(yǔ)句,這就摧毀了語(yǔ)言游戲。[6]

雖然上述維特根斯坦對(duì)矛盾的觀點(diǎn)有很多的錯(cuò)誤,但是它有一個(gè)重要的啟發(fā),且歷史給出了辯護(hù),這就是弗協(xié)調(diào)邏輯。的確,發(fā)展這種邏輯的一個(gè)原因在于它能夠恰當(dāng)?shù)靥幚磴Ｕ撏评?在其中矛盾不能推出任何有害的東西,當(dāng)然維特根斯坦對(duì)于這種未來(lái)的發(fā)展一無(wú)所知。但是在弗協(xié)調(diào)邏輯的前科學(xué)評(píng)論中,維特根斯坦在1930年就預(yù)見(jiàn)了它的發(fā)展,“事實(shí)上,即使在這個(gè)階段,我也預(yù)測(cè)了一個(gè)時(shí)期,將會(huì)出現(xiàn)一個(gè)包含矛盾的演算,人們將會(huì)為從一致性中解放出來(lái)而感到自豪”。[15]332所以維特根斯坦支持這種弗協(xié)調(diào)邏輯的思想,如果語(yǔ)言游戲是弗協(xié)調(diào)的,那么“矛盾摧毀了語(yǔ)言”這種反對(duì)意見(jiàn)將是不充分的。

但是維特根斯坦也不能快速地?cái)[脫批評(píng),在數(shù)學(xué)原理中維特根斯坦接受矛盾確實(shí)是錯(cuò)誤的,因?yàn)閿?shù)學(xué)原理的推導(dǎo)規(guī)則是經(jīng)典邏輯的規(guī)則,矛盾具有很大的破壞性,雖然并不會(huì)導(dǎo)致系統(tǒng)完全無(wú)用,但對(duì)于用這個(gè)系統(tǒng)研究算術(shù)而言,它是無(wú)用的。如果考慮到弗協(xié)調(diào)邏輯,維特根斯坦的思想就另當(dāng)別論了,數(shù)學(xué)原理僅僅是他討論中的一個(gè)例子,類似的討論可以應(yīng)用到算術(shù)的任何其它形式系統(tǒng)上,當(dāng)然包括基于弗協(xié)調(diào)邏輯的形式系統(tǒng),在此系統(tǒng)中,每個(gè)語(yǔ)句或者它的否定都是可推導(dǎo)的,甚至系統(tǒng)也是完全的。在這樣的系統(tǒng)中矛盾的推導(dǎo)被限制在某個(gè)范圍內(nèi),而且也不會(huì)摧毀系統(tǒng)的可應(yīng)用性,就像維特根斯坦在這里所說(shuō)的,在這種系統(tǒng)中哥德?tīng)柌豢膳卸ㄕZ(yǔ)句和它的否定都是可證的。所以即使對(duì)于數(shù)學(xué)原理而言,維特根斯坦是錯(cuò)誤的,但他的觀點(diǎn)運(yùn)用到弗協(xié)調(diào)邏輯時(shí)是正確的。

(二)數(shù)學(xué)命題的意義

不可證明性的證明究竟意味著什么?維特根斯坦嘗試了另一種可能性:他認(rèn)為一個(gè)不可證明性的證明類似于一種特殊的幾何證明——它證明了一個(gè)特定的圖形不能通過(guò)特定的方法產(chǎn)生,例如通過(guò)尺規(guī)作圖不能完成三等分任意角。這種幾何證明包含一個(gè)經(jīng)驗(yàn)的預(yù)測(cè),即提前知道了無(wú)論多么努力都不能用這種方法做出圖形,如果類比不可證明性的結(jié)果,它就迫使研究者放棄尋找這種系統(tǒng)內(nèi)的證明。但是維特根斯坦不認(rèn)可這種類比,他認(rèn)為不可證明性結(jié)果是一個(gè)矛盾,不能用于預(yù)測(cè)(8)A proof of unprovability is as it were a geometrical proof; a proof concerning the geometry of proofs. Quite analogous e.g. to a proof that such-and-such a construction is impossible with ruler and compass. Now such a proof contains an element of prediction, a physical element. For in consequence of such a proof we may say to a man: ‘Don’t exert yourself to find a construction (of the trisection of an angle, say)—it can be proved that it can’t be done’. That is to say: it is essential that the proof of unprovability should be capable of being applied in this way. It must —we might say—be a forcible reason for giving up the search for a proof (i.e. for a construction of such-and-such a kind).A contradiction is unusable as such a prediction.。

如果不可證明性語(yǔ)句不用這種方式解釋,它又意味著什么?維特根斯坦以一種更加一般的方法進(jìn)行了嘗試:數(shù)學(xué)命題的意義是什么?經(jīng)典邏輯學(xué)家會(huì)認(rèn)為命題的意義是由它的真值條件決定的,而維特根斯坦認(rèn)為是由它的證明條件決定的。證明提供了判定命題意義的一個(gè)標(biāo)準(zhǔn),如果還沒(méi)有找到命題的證明,就沒(méi)有根據(jù)去說(shuō)明它的意義;如果找到了一個(gè)新證明,就有了一個(gè)新的標(biāo)準(zhǔn)去理解它;而且證明不能從構(gòu)成證明的概念系統(tǒng)中分離出來(lái),因此語(yǔ)句的意義是由它的證明條件構(gòu)成的將更加準(zhǔn)確。這當(dāng)然是一種直覺(jué)主義的意義解釋,但也是維特根斯坦的觀點(diǎn),因?yàn)樗呀?jīng)用證明來(lái)確定真。所以為了找到命題“P是不可證明的”的意義,就必須尋找它的證明,只有證明才能給出這個(gè)結(jié)論的意義(9)The proposition ‘P is unprovable’ has a different sense afterwards—from before it was proved.If it is proved, then it is the terminal pattern in the proof of unprovability. —If it is unproved, then what is to count as a criterion of its truth is not yet clear, and—we can say—its sense is still veiled. Whether something is rightly called the proposition ‘X is unprovable’ depends on how we prove this proposition. The proof alone shows what counts as the criterion of unprovability. The proof is part of the system of operations, of the game, in which the proposition is used, and shews us its ‘sense’.。接下里,維特根斯坦把此思想運(yùn)用到“P”和“P的不可證性”的證明上。[16]207-227

怎么才能把P看成是已被證明了?維特根斯坦給出了兩種方式:“通過(guò)P的不可證性證明P”和“直接證明P”,即“在元語(yǔ)言中P是不可證明的當(dāng)且僅當(dāng)P”和“在對(duì)象語(yǔ)言中證明P”。在第一種情形中,維特根斯坦僅僅提醒從證明中尋找結(jié)論的意義,他認(rèn)為意義可能會(huì)影響結(jié)論,而且這種證明形式不會(huì)導(dǎo)致證明P,也就是說(shuō)這種證明的構(gòu)造不會(huì)被具體邏輯系統(tǒng)的規(guī)則所影響,因?yàn)樗且环N元語(yǔ)言的證明,是超出羅素系統(tǒng)這種語(yǔ)言游戲之外的證明。在第二種情形中,P在數(shù)學(xué)原理中是可證的,表明這個(gè)證明生效了,無(wú)論P(yáng)意味著什么,都不受解釋“P是不可證的”的影響(10)Now, how am I to take P as having been proved? By a proof of unprovability? Or in some other way? Suppose it is by a proof of unprovability. Now, in order to see what has been proved, look at the proof. Perhaps it has here been proved that such-and-such forms of proof do not lead to P. —Or, suppose P has been proved in a direct way —as I should like to put it—and so in that case there follows the proposition ‘P is unprovable’, and it must now come out how this interpretation of the symbols collides with the fact of the proof, and why it has to be given up here.。

所以維特根斯坦強(qiáng)調(diào)數(shù)學(xué)命題的意義只能在它的證明條件中去尋找,不能分離開(kāi)“證明”來(lái)討論“真”,“真”只是一個(gè)相對(duì)的概念,不能泛泛而談,這本身也是對(duì)柏拉圖主義的反駁。

結(jié)語(yǔ)

綜上,維特根斯坦基于反柏拉圖主義的立場(chǎng),堅(jiān)持“真”即“可證”的觀點(diǎn),以此拒斥哥德?tīng)柕牟煌陚湫远ɡ?但是不能說(shuō)維特根斯坦是錯(cuò)誤的,這只是基于不同哲學(xué)立場(chǎng)所蘊(yùn)含的東西,類似于直覺(jué)主義數(shù)學(xué)不認(rèn)同經(jīng)典數(shù)學(xué)中的很多結(jié)論一樣。維特根斯坦認(rèn)為接受不完備性定理還需要更多的條件,即充分揭露“數(shù)學(xué)真”和“數(shù)學(xué)可證”之間的不同,這個(gè)主題目前仍然沒(méi)有被充分研究。然而這并不意味著維特根斯坦對(duì)不完備性定理的評(píng)論都是正確的,他仍然犯了很多錯(cuò)誤,但這些錯(cuò)誤并不能掩蓋他對(duì)數(shù)學(xué)哲學(xué)的深刻洞見(jiàn),即弗協(xié)調(diào)邏輯和數(shù)學(xué)命題意義的思想。