维特根斯坦论数学基础
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摘要
维特根斯坦认为20世纪初出现的数学基础三大流派之间的不同争论,实际上都是源于对数学基础问题的认识论思考的混淆,即他们没有从语言句法角度对数学基础问题做出真正的澄清。维特根斯坦认为数学本质上并不需要任何基础,无论逻辑还是元数学都不能成为数学的基础。数学是由计算构成的自为的系统,不同的系统之间具有不同的句法规则,数学命题是一种语法规则表述,具有规范性的地位,这种规范性既不来自于逻辑公理,也不是来自于无矛盾证明或人类心灵的直觉力量,而最终在于人类的生活实践之中。
Ludwig Wittgenstein argues that the different debates among three schools of foundations of Mathematics at the beginning of 20 th Century,as a matter of fact,all originate from the cognitive confusion of the foundations of Mathematics,that is to say,philosophers fail to make genuine clarifications from the point of syntax and grammar.He thinks that Mathematics does not need any foundation at all,whether it is logic or Metamathematics.Mathematics is made up of several systems which consists of different calculations.Each system has unique syntactical rules.Mathematical propositions express grammatical rules and they have normative status.This kind of normativity does not come from logical Axioms,Metamathematics or ground intuition,but comes from human practice.
Ludwig Wittgenstein argues that the different debates among three schools of foundations of Mathematics at the beginning of 20 th Century,as a matter of fact,all originate from the cognitive confusion of the foundations of Mathematics,that is to say,philosophers fail to make genuine clarifications from the point of syntax and grammar.He thinks that Mathematics does not need any foundation at all,whether it is logic or Metamathematics.Mathematics is made up of several systems which consists of different calculations.Each system has unique syntactical rules.Mathematical propositions express grammatical rules and they have normative status.This kind of normativity does not come from logical Axioms,Metamathematics or ground intuition,but comes from human practice.
引文
[1][2][14][29][36]Wittgenstein,L.Philosophische Grammatik,Herausgegen von Rush Rhees,Werkausgabe Band 4,Berlin:Suhrkamp Verlag,2015,pp.296-297,297,296,322,381.
[3][德]弗雷格:《算术基础》,王路译,王炳文校,北京:商务印书馆,2007年,第85页。
[4][英]罗素:《数理哲学导论》,晏成书译,北京:商务印书馆,2005年,第19-23页。
[5][6][7][9][15][16][17][20][23][24]Wittgenstein,L.Wittgenstein und der Wiener Kreis,Gesprche,aufgezeichnet von Friedrich Waismann,Werkausgabe Band 3,Berlin:Suhrkamp Verlag,2015,pp.222,222,164-165,223-224,136,119,120,122,149,121.
[8][25]Marion,M.Wittgenstein,Finitism,and The Foundations of Mathematics,Oxford:Clarendon Press,1998,pp.78,103-127.
[10][13]Wittgenstein,L.,Philosophische Bemerkungen,Werkausgabe Band 2,Frankfurt am Main:Suhrkamp Verlag,2015,pp.125,180.
[11][美]斯图亚特·夏皮罗:《数学哲学》,郝兆宽、杨睿之译,上海:复旦大学出版社,2014年,第155页。
[12]Hilbert,D.“On The Infinite”in From Frege to Gdel,ASource Book in Mathematical Logic,1879-1931,Jean Van Heijenoort(eds.)Cambridge,Massachusetts:Harvard University Press,1967,p.367.
[18]Shanker,S.,Wittgenstein and the Turning Point in the Philosophy of Mathematics,Albany:Sate University of New York Press,1987.
[19]维特根斯坦在其数学哲学中一直批判数学柏拉图主义(platonism),该观点主张数学研究的是外在的客观事实,数学命题的真和假主要依赖于这种事实描述和认识,所以数学命题主要是对于这些数学事实的客观描述。数学柏拉图主义者一般认为数学家的主要工作是发现真理,而不是发明或创造。数学柏拉图主义主要代表有数学家哈代(Hardy)以及哥德尔,而坚持认为数学主要是数学家发明和创造的结果的这种观点的主要代表有维特根斯坦和布劳威尔。关于维特根斯坦对于哈代观点的批评可参见,Wittgenstein,L.,Wittgenstein’s Lectures on the Foundations of Mathematics Cambridge,1939,Cora Diamond(eds.),Hassocks,Sussex:The Harvester Press,LTD,1976,pp.139-140.关于维特根斯坦和哥德尔之间的争论问题的有趣探讨,参见Wang,H.“To and From Philosophy-Discussions with G9del and Wittgenstein”,in Synthese,Vol.88:1991.pp.229-277.
[21]Frascolla,P.,Wittgenstein’s Philosophy of Mathematics,London and New York:Routledge,1994,p.101.
[22]Wrigley,M.,“Wittgenstein's Philosophy of Mathematics”,in Philosophical Quarterly,1977,27(106):pp.50-59.
[26][27]Brouwer,L.E.J.,“Mathematics,Science and Language”,in From Kant to Hilbert,A Source Book in the Foundations of Mathematics,Vol.Ⅱ,William Ewald(ed.),Oxford:Clarendon Press,1996,p.1176,177.
[28]Wittgenstein,L.,Tractatus Logico-Philosophicus,Werkausgabe Band 1,Frankfurt am Main:Suhrkamp Verlag,2014,pp.77.
[30]Wittgenstein,L.,The Big Typescript(TS 213),GermanEnglish Scholars’Edition,edited and translated by Luckhardt,C.G.and Maximilian A.E.Aue,Oxford:Basil Blackwell,2005,p.375.
[31][32][34]Wittgenstein,L.,Philosophische Untersuchungen.Werkausgabe Band1,Frankfurt am Main:Suhrkamp Verlag,2014,pp.336-337,337,246-250.
[33]Wittgenstein,L.Wittgenstein’s Lectures on the Foundations of Mathematics Cambridge,1939,Cora Diamond(eds.),Hassocks,Sussex:The Harvester Press,LTD,1976,p.237.
[35]Garavaso,P.,“Wittgenstein’s philosophy of Mathematics,A Reply to Two Objections”,in The Southern Journal of Philosophy,1988,Vol.ⅩⅩⅥ.No.2.
[3][德]弗雷格:《算术基础》,王路译,王炳文校,北京:商务印书馆,2007年,第85页。
[4][英]罗素:《数理哲学导论》,晏成书译,北京:商务印书馆,2005年,第19-23页。
[5][6][7][9][15][16][17][20][23][24]Wittgenstein,L.Wittgenstein und der Wiener Kreis,Gesprche,aufgezeichnet von Friedrich Waismann,Werkausgabe Band 3,Berlin:Suhrkamp Verlag,2015,pp.222,222,164-165,223-224,136,119,120,122,149,121.
[8][25]Marion,M.Wittgenstein,Finitism,and The Foundations of Mathematics,Oxford:Clarendon Press,1998,pp.78,103-127.
[10][13]Wittgenstein,L.,Philosophische Bemerkungen,Werkausgabe Band 2,Frankfurt am Main:Suhrkamp Verlag,2015,pp.125,180.
[11][美]斯图亚特·夏皮罗:《数学哲学》,郝兆宽、杨睿之译,上海:复旦大学出版社,2014年,第155页。
[12]Hilbert,D.“On The Infinite”in From Frege to Gdel,ASource Book in Mathematical Logic,1879-1931,Jean Van Heijenoort(eds.)Cambridge,Massachusetts:Harvard University Press,1967,p.367.
[18]Shanker,S.,Wittgenstein and the Turning Point in the Philosophy of Mathematics,Albany:Sate University of New York Press,1987.
[19]维特根斯坦在其数学哲学中一直批判数学柏拉图主义(platonism),该观点主张数学研究的是外在的客观事实,数学命题的真和假主要依赖于这种事实描述和认识,所以数学命题主要是对于这些数学事实的客观描述。数学柏拉图主义者一般认为数学家的主要工作是发现真理,而不是发明或创造。数学柏拉图主义主要代表有数学家哈代(Hardy)以及哥德尔,而坚持认为数学主要是数学家发明和创造的结果的这种观点的主要代表有维特根斯坦和布劳威尔。关于维特根斯坦对于哈代观点的批评可参见,Wittgenstein,L.,Wittgenstein’s Lectures on the Foundations of Mathematics Cambridge,1939,Cora Diamond(eds.),Hassocks,Sussex:The Harvester Press,LTD,1976,pp.139-140.关于维特根斯坦和哥德尔之间的争论问题的有趣探讨,参见Wang,H.“To and From Philosophy-Discussions with G9del and Wittgenstein”,in Synthese,Vol.88:1991.pp.229-277.
[21]Frascolla,P.,Wittgenstein’s Philosophy of Mathematics,London and New York:Routledge,1994,p.101.
[22]Wrigley,M.,“Wittgenstein's Philosophy of Mathematics”,in Philosophical Quarterly,1977,27(106):pp.50-59.
[26][27]Brouwer,L.E.J.,“Mathematics,Science and Language”,in From Kant to Hilbert,A Source Book in the Foundations of Mathematics,Vol.Ⅱ,William Ewald(ed.),Oxford:Clarendon Press,1996,p.1176,177.
[28]Wittgenstein,L.,Tractatus Logico-Philosophicus,Werkausgabe Band 1,Frankfurt am Main:Suhrkamp Verlag,2014,pp.77.
[30]Wittgenstein,L.,The Big Typescript(TS 213),GermanEnglish Scholars’Edition,edited and translated by Luckhardt,C.G.and Maximilian A.E.Aue,Oxford:Basil Blackwell,2005,p.375.
[31][32][34]Wittgenstein,L.,Philosophische Untersuchungen.Werkausgabe Band1,Frankfurt am Main:Suhrkamp Verlag,2014,pp.336-337,337,246-250.
[33]Wittgenstein,L.Wittgenstein’s Lectures on the Foundations of Mathematics Cambridge,1939,Cora Diamond(eds.),Hassocks,Sussex:The Harvester Press,LTD,1976,p.237.
[35]Garavaso,P.,“Wittgenstein’s philosophy of Mathematics,A Reply to Two Objections”,in The Southern Journal of Philosophy,1988,Vol.ⅩⅩⅥ.No.2.