Condition:

Event and the Historical Condition of Philosophical Thinking


LAI Tsz Yuen


About the author
Lai Tsz Yuen, Ph.D. Candidate, Department of Humanities and Creative Writing, Hong Kong Baptist University, research topic: French mathematical structuralism (Bourbaki, Lautman, Deleuze, Badiou) and the realist/materialist turn of contemporary philosophy. Author of Topography: 12 Interviews with Contemporary Art Institutions in Hong Kong (Hong Kong: Exterior Culture, 2011).

黎子元,香港浸會大學人文及創作系博士候選人,研究課題為法國數學結構主義思潮和當代哲學的實在論/唯物論轉向。《測繪香港藝術地形——12間當代藝術機構訪談》(香港·域外文化2011年)編著者。09年以來從事文化評論和當代藝術策展。



Philosophy is prescribed by conditions that constitute types of truth or generic-procedure. These types are science (more precisely, the matheme), art (more precisely, the poem), politics (more precisely, politics in interiority, or a politics of emancipation) and love (more precisely, the procedure that makes truth of the disjunction of sexuated positions).

Alain Badiou, Conditions, (1992, 2008), p. 23

 

In Alain Badiou’s Manifesto for Philosophy (1989, 1999), the problem of the condition of philosophy has been put on the agenda again.[1] In the particular context of contemporary philosophy ― from the end of the 20th century to present, confronting the ideology of ‘the end of philosophy’ and the current situation that Badiou calls ‘philosophy’s paralysis’ ― the re-examination of this problem, for Badiou, is to reaffirm the possibility and necessity of philosophy by providing it with proper restrictions: ‘Philosophy has begun; it does not exist within all historic configurations; its way of being is discontinuity in time as in space. It must thus be presupposed that it requires particular conditions’.[2] Here, Badiou examines the possibility of philosophy in terms of its conditioned productive process, what he calls ‘generic procedure’, and announces the four generic procedures: the matheme, the poem, political invention and love,[3] from which truth, the destination of philosophy, can be generated and seized by philosophy.[4] In this sense, matheme, poem, politics and love, as truth procedures, are four conditions of philosophy. They can be qualified as conditions because they are the ‘evental origin’ of truth. In these four fields, event, something new and exceptional that supplements a situation or state of things, can take place, so that truth, which cannot be absorbed by the established system of knowledge relative to the situation but, rather, suspends and collapses the orderings of this system, is able (‘able’ here means ‘by chance’) to be generated in terms of the invention of new name (additional signifier) and the configuration of generic procedure corresponding to that particular event: ‘The origin of a truth is of the order of the event’.[5] Thus it can be seen that it is an event happened in the field of matheme, poem, politics or love that defines and configures particular generic procedures. Through these generic procedures philosophy becomes possible and truth is able to be generated. Motivating by an event, philosophy within its conceptual space liberates the immanent movement of thought and relates those heterogeneous truth procedures to the singularity of time. Since philosophy is always ‘a reflecting torsion about that which conditions it’, it is the crises, the precarious conditions that sustains philosophy and makes a philosophy the great one: Philosophy pronounces the conjuncture of truth and its compounding function sets the generic procedures in the dimension of their joint historicity.[6]

 

What does ‘condition’ mean in particular case? Here, I merely focus on the mathematical condition of Badiou’s thought for the sake of convenience in discussion. Concerning Badiou’s philosophy, or more precisely, his ontology presented in his first magnum opus, Being and Event (1988, 2006), the constitution of set theory of Georg Cantor (1845-1918), as an event happened in the history of the 19th century mathematics, is the mathematical condition of his thought.[7] Cantor’s development of ‘transfinitude’ and his two theoretical inventions around this development, the first concerns the infinite multiplicity in positive sense by establishing the evaluation of different orders of infinity while the second concerns the consistency of the infinite in terms of the concept of ‘set’ in order to solve the traditional difficulty of the inconsistencies of the infinite, constitutes a critical turning point in the history of mathematics, a history that has been haunted by controversies on the problem of the one and the multiple since its beginning. From the first theoretical invention, Cantor finds that there must be orders of infinity in excess of the denumerable infinity or the ‘counted infinity’. From the second, he coins the term ‘transfinite’ which are numbers (cardinal numbers, ℵ) that are larger than all finite numbers yet not necessarily absolutely infinite. By developing such a ‘transfinite paradise’ (imitating David Hilbert’s expression), Cantor invents a new form of multiplicity, namely, a multiplicity without one. Based on Cantor’s invention, Badiou in Being and Event conceives and executes a subtraction of ontology from the metaphysics of the one and make his fundamental distinction between consistent multiplicity (multiplicity counted as ones) and inconsistent multiplicity (multiplicity qua multiplicity). This inconsistent multiplicity has been denied, excluded by the metaphysics of the one and considered as ‘nothing’, as the ‘impossible’. By subtraction or by overcoming the metaphysical constraint of thought that claims ‘what is not a being is not a being’ (Leibniz’s formulation), Badiou makes his decision: ‘the one is not’,[8] which means that ‘nothing’ is, and dedicates his philosophical project towards the problem of that ‘impossible’: Can multiplicity be thought for the sake of the multiplicity rather than reducing multiplicity into any one? In the condition of this ‘Cantorian event’, which happened in mathematics yet contained great ontological significance, Badiou is able to deploy inconsistent multiplicity into his ontology and identify ontology with mathematics by announcing his famous statement: ‘Mathematics is ontology’[9] ― the ontological problem of ‘being’ (being qua being) can be ultimately grasped in terms of an ontology/mathematics of the ‘multiplicity’ (multiplicity qua multiplicity).

 

However, it cannot be ignored that Cantor’s set theory has its limitation, e.g., his intuitive definition of ‘set’, simply as an aggregation of separate objects of intuition or thought, leads to contradiction (which has been exposed by Russell’s paradox). In the subsequent development of set theory this definition is rejected by many mathematicians. Badiou is well aware of the distance between Cantor’s naïve set theory and the axiomatic set theory developed in the 20th century (Zermelo–Fraenkel set theory, in short, ZF, now becoming the standard of set theory) and must acknowledge that his ontology (concerning Being and Event) is deeply influenced by Ernst Zermelo and Abraham Fraenkel’s axiomatization of set theory, Nicolas Bourbaki’s naming of the empty set (∅), Paul Cohen’s forcing of the generic sets, etc. Then in what sense can we say that the constitution of Cantor’s set theory is the mathematical condition of Badiou’s philosophy? Here, a more definite meaning of ‘condition’ shall be explicated. The Cantorian event as a condition means that this event defines the problematics (the problem domain) of the post-Cantorian thought. It is equivalent to saying that what conditions Badiou’s thought is not merely those inventions of Cantor’s set theory but its difficulties. And, for Badiou, the latter is definitely more crucial to the post-Cantorian thought including his philosophy. In Cantor’s later period of research, when he considers ‘what sort of multiplicity would constitute a set’ and ‘can there be a set of all number’, he realizes that, concerning the cardinality (the size) of set, his conception of set leads to contradiction. In order to preserve the consistency of his idea of the well-ordered set, he cannot but separate the inconsistent multiplicities, the ‘untotalizable’ multiplicities what he calls the ‘absolutely infinite’, from any consistent multiplicity that would constitute a set,[10] and leave this inconsistent multiplicities to the realm of the absolute (God). This ‘point of the impasse’[11] of Cantor that forces him to go through with his doctrine of the absolute opens up an evental site and thus prescribes the subsequent development, that is, the axiomatization of set theory in the following decades. It also grounds and limits Badiou’s ontology within the horizon of Cantor’s ‘absolutely infinite’. Paradoxically, it is this absolutely infinite that has been excluded by Cantor from his conception of set makes the conception possible.[12] It reveals a topological relation, therein the inconsistent or excessive multiplicities grounds the establishment of the consistent multiplicities, the obstacle or ‘the point of impossibility’ grounds the consistent treatment of the inconsistency in mathematics, the ‘non-being’ grounds a presentation of being in ontology ― in a word, it reveals the position of the ‘not’, the exclusion that functions as the precondition of the mathematical ontology and its operation. Upon this ‘not’ Badiou develops his most important concept: the void. And this topological relation or the dialectic conditioned by topology manifests the profundity, the dynamic of Badiou’s entire philosophical project in Being and Event.

 

Following Badiou’s retrospect of this Cantorian event, we see the historicity of ‘condition’. The constitution of Cantor’s set theory in the late 19th century was singular in the history of mathematics. His invention of ‘set’ provided a language that could be used in the definitions of all kinds of mathematical objects. And its difficulties (paradoxes) forced set theory towards axiomatization in the 20th century, which is ‘an intrinsic necessity’ for the development of set theory itself.[13] Between 1908 and 1940, the task of axiomatization was undertaken by Zermelo (among some other mathematicians), and accomplished by Fraenkel (i.e. the ZF system) as well as by Bourbaki (see ‘Summary of Results’, first published in 1939-1940, and then incorporated in Elements of Mathematics, Vol. 1: Theory of Sets). Thereafter, set theory became a foundational system of modern mathematics. For Badiou, the axiomatization of set theory manifests even more crucial significance in ontology ― based on the axiomatic set theory (the ZF system), there is only one type of presentation of being: the multiple.[14] Such consideration of any multiple as intrinsically multiple of multiples, as the multiple without implying the being-of-the-one, formulates one fundamental proposition in contemporary research of ontology after the end of the metaphysics of the one or the ‘ontotheology’.[15] This fundamental proposition only becomes conceivable in the condition of the establishment of the axiomatic set theory and its deployment of the multiplicity. To sum up in a general theory or with a ‘logical schema’  from the perspective of intellectual history, the Cantorian event, as an exception or ‘singularity’ happened in intellectual history, disturbs the inherent order of the knowledge system, engenders a transformation of its foundation and progression, and splits up a post-evental period from the pre-evental period. (Event is a caesura. Gilles Deleuze provided an excellent explication of this mechanism in Difference and Repetition (1968).) In the post-evental period, specified by its condition, a certain configuration of mathematics/ontology generates. It cannot be contained and legitimatized by the prevailing language and the dominant system of knowledge of the pre-evental period ― such configuration is totally unconceivable from a pre-evental perspective. In this sense these two periods must be divided and something ‘new’ (the post-Cantorian thought, the ontology of the multiplicities) has been invented ― this ‘new’ does not merely bring up a new form but also reconstitutes the foundation (the ‘past’) of the system. This configuration and its development would persist all through the post-evental period until the happening of a next event which is absolutely unpredictable at the moment. The happening of an event must give rise to the transition of conditions and thereafter the transformation of the configuration of thought. According to above logical schema we see that a historical process, namely, the happening of an event, the split, and the transition of the condition, is irreversible (as what ‘historicity’ means), while an event must be verified and articulated retroactively (as what Badiou’s retrospect and reconstitution of the Cantorian event has revealed). It also demonstrates that in any investigation of intellectual history the conventional schema of ‘foundation’ since Immanuel Kant that focuses on the research of the groundwork of knowledge without concerning the function of event and the transition of conditions from now on must be replaced by this schema of ‘condition’. The ‘foundation’ of a knowledge system in specific period is the effect or consequence of the (re-)configuration induced by an event and prescribed by particular conditions: ‘condition’ is primary while the establishment of ‘foundation’ is secondary. This schema introduces new horizon in studies of intellectual history.

 

In his Manifesto for Philosophy and Conditions Badiou has provided explication of how philosophy eclipses in specific period because of its ‘suture’ to only one condition. He defines ‘suture’ as a situation in intellectual history when philosophy ‘delegates its functions to one or other of its conditions, handing over the whole of thought to one generic procedure’, and result in its own suspension or suppression in favor of that procedure.[16] For example, Badiou indicates, the positivist or scientistic suture as the main suture since the 19th century still dominates academic Anglo-Saxon philosophy.[17] And in continental philosophy, after Hegel, philosophy in a period ‘is most often sutured either to the scientific condition or to the political one’, while after Nietzsche and Heidegger, poetry becomes another privileged condition that gives form to what Badiou calls ‘the age of poets’, a period in which ‘all philosophers claim to be poets’.[18] With regard to contemporary philosophy, Badiou puts forth the completion of this situation by proposing a ‘de-suturation’, especially a de-suturation of philosophy from its poetic condition,[19] and proclaims a renaissance of philosophy. This return of philosophy is only possible when the compounding function of philosophy is resumed, such that its four conditions can again be bonded together in their joint historicity. He affirms that crucial events have already taken place in the field of matheme, love, politics, and poem,[20] and correspondingly, conditions of philosophy must be re-defined. Under these most recently defined conditions, reconfigurations of thought becomes possible and necessary in the contemporary era. This task for contemporary philosophy is very similar to the one that gave birth to the western philosophy in ancient Greece, in which philosophy must distinguish itself from ‘sophistry’[21] so as to draw itself out of the age of sutures. Badiou juxtaposes these two critical moments and declares that, for both cases, the philosophical gesture which is able to carry out the anti-sophistic configuration of thought must be a ‘Platonic gesture’.[22] In the Greek era, it was Plato’s insistence in separating a distance between philosophy and the poem that foreclosed the ‘Parmenidian regime’ ― this regime of the relation between philosophy and the poem formed a pre-commencement of philosophy and produced ‘a fusion between the poem’s subjective authority and the validity of utterances deemed to be philosophical’, such that the authenticity of True was guaranteed by the sacred aura of utterance and the equivocity of language ― and executed, in terms of the order of matheme, the interruption of ‘the sacral exercise of validation by narrative’ or by ‘mytheme’: it was mathematics and its ‘literal univocity’ (for example, it could be found in apagogical argument and its ‘imperative of consistency’, which proved to be incompatible with any legitimation grounded in narrative) warranted this interruption, supported the ‘desacralization’ (‘to tear down mystery’s veil’) that ensured the commencement of philosophy, and finally established a regime of discourse grounding on its own inherent and earthly legitimation.[23] Badiou thus concludes, ‘philosophy can only establish itself through the contrasting play of the poem and the matheme, which form its two primordial conditions (the poem, whose authority it must interrupt; and the matheme, whose dignity it must promote).’[24]

 

In the contemporary era, philosophy is sutured with the poetic condition, dominant by Heideggerian nostalgia (recourse to pre-Platonic, poetic language; ‘only a God can save us’)[25] and Wittgensteinian sophistry (‘whereof one cannot speak, thereof one must be silent’; ‘language game’)[26], and caught in a situation of ‘paralysis’, ‘a completion of philosophy’, as well as a ‘nihilism’ that declares ‘the access to being and truth is impossible’[27] (the ‘postmodern’). For the (re-)turn of philosophy, Badiou proposes a desacralization (or anti-sophistry), a de-suturation of philosophy from its poetic condition, and a contrast between the poem and the matheme. The ‘sovereignty of language’ claims the limits of thinking: ‘of what is subtracted from language, there can be no concept, no thought’.[28] This ‘contemporary conviction’ or ‘general dogma’ has excluded that ‘unnamable’ (by language) from thinking and sentenced it as ‘the indiscernible’. For Badiou, the philosophical gesture that is able to break through such ‘prison of language’ must again be a Platonic one, although it has to be conditioned by that event happened in modern mathematics. After the Cantorian event, the problematics of the post-Cantorian thought revolves around the inconsistent multiplicities, the unnamable and the indiscernible that is totally unthinkable for the philosophy sutured with the poetic/linguistic condition. And Cohen’s forcing of the generic sets and ‘generic’ multiplicities makes it possible to produce a concept of the indiscernible and thus opens the access to that ‘impossible-real’.[29] Badiou proclaims, what a contemporary philosopher counterattacks ‘Great Modern Sophistry’ is the following point: ‘Being is essentially multiple’, and therefore ‘our century will have been the century of protest against the One’ ― ‘God is truly dead, as are all the categories that used to depend on it in the order of the thinking of being’.[30] Retroactively, Badiou affirms that ‘the multiple’ had already been contained in Plato’s thinking. Although, inevitably limited by his time, he still reserved the rights to the One (also see Deleuze’s inversion of Platonic philosophy in his two magnum opuses published in 1968, 1969), Plato’s attempt was to ‘ruin the linguistic and rhetorical variance of sophistry from the aporia of an ontology of the multiple’.[31] Conditioned by modern mathematics after Cantor, the Platonic gesture in contemporary philosophy must be a ‘Platonism of the multiple’. Actually this position had already been undertaken by Albert Lautman (1908–1944) in 1930-40s. Lautman’s thought, commented by Badiou as ‘the only great openly Platonic as well as modern thinking’,[32] had constituted a modern Platonism. Badiou concludes, ‘The century and Europe must imperatively be cured of anti-Platonism. Philosophy shall only exist insofar as it proposes, to match the needs of our times, a new step in the history of the category of truth. It is truth which is a new idea in Europe today. And as with Plato, as with Lautman, the novelty of this idea is illuminated in the frequenting of mathematics’.[33]

 
[1] The problem of the condition of philosophy or of knowledge is critical for other philosophers, for example, Immanuel Kant and Michel Foucault. See Michel Foucault, ‘What Is Enlightenment?’ in Essential Works of Michel Foucault, 1954-1984, Volume I: Ethics: Subjectivity and Truth, ed. Paul Rabinow and Nikolas Rose, (New York: New Press, 2003), p. 43-57.

[2] Alain Badiou, Manifesto for Philosophy: Followed by Two Essays: “The (Re)turn of Philosophy Itself” and “Definition of Philosophy”, trans. & ed. Norman Madarasz, (Albany, N.Y.: State University of New York Press, 1999), p. 33.

[3] Ibid., p. 35. Here, concerning the relation between four conditions and philosophy, Badiou indicates: ‘the lack of a single one (condition) gives rise to its (philosophy’s) dissipation, just as the emergence of all four conditioned its apparition’.

[4] Truth is not produced by philosophy but by the generic procedures. Philosophy ‘does not establish any truth but it sets a locus of truths. It configurates the generic procedures, through a welcoming, a sheltering, built up with reference to their disparate simultaneity.’ See ibid., p. 37. The philosophical category of Truth is empty/void. Truth is a hole in sense. See Alain Badiou, Conditions, trans. Steven Corcoran, (London; New York: Continuum, 2008), p. 11, 43.

[5] Alain Badiou, Manifesto for Philosophy: Followed by Two Essays: “The (Re)turn of Philosophy Itself” and “Definition of Philosophy”, p. 36-37.

[6] Ibid., p. 38-39.

[7] See Tzuchien Tho, “What is Post-Cantorian Thought? Transfinitude and the Condition of Philosophy”, in Badiou and Philosophy, ed. Sean Bowden and Simon Duffy, (Edinburgh University Press, 2012), p. 19-38.

[8] Alain Badiou, Being and Event, trans. Oliver Feltham, (London; New York: Continuum, 2005), p. 23.

[9] Alain Badiou, Conditions, p. 111.

[10] See Georg Cantor, ‘Letter to Dedekind’, in From Frege to Gödel, ed. Jean Van Heijenoort, (Cambridge, MA: Harvard University Press, 1967), p. 114. Based on this standard version, Tzuchien Tho corrects the mistakes in English translation of Cantor’s text quoted by Badiou in Being and Event. See Tzuchien Tho, “What is Post-Cantorian Thought? Transfinitude and the Condition of Philosophy”, in Badiou and Philosophy, p. 31.

[11] Alain Badiou, Being and Event, p. 41.

[12] Ibid., p. 42.

[13] Ibid., p. 43.

[14] Ibid., p. 44-45.

[15] Gerrit Jan van der Heiden, Ontology after Ontotheology: Plurality, Event, and Contingency in Contemporary Philosophy, (Pittsburgh, Pennsylvania: Duquesne University Press, 2014).

[16] Alain Badiou, Manifesto for Philosophy: Followed by Two Essays: “The (Re)turn of Philosophy Itself” and “Definition of Philosophy”, p. 61. Badiou’s conclusive thesis is: ‘if philosophy is threatened by suspension, and this perhaps since Hegel, it is because it is captive of a network of sutures to its conditions, especially to its scientific and political conditions, which forbade it from configurating their general compossibility.’ See, ibid., p. 64.

[17] Ibid., p. 62.

[18] Ibid., p. 69-71.

[19] Ibid., p. 67.

[20] They are: set theory (from Cantor to Cohen), Lacanian psychoanalysis, May 68, and Paul Celan’s poems. See Ibid., p. 80-88.

[21] Sophist is the double of philosopher. ‘The ethics of philosophy essentially inheres in retaining the sophist as adversary, in conserving the polemos, or dialectical conflict. The disastrous moment occurs when philosophy declares that the sophist ought not be, the moment when it decrees the annihilation of its Other.’ See Alain Badiou, Conditions, p. 18-20.

[22] Alain Badiou, Manifesto for Philosophy: Followed by Two Essays: “The (Re)turn of Philosophy Itself” and “Definition of Philosophy”, p. 97-98.

[23] Alain Badiou, Conditions, p. 36-38.

[24] Ibid., p. 38. Badiou also indicates: ‘We can also say that the Platonic relation to the poem is a (negative) relation of condition, one that presupposes other conditions (the matheme, politics, love).’

[25] Alain Badiou, Manifesto for Philosophy: Followed by Two Essays: “The (Re)turn of Philosophy Itself” and “Definition of Philosophy”, p. 50-52.

[26] Ibid., p. 94-95, 97-98.

[27] Ibid., p. 56.

[28] Ibid., p. 94.

[29] Ibid., p. 95.

[30] Ibid., p. 103.

[31] Ibid.

[32] Ibid., p. 100.

[33] Ibid., p. 101.

[34] Alain Badiou, ‘Chapter Seven: Philosophy and Mathematics’, in Conditions, p. 93-112.





康托爾集合論:巴迪歐哲學的數學條件


黎子元

作者簡介
黎子元,網絡頻道「01哲學」主編(hk01.com頻道),香港浸會大學人文及創作系哲學博士,研究課題為「超驗唯物主義」。2009年以來從事文化評論和當代藝術策劃及教育工作,編有《測繪香港藝術地形——12間當代藝術機構訪談》(香港·域外文化2011年)。 

 
人們或會認為,從事數理科學的人,應該與世無爭,躲在自己的小樂園做著安安穩的研究,遠離一切世俗的煩擾和流言蜚語。可惜的是,幻想終敵不過事實,從事數理研究的人也不能逃開別人的攻訐,甚至抹黑和中傷。在西方數理科學史上,其中一位不幸者是德國著名的數學家康托爾(Georg Cantor)。康托爾出生在俄國的聖彼德堡,但後來舉家搬到德國,並在1867年,於柏林大學獲得博士學位。他創立了數學的集合論。不過這個理論在當時不單上得不到廣泛的認可,還遭到一大群數學家的攻擊。他為此患上了抑鬱症,由於病情不斷惡化,最後精神失常於精神病院去世。

格奧爾格.康托爾的名字因為當代法國哲學家阿蘭.巴迪歐(Alain Badiou)的「數學本體論」而為更多哲學讀者所知悉。然而在數學領域中,康托爾早已被數學家們視為現代數學的奠基者——康托爾集合論(Cantorian set theory)的建立作為數學思想史上的重要事件,劃分出了一個「後康托爾」時代,並且規限了這個時代的數學思想發展,即集合論的「公理化」(axiomatization)。作為一種「後康托爾思想」,巴迪歐於《存在與事件(1988)》規劃的數學本體論同樣以康托爾集合論作為其「數學條件」。康托爾對現代數學的貢獻是深遠的,這裡我們僅針對當代哲學,特別是巴迪歐哲學,來談談康托爾的集合論。從康托爾的問題意識上看,藉助「集合」概念,他試圖解決數學學科一個長久得不到解決的難題:如何處理無限?

「康托爾樂園」

德國數學家大衛.希爾伯特(David Hilbert,1862 – 1943)是康托爾的堅定擁護者。他說:「沒有人能夠把我們從康托爾創立的樂園中驅逐出去」。這座「康托爾樂園」就奠基在康托爾對於「超限數」(transfinite numbers,數學符號為「ℵ」)的發現與定義之上。通過對集合的研究(以一集合所有子集為元素的新集合,其基數比原集合大),他發現在超出「可數的無限」(the denumerable infinity or the ‘counted infinity’)之外還存在著更多的「無限秩序」(orders of infinity)。他於是將其定義為「超限」,即比一切可數無限都大卻不必作為絕對無限的數字。

這個偉大的思想革命被巴迪歐回溯性地描述為對於一種新形式的「多」(multiplicity)的發明。「無限秩序」概念的提出意味著通過建立起一套規範化的差異秩序,數學家如今可以從積極意義上重新看待並處理「無限」。而藉助「集合」概念建立起來的一致性則解決了關於無限的「非一致性」(inconsistencies)的傳統難題。

樸素集合論的「死路」

然而康托爾的集合論是一種樸素的集合論(naïve set theory)。它藉助直覺將「集合」定義為把各種不同的東西聚集在一起所形成的總體。在後期研究中,康托爾已經發現他的集合論包含了悖謬,例如會碰到「有沒有所有集合的集合」的問題(後來由著名的「羅素悖論」表達)。面對樸素集合論的「死路」(impasse),康托爾垂死掙扎。最後,為了維繫「集合」概念的一致性(這無疑是集合概念最寶貴的特性),他不得不將一切「不一致的無限」(inconsistent multiplicities)排除在無限秩序以外,並且將這些被排除掉的「絕對的無限」皆歸屬於神的領域。通過這個操作,他保住了集合概念的一致性,保住了「良序集合」(well-ordered set)。巴迪歐評論道:「康托爾毫不退縮的將絕對與不一致相關聯。在計算為一(the count-as-one)的操作失敗的地方便站立著神。」

面對樸素集合論的困境,集合論的「公理化」在20世紀(1908 – 1940)逐步開展,最終確立了成為今天數學學科基礎的公理集合論(axiomatic set theory),例如ZF系統(Zermelo–Fraenkel set theory);布爾巴基(Bourbaki)以集合論為基礎的數學統合計劃。而就在康托爾的死胡同,巴迪歐以強力貫穿出一條通路——他的數學本體論的最重要的概念以及基本邏輯模型就是對康托爾的「不可能」的逾越。正是在這個意義上,我們說康托爾的集合論是巴迪歐本體論的條件。

數學家的晚年

1911年,康托爾作為著名外國學者受邀參加蘇格蘭聖安德魯斯大學(University of St. Andrews)五百週年慶典。康托爾希望在慶典上見到羅素(Bertrand Russell)——羅素在新書《數學原則》(Principia Mathematica)中多處引用了康托爾的學說——然而他的希望落空了。一年後,聖安德魯斯大學授予康托爾榮譽博士的學位,可是,疾病纏身的他無法親自到場參加學位授予儀式。

康托爾於1913年退休,在一戰期間飽受貧困和營養不良的折磨。由於戰爭,公眾為他慶祝70歲生日的活動也不得不取消。他人生的最後日子是在療養院度過的。在這裡,於1918年1月6日,他選擇以自殺的方式於精神病院告別世界。


*原載01哲學





巴迪歐:哲學必須與詭辯區分


黎子元


作者簡介
黎子元,網絡頻道「01哲學」主編(hk01.com頻道),香港浸會大學人文及創作系哲學博士,研究課題為「超驗唯物主義」。2009年以來從事文化評論和當代藝術策劃及教育工作,編有《測繪香港藝術地形——12間當代藝術機構訪談》(香港·域外文化2011年)。



巴迪歐的柏拉圖式操作


在《哲學宣言》裡,巴迪歐宣稱,就像在古希臘時代,是數學促成了哲學的誕生,在當代,倘若哲學試圖擺脫20世紀的困境(思想的語言牢籠),就必須重新與數學「縫合」,重新確認哲學思想的數學條件。而要實現這一哲學的轉向/回歸,就必須經由一種柏拉圖式的操作:哲學必須與詭辯術相區分,這種區分建基於以數學的嚴謹、自足來取代話語的神話特質。在這個意義上,巴迪歐的柏拉圖主義就是一種基於數學科學的當代發展而建立起來的、對於存在的最為嚴格的思考。在《存在與事件》中,巴迪歐提出了一整套數學本體論,即以後康托爾集合論為基礎的對於存在之為存在的科學研究。以集合論為數學條件,巴迪歐的基本哲學命題是:存在的呈現方式只有一種,即「多」(there is only one type of presentation of being: the multiple)。

關於多的柏拉圖主義

巴迪歐聲稱,關於「多」的哲學思考已經包含在柏拉圖哲學當中(見巴門尼德篇的結尾部分提出的關於一與多的難題),然而必須以後康托爾集合論為基礎,柏拉圖哲學才能勝任作為一種關於多的柏拉圖主義(Platonism of the multiple)。巴迪歐把柏拉圖難題中的命題倒置為「if the one is not, (the) nothing is」,並且將柏拉圖關於一與多的對立重新配置為「不一致的多(inconsistent multiplicities or multiple-without-one)」與「一致的多的整體(the whole of the conistent multiple of multiple)」之間的對立。這裡的「不一致的多」就是傳統本體論中那不能被呈現的東西,那被否定了的「無(nothing)」,或者說作為無的多(the multiple-nothing)。這個無的位置,就是巴迪歐數學本體論試圖呈現的「存在之為存在」(being qua being)。

巴迪歐哲學發展上的「事件」

巴迪歐的數學本體論提出之後,面對的最為嚴苛的批判,來自另一位同樣嚴格地將數學視為哲學思想條件的法國哲學家——Jean-Toussaint Desanti(1914-2002)。簡而言之,他的批判可以表述如下:是的,巴迪歐的數學本體論確認了存在只有一種呈現方式,即多,但除此之外,他沒有能力給我們更多關於存在的知識。換句話說,巴迪歐的數學本體論過於乾癟,不過是一種極簡的本真的本體論(the minimal intrinsic ontology)。

同時,他質問,如果我們不採用集合論作為數學條件,而採用其他數學理論體系,那麼,我們是否可以建立更多不同的數學本體論呢?Desanti 的批判可以說是巴迪歐哲學發展上的「事件」,直接引發了他的哲學體系的轉向,即《存在與事件》的第二部:Logics of Worlds 的寫作。儘管稱為第二部,但在這個哲學計劃中,巴迪歐的思想早已發生巨大改變。以範疇理論(category theory)為基礎,巴迪歐試圖調和他早期與晚期的本體論,除了呈現存在,亦試圖描述存在如何呈現(appearing)。


*原載01哲學





巴迪歐:數學本體論


蘇永彝


作者簡介
蘇永彝,目前在劍橋大學修讀數學,通讀巴迪歐三部本體論著作的法語版本。

Derek So, he is currently studying mathematics at Cambridge university and has read through Badiou's three major works on ontology: L'Être et l'Événement (1988), Logiques des mondes. L'être et l'événement, 2 (2006), and L'immanence des vérités (2018).



阿蘭.巴迪歐(Alain Badiou)在現代哲學可說是自成一家,與在法國流行的解構主義、後結構主義等浪潮相反。這些流行的思想(跟從海德格的思路)提出顛覆形而上學、本體論在哲學的位置,嘗試思考傳統哲學的外界/他者(outside/other)。巴迪歐卻反其道而行,提出了一套全新的本體論——數學本體論(mathematical ontology)用以抗衡解構主義、後現代等思潮。巴迪歐在Logics of Worlds中統稱這些思想為「民主唯物主義」(democratic materialism),因為他們相信這個世界只有身體與語言(There are only bodies and languages),而不同的身體與語言都擁有同等存在的權利;巴迪歐把自己的哲學稱為唯物辯證,認為這個世界除了只有身體與語言,還有真理(there are only bodies and languages, except that there are truths)。

在一個「懷疑的解釋學」(hermeneutics of suspicion)流行的年代,巴迪歐重提真理不免顯得不合時宜。從德希達、傅柯等到羅蒂、奎恩等,哲學從不同角度懷疑真理的絕對性,而邏輯、哲學、歷史等傳統的真理概念(真理符應論,correspondence theory of truth)的確有很多漏洞。因此,我們對巴迪歐哲學的第一反應都是一種懷疑——在一個不再相信絕對的年代,我們還能談論真理嗎?

但從他自稱為唯物辯證這個舉動中,我們已可以看出巴迪歐對真理的熱誠從何而來。從他最早的著作和參與的社會運動到現在,他都是一個堅定不移的共產主義者。作為路易.阿圖塞(Louis Pierre Althusser)的學生,阿圖塞式的共產哲學對他影響深遠。在Metapolitics一書中,巴迪歐更說所有當代哲學必須從阿圖塞的哲學觀開始。這哲學觀提出馬克思是思想史中重大的思想革命(epistemological break),指出唯物辯證的科學根基。在阿圖塞的思想史觀裡,每一個科學革命都導致一場哲學革命,而當代哲學的使命就是嘗試在哲學領域推行一場受馬克思的革命(即是政治)所造就(condition)的革命。

但阿圖塞的哲學面對眾多困局,而他的學生對此最為清楚不過。巴迪歐一方面繼承阿圖塞哲學的精神,另一方面開始重新理解哲學的功能,建構一套更完整的哲學。《主體理論》(Theory of Subject)是他哲學系統的起點。《主體理論》的譯者把此書稱為巴迪歐「最具創意及激情」的書。此書沒有傳統學者客觀、置身事外的態度,更像阿圖塞所說的「哲學是理論中的階級鬥爭」。共產主義(列寧,馬克思,毛澤東,恩格斯)當然是此書的主角之一。但除了政治之外,《主體理論》也表現了巴迪歐在理論方面的野心。巴迪歐嘗試提出新的理論框架解讀思想史,對黑格爾、馬拉美、希臘悲劇、賀爾德林等作出深入及獨到的分析。《主體理論》的計劃是要改造唯心辯證法以反思主體的唯物面向,建立一套馬克思主義哲學。巴迪歐之後的哲學擴大了這思路。

《存在與事件》

「The statement that mathematics is ontology…is the stroke of light that illuminates the speculative scene」。數學是令巴迪歐能擴大及超越《主體理論》框架的源頭;「數學就是本體論」也是巴迪歐哲學的中心,堪稱他最大的思想突破。數學本體論這哲學觀中隱藏了巴迪歐對二十世紀哲學的批評。二十世紀哲學(無論是海德格還是維根斯坦、德希達、傅柯等)都偏重語言作為表達思想的媒介,也不斷分析語言對思想的限制;海德格更把哲學跟詩詞(尤其是賀爾德林的詩詞)縫合在一起。數學本體論正是借助數學的絕對性,針對語言的流動所做成的多變性來建立哲學。另一方面,數學本體論的本體論也正在挑戰二十世紀哲學宣判形而上學跟本體論已終結的宣言。在巴迪歐眼中,我們不需要放棄本體論,只需建立一套新的本體論。

四種開啟真理的程序

傳統本體論的失敗只是幾代哲學家的空談。形而上學自柏拉圖都是要尋找恆久不變的真理,把這真理設置為哲學的根據。二十世紀哲學正正發現這是不再可能。因此,哲學本身的根據被動搖,而放棄形而上的思考模式變成最自然的出路。巴迪歐接受了現代哲學對本體論的批判,承認傳統的絕對(真理、上帝等)並無根據。但在巴迪歐眼中,無根據並不是一個問題。哲學並無絕對的根據:它的根據在於偶然的事件(event)中。真理的出現要等待「事件」的發生。再進一步,哲學是不能夠發現真理的--能夠發現真理的是科學、政治、藝術跟愛情,而巴迪歐稱上述四者為哲學的四個條件(condition),或真理程序(truth procedure)。哲學必須精通它的條件,但不可以跟任何一個縫合(suture)。哲學不具備真理,但這並不是哲學的「缺憾」,哲學能夠貫穿四個擁有真理的條件。阿圖塞把哲學跟政治縫合,而海德格把哲學跟藝術縫合,都是錯誤的舉動。

數學本體論之集合論

《存在與事件》一書嘗試證明如下的哲學觀︰論證存在、事件的本體。但巴迪歐哲學的根據在於偶然的事件;因此它不會嘗試「證明」它自己的出發點正確與否。就如黑格爾的哲學一樣,出發點的正確與否是要出發後,再回過頭來才能看清楚的。唯一可以指出的是他的哲學的條件--康托爾(Cantor)作為集合論的先驅者。在這事件的條件底下,《存在與事件》用策梅洛-弗蘭克爾集合論(Zermelo–Fraenkel set theory)為基礎建構一套本體論。

這麼看,集合論的直觀概念其實非常簡單(這被稱為樸素集合論):集合就是集合了任何不同元素/成員。我們可以把所有書本變成一個集合,也可以把貓與狗變成另一個集合。用比較正式的語言說(這是費雷格發展的集合論),一個集合就是所有符合任何一個命題的成員。這個直觀對集合的理解替數學提供了一個簡單的基礎:所有數學的結構都是一個集合,只不過有不同命題。代數中的「組」(group)便是符合四個公理的集合、幾何學不同的流形便是符合不同拓撲條件的集合。

這本來很簡單且圓滿的基礎卻被羅素悖論推翻:所有集合作為命題的集合並不存在(set of all sets does not exist)。因此,集合不能夠被定義為符合任何命題的成員。為了發展一套一致(consistent)的公理,數學家形成了策梅洛-弗蘭克爾(Zermelo-Fraenkel)公理。巴迪歐正是用這套公理來思考本體論。

如果用集合論來思考本體論顯得不倫不類,那麼巴迪歐在《存在與事件》第一個沉思中對柏拉圖《巴門尼德》的解讀就表現了集合論的哲學元素。一與多的問題跟集合論關係密切。本體論從來都是覺得表現是多,表現者是一(what presents itself is essentially multiple;what presents itself is essentially one)。巴迪歐卻宣稱一不存在。「一」是一個操作的結果(operational result)。我們可以把任何東西數成一(count-as-one),而這操作的過程不是必要的;我們可以把任何事物數成一。巴迪歐把呈現的多重性(presented multiplicity)稱為情況(situation);呈現的多重性由多重(multiple)跟「數成一」組成。情況擁有這兩部分。但任何這樣的結構都被再分為︰首先,那些多重本身是「一」,它們是一致的多重性(consistent multiplicity);但在一個情況中,我們會回溯性地發現呈現本身不是一,而是多,這就是不一致的多重性。所以我們可以說策梅洛-弗蘭克爾公理是呈現(presentation)的哲學。

按照策梅勒-弗蘭克爾公理,巴迪歐重新思考了很多傳統哲學的基本問題,例如屬於與包括(belonging/inclusion)、同一與差異等。策梅勒-弗蘭克爾集合論的頭五個公理是我們用來理解呈現(presentation)本身被呈現(presented)的概念。但公理只是形式;任何符合公理的都能成為集合。但集合本身的存在卻不能夠被公理本身確定。公理系統與本體存在之間仍然保持距離,我們需要縫合公理與存在(suture-to-being)。這就是第六個公理,亦即空集公理的功能。我們用以理解呈現的那些概念(亦即公理本身)並沒有存在;他們只是形式。但這虛無本身卻是存在的;他的存在只是一個標記—— Ø 。「…nothing is delivered by the law of ideas, but make this nothing be through the assumption of a proper name」只有不能夠被呈現的才是所有存在的起點。這與海德格與德希達對呈現的形而上學的批判不謀而合:用虛無作為存在的開始意味著存在本身不能夠被呈現。

用最簡單的語言說,巴迪歐用策梅勒-弗蘭克爾集合論釐清了本體論的基本概念,以「一不存在」作為整個本體論的基礎。在這樣的基礎上,巴迪歐能夠數學化地把握不同存在的性質。巴迪歐用更高深的集合論來理解真理與事件只不過是這基礎的延伸。基於集合論,巴迪歐把情況分為三類:自然、歷史與中立。事件在歷史情況中出現。用巴迪歐的詞彙,歷史情況中所呈現的多重性是不正常的(abnormal)。(這些詞彙全部都有嚴謹的定義,但我們這裡不仔細分析)。一個完全不正常的多重就是一個事件場地(evental site),可能出現事件的地方。但這場地並非事件本身,而場地的出現也未必意味著事件必須出現。事件不是被任何因素導致的,因此我們不應以因果關係來理解巴迪歐的事件的出現。事件是在某一個情況裡發生的,但不是情況裡的元素可以完全決定的。事件嚴格的定義是一個包含它的場地和它自己的多重。巴迪歐用法國大革命的例子來解釋他的概念:法國大革命這事件包括了當時的歷史現實——從三級會議、法國的經濟情況、雅各賓派,再到馬賽曲、監獄、斷頭台等。但這些都不是革命本身;法國大革命裡的革命一詞不是列清單就能夠被理解的。把它稱為革命是這革命的一部分。因此一件事件包含了他的場地(1789-1794的法國),但必須包括自己在裡面。

介入與忠誠

我們不能夠邏輯地斷定事件究竟有沒有發生。每一個情況都有它實際、具體的現實,而要斷定事件有沒有發生就要介入(intervention)情況裡。忠誠(fidelity)就是去界定情況中跟事件有關及無關。

真理要在事件發生後才能出現。但真理不能夠受限於不同的情況與歷史,否則巴迪歐的整套哲學只會回到一套歷史相對主義。但巴迪歐的真理觀念與傳統哲學截然不同。真理在哲學中通常用以來衡量不同命題的真偽。巴迪歐把這種真理稱作真實性(veracity),與真理(truth)分開。真實性是不同命題的真偽的準則,而真理確實一種存在。真理是忠誠程序(通用真理程序 generic truth procedure)中得出的,是一個存在的一部分。經歷這過程後,它會變成情況中無限的部分。既然是存在的一部分,他就不會隨時代而改變。

《存在與事件》透過重新建立本體論來確立真理、事件、存在、主體的哲學地位,這可說是近代哲學全新的開始。在這本體論的基礎上,巴迪歐向不同的哲學議題進發,在Conditions、Handbook of Inaesthetics、Briefings on Existence與Metapolitics 等書中對政治、藝術、哲學史、精神分析等近代哲學最重要的議題進行分析。但巴迪歐卻在逐漸發現存在與事件的不足之處,這發現最終引導他寫下存在與事件II,亦即《諸世界的邏輯》(Logics of Worlds)。

存在與事件諸世界的邏輯

如果《存在與事件》是本體論,那麼《諸世界的邏輯》就是現象學了。巴迪歐把這兩部書類比作黑格爾的《精神現象學》與《大邏輯》,只不過他先出版了《大邏輯》,然後再處理現象學。巴迪歐反思存在與事件的框架源於兩個主要的原因:在數學上,範疇論(category theory)囊括了集合論作為數學的基礎的優點,甚至比集合論用途更廣泛;集合論只是一個範疇。在哲學上,巴迪歐要處理不同情況形成的可能性:為什麼不同情況可以存在而不變成純粹的混亂呢?是什麼給予一個情況的結構?在《諸世界的邏輯》一書,巴迪歐把情況稱為世界。因此,這些問題都是在問︰一個世界的元素怎麼在世界中出現(appear)?這出現的方法就是這個世界的邏輯。

範疇論在現代數學的用處多不勝數。代數學的不同結構全部都是範疇;代數幾何的概形、代數拓撲學不同拓撲空間全部都形成不同範疇,而範疇論研究不同結構的方法也是現代數學不可缺少的(層、同調、模範範疇……)。對巴迪歐最重要的卻是範疇論在數學邏輯的重要性,尤其是拓撲斯一概念。拓撲斯是一個符合幾個公理的範疇。它的特徵是擁有自身的邏輯。在一個拓撲斯中,我們日常用的邏輯規矩並不一定正確,排中律(law of excluded middle)不一定成立。但這不代表一個拓撲斯的邏輯是隨意的;它的邏輯是根據它的元素建立的,內在於它自己的。例如,集合的範疇是一個拓撲斯,而這範疇符合我們慣用的邏輯--這邏輯卻不是我們放進去的,而是集合本身的數學結構。如果範疇的成員不是集合,它能夠擁有其他數學結構,改變著拓撲斯的邏輯。

在巴迪歐的哲學中,一個世界就是一個拓撲斯(更準確來說,一個格洛騰迪克拓撲斯,Grothendieck topos)。因此,世界的邏輯是一個內在於世界的邏輯。給予世界結構的功能是一個先驗結構(transcendental);每一個世界作為拓撲斯都有它自己的先驗結構,衡量不同元素在世界中出現(appearance)的強度(intensity)。

在這新的現象學基礎上,巴迪歐重新理解存在在世界之內的意思。這也是對事件的另一個解讀。在世界之中,主體面對多重性;但事件與真理是在這多重性中突出的--主體可以肯定或否定這事件的發生。換言之,主體面對事件終須作出選擇。這選擇把整個世界複雜的多重性變作肯定與否定。巴迪歐把這稱為世界中的一點(point)——這點強迫主體作出抉擇。整個世界的存在都聚焦在這一點上--所有存在都要決定自己是肯定一方還是否定一方。用具體例子說,1940年法國被德國佔領,每一個法國人面對這複雜的政治、軍事形勢都要做出選擇:參與法國抵抗運動還是不參與。整個法國的政界都要面對這點,肯定還是否定。

當今的世界毫無激情

現今的世界在巴迪歐眼中卻是一個拒絕面對選擇的年代。消費主義、資本主義和後現代的風俗都鼓勵與世界保持適度的距離,不需要為任何「真理」保持忠誠。巴迪歐把這種世界稱為無調世界(atonic world)。巴迪歐的整套哲學非常反對這種世界。對事件及真理的忠誠就是要抗衡無調世界對存在的冷感與虛無主義,挑戰虛假的安全感與自由。在巴迪歐深奧的哲學與數學之中不難感受到他對生命的熱誠的肯定、思想的喜悅:"I am sometimes told that I see in philosophy only a means to re-establish, against the contemporary apologia of the futile and the everyday, the rights of heroism. Why not?... my wish is to make heroism exist through the affirmative joy which is universally generated by following consequences through. We could say that the epic heroism of the one who gives his life is supplanted by the mathematical heroism of the one who creates life, point by point.(有時有人告訴我,在哲學中,我看到的只是一種手段,來重新確立英雄主義的權利,而不是像現在那樣,對那些徒勞無益的日常生活進行辯解。為什麼不呢?…我的願望是讓英雄主義通過積極的快樂而存在,這種快樂是普遍的,通過以下的結果而產生的。我們可以說,一個獻出生命的人的史詩般的英雄主義被一個逐點創造生命的人的數學英雄主義所取代。)”


*原載01哲學,經作者授權轉載