Ideas of Alain Badiou, by Theme

[French, b.1937, Born in Morocco. Chair of Philosophy at the École Normale Supérieure.]

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1. Philosophy / C. History of Philosophy / 5. Modern Philosophy / c. Modern philosophy mid-period
In ontology, logic dominated language, until logic was mathematized
     Full Idea: From Aristotle to Hegel, logic was the philosophical category of ontology's dominion over language. The mathematization of logic has authorized language to become that which seizes philosophy for itself.
     From: Alain Badiou (Briefings on Existence [1998], 8)
1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / e. Philosophy as reason
Philosophy aims to reveal the grandeur of mathematics
     Full Idea: Philosophy's role consists in informing mathematics of its own speculative grandeur.
     From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.11)
     A reaction: Revealing the grandeur of something sounds more like a rhetorical than a rational exercise. How would you reveal the grandeur of a sunset to someone?
1. Philosophy / D. Nature of Philosophy / 8. Humour
The female body, when taken in its entirety, is the Phallus itself
     Full Idea: The female body, when taken in its entirety, is the Phallus itself.
     From: Alain Badiou (Briefings on Existence [1998])
     A reaction: Too good to pass over, too crazy to file sensibly, too creepy to have been filed under humour, my candidate for the weirdest remark I have ever read in a serious philosopher, but no doubt if you read Lacan etc for long enough it looks deeply wise.
1. Philosophy / E. Nature of Metaphysics / 7. Against Metaphysics
Philosophy has been relieved of physics, cosmology, politics, and now must give up ontology
     Full Idea: Philosophy has been released from, even relieved of, physics, cosmology, and politics, as well as many other things. It is important for it to be released from ontology per se.
     From: Alain Badiou (Briefings on Existence [1998], 3)
     A reaction: A startling proposal, for anyone who thought that ontology was First Philosophy. Badiou wants to hand ontology over to mathematicians, but I am unclear what remains for the philosophers to do.
2. Reason / A. Nature of Reason / 4. Aims of Reason
Consensus is the enemy of thought
     Full Idea: Consensus is the enemy of thought.
     From: Alain Badiou (Briefings on Existence [1998], 2)
     A reaction: A nice slogan for bringing Enlightenment optimists to a halt. I am struck. Do I allow my own thinking to always be diverted towards something which might result in a consensus? Do I actually (horror!) prefer consensus to truth?
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
There is 'transivity' iff membership ∈ also means inclusion ⊆
     Full Idea: 'Transitivity' signifies that all of the elements of the set are also parts of the set. If you have α∈Β, you also have α⊆Β. This correlation of membership and inclusion gives a stability which is the sets' natural being.
     From: Alain Badiou (Briefings on Existence [1998], 11)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The axiom of choice must accept an indeterminate, indefinable, unconstructible set
     Full Idea: The axiom of choice actually amounts to admitting an absolutely indeterminate infinite set whose existence is asserted albeit remaining linguistically indefinable. On the other hand, as a process, it is unconstructible.
     From: Alain Badiou (Briefings on Existence [1998], 2)
     A reaction: If only constructible sets are admitted (see 'V = L') then there is a contradiction.
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Topos theory explains the plurality of possible logics
     Full Idea: Topos theory explains the plurality of possible logics.
     From: Alain Badiou (Briefings on Existence [1998], 14)
     A reaction: This will because logic will have a distinct theory within each 'topos'.
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
Logic is a mathematical account of a universe of relations
     Full Idea: Logic should first and foremost be a mathematical thought of what a universe of relations is.
     From: Alain Badiou (Briefings on Existence [1998], 14)
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
In mathematics, if a problem can be formulated, it will eventually be solved
     Full Idea: Only in mathematics can one unequivocally maintain that if thought can formulate a problem, it can and will solve it, regardless of how long it takes.
     From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.17)
     A reaction: I hope this includes proving the Continuum Hypothesis, and Goldbach's Conjecture. It doesn't seem quite true, but it shows why philosophers of a rationalist persuasion are drawn to mathematics.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
There is no single unified definition of number
     Full Idea: Apparently - and this is quite unlike old Greek times - there is no single unified definition of number.
     From: Alain Badiou (Briefings on Existence [1998], 11)
Numbers are for measuring and for calculating (and the two must be consistent)
     Full Idea: Number is an instance of measuring (distinguishing the more from the less, and calibrating data), ..and a figure for calculating (one counts with numbers), ..and it ought to be a figure of consistency (the compatibility of order and calculation).
     From: Alain Badiou (Briefings on Existence [1998], 11)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Each type of number has its own characteristic procedure of introduction
     Full Idea: There is a heterogeneity of introductory procedures of different classical number types: axiomatic for natural numbers, structural for ordinals, algebraic for negative and rational numbers, topological for reals, mainly geometric for complex numbers.
     From: Alain Badiou (Briefings on Existence [1998], 11)
Must we accept numbers as existing when they no longer consist of units?
     Full Idea: Do we have to confer existence on numbers whose principle is to no longer consist of units?
     From: Alain Badiou (Briefings on Existence [1998], 2)
     A reaction: This very nicely expresses what seems to me perhaps the most important question in the philosophy of mathematics. I am reluctant to accept such 'unitless' numbers, but I then feel hopelessly old-fashioned and naïve. What to do?
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Mathematics shows that thinking is not confined to the finite
     Full Idea: Mathematics teaches us that there is no reason whatsoever to confne thinking within the ambit of finitude.
     From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.19)
     A reaction: This would perhaps make Cantor the greatest thinker who ever lived. It is an exhilarating idea, but we should ward the reader against romping of into unrestrained philosophical thought about infinities. You may be jumping without your Cantorian parachute.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
The undecidability of the Continuum Hypothesis may have ruined or fragmented set theory
     Full Idea: As we have known since Paul Cohen's theorem, the Continuum Hypothesis is intrinsically undecidable. Many believe Cohen's discovery has driven the set-theoretic project into ruin, or 'pluralized' what was once presented as a unified construct.
     From: Alain Badiou (Briefings on Existence [1998], 6)
     A reaction: Badiou thinks the theorem completes set theory, by (roughly) finalising its map.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
If mathematics is a logic of the possible, then questions of existence are not intrinsic to it
     Full Idea: If mathematics is a logic of the possible, then questions of existence are not intrinsic to it (as they are for the Platonist).
     From: Alain Badiou (Briefings on Existence [1998], 7)
     A reaction: See also Idea 12328. I file this to connect it with Hellman's modal (and nominalist) version of structuralism. Could it be that mathematics and modal logic are identical?
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
Platonists like axioms and decisions, Aristotelians like definitions, possibilities and logic
     Full Idea: A Platonist's interest focuses on axioms in which the decision of thought is played out, where an Aristotelian or Leibnizian interest focuses on definitions laying out the representation of possibilities (...and the essence of mathematics is logic).
     From: Alain Badiou (Briefings on Existence [1998], 7)
     A reaction: See Idea 12323 for the significance of the Platonist approach. So logicism is an Aristotelian project? Frege is not a true platonist? I like the notion of 'the representation of possibilities', so will vote for the Aristotelians, against Badiou.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Logic is definitional, but real mathematics is axiomatic
     Full Idea: Logic is definitional, whereas real mathematics is axiomatic.
     From: Alain Badiou (Briefings on Existence [1998], 10)
7. Existence / A. Nature of Existence / 3. Being / a. Nature of Being
There is no Being as a whole, because there is no set of all sets
     Full Idea: The fundamental theorem that 'there does not exist a set of all sets' designates the inexistence of Being as a whole. ...A crucial consequence of this property is that any ontological investigation is irremediably local.
     From: Alain Badiou (Briefings on Existence [1998], 14)
     A reaction: The second thought pushes Badiou into Topos Theory, where the real numbers (for example) have a separate theory in each 'topos'.
Mathematics inscribes being as such
     Full Idea: Mathematics inscribes being as such.
     From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.12)
     A reaction: I don't pretend to understand that, but there is something about the purity and certainty of mathematics that makes us feel we are grappling with the core of existence. Perhaps. The same might be said of stubbing your toe on a bedpost.
7. Existence / A. Nature of Existence / 3. Being / b. Being and existence
Existence is Being itself, but only as our thought decides it
     Full Idea: Existence is precisely Being itself in as much as thought decides it. And that decision orients thought essentially. ...It is when you decide upon what exists that you bind your thought to Being.
     From: Alain Badiou (Briefings on Existence [1998], 2)
     A reaction: [2nd half p.57] Helpful for us non-Heideggerians to see what is going on. Does this mean that Being is Kant's noumenon?
7. Existence / A. Nature of Existence / 3. Being / i. Deflating being
The modern view of Being comes when we reject numbers as merely successions of One
     Full Idea: The saturation and collapse of the Euclidean idea of the being of number as One's procession signs the entry of the thought of Being into modern times.
     From: Alain Badiou (Briefings on Existence [1998], 11)
     A reaction: That is, by allowing that not all numbers are built of units, numbers expand widely enough to embrace everything we think of as Being. The landmark event is the acceptance of the infinite as a number.
The primitive name of Being is the empty set; in a sense, only the empty set 'is'
     Full Idea: In Set Theory, the primitive name of Being is the void, the empty set. The whole hierarchy takes root in it. In a certain sense, it alone 'is'.
     From: Alain Badiou (Briefings on Existence [1998], 6)
     A reaction: This is the key to Badiou's view that ontology is mathematics. David Lewis pursued interesting enquiries in this area.
7. Existence / A. Nature of Existence / 6. Criterion for Existence
It is of the essence of being to appear
     Full Idea: It is of the essence of being to appear.
     From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.16)
     A reaction: Nice slogan. In my humble opinion 'continental' philosophy is well worth reading because, despite the fluffy rhetoric and the shameless egotism and the desire to shock the bourgeoisie, they occasionally make wonderfully thought-provoking remarks.
7. Existence / D. Theories of Reality / 1. Ontologies
Ontology is (and always has been) Cantorian mathematics
     Full Idea: Enlightened by the Cantorian grounding of mathematics, we can assert ontology to be nothing other than mathematics itself. This has been the case ever since its Greek origin.
     From: Alain Badiou (Briefings on Existence [1998], 1)
     A reaction: There seems to be quite a strong feeling among mathematicians that new 'realms of being' are emerging from their researches. Only a Platonist, of course, is likely to find this idea sympathetic.
19. Language / F. Communication / 3. Denial
We must either assert or deny any single predicate of any single subject
     Full Idea: There can be nothing intermediate to an assertion and a denial. We must either assert or deny any single predicate of any single subject.
     From: Alain Badiou (Briefings on Existence [1998], 1011b24)
     A reaction: The first sentence seems to be bivalence, and the second sentence excluded middle.
21. Aesthetics / B. Nature of Art / 8. The Arts / b. Literature
All great poetry is engaged in rivalry with mathematics
     Full Idea: Like every great poet, Mallarmé was engaged in a tacit rivalry with mathematics.
     From: Alain Badiou (Mathematics and Philosophy: grand and little [2004], p.20)
     A reaction: I love these French pronouncements! Would Mallarmé have agreed? If poetry and mathematics are the poles, where is philosophy to be found?
25. Social Practice / E. Policies / 2. Religion in Society
For Enlightenment philosophers, God was no longer involved in politics
     Full Idea: For the philosophers of the Enlightenment politics is strictly the affair of humankind, an immanent practice from which recourse to the All Mighty's providential organization had to be discarded.
     From: Alain Badiou (Briefings on Existence [1998], Prol)
29. Religion / D. Religious Issues / 1. Religious Commitment / a. Religious Belief
The God of religion results from an encounter, not from a proof
     Full Idea: The God of metaphysics makes sense of existing according to a proof, while the God of religion makes sense of living according to an encounter
     From: Alain Badiou (Briefings on Existence [1998], Prol)