"The thesis that I support does
not in any way declare that being is mathematical, which is to say composed of
mathematical objectivities. It is not a thesis about the world but about
discourse. It affirms that mathematics, throughout the entirety of its
historical becoming, pronounces what is expressible of being qua being... All
that we know, and can ever know of being qua being, is set out, through the
mediation of a theory of the pure multiple, by the historical discursivity of
mathematics."
Why start with mathematics?
The appeal to mathematics is not in order to uncover equations hidden in the cosmos, or to create a predictive model by which one can forecast or estimate future events. Mathematics is here used as a descriptive apparatus: a fundamental language by which we can describe our worlds/situations, and more importantly the conditions of possibility for their potential change. Mathematics not as determinism or as a constraint on our actions, but rather as providing the grounds for invention and revolution.
What is ontology?
This is another concept that has had a rich and circuitous philosophical history. For our purposes, we can treat ontology as the study of what "is". For Badiou, ontology is purely equated with set theory tout court; that is, the terrain of "stuff" he is working with are sets as they are abstractly (yet, precisely) defined by the axioms of set theory. In his writings, the terms "ontology", "being qua being", and "Being" (with a capital "B") all equivalently refer to the mathematical notion of sets. Therefore, ontology (the study of what "is") is equated to set theory (the study of sets). What "is" for Badiou-- what "exists" for our thought in its purest form-- are sets and collections of "things", multiples and multiplicities.
Whereas contemporary philosophers have tried to interrogate multiplicities (whether socio-political collections or scientific ones) and critiqued how we separate these multiples out through our use of broken and contingent languages, Badiou announces that one can precisely and rigorously perform this task with the universal language of mathematical set theory. And that by studying the axioms and properties of sets we may achieve a radically new understanding of knowledge and truth.
Badiou asserts that ontology is the necessary starting point for all truth-seeking enquiry. What he ultimately wants to show is that he can use set theoretical ("ontological") structures to demonstrate notions of situation, truth, knowledge, and change. In particular, Badiou wants to demonstrate how modern set theoretical work (especially the work of Paul Cohen) provides philosophy with the formal resources to describe how knowledge always falls short of truth, and the process by which truths are slowly brought into the world via an infinite project undertaken by agents-- or "Subjects"-- of this truth-process.
Already we are using terms ("world", "situation", "truth", "knowledge") for which Badiou will give rigorous definitions. We will cover these in due course, but almost all of these definitions derive directly from the mathematical structures that Badiou employs in the service of constructing his philosophical system. To begin with, we will concentrate on the aforementioned mathematical field of set theory that forms the theoretical framework and ontology of Badiou's Being and Event. Set theory, as we shall see, provides the foundation for all mathematics (from basic arithmetic on up) and is widely considered to be the field that grounds and secures mathematics. As such, when Badiou refers to "mathematics" he is referring to "set theory" (and thereby "ontology"). We will acknowledge that rival theories do exist (for example, category theory, which Badiou will engage with in his Logics of Worlds), however the idea is that by traversing this traditional foundation of set theory we will illuminate both our understanding of Badiou's philosophy and our understanding of the physical sciences.
So, in brief, Mathematics == Set Theory == Ontology.
Before we begin in earnest, I do recommend the following article published by Paul Cohen for Scientific American in 1967. In this article, Cohen describes the remarkable path from Cantorian set theory to his Non-Cantorian set theory in a "non-technical" manner intended for the scientific public. This path is identical to the one that we shall travel here from Cantor to Cohen, and given how Cohen's revolutionary work will be the dominant point of reference for Badiou's project later on, the article provides a nice overview of set theory's history and impact from Cohen himself to the scientific public.
See: Paul Cohen and Reuben Hersh, “Non-Cantorian Set Theory”. Scientific American, Vol. 217, No.6 (1967) pp.104-116
Till next time!
- Dr. G
Next: Defining sets, capturing multiplicities, and constructing numbers.
- Alain
Badiou, Being and Event pg. 8 (italics mine)
Why start with mathematics?
The appeal to mathematics is not in order to uncover equations hidden in the cosmos, or to create a predictive model by which one can forecast or estimate future events. Mathematics is here used as a descriptive apparatus: a fundamental language by which we can describe our worlds/situations, and more importantly the conditions of possibility for their potential change. Mathematics not as determinism or as a constraint on our actions, but rather as providing the grounds for invention and revolution.
The wager of Badiou's philosophical work is
that modern mathematics (specifically set theory and category theory) have
achieved specific resources capable of clarifying traditional philosophical
notions of "truth", "knowledge", "event",
"change", "reality", "moral obligation",
"good", "evil", etc. in revolutionary and radical ways.
While the latter notions have typically been defined by philosophers using
languages chained to their specific time periods and cultures, mathematics
provides a truly universal and egalitarian language by which to investigate
these concepts. Mathematics can thus take us into new speculative regions of
thought, from which we are obligated to work out the real world implications
and consequences. And, as I will later argue, if this same mathematics is
supposed to underlie our physical sciences, then perhaps there may be similar consequences
for our physical theories.
What is ontology?
This is another concept that has had a rich and circuitous philosophical history. For our purposes, we can treat ontology as the study of what "is". For Badiou, ontology is purely equated with set theory tout court; that is, the terrain of "stuff" he is working with are sets as they are abstractly (yet, precisely) defined by the axioms of set theory. In his writings, the terms "ontology", "being qua being", and "Being" (with a capital "B") all equivalently refer to the mathematical notion of sets. Therefore, ontology (the study of what "is") is equated to set theory (the study of sets). What "is" for Badiou-- what "exists" for our thought in its purest form-- are sets and collections of "things", multiples and multiplicities.
Whereas contemporary philosophers have tried to interrogate multiplicities (whether socio-political collections or scientific ones) and critiqued how we separate these multiples out through our use of broken and contingent languages, Badiou announces that one can precisely and rigorously perform this task with the universal language of mathematical set theory. And that by studying the axioms and properties of sets we may achieve a radically new understanding of knowledge and truth.
Badiou asserts that ontology is the necessary starting point for all truth-seeking enquiry. What he ultimately wants to show is that he can use set theoretical ("ontological") structures to demonstrate notions of situation, truth, knowledge, and change. In particular, Badiou wants to demonstrate how modern set theoretical work (especially the work of Paul Cohen) provides philosophy with the formal resources to describe how knowledge always falls short of truth, and the process by which truths are slowly brought into the world via an infinite project undertaken by agents-- or "Subjects"-- of this truth-process.
Already we are using terms ("world", "situation", "truth", "knowledge") for which Badiou will give rigorous definitions. We will cover these in due course, but almost all of these definitions derive directly from the mathematical structures that Badiou employs in the service of constructing his philosophical system. To begin with, we will concentrate on the aforementioned mathematical field of set theory that forms the theoretical framework and ontology of Badiou's Being and Event. Set theory, as we shall see, provides the foundation for all mathematics (from basic arithmetic on up) and is widely considered to be the field that grounds and secures mathematics. As such, when Badiou refers to "mathematics" he is referring to "set theory" (and thereby "ontology"). We will acknowledge that rival theories do exist (for example, category theory, which Badiou will engage with in his Logics of Worlds), however the idea is that by traversing this traditional foundation of set theory we will illuminate both our understanding of Badiou's philosophy and our understanding of the physical sciences.
So, in brief, Mathematics == Set Theory == Ontology.
Before we begin in earnest, I do recommend the following article published by Paul Cohen for Scientific American in 1967. In this article, Cohen describes the remarkable path from Cantorian set theory to his Non-Cantorian set theory in a "non-technical" manner intended for the scientific public. This path is identical to the one that we shall travel here from Cantor to Cohen, and given how Cohen's revolutionary work will be the dominant point of reference for Badiou's project later on, the article provides a nice overview of set theory's history and impact from Cohen himself to the scientific public.
See: Paul Cohen and Reuben Hersh, “Non-Cantorian Set Theory”. Scientific American, Vol. 217, No.6 (1967) pp.104-116
Till next time!
- Dr. G
Next: Defining sets, capturing multiplicities, and constructing numbers.
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