Being and Event, Part 3: Nature & Infinity

This is the reading for September 8th.

Philosophers, especially of the continental persuasion, have a deeply ambivalent relationship with the post-Galilean project of mathematizing nature.  Martin Heidegger, in particular, considers this mathematization to be a key element in the history of the forgetting of being.  Badiou offers an alternative history, one in which the project of mathematization offers a startling realization: that natural multiples, counted as ordinals, reveal the infinity of nature.

Nature: Poetic or Mathematical?

Heidegger pins a great deal on his reading of the Greek word for nature, φύσις.  The pre-conceptual Greek understanding of nature was that of “the opening up which holds sway,” an auto-presentation that unfolds its own limits.  Essentially, in Greek poetry there was a deep understanding of the presence of nature, as the appearing of being.  In Plato’s move from a poetic understanding of nature to the conceptual understanding of nature as Idea (ιδεία), something was lost.  Nature was reduced to a cut-out of human understanding; the sort of “objectivity” that modern physicists strive for is only possible on the basis of this Platonic forgetting.

There are two histories of western thought, then: one based upon a non-conceptual encountering of being through the poem, and another through the Idea, specifically mathematics.  For Heidegger, the mathematical history is what leads to the objectification of nature as nothing other than a source of resources for humans, or a dead object of study.  Ultimately, this leads to the objectification of humans themselves.

Badiou offers an alternate reading of this history.  He points out that the poetic understanding of nature was certainly not a Greek invention; it existed in Egypt, in India, in China and countless other places long before Parmenides wrote.  The real invention of the Greeks was the matheme.  While the poem attempts to dwell in the full presence of being, the matheme says “that being is expressible once a decision of thought subtracts it from any instance of presence.”  He also says of the Greeks, “For the punctual, ecstatic and repetitive figure of the poem they substituted the innovatory accumulation of the matheme.  For presence, which demands an initiatory return, they substituted the subtractive, the void-multiple, which commands a transmissible thinking.”  (126)

Let’s try and rephrase all of this.  The question is, how can we best describe nature?  Perhaps the easiest and most obvious answer is through the hard sciences such as physics, because that is just the assumption of the culture we live in.  But let’s remember that we are discussing ontology here, so the question is: what is going on behind the scenes?  What lies behind the world that we access through formulas and experiments?  So far as Heidegger and Badiou are concerned, we have two options.

On one hand, we have what we might call the manifest image.  The world, in the sense of what lies behind the scenes, is reached through poetry and affect – let’s say feeling and poetic language.  The painter creating a beautiful image of a forest is more in touch with what the world really is than the scientist chopping down a tree to count the rings within.  For this manifest, or poetic, image of the world, the work of the scientist entirely misses the point; chopping the world up into atoms and numbers is something like a failure of imagination and feeling.  As noted above, there is a coldness to the scientific world that ultimately leads to coldness between humans.  Just as we reduce a forest to lumber or climate data, we reduce other humans to things to use.

On the other hand, we have the scientific image.  Here, it is that very coldness that is valued, because ultimately, humans are not at home in the world.  There is a gap between our everyday language and feelings and the world as it is.  From the point of view of the scientific image, the poetic understanding of nature is a cheap nostalgia for a time in which we did not know about various crazy sounding theories in physics, or the vastness of the universe, or the workings of the brain.  While the poetic, manifest image portrays a world in which humans can be at home, the scientific image emphasizes the fact that the world is entirely unconcerned with humans; the world is an “unhomelike” place – it is uncanny.

For Badiou, the manifest poetic image is ultimately something religious.  It is as if we need to learn to “feel” the world in a certain way and speak a certain kind of language.  But math and science interrupt this; rationality forces us to recognize that we are nothing for nature, and nature is nothing for us.  The great advance is that in the scientific image, we can accumulate and transmit new knowledge to any human whatsoever, whereas the poetic image demands an initiation into a historically constituted community.

Badiou does take one thing from Heidegger.  One of the key elements of the Greek understanding of nature – as it is for us – is stability, constancy.  The question is, is there a “stable” category of multiplicity?  There is: the transitive set, what Badiou will call natural multiplicity.  A transitive set is one in which every element both belongs and is included.  So, in transitive set α, ß both belongs and is included:  ß∈α and ß⊂α.  This means that ß is both presented and represented; any sort of singularity is completely excluded.

Natural Infinity

Part of Badiou’s goal is to provide an entirely non-theological account of infinity.  Not because he is an atheist, though of course he is, but because the theological account of infinity is still too beholden to the Greek concept of finitude.  When Christian theologians speak of God’s infinity, they mean that God is beyond comprehension, he is too big for us to understand.  Ultimately, there is something about the infinity of God that is beyond rationality.  It is not only that the Lord works in mysterious ways, but also that his actual existence is also mysterious.  We can say that God’s infinity is simply the result of our own finitude – we are small and have a limited perspective, and since God is large and has an unlimited perspective, we say he is infinite.  In other words, the infinity of God is a negation of our finitude.  Badiou prefers an infinity that is wholly explicable and rational, one that stands on its own without merely being a negation of finitude.

The hurdle that Badiou must jump is Aristotle’s claim that everything has within itself its own limits.  In other words, everything is what is and no more.  So what we need to find is a still more that is infinite, and what’s more, this infinity can not apply to any one thing – it must be an infinity of nature as such.  There are a handful of requirements:  We will start with α, and following a rule, will find another multiple, and then another multiple, and so on.  This process can continue unendingly, but this unending process is not the infinite Badiou wants.  These new multiples are “same multiples,” or small-o others.  Therefore, we need to say that there is a second set that our ruled-based process can never reach.  This second set would be the point of alterity, the capital-O Other.

We have said that natural multiples are transitive; that is, all of their elements both belong and are included.  The rule we are looking for must allow us to take one natural multiple and build more natural multiples on the basis of the first one.

We have been calling transitive sets ordinals, and explaining this will begin explaining natural infinity.  Sets can have properties; and we can say that a given set, α, is the minimal set capable of sustaining property Ψ, and no “smaller” set has that property.  In ordinal sets, “smaller” is roughly equivalent to “belongs to.”  So if ß∈α, then ß is smaller than α.  If α is minimal with regards to Ψ, then none of α’s elements will have Ψ.  We can also say that α is the “αth” multiple in a series, starting from the void.

Now we need to find a rule.  If one takes the pair {α,ß},  the axiom of union tells us that the set of the elements of the elements of a given set exists – its dissemination.  All this means is that we can put them together: the pair’s “union ∪ {α,ß} also exists; as its elements it has the elements of the elements of the pair; that is, the elements of α and ß.  This is the rule we want: “Our rule of passage will then be the following: α –> α ∪ {α}.  The rule produces the union of an ordinal and its singleton, or powerset.  (152)

The key point is that this is actually a new multiple, an other.  Set α is an element of α ∪ {α}, but it is not an element of itself because α∈α is prohibited on pain of paradox.  Set α and α∪{α} differ by one multiple: α itself.  We can shorten α∪{α} to S(α), which we will read as the successor of α.  The rule allows us to pass from an ordinal to its successor.  “This ‘other’ that is the successor, is also a ‘same’ insofar as the successor of an ordinal is also an ordinal.” (152)

“Moreover, there is a precise sense in saying that S(α) is the successor of α, or the ordinal – the ‘still one more’ – which comes immediately ‘after’ α.  No other ordinal ß can actually be placed ‘between’ α and S(α).  According to which law of placement?  To that of belonging, which is a total relation of order between ordinals.  In other words, no ordinal exists such that α∈ß∈S(α).” (153)  Any such formulation would end up saying ß∈ß.

So what we get: α, S(α), S(S(α)), S(S(S(α))), etc.  This is an unending process, but Badiou cautions us:

“Our intuition would readily tell us that we have definitely produce an infinity of ordinals here, thus decided in favor of a natural infinity.  Yet this would be to succumb to the imaginary prestige of Totality.  All the classical philosophers recognized that via this representation of the effect of a rule, I only ever obtain the indefinite of same-others, and not an existing infinity.  On the one hand, each of the existing ordinals thus obtained is, in an intuitive sense, manifestly infinite.  Being the nth successor of the name of the void, it has n elements, all woven from the void alone via the reiteration of forming-into-one.  On the other hand, no axiomatic Idea of the pure multiple authorizes us to form-one out of all the ordinals that the rule of succession allows us to attain . . . the Totality is inaccessible.  There is an abyss here that solely a decision will allow us to bridge.” (153)

This decision will be about non-successor ordinals, limit ordinals, written lim(α).  He also says, “The wager of infinity turns on this discontinuity: a limit ordinal is the place of the Other for the succession of same-others which belong to it.”  “The crucial point is the following: if an ordinal belongs to a limit ordinal, its successor also belongs to that limit ordinal.

“The result of these considerations is that between a limit ordinal and the ordinal ß which belongs to it, an infinity (in the intuitive sense) of ordinals insert themselves.  That is, if ß∈α, and α is a limit, S(ß)∈α and S(S(ß))∈α, and so on.  The limit ordinal is clearly the Other-place in which the other of succession insists on being inscribed.”

A big structural difference between successor and limit ordinals: the first possess a maximum multiple within themselves, whilst the second do not.  In a formulation that sounds an awful lot like the Lacanian graphs of sexuation, he says:

“If a limit ordinal is at stake, the natural multiple whose substructure of being is formalized by such an ordinal is ‘open’ in that its internal order does not contain any maximum term, any closure.  It is the limit ordinal itself which dominates such an order, but it only does so from the exterior: not belonging to itself, it ex-sists from the sequence whose limit it is.” (155)

Back to the idea of the minimal.  This means that an ordinal can be minimal with regard to the property of “limit.”  “What we have here is smallest limit ordinal, ‘below” which, apart from the void, there are solely successor ordinals.”  This multiple is what marks the threshold of infinity: the ω multiple.

Another long quote is in order;

“This proper name, ω, convokes, in the form of a multiple, the first existence supposed by the decision concerning the infinity of being.  It carries out that decision in for the form of a specified pure multiple.  The structural fault which opposes, with natural homogeneity, the order of successors, (hierarchical and closed) to that of limits (open, and sealed by an ex-sistent), finds its border in ω. The definition of infinity is established upon this border.  We will say that an ordinal is infinite if it is ω, or if ω belongs to it.  We will say that an ordinal is finite if it belongs to ω.” (158)

The matheme of infinity, just for reference: lim(ω) & (∀α) [[(α∈ω) & (α ≠ ø)] –> S(α)]

This is what John Mullarkey has to say about this this second point, or limit ordinal:

“This second point will always be something ‘presented elsewhere’ – as Other.  Interestingly, there is the only alterity allowed into Badiou’s thought: ‘Infinity is the Other on the basis of which there is. . . a rule according to which the others are the same.”  The Other is what enables us to say there are ‘still more’ others of the same type, ‘over there’ while we are still counting ‘here.’  It is the Infinity of a here and a there, a double. . . . The Other is also what allows others to be accorded to the same, to be the ‘still-more’ enabling us to traverse the here to the over there.  The Other, basically, is Badiou’s equivalent of synthetic perception. . . . It is what overarches any others and allows them to be made same, for if there were just others and no Others, then their infinitude would be deduced from the finitude of the here and already (by negation).”

Post-Continental Philosophy, pg 112-113

A final comment.  The limit ordinal, or the Other, synthesizes all the previous successors, and actually seals the sequence as infinite.  While merely counting 1+1+1, we end up with an actual infinite, not just an unending process; as Mullarkey says, it is what allows us to say there is “still more over there” while we are counting “here.”  Is this a legitimate leap?


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