[Badiou and Science] 1.4.2 The Surreal Numbers: Part 2
“And this is the problem: is it possible to identify a Number as opposed to sets of Numbers?… Does the numberless throng of Numbers necessarily lead us into ‘those indefinite regions of the swell where all reality is dissolved’? This is where trans-numeric inconsistency summons us to think the cut. Is it possible, in a fabric so dense that nothing any longer numbers it, to cut at a specific point? Can one determine, by cutting, a singular Number?” — Alain Badiou, Number and Numbers, pg. 140
Number (with a capital “N”) is how Badiou designates the “infinite swarm” of numbers that surreal numbers contain and represent, while numbers (with a lower case “n”) are used to refer to specific numbers within that field and to the familiar numbers we typically work with.
So in Badiou’s terms: Number == surreal numbers, numbers == specific subsets of surreal numbers (reals, integers, infintesimals, etc.).
What is a Number?
While the standard and “popular” way to introduce surreal numbers is through the use of “two” paired empty sets and a game of iteration, in the abstract theory of surreals one only needs a single set and one of that set’s subsets to define a surreal number.
Badiou goes through great pains to explore this in Number and Numbers, and with good reason: the idea that surreal numbers would have to necessarily be produced by “two” paired sets would rob the theory of having a foundational status (e.g. how can mathematics start with “two” sets prior to even having a definition of “two”?). Therefore, Badiou reviews how surreal numbers can be produced starting purely with a single empty set (“founded on nothing”) in the same way traditional set theory is, and how subsequent and specific surreal numbers can then be identified with a given set and an already existent subset of that set. So, instead of needing distinct “pairs” of sets to define surreal numbers (as done in most popular treatments of surreal number theory) in general one only needs a single set.
Badiou can now make the bold claim of what Number N — of what any number — is composed of. The “material” of any Number N consists of:
1) an ordinal set W
2) a subset F of W (where F is a part of W)
3) the set R, which is the “difference” between W and F (“W-F”, or “all sets belonging to W that do not belong to F”)
In this way, all numbers are a “set-theoretical triplet” (W, F, R) based on a single ordinal set (W). For a given W, the subset F can be any part of W, from nothing to the entirety of W. Diagrammatically:
Badiou uses some philosophical terminology to enrich this concept (e.g. W is the “matter” of the number N, while F is its “form” and R its “residue”) but all this is meant to show that any and all numbers can be identified by an ordinal set (W) and one of its subsets (F). All numbers are an aspect of Number, and all arithmetic operations (e.g. addition, multiplication) and the structure of order (“greater than”, “less than”) that we expect of numbers are conserved within this field.
Badiou’s general definition of Number is deceptively simple, and is consistent with our set-theoretical axioms thus far:
"Definition: A Number is the conjoint giveness of an ordinal and a part of that ordinal." [Number and Numbers, pg. 102]
All that needs to be recognized is that we can define any surreal number (and, therefore, all numbers familiar to us or not) by an ordinal set and a subset already included in that ordinal: any number N can be written as (W, F) [1].
The ordinals (our basic “counting numbers”) thereby form the backbone of all numbers. And even though the theory of surreal numbers was formed much after the development of set theory, as far as set theory establishes the being of ordinals as sets, surreal numbers remain compatible and consistent with the regulations of set theory as we know them.
The Cut and The Fundamental Theory of Ontology
Now that we have been catapulted to a field exceeding all empirical comprehension, how do we go about identifying specific numbers within the surreal number space?
Were we purely situated within our comfortable field of real numbers, we would traditionally use a method called the “Dedekind cut” to identify unique numbers within that field; a method by which given any two rational numbers, we can identify (or “cut”) a real number situated between them. This method allows the construction of the irrational numbers on the basis of the rational numbers, which then elaborates the real numbers. All real numbers can be defined by a unique Dedekind cut.
Within the surreal number space, a generalized form of the Dedekind cut can be used to similarly identify unique, or “new”, surreal numbers given any two known surreal numbers. In this way, all surreal numbers can be defined by a cut, and between any two given surreal numbers another unique surreal number can always be identified. This powerful ability gives way to Badiou’s “Fundamental Theory of Ontology”:
“This concept of the cut is presented in the following theorem which, articulating the inconsistent swarming of Numbers with the precision and unity of of a punctual cut, well deserves the name of fundamental theorem of the ontology of Number:
Given two sets of Numbers, denoted by B (for ‘from below’) and A (for ‘from above’), such that every Number of set B is smaller than every Number of set A (in the order of Numbers, of course), there always exists one unique Number N of minimal matter situated ‘between’ B and A. ‘Situated between’ means that N is larger than every element of B and smaller than every element of A.” [Number and Numbers, pg. 144]
Once again, the specifics of this technique are beyond the scope of our present project. However, the intuition is straightforward — between any two given numbers, thought can always “cut” a new unique number:
This is a profound and powerful idea: given any two numbers, thought can always precisely identify a new number between them. As Badiou observes:
“…we are at the heart of the mathematics of Number, and what must be put into play in order to think the cut is of a conceptual interest far surpassing mathematical ontology. All truth-procedures proceed via the cut, and here we have the abstract model of every strategy of cutting” [Number and Numbers, pg. 144]
We will cover Badiou’s concept of truth-procedures in due time, but here we begin to see the kernel of this concept: between the staunch given, thought always has the capacity to actively conceive the new.
The axiomatic decisions to inaugurate zero and infinity, coupled with the discovery of the surreal numbers and the ability to “cut” numbers from the dense fabric of Number, reveal thought to hold some astounding capabilities. I wish to hold off on a discussion on what mathematics deems thought capable of until we finish this section on mathematical foundations, however one can already begin to glean Badiou’s reverence for thought to be able to identify novelty and make radical decisions, and for mathematics as the abstract bearer of this machinery:
“The modern instance of this movement attests to to the void and the infinite as materials for the thinking of Number. Nevertheless, none of these concepts can be inferred from experience, nor do they propose themselves to any intuition, or submit to any deduction, even a transcendental one. None of them amounts to the form of an object, or of objectivity. These concepts arise from a decision, whose written form is the axiom; a decision that reveals the opening of a new epoch for the thought of being qua being… [Number and Numbers, pg. 212, italics mine]
While I believe the consequences of surreal numbers to strongly support the structure of his ontology, after Number and Numbers the surreal numbers only receive a fleeting reference in Badiou’s further oeuvre, despite being an implicit reference point for many of Badiou’s philosophical concepts. This being the case, let us explore some of these consequences for Badiou’s philosophy before we round out our exploration of mathematical foundations.
-Dr. G
Next: Philosophical Considerations of Surreal Numbers
Notes
[1] To provide some insight into how the above diagrams correspond to our traditional numbers, we note that sub-diagram 3) can be written (W,0) and sub-diagram 4) as (W,W). For a given ordinal W, (W,0) = -W and (W,W) = +W. Therefore if, for example, W = 8, then (8,0) = -8 while (8,8) = +8.
