[Badiou and Science] 1.4.3 The Surreal Numbers: Part 3
“But [there are] also so many other [numbers than our traditional numbers], which finitude and the wretchedness of our inherited practice of Number keeps from us. How negligible are numbers amongst Numbers! The being of Number exceeds in every direction that which we know how to negotiate. Our strength, however, is that we possess a way of thinking of this excess of being over thought.” — Alain Badiou, Number and Numbers, pg. 157
If we are to take away anything from Badiou’s interpretation of the surreal numbers, it is that which is punctuated in this opening quotation: we posses a way of thinking the excess of being. We can think beyond the given, because the in-between and outside to this given is literally infinite.
We will refine this idea as we go forward and attempt to apply it beyond our limited scope of mathematics, but so far we have been able to see that by starting on the terrain of numbers we posses a way of not only conceptualizing our familiar real numbers in the abstract, but that we are also more than capable of moving beyond the empirical numbers to numbers with as equal a claim to existence. The capacity of thought to accomplish this task cannot be understated.
What additionally cannot be understated, and should be strictly reinforced, is that this identification of the surreal numbers involves a discovery and not a mere operational construction; that is, the surreal numbers exist equally aside their real counterparts in both our ability to identify them all using the exact equivalent set-theoretical notation, and in their existence as a field indifferent to our perceptions of it. We do not “create” the surreal numbers out of figments of our imagination, but rather come upon them through tireless rational work:
“Fictions have no place in the ontological conception of Number… [Numbers] exist and are distinguished by some property. They cannot irrupt from inexistence, in the form of mere names of a lacuna. According to an ontological conception of Numbers, every Number is, none results or is resolved in the name of an operation. We do battle here against a dominant nominalism, and we do so in the field of number, so commonly taken for an operational fiction” [Number and Numbers, pg. 175]
Formless Notation
What should be clear is that the surreal number realm is independent of our traditional language of describing numbers, and that we only tread upon it by way of our set-theoretical notation (based simply on the empty set) and the abstract procedure of the cut.
It is a deeply ingrained force of habit to think of numbers as our traditional marks “1, 2, 3…”, or as in what Badiou deems “the vulgar notation”. From the perspective of ontology, or of Nature, all that exists is the empty set and sets composed of the empty set:
We do not “invent” numbers, but our access to them is mediated by this notation; a notation that is itself regulated by the axioms of set theory. Badiou’s grand conclusion regarding our historical relation to numbers is in allegiance to this position:
“What can we conclude from all this? That Number is neither a trait of the concept (Frege), nor an operational fiction (Peano), nor an empirico-linguistic datum (the vulgar conception), nor a constitutive or transcendental category (Kronecker, or even Kant), nor a grammar or language game (Wittgenstein), nor even an abstraction from our idea of order. Number is a form of multiple-being. More precisely, the numbers we manipulate represent a minuscule sample of being’s infinite abundance when it comes to species of Number…
There is no deduction of Number, but no induction of it either. Language and perceptual experience prove to be inoperative guides where Number is concerned. It is simply a question of being faithful to whatever portion of the inconsistent excess of being, to which our thought occasionally binds itself, comes to be inscribed as a consistent historical trace in the simultaneously interminable and eternal movement of mathematical transformation” [Theoretical Writings: The Being of Number, pgs. 66–67, italics mine]
With respect to the traditional set-theory that we have been exploring, the theory of surreal numbers declares a simple compatibility — as long as the ordinals can be conceptualized as sets, originating in the empty set, the surreal numbers may be enumerated accordingly. All Numbers are described by an ordinal set and a subset of that set.
Return to the Cut
Once again, any meaning we place on these notations and inscriptions comes from the “outside”. We cut out of the dense fabric of Number that which we consider real numbers, fractions, etc. and operationalize them as such:
“ ‘Cut’ here designates the incision of thought in the inconsistent fabric of being, that which Number sections from the ground of Nature. It is a concept of singularity. Perhaps the concept of singularity, at least in the order of being.” [Number and Numbers, pg. 155].
Badiou goes on to elaborate further on this relation between our finite thought and its historical encounter with Numbers:
“Essentially, a Number is a fragment sectioned from a natural multiplicity… The linear order of Numbers, like their algebra, is our way of traversing or investigating their being. This way is laborious and limited. It exhibits Number in a tight network of links, whose three principle categories are succession, limit, and operations. This is where the illusion arises of a structural or combinatory being of Number. But, in reality, the structures are consequences, for our finite thought, of that which is legible in Number as pure multiplicity. They depose Number in a bound presentation which makes us believe that we manipulate it like an object. But Number is not an object.” [Number and Numbers, pg. 211]
To us, we see “real” numbers, “irrational” numbers, “even” numbers, etc., and operationalize them through algebra, calculus, and monetary transactions (e.g. the “illusion” of Number’s “structural or combinatory” being). However, Nature is agnostic to these categories — all She sees is infinite sets and subsets built off the empty set.
Inconsistent Multiplicity
Finally, we can turn to the concept of “inconsistent multiplicity” that will ground Badiou’s ontology; that which he refers to above as “the inconsistent fragment of being”.
When Badiou describes his concept of inconsistent multiplicity in his major works, he routinely uses the example of the infinite ordinal set (ωₒ) as an “inconsistent set” (recall, “multiplicity” is just another way to speak about a set, as sets are composed of “multiples”. So “inconsistent multiple” = “inconsistent set”). As previously discussed, ωₒ is the infinite set that contains all the ordinals. We declare this set as “inconsistent” because we cannot capture the whole set under a single coherent concept — no matter how many of ωₒ’s ordinals we attempt to hold, there are always further ordinals in excess of those we try and grasp for. This excess is built into the very definition of an infinite set.
However, from ωₒ we can “cut out” manageable subsets such as [all ordinals less than 78], [all even numbers up to 1122], [every other prime number from 3 to 53], etc. All these latter sets are “consistent sets”, in the sense that they are well-defined sets that fall under a coherent concept. From within these consistent sets there is no excess, and these types of sets are those that we typically work with and manipulate in practice.
In this way, the infinite ordinal set ωₒ provides a simple notion of inconsistency that Badiou will use as the paradigm for inconsistency going forward.
However, the surreal numbers surely provide the broadest exemplar of this inconsistency. Just as there is no “total” amount of ordinal numbers, there is no “all surreal numbers” as there are infinitely always more numbers that can be identified (as per Badiou’s “Fundamental Theory of Ontology”). So, while we can define what a surreal number ontologically is, we cannot grasp all of them under a single set.
Surreal numbers thereby escape capture under a single coherent concept, and form an inconsistent set. And as even the infinite sets (such as ωₒ) are subsets of the field of surreal numbers, the surreal numbers therefore represent the most fertile ground from which Badiou’s concept of inconsistent multiplicity springs.
Language: A Prelude
All of this sets up one of the most fundamental themes to Badiou’s entire body of work: An inconsistent multiplicity must be made consistent, via a secondary operation, if it is to then become accessible to thought. Consistent sets may be produced from other consistent sets, but all consistent sets ultimately derive from an inconsistent set. And as should now be apparent, we lend this consistency to inconsistent sets through concepts and language. We make consist that which is inconsistent.
With respect to Numbers, this process is simple: we have recourse to the cut. The cut provides the operation by which we produce our coherent subsections of numbers.
Outside of mathematics, this process of bestowing consistency will involve an analogous operation that Badiou defines as the “count-as-one”: an operation that uses language to separate out multiples from an inconsistent multiplicity and grant them consistency. Consistent multiplicities are thereby always the result of a “naming” operation. While at present this may still sound vague, in due time we will see how this is the process by which the act of scientific discovery “sections” and categorizes Nature into sets such as [atoms], [gases], [forces], etc.
To return to this entry’s opening quotation and remarks, there is therefore always an excess of being over thought, or an excess of the inconsistent multiplicity over the consistent multiplicities that we section and name. But rather than this being some vague figment of a language game or a mystical element, mathematics had supplied us with a way of thinking and articulating this excess. This amazing ability becomes the very foundation of Badiou’s work going forward.
I urge interested readers to take a stab at reading Number and Numbers. At this point we have covered most of the major concepts that will enable one to explore this fascinating text, part history lesson and part number theory, and to appreciate both Badiou’s deep understanding of mathematics and his philosophical concerns in approaching surreal numbers.
Coda I: John Conway on His Discovery of Surreal Numbers
There is an absolutely excellent talk given by John Conway at the University of Toronto in 2016 on how he discovered the surreal numbers; a talk at once non-technically educational and deeply personal. One can see the awe Conway still expresses for the discovery he made almost 5 decades ago, being driven to tears when recounting his experience uncovering a realm that no one had ever encountered before, and on describing his eternal reverence for Georg Cantor.
It is crucially important to note that Conway treats his work on surreal numbers as a discovery and not an invention, which he explicitly states in this talk while referring to himself as a Platonist when it comes to numbers, as in the tradition of Kurt Godel and as Badiou does.
Coda II: On Capital
In the course of these dialogues, I will attempt to avoid engaging (too much, at least) with Badiou’s grounded politics; although his philosophy is undoubtedly entwined and sutured to a politics. That being said, Badiou’s closing remarks in Number and Numbers on our contemporary situation when it comes to Capital’s barbaric use of numbers is especially prescient:
“If the reign of number — in opinion polls or votes, in national accounts or private enterprise, in the monetary economy, in the subjectivising evaluation of subjects — cannot be authorized by Number or by the thinking of Number, it is because it follows from the simple law of the situation, which is the law of Capital… In our situation, that of Capital, the reign of number is thus the reign of the unthought slavery of numericality itself. Number, which, so it is claimed, underlies everything of value, is in actual fact a proscription against any thinking of number itself…
The reverse side of the abundance of capital is the rarity of truth, in every order where truth can be attested to: science, art, politics, and love… a supernumerary hazard from which truth originates, always heterogeneous to Capital and therefore to the slavery of the numerical. It is a question, at once, of delivering Number from the tyranny of numbers, and of releasing some truths from it.” [Number and Numbers, pg. 213–214]
Until next time!
- Dr. G
Next: Axioms and Predicates
