[Badiou and Science] 1.4.1 The Surreal Numbers: Part 1
“…along with reals and ordinals, [the surreals] contain an infinitely infinite throng of numbers whose existence no one has conceived of before, and which retroactively make our historical numbers seem like a minuscule deduction from all those abundant varieties of numerical being…” — Alain Badiou, Number and Numbers, pg. 107–108
As we are beginning to see, there is a significant difference between the numbers we typically encounter empirically in the world and those that our thought can grasp. Nevertheless, the construction of these novel numbers remains consistent with the construction of our familiar empirical numbers via set theory, and we are obligated to pursue just how far our thought can go on this (literally) infinite terrain.
John Conway’s Theory of Surreal Numbers (crafted in the 1970’s and popularized by Donald Knuth in his Surreal Numbers: How Two Ex-students Turned on to Pure Mathematics and Found Total Happiness) further expands on the types on number that thought can encounter in this manner.
What I wish to do here is to present the broad characteristics of surreal numbers that are important to our project and to understanding Badiou’s core concepts. Number and Numbers is essentially Badiou’s book on number theory and his investigation of surreal numbers; as such, it is indispensable for understanding Badiou’s theory of the Event and, most significantly, the concept of “inconsistent multiplicity” that grounds his mathematical ontology.
The fine details of how to construct surreal numbers and the many specific types of surreal numbers need not concern us. What is of import are the theoretical consequences of surreal numbers and how it is possible for our thought to obtain them. This first entry will delve into these theoretical aspects, while the next entry will ask the (seemingly innocuous) question of “what is a number?”. A third entry will tie the theory of surreal numbers to some philosophical considerations.
Although we will not replicate the derivation of surreal numbers here, I do urge readers to “play the game” of producing surreal numbers as demonstrated in this superb and brief “Introduction to Surreal Numbers” (just 5 pages long!). Not only is it a fascinating exercise that is relatively straightforward, but it will also allow one to appreciate the iterative nature of constructing these numbers “day by day”.
What I will, however, require is watching this short 8 minute video on how surreal numbers are identified and compare to our typical real numbers. Not only is this video an excellent (albeit simplified) review of how real numbers are constructed from the empty set, but it also describes the basics of how the surreal numbers are similarly produced, easing one into the discussion below.
Surreal Numbers
Akin to traditional set theory, the theory of surreal numbers also starts with the empty set and proceeds from there to produce numbers. An intuitive way to think of surreal numbers is as the field of numbers that both includes and “exceeds” the real numbers. I personally imagine the surreal number field like this, with the expanse extending indefinitely:
The theory of surreal numbers is able to produce all of our familiar numbers (e.g. positive and negative integers, fractions, irrationals) and numbers not captured by traditional set theory (e.g. infinitesimal numbers, transcendental numbers). In fact, as all these numbers can be produced by the infinitely infinite process of iteration upon the empty set (as described in the two introductions to surreal numbers, above), this process produces numbers upon numbers previously unnamed and unheard of. This is best represented in the tree diagram below, starting purely from 0:
While this becomes a dizzying process to follow, what has always stuck with me as just a “piece” of how powerful this theory is has to do with what happens on the “infinite day” (step ω). On this iteration, and quite precisely, the theory is able to identify transcendental numbers such as pi and e:
Again, all of our traditional numbers (integers, reals, negatives) are mere “subsets” of the field of surreal numbers. And no matter what unheard of number we encounter on this field, the relations of order and all algebraic operations are conserved — remarkably allowing one to add, multiply, subtract, divide, etc. not only infinite numbers, but also infinitesimals and any sort of number (making, for example, the phrase “infinity plus one” a coherent statement in this field with a unique number corresponding to it).
The ability of the theory of surreal numbers to preserve these functions of order and basic arithmetic, and extend them to any conceivable or unheard of number, is not only a testament to the power of the theory itself, but also lends credence to the surreals being the ultimate material space for mathematics.
Drowning in Numbers
What results is that surreal numbers are the collection of real numbers and the numbers (such as the infinitesimal ε) that exist “between” and “around” the real numbers, causing every real number to be “surrounded by surreals” closer to it than any other real number. Our familiar real numbers appear to “drown” in their fellow surreals.
We should also note that even though the surreal numbers have an expanse and density that almost defies comprehension, it is theoretically possible to identify each and every one of them using the “paired set” notation seen in the above introductions. This notation can even be further simplified to a single set.
The method of identifying specific numbers within the field of surreals involves a technique known as a “cut” (a generalized version of the famed “Dedekind cut” used to identify specific numbers in the real number field). Again, the details of how to perform this cut are complicated and should not concern us here, however the very ability to perform such a localizing activity will be important to Badiou’s ontology, which we shall discuss in the next entry.
In Surreal Summary
As mentioned above, the mechanics of surreal numbers and their specific constructions need not concern us. While following their process of construction and their capability for arithmetic is fascinating (and very much encouraged for the adventurous reader), all that needs to be understood is that the concept of surreal numbers provides an infinite “outside” to the numbers that we typically encounter either empirically or in everyday mathematical work, and that thought is able to “grasp” these numbers through consistent procedures similar to those that enumerate the real numbers.
In summary, we should take away that the theory of surreal numbers:
• posits that many more numbers — an “infinitely infinite” amount — exist aside our familiar real/empirical numbers
• demonstrates that we cannot capture all these numbers in one fell conceptual swoop, as the process of attempting to do so is infinitely iterative
• facilitates the method of the “cut”, which allows us to localize (or “find”) unique numbers within the surreal field
Now, with this powerful theory at our disposal can we finally define what a “number” is? Stay tuned!
- Dr. G
Next: What is a Number?
Notes
[1] For a more technical, but no less excellent, introduction to surreal numbers, see: Gretchen Grimm, An Introduction to Surreal Numbers (2012)
