There is an ongoing debate among mathematicians and philosophers on the nature of the realm where all mathematical activities are performed. But, before I pose the problem, we need to answer a pertinent question: “A realm? Do you mean like… our minds or what?” No. I mean metaphysical entity mathematicians are studying just like physicists are studying the physical universe. One can intuitively convince oneself how it is possible to think about natural numbers along these lines. Natural numbers are relatively easy to imagine and, for example, it is the law of the realm of natural numbers that Fermat’s Last Theorem,
has no solutions for non-zero and ,
holds, as proved by Prof Sir Andrew Wiles in 1994. Once one realizes what to substitute in place of natural numbers so as to make assertions about the whole mathematical realm, not just natural numbers, the generalization quickly becomes as compelling.
The mathematical realm
The mathematical realm starts with an empty set , a set having no elements. Then, we take its powerset, the set of all subsets of the empty set, namely, the set consisting of the empty set . After that, we have the set of all subsets of the set consisting of the empty set, the set . And so on, until the whole universe, unfolds. In , every mathematical individual can be accommodated, and thus, is the ultimate realm of mathematical practice. But is it?
While the idea of a single, ultimate mathematical universe is tempting and convincing at first glance, there are a plethora of issues with this approach. The problem arises due to Gödel’s incompleteness theorems implying that:
- There are statements which can neither be proved nor disproved using mathematical methods;
- It is impossible to prove that mathematics is consistent, i.e., there might be a statement such that both and are provable.
Thus, any candidate for an ultimate, true mathematical realm must address Gödel’s incompleteness, at least in some way. It follows from Gödel’s theorems that there is a statement we can consistently adjoin to the fundamental laws of mathematics, but, at the same time, we can do the same for its negation . Which one should we choose? We certainly can’t adjoin both of them as that would yield an inconsistent theory! However, we shall not focus on this problem and assume that the truth value of each statement can be verified by some extrinsic means or other methods. Still, can we talk about the mathematical universe? The result of the following limit is well known:
But accepting as an object (so we can write for some set ) is not the same as accepting all its approximations (larger and larger numbers) as objects. Similarly, accepting for all is not the same as accepting . This argument is too weak and is only for analogy purposes. Let me now argue for the contrary.
The realm of complex numbers
Let me quickly describe what is a field in mathematics. A field is a set together with two operations addition () and multiplication () that satisfies certain features so that it behaves similarly to the real numbers, rational numbers, complex numbers, or Galois fields.
Field theory is a part of set theory.
Fields can be ordered with respect to their extensions. For example, the field of rational numbers can be extended to the field or real numbers can be extended by the imaginary number and form the field of complex numbers . What is special about the field of complex numbers is that it is algebraically closed. So, contrary to or , every non-constant polynomial with coefficients in has a root, i.e., there is some such that . For example, while a polynomial
has no roots over , it has two roots over , and, therefore, can be factorized into . Excellent! Now, every field has a theory and for a statement , we write (F satisfies ) whenever is true in . So for example, if a statement expresses that the equation has a solution, then we can say that while the field of complex numbers satisfies or write , the field of real numbers doesn’t satisfy or . Since fields are ordered by extensions, we can extend our language and talk about possibility and necessity. We write and say that satisfies that is possible whenever there is some field extension of such that . Furthermore, we write and say that satisfies that is necessary whenever for all field extensions of , .
The potentialist maximality principle is a feature that a field can admit. The underlying condition is that every statement that is possibly necessary () is already true. In other words, the potentialist maximality principle says: everything that could be true is already true. This is precisely, as we discussed a while ago, the feature the whole universe of sets must have (but in a much more general sense)! So while ‘s existence is questionable, can we find an analogy in the restricted set-theoretic realm, namely, the realm of fields by giving an example of a well-defined field admitting the potentialist maximality principle? Actually, we can.
The field of complex numbers admits the potentialist maximality principle.
And then maybe we can argue:
Hey, I restricted myself to field theory and I can tell you that I can truly pinpoint at the ultimate, complete notion of a number – the notion of a complex number. This is truly complete and ultimate as you can see by analysing modal logic or even by common sense looking at polynomial factorization. Give me a polynomial over complex numbers and I will reduce it into linear factors. That’s it! You can’t do any better!
And then we can go really crazy and say:
In retrospect, what allowed me to find the ultimate notion of a number was restricting mathematical realm to much simpler case of fields. Field theory is not only simpler but also much better understood. I claim that once we come to know of sets as well as we currently know of fields, we will be able to find a perfect candidate for V just like we found one for fields.
Now let me refute this crazy idea and give a multiversist reply. We shall call the multiversist the one who denies an idea of the ultimate, single, divine mathematical universe. I shall abstain from criticizing the fundamental difference between membership relation and the relations we use in field theory (addition and multiplication). However, relaxing our skepticism, even to that extent, does not violate the argument we shall give! Let’s get to it.
The argument of complex numbers is flawed in a number of ways and ultimately incorrect. The whole idea of the universe of sets is to refrain from any limitations as we want to grasp the wholeness of mathematical practice.
Before we move on, let us say a few things about algebra. We shall give two definitions and justify a simple fact.
- A skew-field has exactly the same definitions as a field but does not have to be commutative, i.e., it may be the case that .
- A number system called quaternions extends the complex numbers and is a non-commutative skew-field. The set of quaternions consists of elements of the form such that , , , , , , and .
Fact. The equation has infinitely many (actually continuum many!) solutions over the skew-field of quaternions.
Proof. We claim that all elements of the set are solutions to the equation . Set . Squaring gives us
Note that must be equal to zero (why?) and it follows that . Thus, there is as much solutions to the as there are triples of real numbers.
Let’s continue our argument.
The case of quaternions show that your restriction to fields was unsubstantiated. Proper methods and proper objects of mathematics shouldn’t be as we wish them to be, but rather the other way round. Complex numbers are indeed beautiful and algebraic closure is a nice property, but treating complex numbers as the ultimate notion of a number is like living in a bubble. Just one step ahead and there is a whole different realm where things are not that elegant, but their mathematical significance is irrefutable. Therefore, your complex numbers are just like the axiom V=L where every set is constructible, and so, things are simple. But all set-theorist know that V=L cannot be regarded as an ultimate axiom due to the limitation it poses. Finally, as you can see looking at the number of roots of a polynomial did not matter at all since in skew-fields you can have an infinite number of roots.
So, answering the question in the title, complex numbers tell us that the problem with the status of the mathematical realm is very subtle. At first, when I was thinking about complex numbers as an analog of the mathematical universe.
I started with a pro multiverse argument, got a single universe one that later turned out to be pro multiverse after all. Wow…
I felt like “Ok, so I am now producing a multitude of simple structures to show how erratic mathematics can be, I do this in favor of the multiverse view… Ok, Great! But let’s look at fields, it seems that there is some kind of a complete structure, a complete notion of what we call a number.” After going through a similar train of thought as outlined earlier, I realized that I started with a pro multiverse argument, got a single universe one that later turned out to be pro multiverse after all. Wow… But this was just a common human fallacy to fall in love with elegance and order. It was only me (with the help of years of singleverse indoctrination!) who decided that this and this structure with this and this property I am going to look at today. I could have looked beyond commutativity in the first place and never come across this problem. It is, of course, fine and wise to do mathematics in such idealized environments as complex numbers, V=L, or large cardinal notions. But it is not wise to ignore the whole cosmos of other possibilities.