Are the imaginary numbers real?
Are the imaginary numbers real?
What are the complex numbers, really?
Complex numbers
Given the real numbers, one proceeds to the complex numbers ℂ, motivated by the enticing, yet perhaps terrifying, possibility that the imaginary unit i = √-1 exists as an actual number. One wants to consider complex numbers of the form a + bi, where a and b are real. What is a complex number?
Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.
We can easily construct a natural presentation of the complex field by means of the complex plane. Specifically, since complex numbers are to have the form a + bi for real numbers a and b, let us think of the number a + bi as represented by the pair (a,b), a point in the plane. We may define the usual coordinate-wise addition operation (a,b) + (c,d) = (a + c, b + d), but we use a certain strange multiplication operation, defined by (a,b) · (c,d) = (ac - bd, ad + bc). This definition exactly implements the distributive consequences of i2 = -1 in the product
By identifying a + bi with the point (a,b), we have realized the complex numbers as points in the plane, now called the complex plane, and we understand the complex arithmetic simply as certain elementary operations defined on those points. So there is no terrifying mystery after all in the complex numbers. We can construct them from the real numbers.
Platonism for complex numbers
We formed the complex numbers by extending the real numbers with a solution of the equation z2 + 1 = 0, using the solution z = i. It is a remarkable fact, known as the fundamental theorem of algebra, that by adding this one solution, the complex numbers thereby become algebraically closed: every nontrivial polynomial equation over the complex numbers has a full solution there. The complex numbers are the algebraic closure of the real numbers.
What is a complex number, actually? Imagine that at your death, you are astonished to meet God in Heaven, who informs you, “Yes, you were completely right about platonism for the real numbers—there they are!” He points across the way, and there you see them—the real numbers, each of them a perfect platonic ideal of its kind. You find the numbers π, e, √2, each where you expect them. “But,” God continues, “you were wrong about platonism for the complex numbers; you have to construct them from the real numbers as pairs (a,b), with the parentheses and comma and everything.”
The situation is absurd because we expect that our mathematical ontology should treat similar kinds of mathematical objects similarly; if the real numbers are real, then the complex numbers should be as well. Is this a slippery slope for platonism? Once one admits a real existence for one kind of mathematical object or structure, why not more? Soon, we shall find ourselves in plenitudinous platonism. But what of mathematical structures that might differ in their level of abstraction? Some philosophers propose that the natural numbers have a more definite existence than real numbers, and that while platonism is correct for the natural numbers, it is not for the real numbers. Is the allegory relevant for them? Perhaps not; perhaps the difference in abstraction makes natural numbers and real numbers fundamentally different in kind, unlike the real and complex numbers of the allegory.
Categoricity for the complex field
Like the real numbers, the complex field ℂ admits a categorical characterization. Namely, the complex field is uniquely characterized up to isomorphism as being the algebraic closure of a complete ordered subfield, the real numbers. Any two fields like that are isomorphic, since their real subfields will be isomorphic and this isomorphism will extend to the algebraic closure. The complex field is also characterized up to isomorphism as the unique algebraically closed field of characteristic 0 having size continuum. One can express the concept of having size continuum in second-order logic by asserting that there is a bijection with a subset that is a real continuum.
Thus, each of our familiar number systems—the natural numbers ℕ, the integer ring ℤ, the rational field ℚ, the real field ℝ and the complex field ℂ—admit categorical characterizations. Precisely because of these categorical accounts, we are able to pick out and refer to these structures simply by describing what is true in them, rather than by having to exhibit sample instances of the structures. We don't need to present a particular constructed copy of ℂ to refer to the complex field, because we can just say that we are referring to the algebraically closed field of characteristic 0 having size continuum. To my way of thinking, this ability to refer to structures without needing to exhibit particular instances is a core part of the deep connection between categoricity results in mathematics and the philosophy of structuralism.
A complex challenge for structuralism?
Although one conventionally describes i as “the square root of negative one,” nevertheless one might reply to this, “Which one?” in light of the fact that -i also is such a root:
Indeed, the complex numbers admit an automorphism, an isomorphism of themselves with themselves, induced by swapping i with -i—namely, complex conjugation:
The conjugation map preserves the field structure, since
and therefore the complex field is not a rigid mathematical structure. Since conjugation swaps i and -i, it follows that i can have no structural property in the complex numbers that -i does not also have. So there can be no principled, structuralist reason to pick one of them over the other. Is this a problem for structuralism? It does seem to be a problem for singular terms, since how do we know that the i appearing in my calculations this week is the same number as what will appear in your calculations next week? Perhaps my i is your -i, and we do not even realize it.
If one wants to understand mathematical objects as abstract positions within a structure, as in abstract structuralism, then one must grapple with the fact that in light of the conjugation automorphism, the numbers i and -i play exactly the same roles in this structure (see Shapiro, 2012). The numbers i and -i have the same isomorphism orbit with respect to the complex field, and so in this sense, although distinct, they each play exactly the same structural role in ℂ. This would seem to undermine the idea that mathematical objects are abstract positions in a structure, since we want to regard these as distinct complex numbers.
Furthermore, there is nothing special about the numbers i and -i in this argument. For example, the numbers √2 and -√2 also happen to play the same structural role in the complex field ℂ, because there is an automorphism of ℂ that swaps them (although one uses the axiom of choice to prove this). Contrast this with the real field ℝ, where √2 and -√2 are of course discernible, since one is positive and the other is negative, and the order is definable from the field operations in ℝ via
It follows that the real number field is not definable in the complex field by any assertion in the language of fields. In fact, there is an enormous diversity of automorphisms of the complex field; one may move the real cube root of 2, for example, to one of the nonreal cube roots of 2, such as
Therefore, the numbers
are indiscernible in the complex field—there is no property expressible in the language of fields that will distinguish them. Indeed, except for the rational numbers, every single complex number is part of a nontrivial orbit of automorphic copies, from which it cannot be distinguished in the field structure. So the same issue as with i and -i occurs with every irrational complex number. For this reason, it is problematic to try to identify complex numbers with the abstract positions or roles that the numbers play in the complex field.
Meanwhile, one recovers the uniqueness of the structural roles simply by augmenting the complex numbers with additional natural structure. Specifically, once we augment the complex field ℂ with the standard operators for the real and imaginary parts:
then the expanded structure ⟨ℂ,+,·,Re,Im⟩ becomes rigid, meaning that it has no nontrivial automorphisms. Thus, every complex number plays a unique structural role in this new structure, which is Leibnizian. This additional structure is implicit in the complex plane conception of the complex numbers, which is part of why the number i appears fine as a singular term—it refers to the point (0,1) in the complex plane—whereas -i refers to (0,-1). The complex plane is not merely a field, for it carries along its coordinate information by means of the real-part and imaginary-part operators, making it rigid. In the complex plane, every complex number plays a different role.
Structure as reduct of rigid structure
This situation, where a natural nonrigid structure is made rigid by natural additional structure, is extremely common in mathematics. Examples abound. The additive group of integers ⟨ℤ,+⟩ admits an automorphism by negation, but is made rigid with the multiplicative structure ⟨ℤ,+,·⟩ or the order structure ⟨ℤ,+,<⟩. The rational order ⟨ℚ, < ⟩ is a countable endless dense linear order and therefore highly nonrigid—every point looks the same, and indeed any two finite sets of the same size are order-automorphic—but becomes rigid with the field structure ⟨ℚ,+,·,<⟩. The complex field ⟨ℂ,+,·⟩ has
many automorphisms, but is made rigid by incorporating the coordinate structure. Every group G with at least three elements is nonrigid, but elements are distinguished when the group is given a particular presentation, such as by means of generators and relations or as permutations of a particular set.
The pattern is that a particular nonrigid structure is realized as a reduct substructure of another structure that is rigid, thereby resolving the problem of reference, since we may refer to the objects of the nonrigid structure by reference to their roles in the expanded structure. I claim that this pattern is inherent in mathematical practice. The reason is that precisely because of the reference problem, it is difficult for us ever actually to present or specify a nonrigid structure, except by presenting it as a reduct substructure of a structure in which the objects are individuated. How else are we coherently to specify the structure on those objects in the first place? We don't start with a naked copy of ℂ and then seek to impose an orientation on it that will enable us to resolve i from -i. Rather, we proceed oppositely: instances of mathematical structures are obtained from richer contexts where the objects were already individuated. We might build a copy of ℂ from ordered pairs of real numbers, for example, where we can discern (0,1) from (0,-1) and therefore i from -i in this particular copy of ℂ. Every particular copy of ℂ and indeed every particular mathematical structure of any kind arises similarly from a context in which the objects are individuated.
When using ZFC set theory as a foundation of mathematics, this philosophical observation becomes a mathematical theorem: every set is a reduct substructure of a rigid structure, a structure in which every individual plays a distinct structural role. The reason is that every set is a subset of a transitive set, and every transitive set is rigid with respect to the ∈ membership relation. Indeed, the set-theoretic universe ⟨V,∈⟩ as a whole is rigid—any two objects in the set-theoretic universe are therefore distinguishable as sets and play different set-theoretic roles (see the argument on page 286. Therefore, every mathematical structure that can be realized in set theory at all can be realized as a reduct substructure of a rigid structure. We can refer to distinct individuals in the original structure by the distinct structural roles they play in the larger context.
Continue reading more about this topic in the book:
Lectures on the Philosophy of Mathematics, MIT Press 2021
The fact that it has been proven that certain results in quantum mechanics cannot be explained by the use of real numbers and one must use complex numbers to describe what Nature is doing makes complex numbers a part of reality for me.
https://www.nature.com/articles/s41586-021-04160-4