I vaguely recall that some years ago, there was an MO question about how to distinguish between the complex numbers $i$ and $-i$, and in what sense this might be meaningful.

I also recall that I thought this was a very silly question until a response (either an answer or a comment) from someone who was probably Joel David Hamkins brought me up short and made me think it might be less silly than it appeared to me.

Beyond that, I recall nothing about this post, but I'd like to find it again, and my searches have been fruitless. Does anyone know the post I'm referring to?

  • $\begingroup$ Is it possible that you're thinking of this one? It's about quaternions, not complex numbers, with a nice answer by Matt Emerton. mathoverflow.net/questions/53822 $\endgroup$ – Dan Petersen Apr 5 at 6:12
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    $\begingroup$ @DanPetersen: Thank you, but that's not it. The question I'm remembering (perhaps incorrectly?) appeared (at least to me) to have much less mathematical content than this one (though the answer/comment I'm remembering suggested that there was more mathematical --- or perhaps philosophical --- content than had met my eye). $\endgroup$ – Steven Landsburg Apr 5 at 6:18
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    $\begingroup$ I also seem to vaguely remember it, but cannot find it. I only found a (rather unexciting) related question on math.SE (math.stackexchange.com/questions/177594/how-to-tell-i-from-i). $\endgroup$ – Emil Jeřábek Apr 5 at 9:43
  • $\begingroup$ I'm afraid I can't help, but your description does ring a bell. Perhaps either the question or one of the comments/answers mentioned Gal(C/R)? $\endgroup$ – Yemon Choi Apr 5 at 17:14
  • $\begingroup$ I think somebody once even told me that the physicist's $i$ is the mathematician's $-i$. Looking back, I'm pretty sure they were joking :). Although there are weird places where discrepancies like this come up between physics and math conventions. For instance, I believe the physicists' $su(n)$ is really $\pm i$ times the mathematician's $su(n)$, and I can easily imagine discrepancies arising from whether you choose $i$ or $-i$ to be the relevant factor. $\endgroup$ – Tim Campion Apr 5 at 17:53
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    $\begingroup$ I remember that in Théorie de Hodge that Deligne makes a point of discussing how the choice of $i$ influences various things such as orientations of complex manifolds. So perhaps it's not such silly question. ( I don't know where it is on MO.) $\endgroup$ – Donu Arapura Apr 5 at 18:03
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    $\begingroup$ Why not simply ask this question yourself? If such a question was asked before, this one will be closed as a duplicate, and you will find out what the original question was. If it was not asked, you will still receive answers. $\endgroup$ – Dmitri Pavlov Apr 6 at 13:26
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    $\begingroup$ @DmitriPavlov: The main thing I'm trying to remember is exactly what the question was. $\endgroup$ – Steven Landsburg Apr 6 at 17:27

This question relates directly to issues that are often discussed in the philosophy of mathematics.

According to one of the standard accounts of structuralism, what mathematical objects are at bottom are the structural roles that they play within a mathematical system. One might define the equivalence relation whereby element $a$ in structure $A$ is equivalent to element $b$ in structure $B$ exactly when there is an isomorphism of $A$ to $B$ taking $a$ to $b$. This results in what I have called the isomorphism orbit of the object $a$ in $A$.

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According to some accounts of the philosophy of abstract structuralism, what a mathematical object is, is the structural role that it plays in abstract mathematical structures.

The problem for this view arises with nonrigid mathematical structures such as the complex field $\newcommand\C{\mathbb{C}}\C$.

Here is how I describe the situaiton is my book, Lectures on the Philosopy of Mathematics (MIT Press)

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 $\newcommand\unaryminus{-}\unaryminus i$ also is such a root: $$(\unaryminus i)^2=(\unaryminus 1\cdot i)^2=(\unaryminus 1)^2i^2=i^2=\unaryminus 1.$$ Indeed, the complex numbers admit an automorphism, an isomorphism of themselves with themselves, induced by swapping $i$ with $\unaryminus i$---namely, complex conjugation: $$z=a+bi\qquad\mapsto\qquad\bar z= a-bi.$$ The conjugation map preserves the field structure, since $\overline{y+z}=\bar y+\bar z$ and $\overline{y\cdot z}=\bar y\cdot\bar z$, and therefore the complex field is not a rigid mathematical structure. Since conjugation swaps $i$ and $\unaryminus i$, it follows that $i$ can have no structural property in the complex numbers that $\unaryminus i$ 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 $\unaryminus i$, 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 $\unaryminus i$ play exactly the same roles in this structure (see Shapiro, 2012). The numbers $i$ and $\unaryminus i$ 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.

The point is that one cannot understand the mathematical objects as identical to the structural roles that they play within a system, since $i$ and $\unaryminus i$ play exactly the same role in $\C$ as each other, yet are to be regarded as distinct complex numbers.

In fact, there is nothing special about the numbers $i$ and $\unaryminus i$ in this argument. For example, the numbers $\sqrt{2}$ and $\unaryminus\!\sqrt{2}$ 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 $\newcommand\R{\mathbb{R}}\R$, where $\sqrt{2}$ and $\unaryminus\!\sqrt{2}$ 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 $x\leq y\iff\exists u\ x+u^2=y$. 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 $\sqrt[3]{2}$, for example, to one of the nonreal cube roots of $2$, such as $\sqrt[3]{2}(\sqrt{3}i-1)/2$. Therefore, the numbers $\sqrt[3]{2}$ and $\sqrt[3]{2}(\sqrt{3}i-1)/2$ 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 $\unaryminus i$ 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 may 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: $$\text{Re}(a+bi)=a\qquad\qquad\text{Im}(a+bi)=b,$$ then the expanded structure $\langle\C,+,\cdot,\text{Re},\text{Im}\rangle$ 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 $\unaryminus i$ 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.

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    $\begingroup$ Oh, I see now that this was a meta question, but I answered as though it was on main. Please forgive me. $\endgroup$ – Joel David Hamkins Apr 6 at 19:42
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    $\begingroup$ I am not sure whether this question/answer is appropriate here on Meta---perhaps it should simply be migrated to the main site? $\endgroup$ – Joel David Hamkins Apr 6 at 19:52
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    $\begingroup$ Thanks for this, though for reasons that include nostalgia, I'd still like to find the original question I asked about. $\endgroup$ – Steven Landsburg Apr 6 at 21:38
  • $\begingroup$ I would urge you (or someone) to create a question on main MO with the $i$ versus $-i$ question, and with your answer, which I found very illuminating.(You might want to fix the typo "one may recovers the uniqueness".) In some way, this is similar to the "coarse versus fine moduli space" issue in algebraic geometry. Many problems, for example classifying isomorphism class of elliptic curves (over $\mathbb C$, say), admit a coarse moduli space, but to get a fine moduli space, one must add structure to "kill" automorhisms. Of course, one can get fancier with staicks (but I won't!). $\endgroup$ – Joe Silverman 5 hours ago

Could this be the question you mean? It's not exactly about $i$ versus $-i$, but it mentions it as a motivating example.

"Co-ordinate-free" mathematics for general structures?

Even if it isn't, don't miss the reference there to Shapiro - Identity, indiscernability, and ante rem structuralism: The tale of $i$ and $-i$.


The discussion following Mike Shulman's 2013 post From Set Theory to Type Theory was useful to me to see what to say about this situation type-theoretically while writing section 3.4.3 of my book Modal Homotopy Type Theory.

Mike distinguishes between the introduction of the type of complex numbers as a particular type and as an abstract type. As a particular type, having already constructed a type of real numbers somehow, you might construct it in some concrete way as pairs of real numbers. In this case, you can name the particular pair $\langle0, 1\rangle$ as $i$.

On the other hand, you might characterise the type of complex numbers abstractly, say, as what turns out to be the unique element of the type of algebraically closed fields of characteristic zero and the cardinality of the continuum. This abstract type is not equivalent to the particular version above since there is no means to distinguish roots of $-1$. So then we could modify this type to include the extra structure of being equipped with a designated root of $-1$.

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    $\begingroup$ I've just seen I fell into the same trap as Joel. $\endgroup$ – David Corfield Apr 7 at 6:59
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    $\begingroup$ It is not enough merely to designate a root of -1, since there are many other automorphisms of the complex field, which fix i. You basically have to designate a copy of R inside the field, as well as i. $\endgroup$ – Joel David Hamkins Apr 7 at 8:56
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    $\begingroup$ Right, more is needed. Specifying continuity of automorphisms for C as R^2 is one way of phrasing things. Then specifying i. $\endgroup$ – David Corfield Apr 7 at 10:19

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