"Are there any simple groups that appear as zeros of the zeta function?" by Peter Freyd; why is this consternating to mathematicians?

Question

I would like to understand the "upsetting"-to-mathematicians nature of this question Freyd poses to demonstrate that "any language sufficiently rich that to be defined necessarily allows you to ask questions which don't have any conceivable answer but sound as if they should" (18:28). I get that we aren't supposed to actually solve the question, but is there an "undergrad" or "serious highschooler" breakdown of the math and why its unanswerable? I understand I'm taking a risk by asking as maybe the reason is something so obvious yet I'm missing it...

Here he is on these type of questions:

any language sufficiently rich that to be defined necessarily allows you to ask questions which don't have any conceivable answer but sound as if they should we're used to that in a number of special cases we're used for instance in mathematics except this is a very Applied Mathematics to know that to ask what what frequencies are present in a signal at a given instance it just doesn't make sense you can't I mean the definition of frequency or how you would measure it is such that you can't ask that question without specifying a nonzero interval

And here here is his brief explanation of why this question about zeta zeroes is so frustrating to mathematicians:

here's one which I like to throw at mathematicians as an example but I warn you people can get very upset at this one uh if somebody takes set theory as a serious foundational language for mathematics I can pose to him because of its untyped nature the question are there any simple groups that appear as zeros of the zeta function I've seen people get very angry at this I've seen people forget why I mentioned it say I didn't really ask it I only mentioned it I I've had I've had people come back to me two weeks later swearing at me for bringing up such an absurd question okay that's transparently absurd notice by the way you have no way of answering it at all even if you took it for the moment seriously until you specify what definition of group you have in mind and what definition of complex number and then after somebody specifies them I've challenged people now prove to me that there aren't any simple groups

However, I only have a small introduction to groups, and no serious academic knowledge of the zeta function, so I'm missing out. Is it possible to get a breakdown of the math and moral?

This is from “An Antiphilosophy of Mathematics,” Peter J. Freyd ~18:28 time stamp

Answer 1

Freyd is talking about an idea which in philosophy I think is called a category mistake or category error, and which in mathematics does not seem to have a widely agreed-upon name, but which it makes sense to call a type error by very close analogy with computer science.

Here is a simpler and less technical example of the same idea:

Is brown tall?

Is the answer yes or no? Or does it seem like a question which is ill-formed somehow? Brown is not even the sort of thing which could be called tall or short. It doesn't have a height at all! It's an object in the wrong category for this to be a sensible question.

So, the first layer of what Freyd is talking about is that, with a minimum of technical detail:

• The zeroes of the zeta function are certain special complex numbers, which are, to be slightly more concrete, certain pairs $a + bi$ of real numbers.
• Simple groups are... well, it's complicated to say exactly what they are. You can think of them as certain collections of permutations. The important thing is that they can be defined without any reference to anything that looks like a number. The smallest (edit: non-abelian!) example is the group of rotational symmetries of an icosahedron.

So they are two very different type of objects. There's no meaningful sense in which we can interpret a simple group as a complex number; in computer science terminology (we have to keep importing this because in mathematics none of what we're currently talking about is explicitly given a name) there is no meaningful type conversion from one to the other.

So the first layer is that "are there any simple groups that appear as zeros of the zeta function?" is a type error and so not a sensible question, in the same way that "is brown tall?" is not a sensible question.

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The second layer, and the reason he says this makes mathematicians upset, has to do with set theory. Freyd is alluding to the fact that, technically speaking, in the modern foundations we are supposed to be using to do mathematics, every single mathematical object is something called a set, and without wading into the details of what exactly that means, for any two sets $X$ and $Y$ it is possible to ask questions like whether $X$ and $Y$ have any elements in common.

For a simple example we could consider sets of numbers. E.g. the set of odd numbers $\{ 1, 3, 5, \dots \}$ and the set of prime numbers $\{ 2, 3, 5, \dots \}$ have many elements in common, such as $3$, but the set of odd numbers and the set of even numbers $\{ 2, 4, 6, \dots \}$ don't have any elements in common.

However, in set-theoretic foundations, because any pair of two mathematical objects are sets, it is technically possible to ask, for any two sets of any kind of object whatsoever, whether they have any elements in common. So, Freyd considers the two sets

$$Z = \{ \text{the zeroes of the zeta function} \}$$ $$S = \{ \text{the set of simple groups} \}$$

and asks: do $Z$ and $S$ have any elements in common?

This is frustrating on multiple levels. This should be a type error, but it isn't, because set-theoretic foundations don't technically have a type system at all. Essentially all practicing mathematicians behave as if mathematics does have a type system (see e.g. this blog post for a brief discussion), but technically it appears nowhere in our foundations. This is an awkward and unpleasant disconnect between the way mathematics is supposed to work in theory and the way it is actually done in practice.

The answer to this terrible question in set theory also depends very delicately on the precise choices we made for how to "encode" mathematical objects as sets. Again, without getting into the details of what it means to encode a mathematical object as a set, let's use a computer science analogy: every file on your computer is technically being encoded on some level as a long sequence of $0$s and $1$s. So if I had an image on my computer, and you had an image on your computer, there are at least two things we might mean by "are these exactly the same image?"

1. We might mean, are our images the same size in terms of pixels, and is every pixel of my image the same color as the corresponding pixel of your image?
2. Or we might mean, are our two image files coded by the same sequence of $0$s and $1$s on my computer as they are on your computer?

I would say that the first question is a meaningful notion of what it means for two images to be exactly the same image, and the second one is meaningless. The reason the second one is meaningless is that it depends on the different choices our two computers are making for how to encode images as sequences of $0$s and $1$s - for example mine might be a .png and yours might be a .bmp - and those choices are essentially "arbitrary." They don't affect the "meaning" of an image file, which is its pixels. The situation is the same in mathematics; the sequences of $0$s and $1$s used to encode files on a computer are very much like the sets used to encode mathematical objects (in that the choice of encoding is essentially "arbitrary" and we could make other choices).

(However, note that this second question is not "as meaningless" as the example "is brown tall?" we started with. In that example it didn't seem to make sense to answer yes or no. With this example the question definitely either has a yes or no answer, but not one that matters very much.)

So, Freyd is pointing out that the mathematician ought to be able to easily dismiss his question as meaningless and a type error, but set-theoretic foundations don't allow us to easily do so. Again there is a disconnect between how mathematics is supposed to work in theory (set theory) and how it actually works in practice (an informal type theory which nobody specifies). Mathematicians do not like to be reminded about this! It's a sore spot. Personally I find it fascinating.

For more on these themes, see this see this recent answer I wrote about a cataclysmic event called the foundational crisis in mathematics which is very relevant context for how things ended up this way.

Answer 2

It is upsetting and unanswerable because it is designed to make very little sense, but not none.

It's close to asking "What colour did you eat this morning?", which makes no sense, and everyone can see that, because everyone is familiar with the concept of colour and eating: you eat food, and colours are not food. In the question "Are there any simple groups that appear as zeros of the zeta function?", a layman would not understand what a group or a zero of the zeta function is, but otherwise the question is grammatical, so it sounds as if it makes sense. However, a sufficiently knowledgeable person would know, a (simple) group is an algebraic structure, and the Riemann zeta function takes complex numbers as input, so it is nonsensical to plug a group into the Riemann zeta function and ask whether it is zero. Any mathematician, upon hearing this question, would quickly disregard it as nonsense (as manifested in the comments of this question).

The difference is, in mathematics, fundamentally, every object is a set (at least by modern mainstream convention). A group is an ordered pair, $(G, \cdot)$, where $G$ is a non-empty set and $\cdot: G \times G \to G$ is a function satisfying the group axioms. However, $\cdot$, as a function, is defined as a set, and an ordered pair, is surprisingly also constructed using orderless sets. A similar story goes for complex numbers: every complex number is an ordered pair of real numbers (the operations are given elsewhere, extrinsic to the number itself), so it is a set. In the end, both groups and complex numbers are sets, so you can theoretically typecast a group to a complex number (note, the object, as a set, is unchanged; it is only the context that changes). It'll certainly be meaningless. It'll also almost certainly not be successful, just like how typecasting arbitrary strings into floating-point numbers would almost certainly fail. But who's to say it'll never be successful? Until you have proved either:

• there are no sets that are simultaneously a simple group and a complex number, or
• any simple group that is also a complex number (equal as sets) is not a zero of the Riemann zeta function,

you cannot theoretically answer "no" to the question. This is where one may start to feel upset (or amused, like myself), because the question should be nonsensical, but it isn't. Moreover, answering the question would only be tedious work with absolutely no benefit. It is uselessly meaningful.

The question makes extra less sense when you consider that there are (potentially infinitely) many equivalent but different realizations of mathematical objects as sets, sometimes with no concensus. It is equally valid to define groups as $(\cdot, G)$. Meanwhile, real numbers (whose construction complex numbers' construction relies upon), although usually constructed as equivalence classes of Cauchy sequences of rational numbers, are sometimes constructed as subsets (or pairs of them) (i.e. Dedekind cuts) of rational numbers. One may argue the question is ill-posed because of this. One may also argue the question becomes harder to answer, because there are now more versions of simple groups and complex numbers to compare.

So, what colour did you eat this morning? Well, I ate some oranges.

Answer 3

Because it has an answer

Let $G$ be a simple group. To be a zero of the zeta function, it must be a complex number.

Every simple group is an ordered pair $(S, \cdot)$, where $\cdot$ is a binary operation on $S$. A complex number is an ordered pair of real numbers, and thus both $S$ and $\cdot$ must be real numbers.

$\cdot$ is a binary operation, meaning it is a function, meaning it is a set of ordered pairs. For $\cdot$ to be a real number, it must be an ordered pair of infinite sets of rationals $(L, R)$. By the definition of ordered pair, $(L, R) = \{\{L\}, \{L, R\}\}$. Since $\cdot$ is a set of ordered pairs, this implies that $\{L\}$ must be an ordered pair, meaning $\{L\} = \{\{x\},\{x,y\}\}$ for some $x$ and $y$. So $L = \{x\}$ and $L = \{x,y\}$. But this is impossible since $L$ was supposed to be infinite. Contradiction. So $G$ is not a zero of the Riemann zeta function. $\square$

The answer to "why is this consternating to mathematicians?" is left as an exercise to the reader XD.

(Also, fun fact: the above proof still works even if we only use constructive set theory.)