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Recently, I was watching an interview that John Conway did with Numberphile. By the end of the video, Brady Haran asked John:

If you were to come back a hundred years after your death, what problem would you like to know about ?

John replied:

I would really like to know why the Monster Group exists.

From what I have read, it seems like research has moved on toward exploring the connections between the monster moonshine conjectures and other fields such as conformal field theory. I do know that Conway didn't particularly like the idea of the existence having to do with CFT, e.g.,

Therefore, I am trying to understand what he means by "why". I couldn't really find any papers where he further went into the question, why can't the existence of the group be a coincidence? Have there been any further developments in relation to this question?

Update: So I found another interview, in which John explains a bit further: "I think there is something so fundamental about this object that there should be a simpler definition or a simpler construction"

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    $\begingroup$ When one looks for a satisfying explanation of a mathematical fact, one usually doesn't know how to describe which explanations one would find satisfying before seeing the explanations. So it might not have been possible for Conway to precisely answer your question, let alone for anyone else to do so. $\endgroup$
    – Will Sawin
    Commented Aug 21, 2023 at 19:51
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    $\begingroup$ "Therefore, I am trying to understand what he means by "why" [...] why can't the existence of the group be a coincidence ?" You know what? I think your question is exactly the answer to your question. $\endgroup$
    – Olivier
    Commented Aug 22, 2023 at 8:05
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    $\begingroup$ @CarloBeenakker, your comment seems like a reasonable answer and should be posted as such. Conway wanted an explanation for this exceptional simple group beyond saying that this is the way the full classification comes out. The fact that the group arises naturally as a group of symmetries is an answer Conway may have appreciated. The closing votes are unreasonable. $\endgroup$ Commented Aug 22, 2023 at 9:11
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    $\begingroup$ I agree that technically, the question is opinion-based, but not in a way that I think demands closure. We can't know for sure exactly what Conway meant, but "why" questions of this kind are asked by research mathematicians all the time, always with the intent of eliciting mathematical facts rather than non-mathematical opinions. I am voting to reopen. $\endgroup$ Commented Aug 22, 2023 at 12:22
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    $\begingroup$ I doubt that the question "why does the Monster Group exist?" can receive a mathematical answer. Why do the natural numbers exist? Why do groups exist? But do they "exist", like this keyboard that I am typing on? Why do we exist? Do cats understand humans? I have voted to leave the question closed. $\endgroup$
    – Alex M.
    Commented Aug 22, 2023 at 15:02

2 Answers 2

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The OP wrote:

I am trying to understand what he means by "why".

Although several respondents have expressed skepticism that this question can be answered satisfactorily, I maintain that it's not that hard to understand. For doubters, I would offer the following paraphrase of Conway's statement, "I would really like to know why the Monster Group exists":

I would like a simpler and more natural construction (or existence proof) of the Monster.

Justification for this paraphrase may be found in Conway's own work in finite group theory. For example, what was the point of his paper, A simple construction for the Fischer-Griess monster group? Conway himself wrote, "The main aim of this paper is to present a simplified construction [of the Monster]." If you read his book with Sloane, Sphere Packings, Lattices and Groups, you will similarly find an emphasis on the nicest possible constructions. In Chapter 11, we read:

The Golay code $\mathscr{C}_{24}$, the Steiner system $S(5,8,24)$, and the Mathieu group $M_{24}$ are beautiful combinatorial objects with a great wealth of structure and applications. The MOG [Miracle Octad Generator], and its companion-at-arms the hexacode, are computational tools that enable one to perform mental calculations on these objects with great ease. … There is also a MINIMOG, with companion the tetracode, whcih together perform similar services for $M_{12}$.

In the introduction, we get an example of something that the authors regard as shedding light on the miraculous properties of the Leech lattice:

Considerable light is thrown on these mysteries by the realization that the Leech lattice and the Niemeier lattices can all be obtained very easily from a single lattice, namely $\mathrm{II}_{25,1}$, the unique even unimodular lattice in Lorentzian space $\mathbf{R}^{25,1}$.

Again, ease of construction is regarded as very important. A similar dissatisfaction with ad hoc miraculous constructions and extraneous detail can be found in all of Conway's work, not just his work in group theory; a case in point is his proof of the classification of closed surfaces, which he dubbed the "ZIP proof" or "Zero Irrelevancy Proof."

The sources cited by the OP corroborate this point of view. The IAS article says, "In his view, conformal field theory is too complicated to understand, and thus too complicated to be the only answer." Again, note the emphasis on simplicity versus complexity.

So I think we can safely assume that what Conway was hoping for was that in the next 100 years, a much simpler construction of the Monster would be found.


Regarding the question of why the existence of the Monster couldn't be a "coincidence," I feel confident that Conway would have rebelled against that attitude. It is true that there are some types of mathematical statements that Conway might have conceded are true by accident. As I mentioned in that other MO answer, Conway felt there was some force to the heuristic, probabilistic argument that the Collatz conjecture is true, but not provably so. But in that setting, the culprit is that provable intractability (and lack of structure) is crouching at the door. In the case of the Monster, we have a proof of its existence, and there is tons of structure present. I am sure that Conway would have categorically rejected any argument along the lines of, "Oh well, you know, if you just generate group multiplication tables at random, it's not at all surprising that a simple group of order 808, 017, 424, 794, 512, 875, 886, 459, 904, 961, 710, 757, 005, 754, 368, 000, 000, 000 would emerge just by chance. Let's move on; nothing to see here."

I seem to remember having an exchange with Conway (maybe on the math-fun mailing list many years ago?) in which I raised the possibility that the main theorems of class field theory possess an ineliminable core of miraculous coincidences that can't be "explained" in a purely "conceptual" way, and Conway said that he thought that attitude was too defeatist, and that we shouldn't stop trying to find increasingly simpler proofs of something so beautiful. Certainly, Conway's philosophy of doggedly seeking simpler explanations of mathematically significant facts served him immensely well over the course of his career.


EDIT (October 2023): I just discovered a talk by Richard Borcherds from a couple of years ago. Someone asked a question about the construction of the Monster, and Borcherds' answer may give some insight into what Conway was hoping for.

This is an open research question for anyone who's interested: Find a natural construction of the Monster. Every construction of the Monster we have is a mess. Most of the constructions are essentially simplifications of Griess's original construction, and the problem with all of them is, you take two huge vector spaces which seem to have nothing to do with each other; you take their direct sum, and you put some sort of piecemeal algebra structure on this, and by some amazing fluke it just happens that the Monster is a group of automorphisms. You would never find this construction unless you already suspected the Monster existed and were trying to construct it. What we'd really like is a sort of one-piece construction where you construct it just in one piece. For instance, [for] Conway's group, you take the Leech lattice in 24 dimensions, and that's a very natural single object in 24 dimensions. We don't have anything like that for the Monster yet.

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    $\begingroup$ Watching the Numberphile video again, I see that Conway does explicitly say, at the eight-minute mark, that the Monster group is "obviously not there just by coincidence; it's got too many intriguing properties for it to all be just an accident." Then when Brady complains that 196,883 seems so "arbitrary," Conway objects that it is definitely not arbitrary. $\endgroup$ Commented Sep 16, 2023 at 4:29
  • $\begingroup$ I fully support the idea that not all proofs that something is true are explanations as to why it is true (e.g., the proof of the four color theorem), and also that some theorems are "accidents" (which is how I think of some Godel statements). Great post! $\endgroup$ Commented Oct 3, 2023 at 4:58
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[comment turned into an answer, as suggested by Mikhail Katz]

a 2022 paper by Scott Carnahan, "Why do the symmetries of the monster vertex algebra form a finite simple group?" goes some way towards answering Conway's question, using "19th century group theory".

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    $\begingroup$ Scott Carnahan is an MathOverflow moderator. Perhaps he could add more details to this answer. $\endgroup$ Commented Aug 22, 2023 at 15:34
  • $\begingroup$ Or you could at least give a summary of what this partial answer is? $\endgroup$
    – Kimball
    Commented Aug 23, 2023 at 12:38
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    $\begingroup$ my -- very limited -- understanding is that Carnahan answers the "why" question by "the monster group exists as a finite simple group because it is the automorphism group of the monster vertex algebra". $\endgroup$ Commented Aug 23, 2023 at 13:16
  • $\begingroup$ Just curious: do you know what the comment about 19th century group theory means exactly? $\endgroup$ Commented Aug 23, 2023 at 15:38
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    $\begingroup$ I quote: "our main group-theoretic tools for proving finiteness and simplicity are Sylow’s theorem and the Frattini argument, both from the late 19th century". $\endgroup$ Commented Aug 23, 2023 at 16:10

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