Toilet: Hey Sock! Haven’t seen you for a while! What’s going on?
Socrates: Not much. I have just been travelling a lot, seeing some old friends, but mainly Parmenides.
Toilet: Oh, how nice! Did you guys have a drink?
Socrates: Yes, indeed. Just like the old times – chatting and drinking.
Toilet: What sort of things did you chat about, if you don’t mind me asking?
Socrates: Just the old stuff – Being and Non-Being.
Toilet: What about them?
Socrates: That Being is and Non-Being cannot be.
Toilet: Well of course. That seems pretty tautological. I don’t see any discussion can come out of it.
Socrates: You’re kind of right. Being, by definition, is. Non-Being, by definition, is not. Very true but also not very exciting.
Toilet: On that note, I remembered hearing my friend Albert once talking to himself on a cigarette break “but Truth is sterile”*.
Socrates: What an interesting point, Toilet! But I am not sure what exactly your friend meant. Maybe we can invite him over for coffee sometime.
Toilet: For sure. I will call him later. But for now, I’m just curious what you and Parmenides can make out of this sterile topic.
Socrates: You see, maybe like what your friend Albert had said, Truth is sterile for its completeness and consistency, like the pair of statements “All that is is, and all that is not is not”. If Truth is complete, it does not branch outside of the system. If Truth is consistent, you cannot unsettle it within the system.
Toilet: So something that never grows out of itself nor change within is essentially stationary.
Socrates: Precisely. And that may be what your friend meant by “sterile”. Well, on the other hand, being sterile doesn’t seem to undermine its power.
Toilet: How so? What is this “power” you’re talking about?
Socrates: By “power” I meant whether a system can be used to describe all that can ever be described, if not, how much it can describe.
Toilet: So how would you score the system “Being is; Non-Being cannot be” in terms of its power?
Socrates: All that can be described is, even if that existence is only confined to the domain of description, or thought, or imagination, or whatever. That very thought or image is, whether or not the same object has additional copies outside of the domain of thinking and imagination.
Toilet: You mean if I say “a five-ear cat”, even if it’s a vague image I had in mind that would never be corroborated in any zoo, that very image still is, and I cannot say it is not.
Socrates: That’s not a bad example… So if all that can be described is, all what we ever talk about partake Being. Then “Being is; Non-Being cannot be” applies to anything describable and even those that is but not describable.
Toilet: In that sense the system is very powerful… But then that’s it. I don’t know what else anyone could talk more about it.
Socrates: Ah. My friend Parmenides and I were talking about this little book he wrote called “On Nature”. What you and me have been talking about reminds me of a segment in the book called “The Way of Truth”, which may not generate a lot of conversation. Luckily, there is another segment called “The Way of Opinion”, which is more colorful but also, in his word, “deceptive”.
Toilet: Oh? What is that about?
Socrates: To have a more colorful description of things, you start with dichotomizing phenomena, like, day and night, women and men, heaven and earth, past and future. You can study one part at a time and create more divisions. But then you have to put them back together through systems of statements in terms of the divisions you so created… which could actually be the fun part.
Toilet: In the way you put it, it sounds like some brainy game one play with oneself.
Socrates: In a way, it is. You can put one set of divisions as the bases with which you construct a space, where you can chart the other set of divisions. Through this process, one may understand the relations between all the divisions that oneself makes.
Toilet: Now it sounds more like mathematics to me. Hmm… especially it reminds me of group theory. Like there is a group whose elements are some transformations. This group can now act on Being, where invariant subsets or partitions of Being will emerge under this group of transformations. Then by studying the relationships between the invariant subsets, one may understand the structure of the group of transformations.
Socrates: That sounds like a reasonable association to me! If I follow your analogy, that the action of a group on Being is responsible for the divisions, I am wondering what mechanism you would propose to handle the part where a system of statements needs to stick them back together. Would the same group also be responsible for that?
Toilet: Well that could be difficult. For one thing, action of the same group can never take what is in one division of Being outside of that division. Because we first defined the divisions as subsets of Being who themselves are invariant of the group’s action.
Socrates: So they are still kind of sterile?
Toilet: One may say so! I guess we need additional transformations that are not in the group to make connections across the divisions. In fact, I may need a whole other group to act on Being, so it can move stuff across the divisions made by the original group. Let’s say the former group A, acts on Being and creates invariant subsets {Beinga1, Beinga2, Beinga3, …}. The later group B must have some element, that is, a transformation, which will take some stuff from Beinga1 and map it into, say, Beinga3. Then we have virtually a statement about the relation between Beinga1 and Beinga3.
Socrates: That could work. But you need to make sure this group B would manage to connect every pair of partitions Beingai , Beingaj in a way that reflect the system of relational statements one choose to describe Being. That sounds like a lot of work to find the right group B!
Toilet: It could get messy. But it will worth it. You will have a much more colorful description of Being, than “Being is; Non-Being cannot be”!
Socrates: That sure sounds promising. But as what we have done for the “Being is; Non-Being cannot be”, we need to talk about the power of group A-group B system you are proposing here.
Toilet: I’m more like entertaining the, let’s say, two-group system than proposing anything. But anyway, if group B were to be powerful enough, it must be able to represent all descriptions of Being in terms of partitions induced by group A. Then we probably need to say there is a group C which is the product of two subgroups A and B, and when group C act on Being, there is only one invariant subset of Being that is Being itself.
Socrates: Would the action of group C be asserting that “Being is”?
Toilet: Sure. In some way. But now group C is a much richer system than “Being is; Non-Being cannot be”. Is it not?
Socrates: You would certainly have more to talk about with group C, which has two subgroups A and B. Subgroup A can provide the divisions of Being, and subgroup B provides relational descriptions of Being in terms divisions resulted from the action of A.
Toilet: That encapsulates what we have been talking about.
Socrates: But would subgroup B, acting by itself, also be able to create divisions of Being? Like {Beingb1, Beingb2, Beingb3, …}.
Toilet: Hmm… I can’t see why not. In that case, subgroup A will be getting the job of describing Being in terms of Beingb1, Beingb2, Beingb3, … what matters now is that the product of subgroup B and subgroup A, group C, still leaves Being in one piece.
Socrates: So the divisions can be the relations of relations, and relations themselves can be divisions. It’s all up to interpretation except the part that they have to work with each other to preserve Being.
Toilet: Sounds about right.
Socrates: Back to where subgroup A creates divisions, and B relations among the divisions, the remaining question that’s bothering me is whether you can always find a subgroup B to stick all the pieces back together.
Toilet: I think you can always specify a subgroup B for subgroup A. But you may have to be patient enough in some cases.
Socrates: What do you mean?
Toilet: Well, if the divisions of Being created by a particular subgroup A happen to be infinite, you may never finish specifying subgroup B!
Socrates: Ah, that could be exhausting. But what if subgroup B can specify itself without laboring you, once you give it some initial elements? Would you consider subgroup B specified?
Toilet: You mean to find the generators of subgroup B?
Socrates: Yes, or say, from a few relational statements about Beinga1, Beinga2, Beinga3, … one may be able to derive all possible statements or descriptions of Being.
Toilet: I’ll say that would really depend on what kind of subgroup B really is – how its product is defined. In other words, how do you generate a new element of subgroup B from two old elements of subgroup B.
Socrates: So if I were given an arbitrary set of, say, “rules of generation” for subgroup B, I may or may not find a manageable set of initial elements to represent B, which will keep Being in one pieces. Is that what you have said?
Toilet: Yes, you got my point.
Socrates: Then we just have to find a right set of rules of generation together with some initial elements. Wouldn’t that solve all the problems?
Toilet: That sounds reasonable but sometimes people set up some special rules just to make the game more interesting. For example, they forbid certain elements, that is, transformations to be present in subgroup B.
Socrates: Why would someone do that?
Toilet: That has to do with a special transformation in subgroup A that creates new divisions named “is; is not”.
Socrates: What?! Haven’t we talked about something like “what is not cannot be”? Now are you saying that Non-Being gets represented inside of Being?
Toilet: Look Socrates, you need to relax. We are not talking about Being and Non-Being here. We are only talking about names or labels inside of some divisions of Being, like “is woman; is not woman”, “is day, is not day”, “is future; is not future” etc.
Socrates: You mean they are just names in the sense that “is not future” still is, hence partakes Being.
Toilet: Right!
Socrates: Wouldn’t that cause a whole lotta confusions!?
Toilet: In fact it might. People may want to transfer the rules of consistency for “Being is; Non-Being is not” system, to “is; is not” systems.
Socrates: Rules of consistency? You mean if “Being is” and “Being is not” cannot both be true, nor can “Non-Being is not” and “Non-Being is”.
Toilet: Or say you cannot derive “Being is not” from “Being”, but you can derive it from “Non-Being is”. That is to be consistent.
Socrates: Sure. Now if you transfer that to the subgroup AB system…
Toilet: It would mean that no transformations or elements in subgroup B, when acting on Being, should generate both statements “x is y” and “x is not y” at the same time, directly or indirectly.
Socrates: That certainly makes the game “find subgroups A & B” a lot more challenging and interesting. But wouldn’t that leave you half of Being isolated from the other half of Being under the action of group C, the product of A and B? In other words, group C can no longer make Being the only invariant subset while acting on it, but cut it half and half. That seems to make the ultimate goal of the game unattainable.
Toilet: Oh, that’d be worse than half and half, at least according to my friend Kurt.
Socrates: Oh? You have a lot of interesting friends.
Toilet: Who knows! People just come up with interesting ideas when they are around me.
Socrates: Ha! So what did he say?
Toilet: He said, if you want the consistency rule, you will also have infinite amount of fragments of Being that are not reachable from the two main partitions where one is the other’s negation, by any mechanically generable subgroup B that has enough power to describe Being.
Socrates: I think you’ve lost me.
Toilet: I meant that there would be orphan divisions that cannot be reached by the “is” or “is not” camps of Being under the action of group C, if subgroup B is powerful enough to describe complex relations.
Socrates: I’m not understanding how powerful enough would be enough. What do you mean by complex relations?
Toilet: In the sense that the initial elements and the rules of generation of subgroup B are recursively definable. Also that subgroup B must be able to produce all recursive relations among the divisions of Being.
Socrates: Now what are the recursive things?
Toilet: According to my friend Kurt, you can simply think of them in terms of natural numbers. If you can map something to some natural numbers, then they are recursively definable. Recursive relations are like relations between numbers, like, arithmetical relations. Or say, subgroup B itself could be a bunch of arithmetical transformations look-alike and at the same time natural number look-alike.
Socrates: So you’re saying if we can map subgroup B and the divisions Beinga1, Beinga2, Beinga3, … somehow to some natural numbers, with their arithmetical relations, we will end up with some orphan divisions in Being.
Toilet: That’s the idea!
Socrates: But what are these orphan divisions? What do they represent?
Toilet: Well, you may simply name them “is not transformed from other parts of Being”, which also happens to map to itself under the action of subgroup B.
Socrates: That is very strange!
Toilet: But you can’t say the naming is not legit!
Socrates: So these special fragments are invariant under the action of both subgroup A and subgroup B. They are rather intact pieces of Being.
Toilet: That sounds rather poetic.
Socrates: Anyway, I’m not sure I like all the fragmentation. This consistency rule seems to be creating a lot of trouble. What if we get rid of it?
Toilet: Then all divisions would be reachable. And group C, that is the product of subgroup A and B, can again assert “Being is”. But then you also lose all the fun.
Socrates: Looks like one has to trade between consistency and the power to speaking about Being in completeness. Or say, one has trade between fun and said completeness.
Toilet: That was Kurt’s finding, which he wrote a paper about**.
Socrates: Sounds like a interesting guy! I will have to meet him. Anyways, we’ve gone pretty far on this two-subgroup game of yours. Now tell me whether these groups one may construct are.
Toilet: Hmm. Well if we can construct them, they must be.
Socrates: Then these groups are divisions of Being, which happen to be resulted from their own action on Being, including themselves. Or can they do that?
Toilet: Oh dear Socrates, that’s gonna be a whole other conversation. I’m ready to call it a day.
Socrates: Very well my friend. Until next time.
Toilet: Bye bye then.
(Socrates flushed the Toilet and washed his hands. Toilet was contemplating on whether the algebra of groups may simply not be powerful enough for its analogy. That concludes the “describe-Being game” that was played and recorded by Toilet and Socrates. )
* From the Appendix of “The Myth of Sisyphus” by Albert Camus: “…For a truth also, by its very definition, is sterile. All facts are. In a world where everything is given and nothing is explained, the fecundity of a value or of a metaphysic is a notion devoid of meaning.”
** Kurt Gödel. “On formally undecidable propositions of Principia Mathematica and related systems”. (1931)
Toilet Socrates 