User:Najawin/Sandbox 10

Category Theory is a Lie and Alexander Grothendieck Set Mathematics Back 100 Years
So perhaps I'm just too set in my ways, but I don't fully understand Scrooge's response about linear time invalidity for contradiction. His argument doesn't seem even slightly analogous to me. Let me explain why using some handy dandy commutative continuity diagrams. Here's the terminology you need to understand these diagrams, okay? "VN", where N is a number means that this is the Nth "Valid" story we're considering for the diagram, "IN" means this is the Nth "Invalid" story, "V1->V2" means something like "V1 informs the continuity of V2", or "V2 is trying to be in continuity with V1". "V1-/->V2" means the negation of that, and "V1<->V2" and "V1<-/->V2" means that we erase the time dependence of our previous relations.

So back in the days of yore, we had the following two stories, $$ \begin{array}{lcl} & V1, & I1\\ \end{array} $$. You might ask why I'm not including an arrow here. Because, frankly, my argument has no need for it. V1 and I1 might have related continuities, they might not. (EG: It's hard to deny that Shalka is continuous from the classic series.) My argument does not assume that invalid stories and valid stories are intentionally discontinuous. Indeed, it considers the entire issue a red herring. Now, given these two stories, we can consider another story that references the two of them. $$ \begin{array}{lcl} & V1 & \rightarrow I2 & \leftarrow & I1\\ \end{array} $$. Now, I2 was made invalid, and this, crucially, was because of the connected arrows, because of a pattern present in this diagram, namely, $$ \begin{array}{lcl} & I1 & \rightarrow & ? & \Rightarrow & I1 & \rightarrow & I2\\ \end{array} $$. The sequels/prequels to invalid stories thread changed this. Now, depending on factors other than patterns in these diagrams, both $$ \begin{array}{lcl} & V1 & \rightarrow I2 & \leftarrow & I1\\ \end{array} $$ and $$ \begin{array}{lcl} & V1 & \rightarrow V2 & \leftarrow & I1\\ \end{array} $$ can exist. The R4bp thread concluded with the idea that considering diagrams of the form, $$ \begin{array}{lcl} & V1 & \rightarrow V2 & \leftarrow & I1\\ \end{array} $$ we can replace them with $$ \begin{array}{lcl} & V1 & \rightarrow V2 & \leftarrow & V3\\ \end{array} $$. This is asymmetric. Scrooge suggests the issue is one of looking at the same graph while removing the time element. I wish to submit that this is clearly false. If we're considering diagrams of the form $$ \begin{array}{lcl} & V1 & \rightarrow \quad ? & \leftarrow & I1\\ \end{array} $$ and simply deciding whether "?" is to be valid or invalid, I don't see how changing this diagram to $$ \begin{array}{lcl} & V1 & \leftrightarrow \quad ? & \leftrightarrow & I1\\ \end{array} $$ changes the calculation one whit. (Indeed, it's arguably because of this change that my argument works! We need there to be a symmetry between going from $$ \begin{array}{lcl} & V1 & \leftarrow & I1 & \Rightarrow & V1 & \leftarrow & V2\\ \end{array} $$ to $$ \begin{array}{lcl} & ? & \leftarrow & I1 & \Rightarrow & I2 & \leftarrow & I1\\ \end{array} $$ and there just isn't without assuming time invariance. Otherwise my argument only works on diagrams of the form $$ \begin{array}{lcl} & V1 & \rightarrow I2 & \leftarrow & I1\\ \end{array} $$ which is a rather different argument.)

Perhaps the difference will become more apparent if we chart out his proposed counter example and why I don't think they're similar?

Scrooge suggests we consider the following diagram instead: $$ \begin{array}{lcl} & V2 & \nrightarrow & V3 \\ & \uparrow & \nearrow \\ & V1 \\ \end{array} $$, or, if you prefer, $$ \begin{array}{lcl} & V2 & \nrightarrow & ? \\ & \uparrow & \nearrow \\ & V1 \\ \end{array} $$. And this is clearly non analogous, right? I mean, even if you insist that my hypothetical has to populate the continuity relationship between V1 and I1, a view I straightforwardly reject, even if you apply time invariance to this graph as well (bear with me, this is actually the one issue here, diagonal arrows can't do this on mediawiki, so I'm going to turn it into a box diagram) $$ \begin{array}{lcl} & V2 & \nleftrightarrow & ? \\ & \updownarrow && \updownarrow \\ & V1 & = & V1 \\ \end{array} $$, in this analogy, there's still a giant hole - that both stories already propagating the diagram are already valid, and the "non continuous" arrow is a relationship between one of them and the new story we're considering, not between the previous stories that already are present in the diagram.

Now, one might argue, perhaps we shouldn't privilege past states of the diagram over future ones. That if we fill in those continuity arrows between I1 and V1 from my example we get $$ \begin{array}{lcl} & V1 & \nrightarrow & I1 \\ & \downarrow & \swarrow \\ & ? \\ \end{array} $$ and then adding time invariance to this diagram gives us $$ \begin{array}{lcl} & V1 & \nleftrightarrow & I1 \\ & \updownarrow && \updownarrow \\ & ? & = & ? \\ \end{array} $$ and then when we realize that in his example we clearly agree with validity for the question mark, so we do so here as well, and then this back propagates to I1. So $$ \begin{array}{lcl} & V1 & \nleftrightarrow & V3 \\ & \updownarrow && \updownarrow \\ & V2 & = & V2 \\ \end{array} $$. Potentially then all structures of the form $$ \begin{array}{lcl} & X & \nleftrightarrow & X \\ & \updownarrow && \updownarrow \\ & X & = & X \\ \end{array} $$ are either all valid or invalid?

Perhaps this is what Scrooge is suggesting all along? It's not time invariance wrt media, it's time invariance wrt the decisions we make about these pieces of media. I don't really think he is? I'm just trying to steelman this argument. I don't really see a reading in which the two are analogous otherwise. Like - it's got some weird implications for how we conduct validity debates at the very least. At this point, not being a category theorist, and that's probably the best way to think about this, I think I have to beg off discussion ever so slightly, but my intuition is that this is precisely the reductive version of R4 Scrooge has suggested and people have not largely been fans of. It's that instead of springboarding off of Unearthly Child our web springboards off of every piece of DWU media. That's my guess. (I think this is the first time I've ever wished Amorkuz was here - I think he was an algebraist?)