User:Najawin/Sandbox 10

God I feel bad for anyone reading this at this point

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?

${\begin{array}{lcl}&V1&{\overset {f}{\rightarrow }}&Z&\\&g\downarrow &&\downarrow g'\\&Y&{\underset {f'}{\rightarrow }}&W&\\\end{array}}$