#fixed points

Unstable fixed points are the anxious attachments. Will crash out if you don’t text them back. Expect λ > 0 messages sent to your phone. A bit chaotic if you ask me.

    I have come before you to argue that The Assassination of President Kennedy resulted in the Casting of British Actress Alex Kingston as Professor River Song in the hit BBC television show "Doctor Who".

    On November 22, 1963, the very first episode of Doctor Who was set to hit International Television. However, coincidentally the American President at the time, John F Kennedy, was assassinated, delaying the airing by approximately 90 seconds, allowing the English populace to turn on their TVs, resulting in an amazing viewership for the Doctor Who Pilot. Of the two people watching were Bill and Noreen Moffat, and their two year old son, Steven.

    Because of this untimely exposure to the Sci-Fi show, Steven Moffat would grow up to be a fan of the show, his appreciation culminating when he was hired by BBC in 2003 to begin writing for the show itself.

Sitting in a cafe, a woman stirs her drink with a spoon. The liquid spins around the center slowing to a stop. Inside of that closed system of liquid exists a point where the drink remains unchanged. Sound impossible to predict? That is the implication of Brouwer Fixed Point Theorem which states "Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point." To paraphrase, two spaces that are closed systems which also map onto each other one to one will share fixed points. A more obvious example is a map of the world existing on the planet. Even if you crumple or fold the map, there will always be a point on the paper that matches with the real world it represents. (There are a few caveats for this to work, like no cutting the paper or having holes in the map but for this discussion they aren't immediately relevant.)

How are fixed points relevant to the study of deductionism? I would like to make the argument that fixed points are the entire basis of deduction. In fact you can not properly articulate a working definition of Holmesian deduction without them. If you recall the stirred beverage from before, those are two closed systems that map onto each other one to one. Two? No its just one drink. In fact they are two separate spaces, the drink before the transformation and the drink afterwards. Describing the drink's states as events might make it's relevance to deduction more apparent.

This could apply for, say, a murder or a theft occurring in a mansion. The grounds of the estate, if no parts left or entered the system during the transformation, could be a closed system. What counts as a part of the system? People, murder weapons, objects, passageways, any participant in a subsystem, and anything relevant to the "transformation" between the two events we care about. Do you see how useful this framework could be?

To bring this back to a working definition, I'm going to define Holmesian deduction (or deductionism) as: The application of logic to study an observed event K0 to learn the circumstances of another unobserved event K1 using data points Xn with the goal of understanding the transformation between said events. Traditionally in topology fixed points are represented by X. I would've preferred P for points in my definition but we need that variable for the logic section.

What does this look like in practice? Let's take the scene from "A Study In Pink" from the previous post. A woman lies dead. This is event K0. She was previously in another city alive. That is event K1. Holmes studies the fixed points, i.e.: her ring, coat, umbrella, nails, temperature, outfit, etc. Some of these points changed during the transformation between K1 and K0, some did not. The umbrella did not change, so Holmes was able to deduce new data about the transformation, leading to her itinerary.

Listing these points as X1, X2, X3, and so on, we would have a complete list of every data point available. This allows us to know precisely what we have access to and reexamine the evidence from the same pool of information each time. Through this point based framework we can treat the space and time between the two events as a transformation. With every point accounted for, the solution becomes a matter of paring down impossibilities. I'll go into all of the above over future posts.

To recap, am I saying all of deductionism is rooted in this particular fixed point theorem all of the time? No. We are not always gifted with closed systems (and it would be poor form to train as to rely on perfect situations). However, fixed points are core to deductionism. Brouwer Fixed Point Theorem is widely applicable and is foundational to a lot of topology, so it's a useful starting point. Brouwer in deductionism is a boat to help you understand how deduction relies on fixed points. At some point you reach the other side and have to leave the boat behind. If you want to understand Brouwer Fixed Point Theorem in a broader sense, this video is good for a basic understanding of it's implications.

It’s only now that I’m attempting to write a KHR fic and looking up the timetravel thing in their universe that I realise:

The Vongola famiglia fucking defeated a fixed point in time.

A fixed point in time is when there is one thing that always happens in every timeline. For example, say slipping on a banana peel is a fixed point in time. So in every single universe, I will slip on that banana peel. No ifs and buts, in every possible time line I slip on that banana peel.

In every universe, Millefiore has won against the Vongola. The reason Byakuran has won in each and every universe is because of his ability to see parallel universes and using the information gained to win. Whether he knows it or not, he’s in the process of creating a fixed point of time where it is fixed that he wins.

However, it’s not completed yet. There’s one universe left, and he’s already partway conquering it when the current Vongola Guardians as we know them is summoned to the future. Which wouldn’t be so bad if they hadn’t brought forth a troublesome feature - the Vongola rings and the Pacifiers. Which was an instrumental part in the defeat of Millefiore, which means that this is no longer a fixed point in time because there is no longer only one possibility, which lets the other parallel universes change.

Which means that they shattered a fixed point of time in the making, and like. My mind is blown by this. What does this mean about the rings? Does this mean that the rings can mold time? What the even?