### The Trains

October 25, 2006 Wednesday The Trains I don't know why it looks extra wrinkly.

10:49 PM I'm getting faster at writing the papers. I'm getting clearly with my statements. I'm learning to do more with less time. It's astonishingly actually. I've only spent about 2 hours and am closer to writing this paper than 2 hours ever got be before. I recall the best moment of the day was when I sat down next to this female whom was playing the piano and got to listen. I was hypnotized by the sound. The enthralling melody played on me and sent waves of sensations through out my body. It was amazing. I recall wishing it wouldn't end in one of those moments. But of course the day moves on. The whole vacation that it was made the whole day better. Busy as I have been today... I've done enough of that paperish work to begin thinking about coming back to other things. Though I'm quickly remembering I have Software Engieering matters to tend to. And even biz matters to tend to. I realize I have to be in the proper state of mind in order to film a movie. Sort of, energetic and ready for anything. Yeah, only the completion of all immediate academic work brings me closer to that. That's why I keep waiting on things. I want the school work done before I begin other things. But if I'm burnt out already, then the school work gets pushed back and consequently the movie work does as well. But if I finish it all sooner than later. .. well then I can have some fun. I remember thinking earlier in the night about the encapsulation of mathematics... or more like lateral thinking in mathematics.Take the trains for example.Two trains start at two points on a track, and they have two different velocities. It's a classic problem. Perhaps, you have seen if before. Maybe, on a test or in a physics course or most likely an algebra and above sort of encounter. The thing about it is if you know something about the position function of an object from physics. x = x1 + v1*t + (1/2)a*t^2 you merely have to set up two of those equations (one for each train) and solve for whatever variable you like. The fact is that this model (equational model) describes that system perfectly. Well, as perfectly as an ideal case may be. And with algebra and two of these equations we can see when if and when the trains will hit. All the cases of train interaction are expressed given you can decode the equations and interpret them in real world terms. I believe I was in middle school in a car heading back home from a long car trip when I thought of using these equations to model the trains... Anyway, the thought that dawned on me much more recently. (after linear algebra) was that the same information in this train problem could be pictured completely differently. And express a completely different idea. Suppose, I merely take the number for position and the number for velocity. These are just two numbers. And two numbers make a point. So what is to stop me from plotting this 'train function' on a graph? Now, I'm no longer looking at an equation but where this particular equation fits in it's equation space. Thinking about it this way.. each point in the 2D coor system is a train. Or at least it represents what a train could be given it has some position and some velocity. Now I have two trains in this problem so I have two points on my graph. And suddenly the familiar train problem takes on a whole new wonder. Now, I can see that the number of possibilities of outcomes for these two trains is merely a characteristic of the relationship between these two points. So.. what happened mathematically? There was no algebra step that said, "Okay now just forget about that equation and plot the points" naw, it was merely stepping back and looking at the same situation with a different tool. I'd note that the reason I could do this is that the root information is conveyed.. only my choice of conveying is changed. I say root because in this representation I still need to know what I'm describing. (ie the equation is information too) I was quite excited when I realized this one day for I suddenly saw that I could apply a whole different field of mathematics to those old trains. And then I was left scribbling out results and pondering the consequences for hours. Lovely. But what do I call this sort of step? .... Encapsulation of Mathematical Ideas? ..... Lateral Heuristic? ...... Logical Perspective Change? Eh, who knows.

## 1 comment:

Dear Chris,

It appears that what you have done is a rediscovery of an object called "phase plane". If never encountered this construction before, you have made indeed a remarkable jump of imagination.

How did you do that? Well, it is a more interesting question. I would suggest that your discovery included the following three steps:

1. De-encapsulation of functional dependencies between the parameters in the problem.

2. Change of language (to that of linear or vector algebra), and

3. Re-encapsulation.

However, I would not insist on my explanation because mathematical thinking is a deeply personal matter.

The term "encapsulation" is quite useful for explaining a lot of phenomena of mathematical practice. However encapsulation works only because it is used on par with

de-encapsulation. The two activities should always be viewed together."De-encapsulate and change the language"is one of the routine heuristic rules of mathematical practice - however, it is almost never explicitely formulated. This why your post is so wonderful.I do apologise if I sound too categorical. I have to admit that, indeed, I have rather strong views on the wonderful subject of mathematical heuristics.

Alexandre Borovik

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