The Tar Pit

The blog is a device meant for holding long strings of pointless platitudes that one may review days, months or years later on in his or her life, for his or her amusement, for self-evaluation, or for whatever else. In other words, the blog is, first and foremost among all things, uses and functions, a tool for reflection.

I review my blogs (both of them) periodically. And upon my latest review of my old one, I noticed that... well, sadly enough, there's not much there worth mentioning. Among the multitude of reviews of rock bands, concerts and albums, among the various Facebook-style postings recounting the "événement du jour", way before Facebook was fashionable -- I started blogging all the way back in 2005, mmkay? Among all these, there lie sprinkled the occasional more or less interesting crypto-philomosophical ramblings. Well, among these, very rarely I stumble upon a piece that speaks to me, my future (that is, present, depending from where you look) self, saying: "dude, I don't really know what I'm saying, but I sincerely hope you'll get this the next time you read it".

In 2009, that is, almost a decade and a half ago, I was for some unexplainable reason still studying graph theory. I don't precisely remember that autumn, much like I don't remember the one before or the one after. Maybe nothing of import happened in any of these autumns, which perhaps explains the bland content of said blog... regardless, I was a free kid, lacking the pressures of adult life and as it happens, I chose to spend my life perambulating some of the intellectual realms of discrete mathematics. And on my walk, I decided to revisit an old expression, which led me to an article which spoke to me; thus I will translate it in the following paragraphs.

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This saying, that all roads lead to Rome, has been heard by just about every individual once or twice during his lifetime¹. Apparently this expression doesn't say much: at some point in the past, Rome was the Capital, the navel, the centre and everyone aspired towards Rome, just like today's youngsters aim for Bucharest, emptying entire localities. However, if we strive for a bit we may find deeper meanings to this; such as, for example, that an objective may be achieved in multiple ways: some shorter, albeit not necessarily the best, while others more indirect, but more certain.

From a mathematical standpoint -- after all, a formal description might² help us to form an opinion on the statement's validity -- things aren't that complicated. We leave aside Europe's topology in the years 0 to 100 (while the Empire was still in full development³). Let G(V,E) be the graph of the Roman Empire and let r in G be the node corresponding to Rome. While it seemingly doesn't matter too much if G is oriented or otherwise (after all, roads those times were travelled both ways -- in the actual Rome, not in our abstract, mathematical one), we will take our graph to be oriented and weakly connected, looking through the lens of our statement at the beginning: "all roads lead to Rome", thus "all roads end with Rome⁴". We can express this thusly:

For any u, v in G, and v in TG, then v is r, or otherwise: TG = {r}, where TG is the set of terminal vertices (vertices whose set of successors is the null set) in G.

This leads us to some interesting observations. The first observation would be that, intuitively, r is necessarily an articulation point⁵. In fact we could say this from the beginning, as we knew that r plays an important role in our graph. Furthermore, we can find a second, more profound interpretation of G: quite possibly our graph forms a sort of star topology⁶, with r playing a central role. Moreover, by overlaying a dynamic element (say, a carriage pulled by horses) on top of G, we may impudently state that Rome is a black hole from which nothing, not even light can escape. Of course, it's also possible that my premises are entirely erroneous.

To sum this up, I'd say that the Romans kinda deserved their fate.

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Of course, meanwhile I've visited the underlying mechanisms behind the so-called network effects countless times; and their fractal nature, and their feedback loops -- some indeed prefer to view the Imperial star topology as purely controlled from the centre, when in fact the arrow goes both ways, and in varying degrees of intensity, which is why ultimately Empires dissolve in the worst manner possible, the worse as they are more connected to the centre. What transpired from my old article was merely a hint, but a hint of many things, among which: the systematic misrepresentation of the arrow of causality.

This problem stems from some of the most trivial observations⁷ in life. Say, for example you are doing some work, whether watering some plants, writing some code, jogging, or whatever. This causes something, such as the completion of a task, or the increase of humidity in some place or whatever. This, in purely physical terms, changes the state of your environment from a p1 to a p2, where p is some particular measure that may be used to compute entropy. More importantly yet, that so-called "physical" environment includes you, from which emerges the trivial observation that whatever action you apply upon your external world, "it" isn't changed by your action more than "it changes" you⁸. The arrow of causation, however consistent in a purely Platonic sense, suddenly becomes muddy when it hits the real world.

Or, as a very wise person used to say to me that one day which I remember with crystal clarity, but which I never sat down to recount: you gotta give some, but you also gotta take some. Mother Nature, it abhors singularities.

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