We travelled upstream against the flow. From the sea, we entered the delta and moved forward, always struggling, pushing against the current. The river branched and we chose to follow it west, travelling with no destination in mind, always pitting ourselves against the strongest flow. It branched again and again, until we faced a mountain, a small rapid current running down its slope. We swung our oars with all our strength but the water was too strong for us and we could make little headway. Stepping out, we turned our oars to spades and started digging a new channel. It would take years, but in time a new stream would flow, with a milder slope, and boats will run along it as far as they can.
I think what we could really use sometimes is a comprehensive map of all human knowledge. Not a map of what that knowledge is, mind you, but rather a map of how its bits and pieces are related. While this is a daunting and implausible task, I think it would not be unreasonable to consider how such a thing could be constructed.
I will primarily first focus on knowledge of mathematics, since it's the clearest of all fields of study, and extend the system to other fields by comparison. In essence, I believe that knowledge is intimately tied to its acquisition, or, more precisely, I would claim that it is tied together by its acquisition. We can consider each little bit of knowledge, a theorem or a lemma say, as a point on a giant field. Then, in our construction, we will have an arrow direct us from one point to one or more others if you can learn the next point with some study once you know this one. Of course, most new bits of knowledge require combinations of several previous bits, so let's say one colour will be one possible combination, and another colour will be a different one.
As an example, we might be able to learn integration if we understand infinite sums, areas of rectangles, and graphs. Then, we'd have arrows from those things pointing towards integration, and we might colour these arrows red. Then we might alternatively be able to learn, at least indefinite, integration as the reversal of differentiation, so we might have a green arrow pointing from differentiation to integration. To avoid cyclical definitions, however, we should specify that while there can also be an arrow pointing from integration to differentiation, that arrow would have to be red, not green, to signify that you can't learn integration and differentiation from each other without invoking anything else. In fact, if you consider this entire system, you will notice that there can be no cycles – that is, you can't have A learnt from B, which is learnt from C, which is learnt in turn from A. Having such a cycle would mean that you're using circular logic somewhere, which would automatically render such a bit of knowledge invalid unless it has an alternative source that does not involve itself. What you can have, however, is a bit of knowledge that is based on nothing. The only such bits, however, would be the axioms of mathematics, and equivalent things in other fields.
Next comes perhaps the most important step. We can add weights to the different colours of arrow, with the weight signifying how long it would take to acquire the new bit of knowledge from the previous bits if you follow this route. These could be empirically estimated by having a few people actually learn the information, although these would naturally never go past estimates because of people's varying aptitudes towards different fields or approaches to learning. Nonetheless, with these weights in hand, we can do many things, some merely interesting, some genuinely useful.
On the useful side, we can find the shortest path to learning each bit of knowledge and thus plan out the most efficient route of learning if we have some goal. This could be extremely relevant for empirically designing courses of all levels. For example, bits of knowledge that can all be learned from the same other bits can be grouped into a course or, more widely, a field of study. As another example, knowledge that does not have many requirements but leads to many other possible bits would make good introductory courses.
On the interesting side, it is not uncommon to describe a field of study as "broad" or "deep", and these can now be assigned real definitions. A deep field is one which has a very high weight on at least one bit of knowledge. For example, if it takes a decade of study to learn to grasp some part of string theory this can be considered a very deep field. Conversely, if all you need to understand any particular bit of literature is to know the language and maybe read some commentary, and it only takes a few years in sum, this field might be relatively shallow. A broad field, on the other hand, is one in which the sum of the weights of the deepest (this restriction is needed for the sum to not depend on the number of intermediaries) bits of knowledge is relatively large. For example, if in physics once you knew string theory you knew all physics (wouldn't that be nice?), the field would be relatively narrow. Conversely in literature you may have to take the sum of the weights of all books ever written and all commentary ever written on them for this metric, and this would make it an extremely broad field. Thus, we would have some interesting empirical measures. Others could be made up, perhaps something can be designated "trivia" if it takes very little to learn and isn't a prerequisite for much else, and something might be "easy" if the weights of the arrows to it from its predecessors is low, and "difficult" if the weight is high.
There are wide individual applications as well. A person might track how far she is on the graph, and compare how much of the same knowledge she and a friend possess and how much of his knowledge complements hers. To efficiently solve some problem, you might assign several people who all have the bits of knowledge that are absolutely essential to the problem, and have very little knowledge in common outside of those, so they can each bring a different perspective or focus on a different aspect. For bragging rights you might take the sums of everything you've learned and see if you're more knowledgeable than someone else. We can also see that someone with very broad knowledge would be a good generalist, and perhaps conversationalist, someone with very deep knowledge a good specialist, and perhaps would be suitable for being a scientist or other researcher.
On the whole, putting knowledge in a clear structure such as this would be highly advantageous for both understanding it and making any decisions that involve it in an way. From education to employment decisions to conversation or philosophy, a system such as this could facilitate many things, especially as it itself would not need a high weight to learn.
No comments:
Post a Comment