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Sunday, July 21, 2013

Conceptualize This

This is the first part of a series on concepts.  In this series, I will talk about the way in which people use and misuse concepts, as well as a few things you can do to help aid in the appropriate use of your concepts.  This series is ongoing (so stay tuned for more!)  To start, I’m going to outline what I mean by ‘concept’ and why it is we have them.

Let’s suppose you have a nice enormous bin of gems, all of which come in two colors (yellow or blue), two shapes (square or triangular), and two types of cores (diamond or platinum).  After analyzing some of the gems with your mathematically knowledgeable friends, you determine the following correlations:

99% of the yellow gems are square
97% of the blue gems are triangular
93% of the yellow gems contain diamond cores
94% of the blue gems contain platinum cores

Here is a graphical representation of the correlations between the various properties of the gems.  It is a representation which takes into account ALL of the information gained by you and your intelligent friends.  This means ALL of the information contained in the simple statements of the correlations above, as well as some more that I left out in the “etc.” portion for brevity.

Here, each line connecting two traits represents the likelihood of finding the traits paired together, with thicker lines indicating higher likelihood.  For instance, the fact that the line between “Blue” and “Triangle” is very thick indicates that a blue gem is very likely to also be triangular (and, for simplicity, vice versa).  Meanwhile, the thinness of the line between “Blue” and “Diamond” indicates that a blue gem is very unlikely to contain a diamond core.

What is the usefulness of this map?  It lets us make predictions.  If you tell me that you have a gem which is both yellow and square, I can predict with high certainty that the gem will contain diamond.  In fact, if you only tell me that a particular gem is blue, then I can still have a high certainty that it is triangular and that it contains platinum.  In this way, I can get a very good idea of what a gem contains without having to break it.

But the diagram above is not what goes on inside your brain.  That diagram is just too complicated to be worth it.  What goes on in your brain, and indeed in much of human dialogue, is something like this:

What we do is create concepts, labels, like “YSD-Gem” or “BTP-Gem” to group the various properties.  When we notice that most of the gems are either yellow-and-square-and-diamond or blue-and-triangular-and-platinum, we generate these two groups and give them different names.

Now this map is both less accurate and less precise than the previous map.  However, it is much easier to remember.  There are six lines – six correlations – instead of twelve.  Moreover, the map has been completely split into two separate pieces, which can each be utilized separately.  So any individual application of our map is only going to really utilize three of the lines.  This makes it much easier to use.

But notice the difference between what goes on when we make predictions with the two maps.  Let’s say you told me that you were holding a yellow gem, and asked me to predict whether the gem contained diamond or platinum, and also whether it was square or triangular.  With the first map, I compare the line thickness between “Yellow” and “Diamond” to the line thickness between “Yellow” and “Platinum.”  Since the first is thicker, I guess that the gem contains diamond.  Then to answer the second question, I compare the thickness between “Yellow” and “Square” to the thickness between “Yellow” and “Triangle.”  Since the first is thicker, I guess that the gem is square.  Note that it requires me to look at four different lines, and apply two different comparisons, to answer your questions.

Answering the questions using the second map is easier.  When you tell me that the gem is yellow, I look at the line from “Yellow” to “YSD-Gem” and conclude that the gem is a YSD-Gem.  Then I look at the two lines coming from “YSD-Gem” and conclude that the gem is square and contains diamond.  With this method, I only need to look at three lines and I don’t even need to make any comparisons between lines.  The second map gets you your predictions faster and with less effort.

And the important thing is that this discrepancy scales.  If you there were to add a fourth property of our gems, say weight, in which nearly all yellow and square and diamond-containing gems were light and nearly all blue and triangular and platinum-containing gems were heavy, then the second diagram would only need two additional lines – one from “YSD-Gem” to “Light” and another from “BTP-Gem” to “Heavy.”  The first diagram, however, would need twelve additional lines – one from “Light” to each of the six non-weight properties, and another from “Heavy” to each of the six non-weight properties.  Furthermore, taking one property and guessing all three other properties with the second map would require looking at four different lines and making zero comparisons.  But taking one property and guessing all three other properties with the first map would require looking at six different lines and making three comparisons.

In a more extreme scenario where we have five hundred different properties, the discrepancy is quite large.  For five hundred binary properties, a map of the first type would require nearly 250,000 lines.  But a map of the second type requires only 1,000 lines.  Moreover, a map of the second type allows us to use one property to predict the other 499 properties by looking only at 500 lines and making zero comparisons.  To make such predictions with the first map would require looking at 998 lines and making 499 comparisons.  Using the second map is way easier.  That’s why we use it!

But keep in mind that the second map does not contain all of the relevant information.  All of the relevant information requires twelve lines to represent, and the second map only has six lines.  In fact, the second map does not explicitly contain any of the relevant information – there are no lines that connect the various attributes together.  The second map does not explicitly contain information about the likelihood of a yellow gem being square (in other words, your guesses don’t rely on memorizing the fact that 99% of yellow gems are square).  Instead, a yellow gem is labeled as a YSD-Gem, and then concluded to be square or at least probably square.  But by utilizing the concept of “YSD-Gem,” the second map allows us to guess many properties very quickly and easily.

But despite this, “YSD-Gem” is not a property of the gems (at least, not in the same sense that yellow, square, and diamond-containing are properties of the gems).  This is the mistake that Plato made, with his notion of Platonic ideals.  The “ideal” of a YSD-Gem is something we came up with, to help us make predictions more easily.  But you can express all of the relevant correlations between properties without ever referring to this “YSD-Gem” concept (as the first map does).  When it comes to determining what properties the gems have, it can be useful to use the concept “YSD-Gem,” but it is not strictly necessary.  If we have enough processing power and enough memory, we can instead use the first map to make our predictions.  In this sense, “YSD-Gem” is a conceptual tool that we utilize to help make our lives easier.  It helps us represent tendencies in the world, but it is still a tool.

And yet at the same time, if you find a gem that that is yellow and square but contains platinum instead of diamond, you will still find yourself asking whether it is a YSD-Gem or a BTP-Gem.  But these terms don’t correspond to the same kind of thing as yellow, square, and ‘contains platinum.’  With this particular gem, all of the observables have been determined.  You know that it’s yellow, you know that it’s square, and you know that it contains platinum.  And yet at the same time it feels like there’s an important unanswered question: Is it YSD or BTP?

And two people can disagree about the “right answer” to this important-feeling question even if they agree about all the observables.  The problem is that your mind is so used to using the second map that it doesn’t realize that YSD and BTP are just tools – not observables – and that these tools are there specifically to help predict observables.  Thus, your mind does not realize that once all your observables have been measured, asking “YSD or BTP?” is utterly pointless.  It’s a question about which tool is best to use to achieve something you already have.  And yet, because it’s still one of those nodes on your cognitive map (map 2), it feels like “YSD or BTP?” is just as meaningful and important a question as “Yellow or Blue?”

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