Knowledge Problems

Monday, August 13, 2007

The Fundamental Operation: Projection

To illustrate a further important point, below is an actual object-property table. It shows 10 objects and 5 attributes. You can think of it as representing 10 objects that just came off an assembly line. Each of those objects has 5 properties, for example `\{P_1,P_2,P_3,P_4,P_5\}=\{text{color},text{shape},text{texture},text{size},text{orientation}\}`. Thus, for example, these objects can have three possible colors, `P_1 in \{0,1,2\}`, which might be `\{text{blue},text{red},text{green}\}`. It can be noted that not all of the objects need be distinct, and in fact there are several identical objects in this set. (Can you find them? Just kidding.)


Object Property
  `P_1` `P_2 `P_3` `P_4` `P_5`
`O_1` 1 2 0 1 1
`O_2` 1 2 0 1 1
`O_3` 2 0 0 1 0
`O_4` 0 0 1 2 1
`O_5` 2 1 0 2 1
`O_6` 0 0 1 2 2
`O_7` 2 0 0 1 0
`O_8` 0 1 2 2 1
`O_9` 2 1 0 2 2
`O_{10}` 2 0 0 1 0


The crucial fact that must be observed is that the distinguishability of objects depends on which attributes are considered. For example, in the table above, we can already note that objects `\{O_1,O_2\}` are indistinguishable, as are `\{O_3,O_7,O_{10}\}`. The objects in these sets correspond to the same point in feature space, even if they are physically distinct objects (which we assume they are). However, if we were to further assume that attributes `\{P_1,P_2,P_3\}` are inaccessible, and were therefore able to consider only attributes `P_4` and `P_5`, we would observe (table below) that several other objects now also become indistinguishable, collapsing to the same point in feature space.


Object Property
  `P_4` `P_5`
`O_1` 1 1
`O_2` 1 1
`O_3` 1 0
`O_4` 2 1
`O_5` 2 1
`O_6` 2 2
`O_7` 1 0
`O_8` 2 1
`O_9` 2 2
`O_{10}` 1 0


In particular, we now find that we have the following sets of indistinguishable objects:

`{(\{O_1,O_2\}),(\{O_3,O_7,O_{10}\}),(\{O_4,O_5,O_8\}),(\{O_6,O_9\}):}`

So there are only four discriminable kinds of objects represented in this knowledge system after attributes `\{P_1,P_2,P_3\}` have been dropped and consideration has been restricted to attributes `\{P_4,P_5\}`. These four kinds are called "equivalence classes" because the entities contained in each class are indistinguishable (equivalent) based on the attributes under consideration. The process of "dropping attributes" is called projection and corresponds to an orthogonal (parallel to axes) geometric projection in feature space. In the previous case, we would say that the data is projected onto dimensions `\{P_4,P_5\}`. In general, the projection onto a given set of attributes will produce a collection of equivalence classes containing objects which cannot be disambiguated based on those attributes. This will be extremely important in what follows.

7 Comments:

  • nice

    By Blogger LakewoodShmuck, at Aug 28, 2007, 10:27:00 PM  

  • Curious to see where you go with this.

    Question though, it seems possible to know what the attributes are, without knowing what the object itself is.

    Especially if you postulate infinite properties with infinite variability.

    By Anonymous Daganev, at Aug 29, 2007, 9:40:00 PM  

  • I'm curious too. Not sure what you mean, though, about infinite properties, etc. In such a case, the properties still describe the object, it's just that they cannot be listed. Not a problem. The only commitment is that there is nothing "beyond" the properties, or rather that what lies beyond the properties cannot be the subject of knowledge. In any case, the point is that the knowledge we are talking about can be expressed as the "knowledge of properties". It's a minor commitment, since any theory of knowledge that we can construct will meet this criterion, and there is no sense (literally) in talking about theories of knowledge that we cannot construct.

    By Blogger Big-S Skeptic, at Aug 30, 2007, 8:44:00 AM  

  • Well, I'm just reminded of the movie the "gods must be crazy" and the soda bottle that lands in the middle of the tribe.

    The tribe finds all sorts of properties for the soda bottle, but they never come to the conclusion that its just a left over soda bottle.

    In our case, since there are an infinite amount of properties, and we may never know what all of them are, (since we are limited by our imagination) you can know all -about- the thing, without knowing what the thing itself is.

    By Anonymous Daganev, at Aug 30, 2007, 3:54:00 PM  

  • I loved that movie! I saw it as a teenager, and I laughed so hard I cried. I thought it was so great that I recommended it to a couple of my older teen friends. They hated it, and turned it off after 15 minutes. I was mortified, but I think at that age they needed to see guns and cars and stuff exploding. The humor in Gods Must be Crazy was somehow too challenging for them. Or maybe they were embarrassed by the naked black people. Anyway, that was the last time I recommended a movie to those guys!

    Well, I agree that the list of possible properties is infinite, but that doesn't mean that some finite subset doesn't capture all the information. It might be the case that after the first 10,000 properties, the infinite remainder are just rehashing the same information. I don't think the argument is going to turn on this point, however.

    I am going to formulate the problem of knowledge a bit differently that what I think you are getting at. Stay tuned, and then we can see if we're talking about the same thing or not.

    By Blogger Big-S Skeptic, at Aug 30, 2007, 11:21:00 PM  

  • any plans on coming back?

    By Blogger B. Spinoza, at Jun 29, 2008, 11:53:00 PM  

  • I think about it sometimes. I have the rest of this essay written, more or less, but I'm working on some other projects, and I've kind of stopped reading the blogs anyway (just other demands on my time). How are you, and how are things on the blogoshpere?

    By Blogger Big-S Skeptic, at Jun 30, 2008, 7:42:00 AM  

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