Pat, (01)
Thanks for addressing my complexity point. (02)
On Tue, Oct 7, 2008 at 11:26 PM, Pat Hayes <phayes@xxxxxxx> wrote:
>
> On Oct 7, 2008, at 2:44 AM, Rob Freeman wrote:
>
>> My argument has been that for many systems there are more ways of
>> generalizing examples than there are examples. This is a game changing
>> assumption. Up to now it has always been assumed that there will be
>> fewer generalizations than examples (rules, classes, etc.) At worst
>> that you need to make one generalization per example (analogy.) If it
>> is possible to make more generalizations than examples, the game
>> changes completely. No one set of generalizations will suffice.
>
> OK, that sounds really interesting. But Im having trouble understanding it.
> Can you make the point more concrete, perhaps with an example/sketch? What
> leads you to this conclusion, that there will be this overwhelming number of
> generalizations in many systems? What kind of systems, and why will they
> have the top-heavy quality? (03)
It is easy to prove there could be more meaningful combinations of n
elements than n, for any system. The maximum number is something like
the binomial coefficient {n choose k} = {n!}/{k!(n-k)!}. With
variations according whether the order of elements matters etc. (04)
The only question is whether all these possible sets are distinct in
meaningful ways. (05)
Let me turn the question around. How do we know they are not? Has
anyone ever looked into it? (06)
>> The only way to model that would be to base analysis on sets of examples.
>
> There is an old idea (due I believe to Whitehead in the early 20th century)
> that may be relevant here. If all you have are categories (generalizations,
> classes,..), then you can define individuals to be maximal sets of
> intersecting categories. Mathematically, they are ideals or ultrafilters
> (same thing looked at upside-down.) Its a very elegant and powerful
> technique: I've used for example to show that any notion of 'context' that
> satisfies some basic assumptions can be reduced to sets of individual
> context-points. It can be used to define time-point in terms of
> time-interval, and other apparently impossible "backward" creations of
> individual- or concrete- ish objects from clouds of vaguer ones. (07)
Right, yes, basing categories on sets has a history. We can feel
comfortable thinking of categories in this way. Others have done it.
Using sets to represent categories is not a crazy proposal at all. The
question is whether you can ignore sets and work only with categories. (08)
It is really a question of complexity. For a given set of examples if
there are fewer distinct sets than examples you will be able to label
all the sets and get along fine with the labels. On the other hand if
there are more distinct sets than elements you will have no
alternative but to work directly with the sets. (09)
-Rob (010)
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