On Oct 7, 2008, at 11:59 PM, Rob Freeman wrote: Pat, Thanks for addressing my complexity point. 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 topheavy quality?
It is easy to prove there could be more meaningful combinations of n elements than n, for any system.
More combinations, of course. This is why for example classical higherorder logic is undecidable, because its semantics requires that its interpretations contain all relations in this mathematical sense (and when the base set is infinite, this makes the set of relations much more infinite.) But (1) you have to be more specific about what you mean by a 'system' in order to see what this means in practice; and (2) not all combinations are 'meaningful'.
The maximum number is something like the binomial coefficient {n choose k} = {n!}/{k!(nk)!}. With variations according whether the order of elements matters etc.
For sets its just 2n The only question is whether all these possible sets are distinct in meaningful ways.
No, that is not the only question. A more important question is, so what? Why does this fact of combinatory mathematics matter to a given system? Take Common Logic, for example (which isn't a system, exactly, but never mind): the number of possible relations over a universe is a much larger cardinality than the universe (see above), but this doesn't matter because the logic isn't obliged to consider all of these relations, only enough to provide denotations for all the relationnaming terms in the language. So this just isn't an issue in CL. Let me turn the question around. How do we know they are not? Has anyone ever looked into it?
Well, the question as posed isn't precise enough to look into. What kind of system are you talking about? For the case of cardinality in logical interpretations, the answer is certainly yes, in fact its been a major focus of work in model theory for the past decade.
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 upsidedown.) 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
contextpoints. It can be used to define timepoint in terms of
timeinterval, and other apparently impossible "backward" creations of
individual or concrete ish objects from clouds of vaguer ones.
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.
Um...no, its not. In fact, virtually all of modern mathematics is 'based on sets'. The question is whether you can ignore sets and work only with categories.
? Why is that an interesting question or an interesting ambition? What do you gain by ignoring sets? 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.
I completely fail to follow you here. Of course there are not fewer sets than examples, since there is a singleton set for each example.
Pat
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