next up previous
Next: The wants and actions Up: What kinds of approximate Previous: What if?

Mount Everest.


It is clear that we cannot hope to formulate a useful definition of Mount Everest that would tell about every rock whether it is part of the mountain. However, we might suppose that there is a truth of the matter for every rock even though we cannot know it. Our axioms would then be weaker than the truth. The question would be settled for some rocks and not for others.

Not even this is appropriate. The concept of the territory of Mount Everest may be further refined in the future--and refined in incompatible ways by different people. If we suppose that there is a truth about what rocks are part of the mountain, then the people refining it in different ways would get to argue fruitlessly about which definition is getting closer to the truth. On the other hand, there is a truth of the matter, which may someday be discovered, about whether Mallory and Irvine reached the summit in 1924.

Consider two theories of mountain climbing, T1 and T2. Besides these theories, there is T3 based on plate tectonics that tells us that Everest is still getting higher.

In the simpler theory T1, there is a list of names of mountains paired with lists of climbing expeditions or names of climbers. As a logical theory it would have sentences like.


The larger theory T2 contains routes up the mountain of the various parties. Routes are approximate entities.

T1 is an approximation to T2, but T1, may be regarded as not approximate at all. In particular, it can be complete, e.g. it decides any sentence in its limited language.

T1 and T2 may be related using formalized contexts as in [McC93a] or [MB97], but we won't do that here.

One approximate theory may be less approximate than another. We want to discuss relations between sentences in a theory and sentences in a less approximate theory. It makes the ideas neater if we imagine that there are true and complete theories of the world even if they're not finitely expressible and none of the language of any of them is known. This permits regarding approximating the world as a case of one theory approximating another. If this is too platonic for your taste, you can regard approximating the world as a different notion than that of one theory approximating another.

There are other approximate theories involving Mount Everest.

One such theory that lists names of mountains and the continents containing them. Thus we have tex2html_wrap_inline405 . A less approximate theory gives countries, e.g. tex2html_wrap_inline407 . A still less approximate theory gives locations, e.g. tex2html_wrap_inline409 .

next up previous
Next: The wants and actions Up: What kinds of approximate Previous: What if?

John McCarthy
Wed Feb 2 15:59:04 PST 2000