Rob, (01)
This discussion leads to many issues about mappings: (02)
RF>> The work of Schmidhuber and Hutter avoid these "pitfalls"
>>> [of logic] by ignoring formal logic and seeking a basis for
>>> meaning in prediction/probability. (03)
JFS>> That's like a carpenter who struck his thumb with a hammer and
>> later avoids the pitfalls of hammers by using nothing but saws. (04)
RF> I think of it in terms of a different metaphor. I think of it
> as a cartographer who finds projections to a plane are always
> distorted, and so decides not to limit herself to flat maps. (05)
Interesting comparison. It just so happens that C. S. Peirce was
not only the person who invented the algebraic notation for FOL
(in 1880-85), he also invented the quincuncial projection of the
earth to a plane (1879). See the Wikipedia entry: (06)
http://en.wikipedia.org/wiki/Quincuncial_map (07)
Unlike the common Mercator projection, which exaggerates the
areas of Canada, Alaska, and Greenland, the Q. projection
preserves relative areas. Although it has some distortions,
it has the very important property that great circle routes
(for shipping and airlines) can be drawn with a straight edge
on that map (to an approximation with negligible error for
navigational purposes). (08)
This example shows why a person who rejects flat maps would
overlook an important application. By that, I do *not* claim
that people should limit themselves to flat maps, but that
they should "never say never" to any point of view. (09)
As for category theory, its basic focus is on maps of all
possible kinds. There are an infinity of ways of mapping one
shape to another with many different kinds of distortions,
some of which highlight different aspects of the things that
are being mapped. (010)
Similarly, there are many different ways of mapping language
to logic or logic to language. The English words 'and', 'or',
'not', 'some', and 'every' are landmarks that a mapping might
preserve, highlight, or distort. But many different ways of
mapping the content words of language will also preserve,
highlight, or distort other aspects for various purposes. (011)
Instead of "ignoring logic", I suggest that people keep in
mind that it shows up as those ubiquitous landmarks in English
and other natural languages. Languages wouldn't have those
words if people didn't find logical aspects important. (012)
John (013)
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