Re: Universal grammar

In article <1161318846.624985.11180@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Rob Freeman <groups@xxxxxxxxxxxxxxxxxxx> wrote:
Hans Aberg wrote:
In article <1161269148.965379.301930@xxxxxxxxxxxxxxxxxxxxxxxxxxx>, "Rob
Freeman" <groups@xxxxxxxxxxxxxxxxxxx> wrote:

Is it possible to reduce all of mathematics to logic?

In principle, yes, of the part of mathematics that deals with theorems and
proofs, though there are practical problems when trying to implement it
into a computer. Working math also consists much of human cognitive
information, which is then lost, in as much it is not described in formal
logical terms. The best hope for the immediate future is though a program
that aids the human in writing proofs, proving things that might be
technical but not depending so much on cognition and is not overly
structured.

So you want to factor some "human" factor X out of mathematical proofs?

I find the idea that there is an indefinable "human" element in
mathematical proofs quite radical. This is something quite new, and
unique to your own work, isn't it?

It is unclear whether there is a human element in the proof
itself. However, finding proofs is a very definitely human
process, even in situations in which we know that a machine
can find it in a systematic manner. Even with our "high
speed" machines, that is much too slow.

Also, it has been shown that, for any k, there are theorems
for which the shortest mathematical proof is at least k times
as long as a metamathematical proof.

Still, it is not inconsistent with Goedel's own interpretation, which
was that absolute "truth" existed, but that it must be outside of maths
itself.

Some aspects of truth exist, but not all which people want.

I still find it ironic that people should interpret the failure to
observe something (absolute "truth") as evidence that it does still
exist, but that it exists only in some sense which is innate to humans
(or beyond.) Why are we moved to push everything we don't understand
into the "human" category? And then we bemoan the fact that we are
unable to understand humans!

In this case the mystery of the missing "human" element would cease to
exist, if once we accepted absolute "truth" does not exist.

Is it possible to reduce all patterns to logic?

What to you mean by pattern? A language formally is just a set of valid
word (or token) sequences.

I think the question applies quite generally. We could start by
applying it to patterns defined as "sets of token sequences". If the
sequences are "valid" it is easy. The logical specification is just the
set of rules which specify "validity".

But what if the sequence is not "valid"? Another way of phrasing the
question might be--does there exist a sequence of tokens which is
non-"valid" for all possible sets of rules?

If you assume simple logic is included, and insist that there
be some statement which can be shown to be valid and which
cannot also be shown to be non-valid, any contradiction would
suffice. But if you are just given a finite sequence of
tokens, one could make this sequence valid, but nothing
useful derived from it.

For example, if one starts with the sequence of tokens
This statement is false., this could be declared a true
proposition, but other basic propositions, formulas,
etc., would have every "character" start with the token
string xyz.

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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