Path: utzoo!utgpu!jarvis.csri.toronto.edu!rutgers!usc!wuarchive!mailrus!purdue!mentor.cc.purdue.edu!l.cc.purdue.edu!cik
From: cik@l.cc.purdue.edu (Herman Rubin)
Newsgroups: comp.ai
Subject: Re: Can humans "understand" mathematics
Summary: One can understand, but not by translating
Message-ID: <1789@l.cc.purdue.edu>
Date: 17 Dec 89 12:12:14 GMT
References: <3120@uceng.uc.edu> <130200002@peg>
Organization: Purdue University Statistics Department
Lines: 36

In article <130200002@peg>, ggast@peg.UUCP writes:
> 
> 
>     I agree entirely that Searle is missing something.Everyone who has 
>     learnt  a foreign language will know immediately what I  mean.When 
>     you  start learning a foreign language your thought process is  in 
>     your  mother  language, and each word of a sentence  you  want  to 
>     speak  in the foreign language you have to translate according  to 
>     memorized  rules  .This is a difficult and tedious  process,  just 
>     like learning to play piano where you have to translate notes into 
>     keystrokes consciously.

This may be why there is so little understanding.  Certainly there is some
of this; even in one's own language, when one looks up words in a dictionary.
But understanding is not obtained in this way.  Even in languages which I
know poorly, there is considerable amalgamation in my thought processes with
my native English.  This produces difficulty mostly in trying to use the 
other language, as the translation process runs in the background, it at
all.

Understanding mathematics is somewhat different.  The ability to use
mathematical objects even with great facility has nothing to do with
understanding.  I maintain that it can even detract from understanding,
and make it difficult to acquire this understanding.  It is almost as if
acquiring the skills in manipulation, proof, etc., makes it difficult to
learn what is at the basis.  What makes it harder is that sometimes there 
is no known basis, and that it is mere accident.

A mathematical concept can even be learned without learning to manipulate.
It corresponds to the "leaning with understanding" a foreign language.  The
biggest problems that people, especially non-mathematicians, have is using
mathematics as a language to precisely express their ideas.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907
Phone: (317)494-6054
hrubin@l.cc.purdue.edu (Internet, bitnet, UUCP)