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ed=3Da) Mathematics has benefitted greatly from computation. The most=20
obvious examples are things like the four-color proof, which could
not have been done without a computer
We always knew the Four Color Theorem was true (or strongly suspect=
). The beauty of the theorem is that the statement of the theorem =ca=3D
n be explained to anyone; it is there before your eyes. But the pr=oo=3D
f by computer is completely opaque and almost worthless. Even the=v=3D
ery long proof of Fermat's Theorem is less than satisfying. What o=ne=3D
wants is a short, elegant proof that stands before one's eyes as a =s=3D
ingle object and can be taken in at a glance.e =3D
=20
Moreover, see the reference to the work by Clifford Truesdell in th=
references to my paper Some Unintended Effects of Computers at http=:/=3D
/tinyurl.com/dqvj8 , which contains some additional remarks on the =ef=3D
fects of computers.s
=20
Tom Wayburn, Houston, Texas, http://web.wt.net/~twayburn=3D20
=20
=20
http://www.google.com/search?q=3D3Dhaken+appel
but less-obvious effects are pervasive. Math (of the sort that
mathematcians do) is all about proofs ... and the standard of
proof involved in /proving/ the correctness of computer programs i=
sdramatically higher than the old-style standards of the mathematic=
m=3Dcommunity.
=3D20
a') Conversely, computing has benefitted greatly from mathematics.
The algorithms you need for doing things on a computer are oftenti=
esw=3D
different in detail from the algorithms you need for doing things =
ithe=3D
pencil and paper ... but only in detail. The general thought-proc=
ssesa=3D
that go into designing a fiendishly clever algorithm are fundament=
llyn
the same.
=3D20
=3D3D=3D3D=3D3D=3D3D=3D3D=3D3D=3D3D=3D3D=3D3D=3D3D
=3D20
b) At the other end of the spectrum, as computers and technology i=
ogeneral become more sophisticated, the more they look like magic t=
c=3Dthe casual user.
=3D20
Belief in magic is extremely corrosive to rational thought. In a
democracy, this is a very very serious problem.
=3D20
On a smaller scale, but still a topic of concern, I don't like the
fact that a lot of mass-market software tells the user what to do,
not vice versa. The nightmare scenario of the sorcerer's apprenti=
es=3D
goes back ~2000 years ... not to mention Faust and probably 100
science-fiction dystopia stories per year for the last 100 years.
I want to be telling my machines what to do, not vice versa!
=3D20
This makes contact with Michael E's lament that students tend to a=
sumee=3D
the powerpoint format must be the right report format, and the exc=
ln=3D
"chart" format must be the right "chart" format. It is crucial to
teach the students that they are in charge ... and they have perso=
alo
responsibility for the final product.
=3D20
I see this in my student pilots. The ones that can afford fancy
electronics (GPS moving map etc.) would -- if I let them -- come t=
t=3Drely on that stuff too much. I counter this initally by asking lo=
st
of hypothetical but pointed questions about what they would do if
this-or-that instrument failed. Later I make the lesson entirely
non-hypothetical by turning off all the fancy stuff when they leas=
cexpect it, giving them the "opportunity" to get us home using basi=
a=3Dstick-and-rudder skills. Sometimes they are initially irate about
this, but a couple days later they call me up and say "let's go do
that again". It is easy for the flight instructor to motivate the
students, because these are clearly life-and-death issues. It is
harder to achieve the same level of motivation in intro physics cl=
ss.
=20