The following is a myth based on a true story. It contains inaccuracies that exist only because the story would be long, complicated and incomprehensible to most if I told it accurately. I have tried to correct some of the inaccuracies in the footnotes for you purists out there. Links are to the History of Mathematics page.
I think it was Benjamin Franklin,
who once said, "The only bad mistakes
are the ones you don't learn from."
A lesson for us all.
That being the case, I shall tell you the story of a good math
mistake,
because we learned from this one.
In fact, we learned so much
that this math mistake is not only good,
it is great, perhaps the greatest
math mistake of the century.
It was made by David
Hilbert, early in the century.
Hilbert was a great mathematician, who
mastered all the fields of math there were,
because you could still do that
back then.
So it is surprising that such a genius could make such a
blunder.
Hilbert wanted to solve every arithmetic problem there is(1).
Or,
rather, find a method that could be used to solve every one.
It seemed
obvious that they were all solvable!
That was Hilbert's big mistake.
"Yeah, how hard can it be?", said Bertrand
Russell.
"1 + 1 = 2, everything follows from that", said Alfred
North Whitehead.
Many years passed before Principia Mathematica
came about(2)(it
was harder than they thought),
"but Hilbert is right, and here's the proof",
said they.
"Not so fast", said a tall gangly Kurt
Gödel in the back row with an Austrian accent.
"The problem of solving
every arithmetic problem is itself an arithmetic problem,
and proving that
all arithmetic is solvable is also an arithmetic problem;
Hence,
proving all arithmetic is not solvable is also an arithmetic
problem.
And, if this last problem is solved then we have proven
arithmetic false,
but if we cannot solve this last problem, then arithmetic
is, by counter example, incomplete."
Kurt Gödel's incompleteness theory proved Hilbert wrong(3).
Russell
and Whitehead took up Philosophy,
which made them much more popular,
since
no one understood Principia anyway.
A few years later a British track star with the unlikely name of Al
Turing asked an unlikely question(4):
"What
if we limited math to what could be done by a computer, would it be subjected to
Gödel's restrictions?"
"What's a computer?", they responded.
"Well, it is
a machine that can do math for you.", said Al.
"Interesting!", they
responded.
"You see if you limit the eigenstates to what is mechanical you
can use a similar trick Gödel used...."
"What kind of a machine?"
"Just
any computing machine, now as I was saying..."
"Wait a second, are you saying
it is possible to build a machine to do math for us?"
"Yes, and with it I can
prove..."
"Hold on a minute," they responded, "Are you telling us that we
have rooms filled with accountants using slide rules and abacuses figuring out
all our companies finances and all this time we could have just used some bloody
machine do it for us?"
"Well they can't do everything, as I was about to show
you..."
"Where can we get one of these computers?"
"Oh forget it!",
responded Al dejectedly.
What Turing was trying to say was this:
In Kurt Gödel's theory, he used a
technique of enumerating every arithmetic problem there is.
1 + 1 = 2 is
enumerated to 45236, etc.
Using the same technique, we can create a
theoretical operating system,
for a theoretical device that can do math
automatically.
The "Turing Machine" as described was completely impractical
and could never be built.
But, it did not rule out the possibility of a
practical design.
A problem for someone else.
The someone else was Turing's professor, John
Von Neumann(5).
He
actually attended Gödel's original lecture.
And, may be the only man, besides
Hilbert, to understand it at the time.
Since they all spoke German.
"A computer you say...Hmm, very interesting.", said Von Neumann,
"I have
not seen one, but if I were to build one, I would need a input device of some
sort."
"Oooh", they said.
"And some kind of control mechanism, electrical
if possible."
"Ahhh", they said.
"It would be better if it were binary,
that way memory could be created easily"
"Wow, memory"
"And of course you
would need a means of receiving output, like a television."
"What's a
television?"
The (ahem!) "Von Neumann Machine" was born.
John may be famous for
many things,
Humility was not one of them.
Luckily "computer" had a better
ring to it.
Later, Von Neumann did actually help design the first computer,
with a
couple of other "John"s named Eckert
and Mauchly(6).
But,
just in case the computer started taking control of the world like in those
sci-fi stories,
Von Neumann helped design the Atom Bomb the previous
year.
He then invented Game Theory, which proved that the bomb should never
be used,
unless you are really really mad(7).
Same
goes for computers for that matter.
Eckert and Mauchly went on to actually build the first programmable computer
in Philadelphia.
Just across the bridge from New Jersey where Kurt Gödel
lived(8).
Small
world.
Later we learned Alan Turing built a programmable electronic computer
predating the one in Philly(9).
It
helped win the war, but Turing could not take credit or make any money from his
invention.
It was Top Secret.
Alan Turing committed suicide(10),
after a sex scandal. (He was gay -- long before it became trendy.)
John Von
Neumann died of radiation(11),
after standing too close to the Atom Bomb.
Kurt Gödel, after mathematically
proving food is bad for you, died of starvation(12)
.
They were all completely crazy when they left this mortal
realm...
Computers can do that.
David Hilbert lived a long and happy life(13),
having never lived to see a computer.
Never knowing how big the consequences
of his great little mistake were.
We unfortunately were not so lucky.
The rest, as they say, is history.
Later Bill Gates came along and took over the world,
Where is Von
Neumann's bomb when you really need it?