The Greatest Math Mistake of the Century

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?

1. Hilbert's challenge can be found in his famous 1901 lecture on 23 unsolved problems in Mathematics. Hilbert was a founder of the Formalist school of mathematics which believed it was possible to solve every math problem. I use the term "arithmetic" where in reality it was a general class of logic structure called formal systems, what we call "arithmetic" being an example of a formal system, others include set theory, linear algebra, polynomial algebra, symbolic logic, and all computer languages. Noam Chomsky of MIT believes that human languages may also be formal in structure.
2. Principia Mathematica was started in 1910, finished in 1913
3. Kurt Gödel gave his lecture in 1931.
4. Turing's work On Computable Numbers was published in 1936.
5. Von Neumann taught at Princeton University in the late 30's while Turing was a grad student, he worked on designs of "Von Neumann Machines" through the early 1940's. Other Pre-realization computer pioneers included Norbert Weiner, Howard Aiken, Alonzo Church, and Alwin Walther of Germany. The latter working with computer pioneer Konrad Zuse. The two worked for Germany during W.W.II completely unaware of the work of Turing or Von Neumann.
6. In truth, Von Neumann merely played an advisory role on ENIAC. His role on UNIVAC was more significant.
7. Von Neumann worked for the Manhattan Project at the end of W.W.II, his game theory argued strategies for the Cold War that inspired the movies Dr. Strangelove and War Games.
8. During the 40's and early 50's, Gödel worked with Albert Einstein at Princeton University. He was portrayed in the 1996 movie IQ by an actor who looked nothing like him.
9. Turing's digital electronic computer "Colossus" was completed in Britain in 1943, followed by Colossus II in 1944. Other computers predating ENIAC were built by Charles Babbage and Ada Lovelace (Britain,1842, mechanical, never finished), Herman Hollerith (America,1890, Mechanical), Vannevar Bush (America,1930 analog electro-mechanical), Konrad Zuse (Germany,1939, digital electro-mechanical), Helmut Hoelzer (Germany,1941, analog electronic), Howard Aiken and Grace Hopper (America, 1942, electro-mechanical), and John Atanasoff and Clifford Berry (America, 1942, digital electronic, never finished).
10. Died 1954 in England.
11. Died 1957 of Brain Cancer in Washington D.C.
12. Died 1978 in Princeton, New Jersey.
13. Hilbert retired in 1930 and despite the war, died peacefully in Germany in 1943.

Details about the significance of Gödel's effect on Hilbert's school of Formalism can be found in the book Pi in the Sky by James L. Barrow.
The connection between Gödel's theory and Turing's theory is explained in Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. A more technical explanation can be found in The Emperor's New Mind by Roger Penrose.
For an entertaining book on the history of computers check out The Devouring Fungus by Karla Jennings.