FOM: General Intellectual Interest

[Jerry Seligman]

On Fri, 26 Mar 1999 at 17:18:58 -0600, Pat Hayes wrote:

>Do you think that the West End success of a play
>about Alan Turing leads directly from his mathematical work, or requires a
>widespread comprehension of it? If so, I think you are living in a
>mathematical logician's paradise.

Perhaps it would be useful to recall the passage from Hugh Whitmore's
play that deals directly with f.o.m.  Do those who are sceptical of
(existing) popularisations of f.o.m. think that this exposition does more
harm than good?


[ Alan Turing, in his twenties, is being interviewed for a job at Bletchley
Park, the UK government code-breaking centre, during the war.  His
interviewer is Dillwyn Knox, a classical scholar.]

KNOX: I've been furnished with some details of your work, Mr Turing, most
of which I have to tell you I find almost totally incomprehensible.

TURING: That's hardly surprising.

KNOX: I knew quite a bit about mathematics when I was young, but this is
--- well, baffling.

[studies file]

Um --- for example ...  this thing here: 'On Computable Numbers with an
Application to the Ent-scheid-ungs-prob-lem'.

[ He raises his head and looks at TURING. ]

Perhaps you could tell me about it.

TURING: Tell you what?

KNOX: Well, anything --- a few words of explanation --- in general terms.

[ TURING looks at him.  Brief pause.  ]

TURING: A few words of explanation?

KNOX: Yes.

TURING: In general terms?

KNOX: If possible.

TURING: It's about right and wrong.  In general terms.  It's a technical
p-p-paper in mathematical logic, but it's also about the difficulty of
telling right from wrong.

[brief pause]

People think --- most people think --- that in mathematics we always know
what is right and what is wrong.  Not so.  Not any more.  It's a problem
that's occupied mathematicians for forty or fifty years.  How can you tell
right from wrong?  You know Bertrand Russell?

KNOX: We've met.

TURING: Well he's written an immense book about it: 'Principia
Mathematica.'  His idea was to break down all mathematical concepts and
arguments into little bits and then show that they could be derived from
pure logic, but it didn't quite work out that way.  After many years of
intensive work, all he was able to do was to show that it's terribly
difficult to do anything of the kind.  But it was an important book.
Important and influential.  It influenced both David Hilbert and Kurt
Gödel.

[a brief digression]

You could, I suppose, make a comparison between these preoccupations and
what physicists call 'splitting the atom'.  As analysing the physical atom
has led to the discovery of a new kind of physics, so the attempt to
analyse these mathematical atoms has led to a new kind of mathematics.

[resuming the main thread of his explanation]

Hilbert took the whole thing a stage further.  I don't suppose his name
means much to you --- if anything --- but there we are, that's the way of
the world; people never seem to hear of the really great mathematicians.
Hilbert looked at the problem from a completely different angle, and he
said, if we are going to have any fundamental system for mathematics ---
like the one Russell was trying to work out --- it must satisfy three basic
requirements: consistency, completeness, and decidability.

Consistency means that you won't ever get a contradiction in your own
system; in other words, you'll never be able to f-f-follow the rules of
your system and end up showing that two and two make five.  Completeness
means that if any statement is true, there must be some way of proving it
by using the rules of your system.  And decidability means, uh ---
decidability means that there must exist some method, some d-d-definite
procedure or test, which can be applied to any given assertion and which
will decide whether or not that assertion is provable.

Hilbert thought that this was a very reasonable set of requirements to
impose; but within a few years Kurt Gödel showed that no system for
mathematics could be both consistent and complete.

He did this by constructing a mathematical assertion that said --- in
effect: 'This assertion cannot be proved.'

Well --- either it can be proved or it can't.  If it can be proved, we have
a contradiction, and the system is inconsistent.  If it cannot be proved,
then the assertion is true --- but it can't be proved, which means that the
system is incomplete.  Thus mathematics is either inconsistent or it's
incomplete.

It's a beautiful theorem, quite b-b-beautiful.  Gödel's theorem is the most
beautiful thing I know.  But the question of decidability was still
unsolved.  Hilbert had, as I said, thought that there should be a clearly
defined method for deciding whether or not mathematical assertions were
provable.  The decision problem he called it.  'The Entscheidungsproblem'.
In my paper on computable numbers I showed that there can be no one method
that will work for all questions.  Solving mathematical problems requires
an infinite supply of new ideas.  It was, of course, a monumental task to
prove such a thing.  One needed to examine the provability of all
mathematical assertions past, present, and future.  How on earth could this
be done?  One word gave me the clue.  People had been talking about the
possibility of a mechanical method, a method that could be applied
mechanically to solving mathematical problems without requiring any human
intervention or ingenuity.  Machine!  --- that was the crucial word.  I
conceived the idea of a machine, a Turing Machine, that would be able to
scan mathematical symbols --- to read them if you like --- to read a
mathematical assertion and to arrive at the verdict as to whether or not
that assertion is provable.  With this concept I was able to prove that
Hilbert was wrong.  My idea worked.

KNOX: You actually built this machine?

TURING: No, of course not.  It was a machine of the imagination, like one
of Einstein's thought experiments.  Building it wasn't important; it's a
perfectly clear idea, after all.

KNOX: Yes, I see; well, I don't, but I see something.  I think.

[ He looks at TURING. ]

Forgive me for asking a crass and naive question --- but what is the point
of devising a machine that cannot be built in order to prove that there are
certain mathematical statements that cannot be proved?  Is there any
practical value in all this?

TURING: The possibilities are boundless --- this is only the beginning.  In
my paper I explain how a special kind of Turing Machine --- I call it the
Universal Machine --- could carry out any mental process whatsoever.

Jeremy M. Seligman
Department of Philosophy,
The University of Auckland, Private Bag 92019, Auckland, New Zealand
Tel: +64-9-373-7599 xtn. 7992, Fax:  +64-9-373-7408, Time Zone: GMT +13 hours