random

[Tim Peters]

[David C. Ullrich, on http://www.cs.umaine.edu/~chaitin/cmu.html]

> ...
> Very interesting stuff, without a doubt. He wants to say that
> his point of view makes incompleteness seems inevitable he
> may well be right. But when he gets into statements like
>
> "there is no reason that individual bits [of Omega] are 0 or 1!"
>
> it's just too... well I better omit the adjective and just say
> that "there is no reason that individual bits are 0 or 1" is not
> a mathematical statement amenable to proof.

It *sounds* powerful, though <wink>.  I don't really object, because it
conveys *something* of the story in a way that would otherwise be over his
audience's head.

For example, over the last few weeks we've had several examples on c.l.py of
printing the exact decimal value of a machine binary floating-point value
with a non-zero fractional part.  I don't know whether anyone noticed, but
every such example ended, in decimal, with the digit 5.  The natural
inclination of anyone with even a slight bent toward mathematical thought is
to ask "Hmm!  Was that just luck, or is there 'a reason' that's so?".  With
a little thought, and even in the absence of a formal notion of axioms or
proof, I expect many people could get close to proving that 5 is no
accident -- it must be a 5, "it makes sense", "there's a reason for it".
And, indeed, that's the only kind of problem most people get exposed to in
school:  "prove or disprove".

It's at least a minor revelation then that some questions can't be answered
that way (by appeal to principle or proof -- "a reason"), that brute force
can be the only answer there is.  But *phrasing* it as "no reason" is too
close to "senseless" for my tastes, so I'm sure it misleads people too.

> Preciate the reference - I definitely learned one thing, at least:
>
> "Now what is the reaction of the world to this work?! Well, I think
> it's fair to say that the only people who like what I'm doing are
> physicists!"
>
> Not having studied his stuff or people's reactions to it I thought
> it was just me (seriously). (Um: It's _not_ true that mathematicians
> don't like his work, I know of plenty of mathematicians who find it
> fascinating. What people don't like is his [adjective] claims
> about what his work _means_...)

I agree:  it's the popular presentation, not the work, that people object
to.  I read that Gödel delayed presenting his incompleteness theorem (surely
a much greater shock at the time!) because he feared nobody would believe
him, so worried over every little detail of the arithmetic coding to make
the proof unassailable first.  But by all accounts I've read, the ideas
behind the proof were compelling the instant logicians heard about them, and
despite that it wasn't what they were hoping to hear, and most skipped over
the laborious coding details as curious drudgery.  Mathematicians are quite
calm in the face of crisis <wink>.

OTOH, Gödel kept what he thought his result "meant" to himself.  Take
someone who learned it from the philosophy dept and another from the math
dept, lock them in the same room, and see whether they have *anything* in
common <wink>.  But Rudy Rucker reported that in private conversation,
Gödel's beliefs were actually closer to the former, that the result had
profound implications beyond just pointing out a limitation of formal proof
systems.  I wish he would have written more about that; at least I won't
have the same regret with Chaitin.

> The (or in any case _a_) point being things like
>
> "I think there may be political reasons, but I think there are also
> legitimate conceptual reasons, because these are ideas that
> are so foreign, the idea of randomness or of things that are true by
> accident is so foreign to a mathematician or a logician,
> that it's a nightmare! This is their worst nightmare come true! I
> think they would prefer not to think about it."
>
> He insists that the world of mathematics is all upset over his
> well-defined but unknowable Omega.  If that irritates people what
> irritates them is not the well-defined but unknowable character
> of those bits, it's the idea that the idea of something being
> well-defined but unknowable started with him.  It's no big deal,
> people have been dealing with "unknowables" for a long time.

Wholly agreed.  Some people were driven to despair and even suicide by
Gödel's results at the time.  Everything since (and there's been a lot) has
been greeted with "Wow -- here's yet *another* thing we can't know.  Cool!".

the-first-time-was-a-major-surprise-but-that-was-70-years-
    ago-ly y'rs  - tim