From: hal@finney.org
Date: Thu Apr 26 2001 - 14:26:02 MDT
I want to go back to this very old thread, which I have continued to
think about, and add a couple of points.
Earlier I wrote a response to Lee's puzzle about whether it makes sense
to say that Goldbach's conjecture is unprovable:
> Goldbach's Conjecture is that every even number is the sum of two primes.
> If it is false, there must be an even number which is not the sum of
> two primes. If such a number exists, then by showing it we could show
> that GC is false and GC would not be unprovable (that is, it would be
> provably false). For GC to be unprovable, therefore, would mean that
> no such number exists. But this is simply the statement of GC itself,
> so to show it to be unprovable (undecidable) is to show it to be true.
> Hence you can't show GC to be unprovable.
Other people gave a similar response.
I wrote earlier about Euclidean geometry minus the parallel postulate
as a specific historical example of an issue where provability was very
much in question. Most of Euclid's axioms seemed completely self-evident:
two points determine a line, and so on. But the parallel postulate, that
through every point not on a line there is exactly one line parallel to
the given line, never seemed as obvious. (This is especially true because
Euclid's actual parallel postulate was worded in a more complicated way
[1], although it was mathematically equivalent.) It didn't fit well
with the other postulates.
So there were a number of attempts over the years to eliminate the PP by
showing that it could be derived from the other axioms, but none of them
succeeded. In the end it was shown that the PP was independent of the
other axioms, and in fact you can get non-contradictory and interesting
versions of geometry by assuming either that there are multiple parallel
lines through a point or no parallel lines.
Now, here is how the PP is like GC. Consider a line and a point above
the line. Let there be a 2nd line through the point, not parallel but
almost. Say it makes a 5 degree angle with the base line. Then the
intersection point is well off to the side.
Rotate the top line so that it moves through the parallel angle.
The intersection point moves off, faster and faster. As we pass through
the angle of parallelism, the intersection point zooms off to infinity.
At the same instant, a new intersection point appears at infinity on
the opposite side, and zooms towards us, slowing down progressively.
Let's suppose we want to ask whether a weaker version of the PP is true,
which is whether or not there is at least one parallel line, versus
whether there are no parallel lines. We know in fact that this question
is unprovable. But here is an argument for why it "can't" be unprovable:
The PP states that there is an angle at which there are no points of
intersection as we do the rotation above. If it is false, as we rotate,
the 2nd intersection point exists before the first one disappears.
Therefore there must be some maximum distance away that an intersection
point can reach; beyond that distance the 2nd intersection point will be
closer. The PP is therefore a question of whether such a point of maximum
distance exists, or whether that point is in effect infinitely far away.
This point is like the number which is the counter-example to the GC
in the argument above. Now I am going to follow sentence-by-sentence
paralleling the GC argument above.
"If such a point exists, then by showing it we could show that the PP is
false, so the PP would not be unprovable. For the PP to be unprovable,
therefore, would mean that no such point exists. But this is simply
the statement of the (weak) PP itself, so to show it to be unproveable
(undecideable) is to show it to be true. Hence you can't show the PP
to be unprovable."
In other words, just as the falsity of the GC would seem to imply that
there must be a specific number which ISN'T the sum of two primes, so
the falsity of the PP would seem to imply that there must be a specific
distance to a point which is the farthest possible point of intersection.
For either proposition to be unprovable would seem to imply that there
is no such number or point, which would mean that they are true.
In the case of the PP, we can measure the distance to the intersection
point as the lines approach parallelism. We can see it getting farther
and farther away, while confirming that no intersection appears within
the same distance in the other direction. This is like experimentally
verifying the GC by testing it with larger and larger numbers.
The question is, will the point get to infinity? And can we keep
finding even numbers that don't violate the GC, all the way to infinity?
It's really the same kind of question.
We know that the reasoning is false in the case of the PP; there exist
consistent geometries where there are no parallel lines. We might
even live in a universe where that is the case; it's possible that if
we actually tried the rotating line experiment, we would find that the
intersection point never gets farther than X billion light years away.
Given that this seemingly valid reasoning is actually mistaken with
regard to the parallel postulate, I think we need to be very cautious
about accepting it with regard to Goldbach's Conjecture. The two cases
are more alike than might seem at first.
Hal
[1] Euclid's actual Parallel Postulate: If A and D are points on the same
side of segment(BC) such that measure(angle(ABC)) + measure(angle(BCD)) <
pi, then ray(BA) intersects ray(CD).
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