Hal Finney wrote:
>
> Dan Clemmensen, <Dan@Clemmensen.ShireNet.com>, writes:
[SNIP of Godel undecidabiliy permitting the addition of the CH or
its contradiction as an axiom]
> > How am I supposed to go from this to Penrose's conclusion? Sounds
> > crazy to me.
>
> It would be better for someone who is more familiar with Penrose's argument
> to say whether I have accurately described it. However I think the idea
> here is that we aren't supposed to be just playing with formal systems
> when we do math. We are seeking after truth.
>
> The mathematics we create is intended to accurately represent the
> behavior of abstract mathematical objects which we are thinking of,
> such as the points and lines of plane geometry, or the integers under
> the rules of arithmetic. We set up axioms which are intended to
> caputre all the relevant properties of these objects. "Truth" in this
> context means that the axioms accurately describe the mathematical
> objects we have in mind.
>
> We have to go outside the axiomatic system to determine the truth of a
> proposed axiom. We have to fall back on our general reasoning power,
> persuasive arguments, looking at the conclusions which flow from different
> assumptions, and seeing if they seem plausible. It's almost more like
> political argument than the kind of logic we associate with mathematics.
>
> I just skimmed Penrose's book, but my impression is that he claims that
> mathematicians have an instinctive ability to recognize mathematical
> truth. Machines, which are deterministic computer programs and therefore
> can be in principle be described as formal systems, are limited by Godel's
> theorem and are unable to recognize certain truths. Since people are
> (according to Penrose) unlimited in their ability to recognize mathematical
> proof, this would mean that they are not formal systems and therefore no
> machine can do mathematics in the way a person does. (And hence, among
> other things, uploading and AI are impossible.)
>
> In the case of the CH, as I quoted from Rucker's book there has not been
> a killer argument made either way. It could be that our understanding of
> infinities is so poor that choosing the axiom one way or the other will
> both produce mathematics which seems OK. The properties of the infinite
> objects will be different, but either one could fit into our sloppy
> mental model of how an infinite object should behave. If so this would
> seem to contradict Penrose's premise.
>
> Hal
Indeed, this is hardly the first time this has come up. The "parallel
postulate"
was in the same situation at one time that you describe for the CH
today.
Mathematicians tried to show that the parallel postulate could be proven
from Euclid's other postulates, and failed. The result was the discovery
of non-euclidian geometries in which the paralel postulate does not
hold.
We now commonly describe the "true geometry of the universe" as being
non-euclidean. IMO this constitutes a dramatic counterexample to
Penrose's
idea that humans have an instinct for mathematical truth, unless we wish
to believe the instinct is less than 500 years old.