Not by that name. Are you referring to Penrose’s belief that a computer cannot simulate human intelligence, and/or that the current “laws” of physics are insufficient to explain human intelligence?
Yeah that’s right, although the stuff about not being at a sufficiently developed stage of science is not entirely relevent to my dissertation.
I’m working towards the conclusion that the kind of informal mathematical visualisation/understanding that Penrose thinks we posess is nothing more than imprecise descriptions of underlying formal procedures.
There’s nothing wrong with his use of Godel’s theorem, although really he uses Turing’s proof of the halting problem. He just uses the theorem as a function really.
Hi debaser, I’d be interested in hearing more about your dissertation. I’ve lightly dabbled in Shadows of the Mind and want to read more Penrose.
Hi Obw,
A good place to start might be Minds, Machines and Godel by John Lucas. It is significantly shorter than any of Penrose’s books but still contains the essential argument. Also you can download it free of charge. Penrose simply took Lucas’ paper and threw in what he thinks is supporting evidence of the essential argument and offered a tentative explanation in tems of quantum mechanics. Penrose’s account is definitely more thorough and exacting but to get to grips with it in a short time go with the Lucas paper:
univ.trieste.it/%7Eetica/2003_1/index.html
also this website contains benacerafs argument against the hypothesis. Not much has changed since the early days to be honest.
what is the ‘esscential argument’ in Gödel’s theorem?..
I don’t understand anything but lingüistic examples… 
I think I really suck at logic and the propositional language like…
’ if G is a statement inside this system, it can’t be proved and blah blah’ 
heavenly demonic
sorry i was referring to Roger Penrose’s essential argument concerning Godel’s theorem. Godel’s theorem is not something you can just turn up, look at and understand. Linguistic examples can only get you so far unfortunately… I would suggest reading up on the developments in maths and logic that led up the theorem. It is closely related to Cantor’s diagonal argument - if you look at this you might begin to understand how something can be true but not provable in a system. The diagonal argument proves that given the notion of infinity there must be two kinds of infinity (cardinalities). The conclusion isn’t so important but the method of proving it is. This method is integral to Godels theorem and Turings proof of the halting problem.
I’m not an expert on this however - hence my original post. I was hoping someone would have some knowledge i could use.
Thanks- looks very interesting.
Hi Bill, earlier on in this thread you described the notion of mathematics being concerned with truth as being ‘pre-godel’. Could you expand on this at all? Penrose claims to be a platonist even though he knows Godel’s theorem as well as anyone. Is he entitled to his opinion or is he downright wrong?
I’d really appreciate any input at this point.
Hi to all,
Debasser, Godel himself was a Platonist and the formal systems devised to mimic the Integers were the failure. I don’t see a conflict.
I am curious about your statement about the Diagonal Theorem. I do not recall seeing any reference to it in the book “Godel’s Proof” by Ernst Nagel and James Newman (edited by our old friend Douglas Hofstadter) and I do not know why it would be in the proof.
Finally, I do not understand why you think that Godel’s Theorems are difficult to understand.
The theorem states that inside Principia Mathematica (basically a formalized structure, devised by Russell and Whitehead, that mimicked the integers with addition and multiplication) there exist true statements that can not be proved. Additionally, if all statements within this formalized system could be proved then the system must be inconsistent.
The proof itself is somewhat complex with over 49 definitions, if I recall correctly, but the structure of the proof is fairly straight forward. (I particularly enjoyed the Godel numbering system for sentences).
Hi to Obw,
After a more critical reading I have to agree with you about some of the shortcomings. And unfortunately what I initially saw as intellectual whimsy, is losing much of its’ humor. It’s probably best to read this book as an informal introduction to self referential statements and hierarchies.
Hi Ed, I agree that a linguistic expression of the theorem can be understood easily. The problem is you are only expressing the result not the why or wherefore.
The method involved with cantors diagonal argument is extremely important to the Godel result:
This is really the halting problem which corelates with GIT. With an algortithm A designed to determine whether a turing machine Cq terminates…
If A(q,n)converges the Cq(n)diverges
let q=n [this is from cantors diagonal argument]
If A(n,n)converges then Cn(n)diverges
let A(n,n) = Ck(n) [ie the kth turing machine, from church’s thesis]
and let n = k [again from cantors diagonal argument]
A(k,k) = Ck(k)
If A(k,k)converges the Ck(k) diverges
[A(k,k)=Ck(k)]
If Ck(k)converges then Ck(k) diverges
you can accept the moves from Cantors argument on faith but i just thought it would give a clearer idea if the reason behind it was known.
Anyway I’m not here to say what’s what… i’m not an expert. Can you say more about Godel being a platonist? It seems strane that his result ended an era of math platonists if he was one himself.
Hi debaser,
Sorry I did not get back sooner.
It is my assumption that Godel and the other Platonists regard numbers as platonic ideals. The number 1 is the abstraction of the concept of one of something, and 2 is the abstraction of 2 of something and so on. From a purely Platonic point of view, numbers are a pre existing ideal that we may discover. More pragmatically, numbers are the way its’ taught in grade school.
Hilbert, Russell, and Whitehead wanted to change that to a system based on axioms. Effectively, 0 and 1 are meaningless signs and axioms were invented to account for successor elements.
The Platonists surely would not want the number 2 to be the conclusion of a theorem based on the meaningless signs of 0 and 1 and some axiom schema.
I love your math by the way, and the more you write the happier I will be!!!
However, I still can not find any meaningful reference to the Diagonal Theorem in the book Godel’s Proof. There are two references to Cantor but they are not directly reflected in the proof. I am trying to finish DeTropes’ reference which does address the Diagonal Theorem . However it is only in connection with a very general discussion about the length of proofs in a formal language. (Maybe I will find something later on. I’m only at chapter 3).
(my italics)
Good god man.
Are you serious?
Hi detrope,
The short answer is yes.
Even though I have been modest about my mathematical talent, and I am 57 years old (I now get 3 bonus points on my Wonderlick intelligence scores), I like this stuff and it comes easily to me.
What?!
This stuff comes easily to you and you haven’t offered the meaning to the universe or even a simple philosophy yet? You’ve been a member for how many years now, Ed?
That’s uncool, man. That’s holding back. People like you need to do the thinking and we need to know what you think.
So let’s hear it. What are you? Structuralist, Transcendentalist, Positivist, Rationalist? There’s a whole menu to pick from.
Hi detrope:
I don’t know if you are serious, or just giving me shit.
If you are serious, I would like to point out that I have made some spectacular posts on ILP. No one reads them of course (except zeno in days gone by, and possibly Obw now). I had to beg To Wander is to Wonder to read my post on Evolutionary Mechanisms (possibly my best post), and my series on Models is intended to be a group discussion on the limitations of mankinds’ ability to conceptualize his world.
Are you really serious?
Well Ed, now I feel like an ass because I cannot recall any of your “philosophical” posts or threads and only remember ones dealing with mathematics and such.
This is partly due to my not reading most of your posts…but this in no ways means I am not interested in your posts. It is a matter of consequence and irony, as I prefer to read intelligible posts like your own.
I am sorry, and I was only punching you in the arm with the post above.
[goes and sits in a corner]
Hi détrop,
Thanks for the response.
Your post did get me thinking about what we each have to offer. I am not as articulate as a number of other posters, nor am I as well read. In particular I think that you are much better read than I. If we listen we can all learn something.
Any way thanks for your response.
Hi debaser,
Speaking of listening, I now believe that the halting problem technique probably could be of central use in a proof of GIT. On page 39 of détrop’s reference there is Theorem 3 which reads: No Consistent, sufficiently strong, axiomatized formally theory of arithmetic is decidable. Use of the halting problem technique plays a central role in this proof and it seems obvious to me that it could be central in an alternative proof of GIT. Apparently there is more than one way to skin a cat.
Yes I have always done. I’ve previously mentioned Ed3 as one of my favourite posters. It is a sad fact that anything remotely tricky to get into is likely to be ignored by most.
Hi Ed,
I took a while getting back to this but i just read your post from a few days ago…
Thhis sounds dangerously close to Set theory which Russell turned upside down with his paradox, and Godel subsequently proved why Russell was able to do that and why any similar system would be doomed to the same fate.
It seems stange that he would be a platonist when he proved that it would be impossible to have a complete system to describe a platonic realm. Well, i suppose it’s not that strange… I would have thought he would have addressed mathematics as symbol manipulation. Do you have any references for me to follow up?
That’s funny, Ed.
Don’t you wish.
[laughing]
I can barely spell calculas.
I bet you have a garage full of inventions and levers and devices not unlike a hydromechanized-quantatronic stabalizer, don’t you?