From: Gerald O'Connell <goc.nul> Date: Mon, 7 Nov 2011 01:07:00 +0000 Archived: Mon, 07 Nov 2011 07:44:04 -0500 Subject: Re: Can't Stop Seeing UFOs >From: Michael Tarbell <mtarbell.nul> >To: post.nul >Date: Wed, 02 Nov 2011 16:50:48 -0700 >Subject: Re: Can't Stop Seeing UFOs >>From: Gerald O'Connell <goc.nul> >>Date: Wed, 2 Nov 2011 15:34:23 +0000 >>To: post.nul >>Subject: Re: Can't Stop Seeing UFOs >>>From: Ray Dickenson <r.dickenson.nul> >>>To: <post.nul> >>>Date: Tue, 1 Nov 2011 20:00:17 -0000 >>>Subject: Re: Can't Stop Seeing UFOs >>>Godel's theorem says that mathematics is merely human, and >>>anyway deeply flawed). >>Godel's incompleteness theorems actually show that it is claims >>for the completeness and consistency of axiomatic systems that >>are deeply flawed, not mathematics itself. Godel discredited the >>lingering neoplatonism that underpins the view that mathematics >>represents some sort of perfect, absolute truth. Post Godel >>mathematics still works perfectly well, but we need to have a >>much more sophisticated view as to its limitations and the >>claims that can made by it or supported by it. ><snip> >>There are all sorts of interesting implications. The ones that >>interest me revolve around the status of axioms in logical >>structures. Think of physics as a coherent set of interconnected >>logical structures. The fundamental laws upon which these >>structures are built act as the meta-system's axioms. The >>validity of the whole structure cannot be separated from the >>validity of the axioms. If the axioms can be shown to be >>variable, then the ability of the structure to deliver absolute >>certainty is compromised. So certainty is shown to be non- >>absolute and to be conditional upon a given particular 'state' >>of the axioms, and those 'states' can vary. >>Implication: if these states can vary then, theoretically at >>least, the 'laws of nature' can be changed by human action. >>Raise this point with physicists and there will be a stampede to >>point out why this cannot happen. Inductive reasoning leads me >>to conclude that the only reason it cannot happen is that we >>haven't yet worked out how to make it happen. >>This is not the same as saying that the laws of physics/nature >>are flawed. They work pretty well. It's just that people haven't >>properly come to terms with what they amount to, and that they >>cannot be relied upon to deliver absolute truths and absolute >>certainty in quite the way that those people would like. >Hi Gerald, >This is intriguing, but I think it may be expanding Godel's >result beyond its scope, as well as conflating two distinct >activities, namely mathematics (which produces proofs) and >physics (which does not, although it incorporates mathematical >proofs to support hypotheses). >For example, the axioms of geometry were effectively "shown to >be variable" by the introducing the concept of curved space, but >by no means did this compromise our certainty in the set of >results derived from the original Euclidean postulates. Whether >space is curved or not is an _empirical_ question, properly in >the domain of physics, which may be resolved to greater or >lesser levels of confidence, but will not be 'proven' in the >same sense that the Pythagorean Theorem has been proven, with >absolute certitude, in the context of the axioms of Euclidean >geometry. >Godel showed that for any sufficiently robust formal language >(including the mathematics with which we codify physics), there >can be true statements expressed in that language that are >unprovable. This clearly means that any such formal system is >incomplete (hence the name of the theorem), but that is not the >same thing as "uncertainty", in the sense of doubt being cast >upon established proofs. >As to whether "the 'laws of nature' can be changed by human >action", I would submit that, if true, physicists would simply >generalize nature's 'laws' into a broader context that >incorporates human influence. But I would dispute that human >action can invalidate mathematical proofs (except trivially, by >exposing an error in the derivation). In this sense I think that >mathematics is legitimately categorized as metaphysical. Good analysis Mike. I don't disagree with it at all. However, I do feel that there are some profound implications that follow on from Godel's theorem. It does offer a formal and rigorous demonstration of the fact that logical systems will always be pulling themselves up by their own bootstraps. They can't deliver 'truth' independently of the axioms on which they are built. Your point about certainty 'in the context of the axioms of Euclidean geometry' is a neat instantiation of this. Change the axiomatic baseline and we suddenly find, for example that there are more than 180 degrees in a triangle. Part of my point was that it is all too easy to overlook this fact. My feeling is that in the culture of science there can be a tendency to do this - largely based on the extraordinary success of post-enlightenment thinking in describing and predicting events in the world as we are able to perceive it. Mathematical 'truths' have been so central to this success that it becomes easy to forget the distinction that you (correctly) point out between mathematical proofs and scientific hypotheses, and equally easy to assume that scientifically derived 'laws of nature' are rather more fixed than they really are. It is clear, Mike, that you don't fall foul of this syndrome! If you'll forgive me straying into even deeper waters, another aspect of Godel's theorem that has intrigued me for some time has been what it has to say about the consistency of axiomatic systems. We take for granted (as a hidden assumption, a kind of 'meta-axiom') the need for consistency in our mathematical and scientific thinking, and it goes against the grain to question this. I've thought about this issue from time to time, and wondered about the effects on our ability to describe the world if: a) we demand that our descriptions are consistent, and b) the world actually turns out to be inconsistent. It's just a thought, and a wild one at that, but has Godel unwittingly uncovered a clue as to what we might need to give up in order to match our description to the world? Evidence suggesting that the laws of nature might vary across space and time hasn't helped me get this troublesome thought out of my head... -- Gerald O'Connell http://www.saatchionline.com/gacoc Listen to 'Strange Days... Indeed' - The PodCast At: http://www.virtuallystrange.net/ufo/sdi/program/ These contents above are copyright of the author and UFO UpDates - Toronto. They may not be reproduced without the express permission of both parties and are intended for educational use only.
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