[om-list] Schroedinger's Cat Is All Locked Up

[Thomas L. Packer at home]

2002.09.08/Sun
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Hello Jared Norman

    It was good to see you yesterday "at Peach Days" in Brigham City, and to bump into your wife and two kids at the same time.  Did they watch the parade without you?

    So, yes, I'm at BYU now.  I've wanted to go into AI and knowledge modelling for the longest time, and I feel as though BYU is a good place to do so, but when I applied they had no professors who specialised in knowledge modelling.  Being able to tailor your degree to a particular subject within a discipline depends a lot on what professors are around to become your adviser.

    The closest professor-supported topics BYU had when I started examining them were machine learning and data extraction, one of which I thought I could probably fudge enough, as a degree topic, to cover knowledge modelling, especially data extraction since it involves natural language processing and, in my opinion, any good language processing algorithm would include a semantic model somewhere.  So in the letter-of-intent part of my application I named "data extraction" as my intended area of research, emphasising the information integration part of it.

    I await their decision.  I will hopefully know in a month or two that I was accepted into the program.

    In the mean time, I have discovered that BYU's CS department just hired two new professors.  One is a statistical linguist, which means that, as far as computational linguistics is concerned, I will have more options than I first thought.  I have not heard as much about the other professor, but it sounds like he's into A.I. of one flavour or another.  This may mean I have much more options that I first thought.

    The silly thing about me is this: I'm interested in too many things.  Just as I think I have decided on a topic, I want to change my mind.  I personally believe that all these interesting subjects are necessary steps toward my big goal of making good A.I. and cognitive modelling algorithms, but my lack of focus during my undergraduate years makes it hard for me to get into a degree program or to get a decent job.

    The rest of this letter is about why I want to change my mind this time.

    Just last night, as I was falling asleep, an idea made me want to forget about doing anything with language for a while, and go back to one of my alternative interests: the logical foundations of mathematics and the mathematical foundations of computer science and AI, which seem to be closely tied together as far as theoretical computer scientists are concerned.  At BYU, computer science is full of theory, and I love it.

    Before I can explain why I was too excited to fall asleep last night, I need to insert a few nested "flash backs".

    A few years ago, probably while I was reading about a fascinating subject like the logical foundations of mathematics, trying to develop what I call the "physical epistemology" part of "Em-Veh", a thought hit me.

    I had already decided years before that there were only two types of physical learning and two corresponding types of physical proof: induction and deduction.  This means that any idea you have, if it's provable, is provable using induction, deduction, or a mixture of both.  I think the whole world will agree with that.  I know my grandpa used this idea a generation or two ago when he confounded an evolutionist professor at BYU who had come into his office to argue with him about evolution, only my grandpa called them "synthetic" and "analytic" forms of proof.

    Then several years ago, in my first class on logic at WSU, I decided that deduction was dependent on induction.  That seemed interesting, but not a big deal.  Even though you could not prove anything solely by deduction, you could start with induction, build up a few principles, and then prove things by deduction.  That would mean that any idea you have, if it's provable, is provable either directly or indirectly by induction alone, since you can't use just deduction anymore.

    The thought that struck me as being truly significant about three years ago was this: "What if induction were also dependent on deduction, in some analogous, beautifully symmetrical way?"  (I'm always looking for symmetries when I think about Em-Veh.  It helps me find missing pieces I would not have come up with on my own.)  That would mean there's nothing left.  I had first removed deduction as a possibility for the foundations of proof, and then I was asking if I could remove induction as well.

    The question was staggering.  The implications would be enormous.  For one thing, it would mean that it is impossible to learn or even to think if you are just a physical thing, like a computer or like most people believe human mind/brains are.  In order to put knowledge or truth into a system (envision a box drawn on a white-board) you have to put truth into that box from the outside (envision an arrow from outside to inside).  The box is physical epistemology and it is seen as a closed system by physicalist monists / humanists.  But their closed box is empty until some sort of truth is injected into it from the outside.  But what is outside of a physical box?

    To a Mormon, this question has an immediate answer in the form of spirits, The Spirit, and the Light of Christ.

    After I asked myself the question "What if induction were dependent on deduction?", I thought about it in the context of neurology and organic learning.  "How do people use sensations to build up principles inductively?"  After a few minutes I was convinced that this was true: Induction *is* dependent on deduction.  

    The answer was so profound, I just sat for a while -- basking in the glow of my own intelligence ... not.  I then knew where that thought had come from, and where every other thought came from, with more conviction than I previously had in church.

    To formally prove this idea for all the world to see and believe would be very hard indeed, but to bridge the mind of my incredulous reader to such an outrageous idea would be even harder, despite having a "proof" for it.  Imagine that you were told that some guy from Honeyville had a formal "proof" that there was no such thing as "proof".  You get the idea.

    So I sort of put it on the back burner with all my other nodes and internodes and meta-internodes, not knowing when I might have the chance to attempt it.

    Three years passed, and learning glommed onto learning, and ideas onto ideas, and once a certain idea called the Incompleteness Theorem finally glommed onto my own three-year-old idea in the right way last night, I was ready to go back to it.

    For those of you who have not studied the Incompleteness Theorem in your Foundations of Mathematics courses, allow me to attempt an explanation.

    You've probably heard of Kurt Gödel, of "Gödel, Escher, Bach" fame.  "Gödel, Escher, Bach: The Eternal Golden Braid" is a cult classic among A.I. enthusiasts; a book written by Douglas Hofstadter a few decades ago about the uses paradoxes and seaming contradictions found in certain repetitive patterns, including infinite patterns, such as in the mathematical proofs of Kurt Gödel, the never-ending staircases in the art of M. C. Escher, and the intricate and complex music of J. S. Bach.  You'd enjoy this book if you haven't already read it.  My artist sister Laura probably would, too, simply because of her interest in Escher.

    Well, Kurt Gödel wrote a mathematical proof early last century that disrupted all of mathematics and the computer science to follow: it's called the Incompleteness Theorem.  It's a theorem that sort of turned Russell and Whitehead's famous "Principia Mathematica" on its ear either right after they published it, or as they were writing it, I forget which.

    The readers' digest version of the proof would say something like this: Given some logical system advanced enough to contain the natural numbers, there will always be certain ideas that are "true" within that system but which cannot be proved by that system.

    For my field, apparently one of the implications of this theorem is, it is impossible to write a computer program able to solve all computable problems.

    What does this have to do with my three-year-old idea?  Well, I've often thought the idea had some connection with the Incompleteness Theorem, but I didn't know what, and I don't think the proof of my idea would resemble Gödel's proof at all.  But whenever I would hear people talk about Gödel and his theorems I would always get this burning inside me, like, "I need to do something like that.".  Until yesterday, I had assumed that maybe these feelings meant that I should disprove his proof.  Gödel's proof is such a distasteful proof after all.  I mean, who likes to be told that his chosen field of mathematics, logic, or the A.I. based on them is inherently limited, even in theory, to say nothing of putting it into practice, and that there's no way around this limitation.

    Image this yourself:  You're in the business of proving things.  Then out of the blue you are told, by a brilliant, and yet nervous, anorexic mathematician, that there are certain things in your logical system that you cannot prove nor disprove, even though they are definitely true or false and they are fundamental parts of your logical system.

    It had been many months since I had thought about this Incompleteness Theorem because the practicalities of life have been "distracting" me.  But luckily for me in my new theoretical setting I have a young, enthusiastic, and interesting professor named Dr. Dan Ventura who teaches one of my computer science classes this semester and who did his doctoral dissertation on something like A.I. and machine learning.  Some of the members of my class know this and are more interested in these advanced subjects than the actual class topic.

    A key moment happened during Friday's lecture, right before these class members totally side-tracked his lecture from introductory computational theory (the class topic) to quantum computers (one of his pet topics).  At this moment, Dr. Ventura summarised Gödel's Incompleteness Theorem as an answer to someone's question about why no one can build a program that can compute everything computable.  I've heard several summaries of this theorem, but they way he ended it really woke me up.  After saying that no logical system can prove every part of itself, he added: "More information needs to come from outside the system.", and while saying this he drew an arrow on the white board from the outside of a box to the inside.

    This totally woke me up because that last statement was totally analogous to the implications of my little idea, and I had never before heard anyone say anything so familiar in this context.  While I was quite amazed with this idea, the class went on with it's own ideas unrelated to the subject of induction and deduction.  Dr. Ventura's box soon turned into "the box that contained Schrödinger's Cat", since his new topic was quantum computers.  Interesting class ...

    So, I didn't have much time to ponder the significance of my feelings for a couple of days.  It was last night while I was trying to fall asleep when it hit me: this idea of mine is a super-set of the Incompleteness Theorem, it is an extension of the Incompleteness Theorem, it is bigger and more inclusive.  If I can prove this idea, I will have proven not only that there are certain ideas that are true and improvable in any logical system, I will have proven that *all* ideas in any logical system, no matter what kind of logical system, no matter what kind of ideas, cannot be proved by that logical system.  Furthermore, this hypothetical logical system includes all of human thought, inasmuch as it is physical or performed by physical mechanisms, such as the brain, a computer, or society as a whole.  

    Further-furthermore, it will prove that, if we as human beings have ideas that are true or that seem true simply because of logic, the way we reached those ideas was one big circular argument ... and every body knows that circular arguments are false, right?

    If I go for it and really prove this theorem, which we could call the "Ultra-Incompleteness Theorem", then one of the smaller things it will do for logicians and thinkers and debaters is to totally confuse them about circular arguments, because if we are bound to a physical i.e. a logical system, then there is no valid argument which is *not* circular.  Are you starting to see how outrageously exciting the idea is, and why I could not fall asleep last night?

    The funny thing is, I've known how revolutionary and important this idea might be for a few years now, but it is of course so off-the-wall that I didn't think I had a chance in the world to try to publish it.  I would have to have some way of bridging the world's minds from "common sense" to a whole new world where your thoughts are not your own.  It seemed impossible until last night when I decided that the world already had that bridge in the form of Gödel's Incompleteness Theorem.

    In addition, I've never been on the brink of starting a graduate-level degree at a semi-respectable university until now.  

    So, last night I thought, "Maybe it's time for me to practice writing formal proofs instead of building a semantic-based search engine.".

    I'm a little scared as I write this letter because I *want* to prove it, I think I *can* prove it, but I know the world will not like it, and may not even give the idea a chance.  Who knows, maybe I will even be persecuted for such a radical idea.  But I feel it's so significant and so true, and I just need to prove it.  As I look back over the past few years, I realise that much of my other thoughts and programming ideas are based on this idea, so I feel like maybe it's necessary that I try to prove it.  I'll be holding many of my future thoughts back because their acceptance is dependent on the acceptance of the "Ultra-Incompleteness Theorem".

    We'll see.

    Well, hope you check your email.  If not, at least my parents might read this. :-)

ciao,
tomp

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Víðar sum quem nihil obstat.
www.Ontolog.Com
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