Chaitin -“The Limits of Reason”

Gregory Chaitin has written an article “The Limits of Reason” in the latest Scientific American (March 2006) (the obligatory link to scientific american is www.sciam.com). As an aside, Australian newsagencies sell editions of Scientific American about 2 months after their publication date, but I receive it promptly in the mail as I subscribe to it – very nice.

Chaitin writes that there cannot be any “theory of everything” in mathematics as there exist pieces of mathematics (eg his number Omega) that cannot be effectively compressed into a theory. While I don’t fully understand his article (I’ve only read through it quickly), I recall his previous article introducing Omega in Scientific American a few years ago made sense.

In the “Overview box”, the article is summed up (in the first dot point) by:

Kurt Godel demonstrated that mathematics is necessarily incomplete, containing true statement that cannot be formally proved. A remarkable number known as omega reveals even greater incompleteness by providing an infinite number of theorems that cannot be proved by any finite system of axioms. A “theory of everything” for mathematics is therefore impossible.

Interesting! Previously I’ve felt that mathematics could be seen as a beautiful gem, that different aspects of it were like different facets of the gem, and that there was an overall goal of unifying all the different aspects of mathematics, but I havn’t felt like this for a few years. Now I’m more interesting in constructing bits of mathematics.

7 Responses to Chaitin -“The Limits of Reason”

  1. GMB says:

    It doesn’t really matter if he’s right. That would be my main point. Maths is still a beautiful gem.

    I’ve come to the realisation that one of the major sources of dogmatism is the felt need to prove things to the nth degree by deductive reasoning or else trash them. I think that this has some historical implications.

    How you doing Sasha. I followed your link from Catallaxy.

    Fear not. I won’t fill up your site with polemics.

  2. Sacha says:

    No probs. Yes – Chaitin’s work is very interesting – and maths is even more interesting with results such as his. Who knows what people will come up with next?

  3. GMB says:

    “Who knows what PEOPLE will come up with next?

    People hey? What about you Sacha? Why not you? You going to come up with stuff?

    There is certainly a lot of neglected stuff that needs attending to.

  4. Sacha says:

    “People” includes me 🙂 Hopefully I’ll come up with something extra-unusual or solve a problem such as how turbulence emerges from the Navier-Stokes equations (the equations of fluid flow). Or even coming up with a full theory of quantum gravity. Ha! Now that would be fun.

    While pure maths is quite interesting, it often feels a bit too disconnected from people – I’d like to apply maths to things/problems that people are interested in. Actually, I feel that a bit about theoretical physics/pure mathematics – maybe I’m too interested in people and things in society. But maybe this the problem with whatever job you do – you always have other interests so it’s all about balancing everything.

  5. GMB says:

    How familiar are you with multivalence logic? That is to say fuzzy logic? It has more applicability to the problems of PEOPLE then bivalence logic as it is more analogous to the wider scope of human reason as opposed to bi-valence or traditional logic.

    Its being dismissed as a fad by ideologues. And as I said it is the over-reliance on deductive reasoning that is a major source of dogmatism. So that now “fuzzy’ logic is mostly just being used on end products. Not even on the process to make these end products. People get stooged by the terminology.

    If you can apply this science to areas that no-one would even imagine that heretofore it could be applied then you will be making great breakthroughs. Since there is a disconnect between the sort of mindset that programmers and mathematicians have and the sort of people that would involve themselves with fuzzy logic.

    Plus the commercial possibilities are endless. At our factory we have specific software for the lines that pump out the bagged goods. If I could get it running on a “forgiving” fuzzy logic alogarithm then one line could pump out the work that two currently does.

    There has to be a certain ‘economy of scale’ here. Where you just go on the general principle that you apply it in all areas where it can be applied and then try and apply it to areas where at first you think it CAN’T be applied. And if you are successful in the latter you Sacha will have your breakthrough.

  6. Sacha says:

    I havn’t read about multivalent logic apart from the defn of fuzzy logic – it sounds interesting. Agree that people are very “nonbinary” in the way they think – mmmmmm – I’ll have to read up on this – espec if there are potential applications!

  7. GMB says:

    Fuzzy logic overcomes the problems of deductive logic. But we need the discipline and dexterity of the mathematicians and the computer modellers to be able to get real economies of scale out of this sort of stuff.

    And this is where the disconnect lies. Because they often are just the sort of people to dismiss this stuff as faddish and junk science.

    Just to see the slight difference this way of thinking (thinking with the analogy of Venn Diagrams and fuzzy logic in mind) just to give you a feeling for the lesser rigidity of it check out the way I’m debating amongst the Randians on this website.

    See if you can see a slightly different approach:

    http://rebirthofreason.com/cgi-bin/SHQ/SHQ_User.cgi?Function=FullPostList&UserInfoNumber=2289

    The altruism ones might show you just a hint of what I’m driving at.

    To make thosee breakthroughs that I’m confident you’ll make you have to go for a similiar leap into the unkown that those guys did when they started working with negative integers.

    Its like at some stage they figured they were doing ridiculous and vain stuff because the maths was running ahead of things in the real world that they’d figured out could even justify negative integer multiplication for example.

    But they’d built up a certain process of working through the problems. And at some times by following this sort of order of operations they’d be sort of leaping into the dark. Not putting their faith in the reality of the numbers but putting their faith in the process.

    So you have the start of a problem firmly grounded in reality, then in the middle of the problem its a sort of a leap, but then they find they are getting answers, grounded in reality coming out at the other end of the process. And its the middle part that they would have thought (at the time) was something akin to voodoo.

    This is sort of what I’m asking you to do Sacha. In the interests of you making all kinds of breakthroughs and showering great gifts on the world.

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