There’s Something about Gödel, Chs 10 & 11

After a couple of rather unsatisfactory philosophical chapters, Berto does quite a bit better in the next two. (Which is good, because these comments were getting unseasonably grumpy!)

Ch. 10 is called “Mathematical Faith”, and discusses what we should learn from the unprovability of consistency of a theory (a consistent theory containing enough arithmetic) within that theory. Does it show that the mathematician has to rely on blind faith in some worrying sense? To which the right answer is “no!”. Here Berto pretty closely follows a good discussion by Franzen. There’s nothing that adds much to Franzen’s similarly introductory discussion, but equally Berto doesn’t go astray.

Ch. 11 is on the Lucas/Penrose argument. Again Berto’s discussion is sane and sensible. The ur-Lucas argument is sabotaged by the familiar Putnam riposte. Souped up versions are sabotaged by souped up versions of that riposte. But it remains that something can be learnt from Gödel incompleteness about the nature of the mind — namely the disjunctive conclusion of Gödel’s Gibbs lecture (prefigured also in Benacerraf’s old discussion). This is rapidly done, though: for example, what on earth will the beginner with the thin background Berto is officially presupposing make of the invocation of transfinite ordinals on p. 183? Still, this chapter could make for helpful introductory reading for some students working towards on an essay on this topic.

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2 Responses to There’s Something about Gödel, Chs 10 & 11

  1. Rowsety Moid says:

    I suspect that you may be heartily sick of the Lucas/Penrose debate by now, but from my point of view, as an observer, there doesn’t seem to be a clear explanation of what Penrose gets wrong that is also agreed to be the right explanation. I’ve seen a number of different claims over the years.

    In Torkel Franzen’s Incomplete Guide, he ends up saying that commentaries on Penrose’s “second argument” “differ considerably both in their attempted reconstructions of the argument and in their diagnoses of where the error lies, and then he dismisses the argument because Penrose says he does not regard it as the ‘real’ Godelian reason.

    That’s cheating. There is an intellectual obligation to consider the best versions of arguments, rather than dismiss an argument because of something the person who happens to be making it happens to say. This illustrates a problem that often comes up in comments on Penrose. Is the aim to find out the truth about the issues, or is it to show that a particular person – Penrose – is wrong? It ought to be the former, but often it slips from that into the latter.

    You mention the “Putnam riposte”. I thought that might be what Putnam said in his N Y Times review of Shadows of the Mind, that we might not understand the program that simulates the relevant mathematical capabilities:

    … the possibility exists that each of the rules that a mathematician relies on, explicitly or implicitly, can be known to be sound, that there is a program that generates all these rules and only these rules, and that this program nonetheless cannot be rendered sufficiently perspicuous for us to know that that is what it does. …

    Mr. Penrose almost discusses the possibility of a program that can capture our mathematical capabilities without our being able to understand it, but in fact he misses it.

    In your book (2008 version, p 260) it seems to be that we might not even know it is.

    If I look at what seems to correspond to Putnam 1960 in his Mind, Language and Reality v 2, it seems to be that it’s “unlikely” that one can prove T (the relevant turing machine) is consistent “if T is very complicated” — which I can then see may be what you meant on p 260.

    But if that is Penrose’s error, why doesn’t everyone just say so? Why do they so often write long discussions instead, including ones that seem to point to something else as supposedly being the error?

  2. Compliments on a very nice a useful website!

    If I am allowed, I would like to ask an impertinent question.

    Concerning logic and philosophy, I am not a professional by any stretch. By education I am a physicist and work as applied mathematician and engineer. I came to study this literature recently as I kept pursuing a decade of research coming from as far away as finance and going through the development of constructive methods in probability theory.
    Wanting to understand issues of numerical robustness of certain methods I had developed, I wrote a few papers on constructive methods in the theory of stochastic differential equations and stochastic integrals using renormalization group methods, in the constructive style of mathematical physicists.

    As I kept working on a book to collate this work, I realized that – to be fully constructive – I had to push the analysis down to the Foundations of Mathematics. A random real number itself is a non-constructive concept. To accept real random numbers in a Constructive Mathematics context, one clearly needs to go beyond recursive Arithmetics and accept stochastic recursion as a primitive concept. In light of quantum computing this is physically possible, so I think there is hope and I am making good progress at understanding my problem now.

    But the more I try to understand Godel, the Hilbert program and Classical Mathematics, the more I am puzzled by all this. The only philosopher I can relate to is Wittgenstein. What he says I think makes perfect sense and is very plausible, even more so than Brower’s intuitionism.

    To explain Wittgenstein’s thought the way I intend it, Mathematics is absolute and objective only if the language on which it is based is interpretable by physical computing machines as a programming language. Of course the human brain doesn’t carry out Arithmetics functions well, it excels at a form of pattern recognition which is not entirely machine interpretable. Pattern recognition is very useful for the purpose of inventing new mathematics, but it is not itself formal Mathematics unless it is mappable in a faithful way to machine code. Pattern recognition is also the key to all mystic knowledge, which of course is not Mathematics because it is subjective, non quantitative, etc..

    In particular, Aristotle was absolutely right distinguishing between potential and actual infinite, saying mathematical discourse should only dwell on the former. I disagree with Brower when he says it is legitimate to refer to transfinite ordinals or build functions dependent on the truth of the Goldbach conjecture. I think Aristotle [and Wittgenstein] got it just right and I am puzzled as of why they have been dismissed so badly.

    As any programmer would immediately grasp, any logic formula that refers to actual infinites is ill posed as a logic proposition because it is not verifiable. It acquires a meaning only when it is proved as a Theorem. Without a proof, a proposition with quantifiers over actually infinite classes lives in the limbo of abstract nonsense.

    All this may seem conservative and old fashioned, but it makes perfect sense to me. I would go even further and say that anyone who wrote over a million lines of code professionally would find this natural and intuitive. The axiom of choice or transfinite ordinals may be intuitive to others of course, intuition is subjective. As far as engineers are concerned, the Classical Mathematics story is however really out of touch with reality.

    If I am allowed an impertinent and a bit provocatory question to you and the guests on this site, I would ask if you think I am wrong. It’s quite probable you think so, but can you mention any reason, technical or philosophical or practical, anything at all, why an engineer should pay any interest to Godel’s work?

    Perhaps there is no reason and it doesn’t matter, really. As long as a division of labour between mathematicians and engineers persist, what matters is that the former are educated to appreciate the fine points of Godel’s philosophy. Is social division of labour through linguistic division what this is all about?

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