on December 24, 2007
It might justly be asked what importance Gödel's proof has for our work. For a piece of mathematics cannot solve problems of the sort that trouble us.--The answer is that the situation, into which such a proof brings us, is of interest to us. 'What are we to say now?'--That is our theme. However queer it sounds, my task as far as concerns Gödel's proof seems merely to consist in making clear what such a proposition as: 'Suppose this could be proved' means in mathematics.'
Wittgenstein 'Remarks on the Foundations of Mathematics'
p337(1956) (written in 1937).
'My theorems only show that the MECHANIZATION of mathematics, ie., the elimination of the mind and of ABSTRACT entities, is impossible, if one wants to have a satisfactory foundation and system of mathematics. I have not proved that there are mathematical questions that are undecidable for the human mind, but only that there is no MACHINE (or BLIND FORMALISM) that can decide all number-theoretic questions, (even of a very special kind)....It is not the structure itself of the deductive systems which is being threatened with a brakedown, but only a certain INTERPRETATION of it, namely its interpretation as a blind formalism.'
Gödel "Collected Works" Vol 5, p 176-177.(2003)
'Superstition is nothing but belief in the causal nexus.' Wittgenstein TLP 5.1361
"Now if it is not the causal connections which we are concerned with, then the activities of the mind lie open before us." Wittgenstein "The Blue Book' p6 (1933)
'We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched. Of course, there are then no questions left, and this itself is the answer.' Wittgenstein TLP 6.52 (1922)
I have read some 50 reviews here and on the net (that by quantum physicist David Deutsch was perhaps the best) and none of them provide a satisfying framework, so I will try to give novel comments that will be useful, not only for this book but for any book in the behavioral sciences (which can include ANY book, if one grasps the ramifications).
Like his classic Gödel, Escher, Bach: the Eternal Golden Braid, and many of his other writings, this book by Hofstadter (H) tries to find correlations or connections or analogies that shed light on consciousness and all of human experience. As in GEB, he spends a great deal of time explaining and drawing analogies with the famous 'incompleteness' theorems of Gödel, the 'recursive' art of Escher and the 'paradoxes' of language (though, as with most people, he does not see the need for quotes, and this is the core of the problem). The idea is that their seemingly bizarre consequences are due to 'strange loops' and that such loops are in some way operative in our brain. In particular, they may 'give rise' to our self, which he seems roughly to equate with consciousness and thinking. As with everyone, when he starts to talk about how his mind works, he goes seriously astray. I suggest that it is in finding the reasons for this that the interest in this book, and most general commentary on behavior, lies.
I will contrast the ideas of ISL with those of the philosopher (armchair psychologist) Ludwig Wittgenstein (W), whose commentaries on psychology, written from 1912 to 1951, have never been surpassed for their depth and clarity. He is an unacknowledged pioneer in evolutionary psychology (EP) and developer of the modern concept of intentionality. He noted that the fundamental problem in philosophy is that we do not see our automatic innate mental processes. He gave many illustrations (one can regard the entire 20,000 pages of his nachlass as an illustration), some of them for words like 'is' and 'this', and noted that all the really basic issues usually slip by without comment. A major point which he developed was that nearly all of our intentionality ( roughly, our evolutionary psychology (EP), rationality or personality) is invisible to us and such parts as enter our consciousness are largely epiphenomenal (ie, irrelevant to our behavior). The fact that nobody can describe their mental processes in any satisfying way, that this is universal , that these processes are rapid and automatic and very complex, tells us that they are part of the 'hidden' cognitive modules (templates or inference engines) that have been gradually fixed in animal DNA over more than 500 million years.
As in virtually all writing which tries to explain behavior (philosophy, psychology, sociology, anthropology, history, politics, theology, and even, as with H, math and physics ) , I am a Strange Loop (ISL) commits this kind of error (oblivion to our automaticity) continually and this produces the puzzles which it then tries to solve. The title of ISL comprises words we all know, but as W noted, word uses can be seen as families of language games (grammar) which have many senses (uses or meanings), each with its own contexts. We know what these are in practice but if we try describing them or philosophizing (theorizing) about them, we nearly always go astray and say things that may appear to have sense but lack the context to give them sense. It never crosses Hofstadter's mind that both 'strange' and 'loop' are out of context and lack any clear sense (to say nothing about 'I' and 'am'!). If you go to Wikipedia, you find many uses (games as W often said) for these words and if you look around in ISL you will find them referred to as if they were all one. Likewise for 'consciousness', 'reality', 'paradox', 'recursive', 'self referential', etc. So, we are hopelessly adrift from the very first page, as I expected from the title. A loop in a rope can have a very clear sense and likewise a diagram of a steam engine governor feedback loop, but what about loops in mathematics and the mind? . H does not see the 'strangest loop' of all'that we use our consciousness, self and will to deny themselves!
Regarding Gödel's famous theorems, in what sense can they be loops? What they are almost universally supposed to show is that certain basic kinds of mathematical systems are incomplete in the sense that there are 'true' theorems of the system whose 'truth' (the unfortunate word mathematicians commonly substitute for validity) or 'falsity (invalidity) cannot be proven in the system. Though H does not tell you, these theorems are logically equivalent to Turing's 'incompleteness' solution of the famous halting problem for computers performing some arbitrary calculation. He spends a lot of time explaining Gödel's original proof, but fails to mention that others subsequently found vastly shorter and simpler proofs of 'incompleteness' in math and proved many related concepts. The one he does briefly mention is that of contemporary mathematician Gregory Chaitin'an originator with Kolmogorov and others of Algorithmic Information Theory-- who has shown that such 'incompleteness' or 'randomness' (Chaitin's term-- though this is another game), is much more extensive than long thought, but does not tell you that both Gödel's and Turing's results are corollaries to Chaitin's theorem and an instance of 'algorithmic randomness'. You should refer to Chaitin's recent writings such as 'The Omega Number(2005)', as Hofstadter's only ref. to Chaitin is 20 years old (though Chaitin has no more grasp of the larger issues here --ie, innate intentionality as the source of the language games in math-- than does H and shares the 'Universe is a Computer' fantasy as well).
Hofstadter takes this 'incompleteness' (another word (conceptual) game out of context) to mean that the system is self referential or 'loopy' and 'strange'. It is not made clear why having theorems that seem to be (or are) true (ie, valid) in the system, but not provable in it, makes it a loop nor why this qualifies as strange nor why this has any relationship to anything else.
It was shown quite convincingly by Wittgenstein in the 1930's (ie, shortly after Gödel's proof) that the best way to look at this situation is as a typical language game (though a new one for math at the time)'ie, the 'true but unprovable' theorems are 'true' in a different sense (since they require new axioms to prove them). They belong to a different system, or as we ought now to say, to a different intentional context. No incompleteness, no loops, no self reference and definitely not strange! W: 'Gödel's proposition, which asserts something about itself, does not mention itself' and 'Could it be said: Gödel says that one must also be able to trust a mathematical proof when one wants to conceive it practically, as the proof that the propositional pattern can be constructed according to the rules of proof? Or: a mathematical proposition must be capable of being conceived as a proposition of a geometry which is actually applicable to itself. And if one does this it comes out that in certain cases it is not possible to rely on a proof.' (RFM p336). These remarks barely give a hint at the depth of W's insights into mathematical intentionality, which began with his first writings in 1912 but was most evident in his writings in the 30's and 40's. W is regarded as a difficult and opaque writer due to his aphoristic, telegraphic style, but if one starts with his only textbook style work'the Blue and Brown Books --and understands that he is explaining how our evolved higher order thought works, it will all become clear to the persistent.
W lectured on these issues in the 1930's and this has been documented in several of his books. There are further comments in German in his nachlass (some of it formerly available only on a $1000 cdrom but now, like nearly all his works, on p2p). Canadian philosopher Victor Rodych has recently written two articles on W and Gödel in the journal Erkenntnis and 4 others on W and math, which I believe constitute a definitive summary of W and the foundations of math. He lays to rest the previously popular notion that W did not understand incompleteness (and much else concerning the psychology of math). In fact, so far as I can see W is one of very few to this day (and NOT including Gödel!'though see his penetrating comment quoted above) who does.
In any case, it would seem that the fact that Gödel's result has had zero impact on math (except to stop people from trying to prove completeness!) should have alerted H to its triviality and the 'strangeness' of trying to make it a basis for anything. I suggest that it be regarded as another conceptual game that shows us the boundaries of our psychology. Of course, all of math, physics, and human behavior can usefully be taken this way.
H spends a lot of time on Whitehead and Russell's 'Principia Mathematica', since it led to Gödel's work. W had gone from Russell's beginning logic student to his teacher in about a year, and Russell had picked him to rewrite the Principia. But W showed that the idea of founding math (or rationality) on logic was a profound mistake. W is one of the world's most famous philosophers and made extensive commentaries on Gödel and the foundations of mathematics and the mind; is a pioneer in EP (though nobody seems to realize this); the discoverer of the basic outline and functioning of higher order thought and much else, and it is amazing that Dennett &H, after half a century of study, are completely oblivious to the thoughts of the greatest natural psychologist of all time (though they have 6 billion for company
The Eternal Golden Braid is not realized by H to be our innate Evolutionary Psychology, now, 150 years late (ie, since Darwin), becoming a burgeoning field that is fusing psychology, cognitive science, economics, sociology, anthropology, political science, religion, music (see eg, G. Mazzola's 'The Topos of Music'), art, math, physics and literature. H has ignored the vast majority of the insights from philosophy, quantum physics, probability, meditation, EP , cognitive psychology and psychedelics.
In my estimation, neither H nor anyone else has provided a convincing reason to reject the Chinese room argument (the most famous article in this field) that computers don't think (NOT that they cannot ever do something that we might want to call thinking'which Searle admits). And Searle has (in my view) organized and extended W's work in books such as 'The Construction of Social Reality' and 'Rationality in Action'-- brilliant summations of the organization of HOT (higher order thought'ie, intentionality)'rare philosophy books you can even make perfect sense of once you translate a little jargon into English! H, D and countless others in cognitive science and AI are incensed with Searle because he had the temerity to challenge (destroy- I would say) their core philosophy 'the Computational Theory of Mind (CTM) almost 30 years ago and continues to point this out. Of course they (nearly) all reject the Chinese room or simply ignore it, but the argument is, in the view of many, unanswerable. The recent article by Shani (Minds and Machines V15, p207-228(2005)) is a nice summary of the situation with references to the excellent work of Bickhard on this issue. Bickhard has also developed a seemingly more realistic theory of mind that uses nonequilibrium thermodynamics, in place of Hofstadter's concepts of intentional psychology used outside the contexts necessary to give them sense. '
Few realize that W again anticipated everyone on these issues with numerous comments on what we now call CTM, AI or machine intelligence, and even did thought experiments with persons doing 'translations' into Chinese. I had noticed this (and countless other close parallels with Searle's work) when I came upon Diane Proudfoot's paper on W and the Chinese Room in the book 'Views into the Chinese Room' (2005). One can also find many gems related to these issues in Cora Diamond's edition of the notes taken in W's early lectures on math 'Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge 1934(1976). W's own 'Remarks on the Foundations of Mathematics' covers similar ground. One of the very few who has surveyed W's views on this in detail is Christopher Gefwert, whose excellent book 'Wittgenstein on Minds, Machines and Mathematics' (1995), is universally ignored. Though he was writing before there was any serious thought concerning electronic computers or robots, W realized that the basic issue here is very simple---computers lack a psychology (and even 70 years later we have barely a clue how to give them one), and as usual he summed it all up in his unique aphoristic way ' But a machine surely cannot think!--Is that an empirical statement? No. We only say of a human being and what is like one that it thinks. We also say it of dolls and no doubt of spirits too. Look at the word "to think" as a tool.' (Philosophical Investigations p113). Out of context, many of W's comments may appear insipid or just wrong, but the perspicacious will find that they usually repay prolonged reflection'he was nobody's fool.
Hofstadter, in all his writings, follows the common trend and makes much of 'paradoxes', but ny symbolic system we have ( ie, language, math, art, music, games etc) will always have areas of conflict, insoluble or counterintuitive problems or ill definitions. Hence, we have Gödel's theorems, the liars paradox, inconsistencies in set theory, prisoner's dilemmas, Schrodinger's dead/live cat, Newcomb's problem, Anthropic principles, Bayesian statistics, notes you can't sound together or colors you can't mix together and rules that can't be used in the same game.
Virtually none of those writing the hundreds of articles and countless books on these issues which appear yearly seem aware they are studying the limits of our innate psychology and that Wittgenstein usually anticipated them by over half a century. Typically, he took the issue of paradox to the limit, pointing to the common occurrence of paradox in our thinking, and insisted that even inconsistencies were not a problem (though Turing, attending his classes, disagreed), and predicted the appearance of inconsistent logical systems. Decades later, dialetheic logics were invented and Priest in his recent book on them has called W's views prescient. If you want a good recent review of some of the many types of language paradoxes (though with no awareness that W pioneered this in the 1930's and largely innocent of any grasp of intentional context) see Rosenkranz and Sarkohi's 'Platitudes Against Paradox' in Erkenntnis V65, p319-41(2006). Appearance of many W related articles in this journal is most appropriate as it was founded in the 30's by logical positivists whose bible was W's Tractus Logico Philosophicus. Of course, there is also a journal devoted to W and named after his most famous work''Philosophical Investigations'.
W clearly and repeatedly noted the underdetermination of all our concepts (eg, see his comments on addition and the completion of series in Remarks on the Foundations of Mathematics), which mandated their becoming innate (ie, evolution had to solve this problem by sacrificing countless quadrillions of creatures whose genes did not make the right choices). Nowadays this is commonly called the problem of combinatorial explosion and often pointed to by evolutionary psychologists as compelling evidence for innateness, unaware that W anticipated them by over 50 years.
Before any 'explanations'(really just clear descriptions, as W noted) are possible, it has to be clear that the origins of our behavior lie in the axioms of our innate psychology, which are the basis for all understanding, and that philosophy, math, literature, science, and society are their cultural extensions.
Dennett (and anyone who is tempted to follow him'ie, everyone) is forced into even more bizarre claims by his skepticism (for I claim it is a thinly veiled secret of all reductionists that they are skeptics at heart'ie, they must deny the 'reality' of everything). In his book 'The Intentional Stance' and other writings he tries to eliminate this bothersome psychology that puts animals in a different class from computers and the universe by including our innate evolved intentionality with the derived intentionality of our cultural creations (ie, thermometers, pc's and airplanes) by noting that it's our genes, and so ultimately nature (ie, the universe), and not we that 'really' has intentionality, and so it's all 'derived'. Clearly something is gravely amiss here! One thinks immediately that it must then also be true that since nature and genes produce our physiology, there must be no substantive difference between our heart and an artificial one we make from plastic. For the grandest reductionist comedy in recent years see Wolfram's 'A New Kind of Science' which shows us that all is 'computation'--ie, he eliminates psychology by definition.
One sees that Dennett does not grasp the basic issues of intentionality by the title of his book. Our psychology is not a stance or attribution or posit about ourself, or other beings mental lives, any more than it's a 'stance' that they possess bodies. A young child or a dog does not guess or suppose and does not and could not learn that people and animals are agents with minds and desires and that they are fundamentally different from trees and rocks and lakes. They know (live) these concepts (shared psychology) from birth and if they weaken, death or madness supervene.
This brings us again to W who saw that reductionist attempts to base understanding on logic or math or physics were incoherent. We can only see from the standpoint of our innate psychology, of which they are all extensions. Our psychology is arbitrary only in the sense that one can imagine ways in which it might be different, and this is the point of W inventing odd examples of language games (ie, alternative concepts (grammars) or forms of life). In doing so, we see the boundaries of our psychology. The best discussion I have seen on W's imaginary scenarios is that of Andrew Peach in PI 24:p299-327(2004).
W said many times in many ways that we must overcome our craving for 'clarity' , the idea of thought underlaid by 'crystalline logic', the discovery of which will 'explain' our behavior and our world and change our view of what it is to be human.
'The more narrowly we examine actual language, the sharper becomes the conflict between it and our requirement. (For the crystalline purity of logic was, of course, not a result of investigation: it was a requirement.)'PI 107
On his return to philosophy in 1930 he said:
'The wrong conception which I want to object to in this connexion is the following, that we can discover something wholly new. That is a mistake. The truth of the matter is that we have already got everything, and that we have got it actually present; we need not wait for anything. We make our moves in the realm of the grammar of our ordinary language, and this grammar is already there. Thus, we have already got everything and need not wait for the future.' (Waismann 'Ludwig Wittgenstein and the Vienna Circle (1979) p183
and in his Zettel P 312-314
'Here we come up against a remarkable and characteristic phenomenon in philosophical investigation: the difficulty---I might say---is not that of finding the solution but rather that of recognizing as the solution something that looks as if it were only a preliminary to it. 'We have already said everything.---Not anything that follows from this, no this itself is the solution!'
'This is connected, I believe, with our wrongly expecting an explanation, whereas the solution of the difficulty is a description, if we give it the right place in our considerations. If we dwell upon it, and do not try to get beyond it.'
Some might also find it useful to read 'Why there is no deductive logic of practical reason' in Searle's superb 'Rationality in Action' (2001). Just substitute his infelicitous phrases 'impose conditions of satisfaction on conditions of satisfaction' by 'relate mental states to the world by moving muscles''ie, talking, writing and doing, and his 'mind to world' and 'world to mind directions of fit' by 'cause originates in the world' and 'cause originates in the mind'.
Another basic flaw in H (and throughout scientific discourse, which includes philosophy since it is armchair psychology) concerns the notions of explanations or causes. We have few problems understanding how these concepts work in their normal contexts but philosophy is not a normal context. They are just other families of concepts (often called grammar or language games by W and roughly equivalent to cognitive modules, inference engines, templates or algorithms) comprising our EP (roughly, our intentionality) but, out of context, we feel compelled to project them onto the world and see 'cause' as a universal law of nature that determines events. As W said, we need to recognize clear descriptions as answers which terminate the search for ultimate 'explanations'.
This gets us back to my comment on WHY people go astray when they try to 'explain' things. Again, this connects intimately with judgements, decision theory, subjective probability, logic, quantum mechanics, uncertainty, information theory, Bayesian reasoning, the Wason test, the Anthropic principle (Bostrum 'The Anthropic Principle'(2002)) and behavioral economics. In his pre-Tractatus writings, Wittgenstein commented that 'The idea of causal necessity is not A superstition but the SOURCE of superstition'. I suggest that this seemingly trite remark is one of his most profound 'W was not given to platitude nor to carelessness. What is the 'cause' of the Big Bang or an electron being at a particular 'place' or of 'randomness' or chaos or the 'law' of gravitation? But there are descriptions which can serve as answers.
Thus, H feels all actions must be caused and 'material' and so, with his pal D and the merry band of reductionist materialists, denies will, self and consciousness.
This is especially odd in H's case as he started out a physicist and his father won the Nobel prize in physics so one might think he would be aware of the famous papers of Einstein, Podolsky and Rosen and of von Neumann in the 20's and 30's, in which they explained how quantum mechanics did not make sense without human consciousness (and a digital abstraction won't do at all). In this same period others including Jeffreys and de Finetti showed that probability only made sense as a subjective (ie, psychological) method and Wittgenstein's close friends John Maynard Keynes and Frank Ramsey first clearly equated logic with rationality, and Popper and others noted the equivalence of logic and probability and their common roots in rationality. There is a vast literature on interrelationships of these disciplines and the gradual growth of understanding that they are all facets of our innate psychology. Those interested might start with Ton Sales article in the Handbook of Philosophical Logic 2nd Ed. Vol 9 (2002) since it will also introduce them to this excellent source, now extending to 14 Volumes (the first 9 on p2p).
There is a vast literature on causes and explanations so I will only refer to Jeffrey Hershfield's excellent article 'Cognitivism and Explanatory Relativity' in Canadian J. of Philosophy V28 p505-26(1998) and to Garfinkel's book 'Forms of Explanation'(1981). This literature is rapidly fusing with those on epistemology, probability, logic, game theory, behavioral economics, and the philosophy of science, which seem almost completely unknown to H. Out of the hundreds of recent books and thousands of articles, one can start on this with Nancy Cartwright's books, which provide a partial antidote to the 'Physics and Math Rule the Universe' delusion. Or, one can just follow the links between rationality, causality, probability, information, laws of nature, quantum mechanics, determinism, etc in Wikipedia and the online Stanford Encyclopedia of Philosophy, for decades (or, with W's comments in mind, maybe only days) before one realizes he got it right and that we do not get clearer about our psychological 'reality' by studying nature. One way to look at ISL is that its faults remind us that scientific laws and explanations are frail and ambiguous extensions of our innate psychology and not, as H would have it, the reverse.
It is a curious and rarely noticed fact that the severe reductionists first deny psychology, but, in order to account for it (since there is clearly SOMETHING that generates our mental and social life), they are forced into camp with the blank slaters (all of us before we get educated), who ascribe psychology to culture or to very general aspects of our intelligence (ie, our intentionality is learned) as opposed to an innate set of functions. H and D say that self, consciousness, will, etc are illusions'merely 'abstract patterns' (the 'spirit' or 'soul' of the Church of Fundamentalist Naturalism). They believe that our 'program' can be digitized and put into computers, which thereby acquire psychology, and that 'believing' in 'mental phenomena' is just like believing in magic (but our psychology is not composed of beliefs'which are only its extensions-- and nature is magical). I suggest it is critical to see why they never consider that 'patterns'(another lovely language game!) in computers are magical or illusory. And, even if we allow that the reductionist program is really coherent and not circular (eg, we are too polite to point out 'as do W and Searle and many others'that it has NO TEST for it's most critical assertions and requires the NORMAL functioning of will, self, reality, consciousness etc, to be understood), can we not reasonably say 'well Doug and Dan, a rose by any other name smells as sweet!' I don't think reductionists see that even were it true that we could put our mental life in algorithms running in silicon (or-- in Searle's famous example'in a stack of beer cans), we still have the same 'hard problem of consciousness': how do mental phenomena emerge from brute matter? This would add yet another mystery with no obvious way to recognize an answer'what does it mean (why is it possible) to encode 'emergent properties' as 'algorithms'? If we can make sense out of the idea that the mind or the universe is a computer (ie, can say clearly what counts for and against the idea), what will follow if it is or it isn't?
It's dripping with irony that D's most recent book is on the EP of religion, but he cannot see his own materialism as a religion (ie, it's likewise due to innate conceptual biases). Timothy O'Connor has written (Metaphilosophy V36,p436-448(2005)) a superb article on D's Fundamentalist Naturalism.
Emergence of 'higher order properties' from 'inert matter' (more language games!) is indeed baffling, but it applies to everything in the universe, and not just to psychology. Our brains had no reason (ie, there are no selective forces operative) to evolve an advanced level of understanding of themselves or the universe, and it would be too genetically costly to do so. What selective advantage could there have been in seeing our own thought processes? The brain, like the heart, was selected to function rapidly and automatically and only a minute part of its operations are available to awareness and subject to conscious control. Many think there is no possibility of an 'ultimate understanding' and W tells us this idea is nonsense (and if not then what test will tell us that we have reached it)'Perhaps the last word belongs to Wittgenstein. Though his ideas changed greatly, there are many indications that he grasped the essentials of his mature philosophy in his earliest musings. It is a defensible thesis that the structure and limits of our intentional psychology were behind his early positivism and atomism. So, let us end with the famous first and last sentences of his Tractatus, seen as summarizing his view that the limits of our innate psychology are the limits of our understanding. 'The world is everything that is the case.' 'Concerning that of which we cannot speak, we must remain silent.'