On semi-hidden agendas and the misuse of Gödel's theorems

An old friend of mine, a philosophy PhD with a background in literature and theology, told me that it was the work of Kurt Gödel which had drawn her to philosophy. She, like many others (including myself at one stage), saw his famous theorems as vindicating a non-reductionist and perhaps even religious view of the world.

I don't think this view is correct. A mythic narrative has been built up around Gödel's work which needs to be treated with great skepticism. Certainly his work could be seen to have decisively undermined one aspect of the so-called Hilbert program, but what reason was there to think it likely that a formalized version of arithmetic could be complete or have the necessary resources to prove itself consistent in the first place? And why does this matter so much anyway (except perhaps to mathematicians who might have been seeking a certain kind of certainty or security or self-containedness for (fundamental parts of) their discipline)?

I concede that there are many deep and interesting questions which work by Gödel and others (Turing, Church, Post, et al.) opens up in metamathematics, computer science and other areas. But I question its relevance (or at least its direct applicability) to general human questions.

I say these things to give a sense of my general stance and it is not my intention here to elaborate or defend a comprehensive point of view. Rather, I just want to sketch out my reaction to some posts at Scientia Salon by Marko Vojinovic whose PhD is in theoretical physics but whose interests appear to be largely philosophical. In January a two-part essay of his appeared which uses Gödel's work in what I see as an inappropriate way. [Part I here and Part II here].

Initially I was going to write a more harshly anti-Vojinovic piece but, rereading the pieces and some comments and responses I have tempered my original views somewhat. On a personal level, I think I would like Vojinovic. He writes well and engages readily with his critics, often in a disarming way. He is clearly intelligent and knowledgeable and interested in interesting things.

Nonetheless it remains crystal clear that he has a (not particularly hidden) agenda. In fact his contributions to SciSal (including comments on posts by others) could be seen to provide (further) evidence in support of points I have made in the past about (sometimes hidden) ideological or religious agendas lying behind much philosophical discourse.

The basic pattern is a common one. One has a fundamental view of how things are which one has arrived at for unknown, obscure or unknowable reasons, and one deploys one's intellect and knowledge of science, maths, logic, etc. (i.e. one's expertise) to make a case for or defend the plausibility of one's intuitively held views.

On one level, this sounds fine. And it is fine. Conjectures and refutations. Popper et al.. We can't escape it. Much of science and philosophy and ordinary argument is like this, and that's okay.

But there's a continuum involved (or a multitude of personal continua). My point is that if religion and maybe some other kinds of ideological commitment are too strongly involved the whole process becomes problematic.

Certain aspects of my own personal history give me an insight into these issues. I used to be religious and, just like Duhem (a classic case if ever there was one), I felt that I had an epistemic head start on others because I knew that the true theories of science etc. had to be compatible with my beliefs and, plainly, many popular theories and interpretations were not. Moreover, I was strongly motivated to argue my case (for obvious reasons). I even saw a career in academia (in philosophy, for example) as a possible way of promoting these ideas (and doing good, because the ideas were good and true).

When you are getting too close to the apologetics end of the continuum, expertise is deployed merely to serve the argument. And, as every high-school debater knows, any intelligent person can make a plausible-seeming case for any half-plausible proposition. In the light of these facts and, given the overwhelming amount of stuff out there which might call for our attention, lines just have to be drawn.

It's a personal thing, where one draws the line. I would exclude Pierre Duhem but include Popper (say) – despite the latter's belief in an essential truth underlying religion and his commitment to (something very like) Cartesian dualism – in my personal list of thinkers worth reading for not purely historical reasons.

So where does Marko Vojinovic's work lie on this continuum which runs from careful, exploratory argument to polemics and apologetics? His writing (at least at Scientia Salon) tends to the latter end of the spectrum, I would say. The essays are certainly tendentious.

In a September article at SciSal, he argued against determinism but his argument relied on so many contested assumptions that, even if his reasoning was valid, the soundness of the argument was far from assured. Vojinovic also claimed somewhat surprisingly – and very revealingly – that his conclusions "open[ed] the door for the compatibility between the laws of physics on one side, and a whole plethora of concepts like free will, strong emergence, qualia, even religion – on the other. But these are all topics for some other articles..."

Then, early in January, the two-part essay on reductionism and emergence appeared, with even more extravagant claims in the final paragraph, claims which one sympathetic commenter interpreted as "a tactical error".

Here is the final paragraph in its entirety:

"Giving up the idea of reductionism essentially amounts to accepting strong emergence as a fundamental property of Nature — a physical system might display behavior that is more than the behavior of the sum of its parts. Proponents of reductionism might find this at odds with their favorite ideology (physicalism, naturalism, atheism, etc.), but there are actual examples of strong emergence in Nature, the arrow of time being the most prominent one. It would be interesting to see how many people would actually agree to change their minds when faced with this kind of approach, as giving up reductionism generally weakens the arguments that a physicalist may have against dualism, a naturalist against the supernatural, an atheist against religion, etc. Philosophy teaches one to keep an open mind, while science teaches one to appreciate the seriousness of experimental evidence. When these two combine to demonstrate that certain parts of a physicalist/naturalist/atheist belief system are just unfounded prejudices, even downright wrong, it would be interesting to see how many people will actually give them up. After all, these are precisely the people who boast about both open-mindedness and the scientific method, and invoke them to criticize dualists/supernaturalists/theists. Now they are challenged with giving up one of their cherished beliefs, and I would like to see how truly open-minded and scientific they can be in such a situation."

Overall it seems clear that Vojinovic's philosophico-religious preoccupations have led him to deploy some very idiosyncratic definitions of key words and to make false (or at least very misleading) claims, claims which he has sometimes backed away from when challenged in the comments section – like the claim that his concerns were ontological rather than merely epistemic. (How can Gödel's work be used to make ontological claims if it is all about what we can know and what we can prove within the context of a formal system?)

I agree with those who call into question Vojinovic's use of Gödel's work (and in general the appropriateness of applying Gödel's ideas to scientific theories) and who talk about the red herrings, etc. which the author's use of Gödel has generated.* Gödel's results are specifically about what we can know and prove within certain strictly formal contexts, i.e. it's not about normal scientific thinking. You can axiomatise certain theories, sure, but such formal structures are entirely provisional from a scientific point of view and subservient in the end to empirical considerations.

In his enthusiasm to make his case, Vojinovic even appears to misrepresent Gödel's basic claims. He writes in a comment: "The moral of the [sic] Gödel’s theorem is that there is a difference between truth and provability, in a given axiomatic system. … Gödel’s theorem establishes the existence of statements that are (a) unprovable within a given axiomatic system, and (b) also “true” in that axiomatic system, given any notion of truth the axiomatic system may be compatible with."

"No," replies one commenter, "there is only one notion of truth in an axiomatic system. That is, provability from the axioms. Yes, there are unprovable statements in ZF [Zermelo-Fraenkel set theory (in effect, first-order logic enhanced to encompass arithmetic: Gödel's original paper referred to Russell and Whitehead's system)]. Such a statement will be either undecidable or provably false. An undecidable statement is true in some models of ZF, and false in others. Gödel proved the existence of undecidable statements in ZF, assuming consistency of ZF. Those statements have a metaphysical interpretation in which they are true, but they are certainly not true in ZF, as Gödel proved that there are models of ZF in which they are false."

Conclusion: "Almost everything Marko says about Gödel is wrong."

Hyperbole perhaps. But the main point is that Vojinovic overreaches by trying to apply Gödel's findings to a hypothetical axiomatised 'final theory'. Science is never about certainty in the way logic or pure mathematics is about certainty – and doesn't aspire to this kind of certainly. This simple fact, I think, undercuts Vojinovic's whole argument.

Sure, the validity of arguments on scientific questions matters, but the primary focus is not on validity (as it is in logic, etc.) but on soundness. The axioms, in other words, must all be 'true' – and this is something we can never affirm with absolute confidence. They are always going to be provisional and revisable (in the face of new evidence); and the formal system or theoretical context in which they are embedded is likewise always going to be provisional, even if it functions successfully as – and so is thought to represent – a 'final theory'.



* "... Marko wants to prove that Weinberg’s reductionism is wrong. There are several problems. If Goldbach’s conjecture turns out to be undecidable, then why would that have any physical implications? And if it did, then some physical experiment ought to decide it, thereby eliminating the issue as any impediment to reductionism. So there is no actual connection between Gödel and reductionism, except to confuse readers with red herrings." [From Part II comment thread.]

The wider significance of Gödel's Incompleteness Theorem

Torkel Franzén [1950-2006] devoted a lot of time and energy to playing down the wider significance of Gödel's Incompleteness Theorem.* And there is no doubt that a great many extravagant claims about its significance have been made, usually along the lines that Gödel has demonstrated some fatal limitation in what scientific research can achieve, and an equal and opposite conclusion about human spirituality and artistic creativity.

I don't want to suggest that all claims for the general (as distinct from the mathematical and logical) significance of Gödel's work are mistaken: there is genuine disagreement between very knowledgeable people on the matter. But Franzén's position is widely respected as being scrupulously rigorous and based on a thorough understanding of the logical and mathematical concepts involved.

I personally found his views refreshingly straightforward when I came across them a couple of years ago. I guess I had read one too many of those contentious claims and wanted to develop a better understanding before looking again at more general questions.

Let me say, however, that I think there is abiding interest in Gödel's results, for example, in the contrast between formal systems like first-order logic (which he proved to be complete**) and stronger systems like the one outlined in Russell and Whitehead's Principia Mathematica (which he proved to be incomplete). Franzén points out that 'the incompleteness of any sufficiently strong consistent axiomatic theory ... concerns only what may be called the arithmetical component of the theory. A formal system has such a component if it is possible to interpret some of its statements as statements about the natural numbers, in such a way that the system proves some of the basic principles of arithmetic.'

We know that the natural numbers have surprising (not to say mysterious) properties, and I am tempted to say that the gulf between the first-order predicate calculus and stronger systems which Gödel's completeness and incompleteness theorems establish underscores an informal distinction between the pedestrian logic of commonsense and everyday life (which holds no surprises, life's surprises arising from complex concatenations of events rather than from our naive analyses thereof) and mathematical thinking.

Be that as it may, Gödel's work has another, perhaps more solid, claim to significance which flows from the discovery that a class of functions (recursive) which Gödel defined in the course of elaborating his famous proof turns out to be equivalent to some apparently quite different concepts developed in subsequent years (in particular by Alonzo Church, Alan Turing and Emil Post). David Berlinski writes:

'The idea of an algorithm had been resident in the consciousness of the world's mathematicians at least since the seventeenth century; and now, in the third [sic***] decade of the twentieth century, an idea lacking precise explication was endowed with four different definitions, rather as if an attractive but odd woman were to receive four different proposals of marriage where previously she had received none. The four quite different definitions ... were provided by Gödel, Church, Turing and Post. Gödel had written of a certain class of functions; Church of a calculus of conversion; and Turing and Post had both imagined machines capable of manipulating symbols drawn from a finite alphabet. What gives this story its dramatic unity is the fact that by the end of the decade it had become clear to the small coterie of competent logicians that the definitions were, in fact, equivalent in the sense that they defined one concept by means of four verbal flourishes. Gödel's recursive functions were precisely those functions that could be realized by lambda-conversion; and the operations performed by those functions were precisely those that could be executed by a Turing machine or a Post machine. These equivalencies, logicians were able first to imagine and then to demonstrate.

... A concept indifferent to the details of its formulation, Gödel asserted, is absolute. And in commenting on the concept to an audience of logicians, he remarked that the fact that only one concept had emerged from four definitions was something of an epistemological "miracle".' ****

I don't know about a miracle, but the equivalence of the various definitions is certainly suggestive that the underlying concept has a certain robustness and depth.



* This article (pdf), written just before his untimely death, gives a concise statement of his point of view.

** A set of axioms is complete if for any statement in the axioms' language either that statement or its negation is provable from the axioms.

*** This is a strange error for a mathematician to make - we're talking of the 1930s!

**** The Advent of the Algorithm (Harcourt 2001), pp. 205-6.