About Rationally Speaking

Rationally Speaking is a blog maintained by Prof. Massimo Pigliucci, a philosopher at the City University of New York. The blog reflects the Enlightenment figure Marquis de Condorcet's idea of what a public intellectual (yes, we know, that's such a bad word) ought to be: someone who devotes himself to "the tracking down of prejudices in the hiding places where priests, the schools, the government, and all long-established institutions had gathered and protected them." You're welcome. Please notice that the contents of this blog can be reprinted under the standard Creative Commons license.

Thursday, October 23, 2008

No theory of everything is possible

Pierre-Simon de Laplace, the 18th century French astronomer who proposed one of the early theories of the formation of the solar system, famously postulated a “Demon” who had enough information to know what would happen in any place in the universe at any time. It was the height of mechanistic and deterministic hubris in science, and it seemed that it was only a matter of time before physicists would find out everything there was to find out about the way the world works

That brand of naive hubris has been dealt several blows during the 20th century, beginning with the cautionary arguments of philosophers of science concerned with the epistemic limits to human knowledge, and continuing with scientists themselves demonstrating that nature imposes severe constraints on our ability to make predictions. To name a few examples, relativity theory imposes limits to how fast information can be transferred (the speed of light); chaos theory tells us that the behavior of complex non-linear systems cannot be predicted after a few time steps, despite the fact that these systems are deterministic; quantum mechanics says that we cannot measure all the properties of a particle at the same time (Heisenberg’s principle); and complex systems theory has established the principle of intractability, which shows that the behavior of some physical systems cannot be predicted before actual observation of such systems.

Nonetheless, many physicists still talk about a “theory of everything,” a rather grandiose way to refer to a mathematical theory that unifies the fundamental forces of nature into one (hopefully simple) equation. The increasingly acrimonious debate about string theory and whether it can unify the so far disjunct theories of general relativity and quantum mechanics has been the crux of research in fundamental physics for decades now. (Amusingly, the skeptics have been very active recently, with books with openly provocative titles, like Not Even Wrong.)

Well, call off the search for a theory of everything. Physicist David Wolpert, in an article published in the prestigious Physica D (vol. 237, pp. 1257–1281, 2008), has shown that -- at best -- we can achieve a theory of almost everything. Wolpert’s work is very technical, but its implications are spectacular. Unlike the above mentioned limits to knowledge, which come out of empirical disciplines, Wolpert used logic to prove his point, following in the steps of the famous incompleteness theorem demonstrated by Kurt Godel in 1931. (An accessible summary of Wolpert’s discovery can be found in an article by P.-M. Binder in Nature, 16 October 2008.)

Basically, Wolpert -- building on previous work by Alan Turing -- formalized a description of “inference machines,” i.e. machines capable of arriving at inferences about the world (human beings are one example of such machines). Wolpert focused on what he calls strong inference, the ability of one machine to predict the totality of conclusions arrived at by another similar machine. Wolpert then logically proved the following two conclusions: a) For every machine capable of conducting strong inferences on the totality of the laws of physics there will be a second machine that cannot be strongly inferred from the first one; b) Given any pair of such machines, they cannot be strongly inferred from each other.

An important point to be appreciated is that Wolpert’s demonstration is completely independent of the computational characteristics of the machines, as well as of the details of the particular laws of physics to be uncovered. This is a general result based on logic, not one contingent on technology or the particular kind of universe under investigation. In a bit plainer terms, this means that there are absolute, logical limits to the ability of any method for acquiring knowledge (including, obviously, human science) to produce a comprehensive theory of the world -- i.e., no true theory of everything is actually possible, say bye bye to Laplace’s Demon, and by implication to the idea of determinism.

Before pseudoscientists, creationists, mysticists and assorted charlatans start jumping up and down with joy and declare the end of science, however, let me add the following. First, science still remains by far the best (one could argue the only) way to understand the world, and the fact that its power is limited by the characteristics of the human mind, those of the physical universe, and by the laws of logic is just something that we have to live with. No “alternative” approach has come even close to doing any better. Second, it is a scientist -- not a parapsychologist, a creationist or a mystic -- who has demonstrated the new theorem, which both reinforces the point that alternative forms of knowledge about the world don’t actually produce knowledge and that scientists, unlike practitioners of nonsense, relish the challenges posed by the world as it really is, as opposed to how we would wish it to be. Besides, the next time you hear a pseudoscientist blabber about quantum telepathy, ask him if he knows about Wolpert’s theorem -- and savor the blank stare that will surely follow.


  1. Hi Massimo -- Is there any hope that a mere mortal (say, a biologist who does not have a degree in either physics or philosophy) could follow Wolpert's argument? If so, I'll download the paper and have a look.

  2. feander,

    it depends on your background in formal logic and math. I suggest a look at the Nature summary first, then try to dive into the original.

  3. Contrary to your fears over newagers and the like, does not Wolpert's conclusion imply the logical impossibility of an omniscient god? :-)

  4. Sure, unless you conceive of god as being outside the laws of logic, which apparently he is according to some people regularly commenting on this blog...

  5. Interesting, I have still to check this out.

    But I already have a comment -- not knowing what I'm talking about has not stopped me before, now has it? :-)

    You say Wolpert uses the rules of logic to arrive at his conclusions. Fine, but as far as I understand there are things in nature (at the quantum level) that break the laws of logic. Isn't it the case? Any physicist could confirm or dispel that notion of mine?

    E.g. in our macro world things can not be A and non-A. I have the impression, from my limited knowledge of the matter, that this can happen in quantum systems -- superimposed states, stuff like that.

    Someone correct me, please.

  6. J,

    you have a point, but when I said "logic" I actually meant math (to me, as a philosopher, math is a branch of logic). His theorem deals with mathematical demonstrations of the impossibility of achieving total formal description of logical systems, so his conclusions are actually independent of the specific physical laws of our universe. At least, that's my best understanding of it.

  7. Massimo, Godel was not a scientist, he was a mathematician. Godel's levels of uncertainty have been known to be a hazard for Philosophical Materialism for nearly 80 years.

    For example, from Boyer's "A History of Mathematics", 1968, the following:

    "In its implications the discovery by Godel of undecidable propositions is as disturbing as was the disclosure of Hippasus of incommensurable magnitudes. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena can be deduced."

    What Godel actually implies is that no single level of intellectual endeavor can be completely verified within itself; it requires a meta-level for the perspective; similarly, the meta-level cannot be verified, except by a meta-meta-level, and so on ad infinitum.

    What Wolpert did has been done before. In fact, one of the demonstrative proofs of Godel's theorems is a single machine example, done years ago.

    This is not so much an actual problem for science as it is for Philosophical Materialism (or Naturalism as some prefer to call it). Empiricism voluntarily adopts a functional materialism, which is not the same as Philosophical Materialism. Functional materialism merely claims to test what it can, and forgo opinion on that which it can't. Philosophical Materialism makes the jump to declare that there is no reality beyond that available to empiricism, a claim that is unprovable, nonempirical, and purely philosophical.

    Philosophical Materialism has rigid adherents who behave dogmatically just as the ecclesiasts they decry; it has become religious itself.

    What Godel shows is that there is a "next level" requirement for full validation and comprehension of logic, of our sensory, empirical inputs, and by extension, of any knowledge system.

  8. Stan,

    I think there are significant differences between Godel's theorem and Wolpert's results, the latter being if you will an amplification of the kind of result achieved by the first.

    However, I do agree that the real problem is for the position of philosophical materialism, not for science per se. Then again, most scientists seem to be philosophical materialists, though many are not aware of it...

  9. I'm just fantasize about all billions for "The God Particle" and the theory of everything turning to research more serious problems as global warming, and to the evolutionary biology!. Why not?

  10. Wow! You guys are geniuses.

  11. I wonder what Einstein would think of it? He said such things as: "So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality."

  12. @J.,

    Quantum mechanics does not "break" logic. It just broke our rather simplistic notions of how the universe operates. The weird properties we observe are the result of particles being described by wave functions.


    To me, this result is more intuitively satisfying than the hypothetical theories of everything. Every so often I hear something along the lines of "it is the goal of physicists to develop a theory from which we can derive the fundamental constants, not just measure their seemingly arbitrary value." Which is simply absurd (to me at least). I can imagine that we will unify and significantly reduce the number of arbitrary constants, but eliminating them completely seems an impossible task. Every system has basic axioms that by definition cannot be proven - you have to start with something. I don't see why physics should be any different.

  13. "Sure, unless you conceive of god as being outside the laws of logic, which apparently he is according to some people regularly commenting on this blog..."

    I am away at the moment.

    One only gets out of logic what you put into it. If one happens to be extremely truthful, and only pose perfectly honest and straightforward questions/ formulas...no problem.

    But if a person's world view is genuinely screwy, and to them everything is corrupted and defiled, nothing that leads to a greater or clearer truth is ever going to come out of their quest.

    what can I say, ?

  14. example, (logic is what you make of it)

    my husbands uncle has been married like 4 times. What would the logical approach then be for (a) why so many wives, and (b) in which cases was it ..er, "logical" for Uncle B to leave wife one, two or three? (...she burned the eggs and beat the kids? :)

    speaking from a logical pov, this person likely would have saved the world and themselves a lot of misery by not ever getting married at all. But then, that is just not very compassionate is it.

    Life is messy tho, isn't it. Everything doesn't come at us in nice neat (logical) little packages.

  15. "[S]cience still remains by far the best (one could argue the only) way to understand the world".

    One *could* argue that science is the only way to understand the world, but one would succeed only in making oneself appear ridiculous by doing so.

    While I agree that anti-science fundamentalists are ludicrous, pro-science fundamentalists who worship at the altar of scientism are no less so. In a "fundamental" way, such individuals as Jerry Falwell and Richard Dawkins exhibit more commonalities than differences. Each is shrill and certain in the absolute truth of his belief-system (and science, relying as it does upon human perception and concepts, *is* a belief-system, just a more empirically based, practically and materially minded one).

    Even more tellingly, both the Falwells and the Dawkinses of the world have an "emotional* need for such a belief-system in order to make sense of that world. If that need were not primarily emotional in nature, then the tone of their public comments on the subject would be much different, I think.

    It is a shame to see Massimo share in that emotional shrillness at the end of his otherwise interesting and worthwhile post. His somewhat puerile gibes at such straw men as "mystics", in particular, suggest insecurity and weakness, and not the impression of calm certainty and *ataraxia* that those who truly feel unthreatened would convey.

  16. Corneliu,

    I don't think anyone can reasonably accuse me of scientism, just read what I wrote about it in my Denying Evolution book (whose subtitle was "Creationism, Scientism and the Nature of Science").

    As for Dawkins, I ain't a fan either, but frankly I get irritated at brash comparisons with people like Jerry Falwell, they are not even in the same league.

    Finally, whether science is the best or only way to learn about nature is a matter of comparison with the alternatives, not an a priori argument: what sort of non-science way do you suggest we use to learn about the natural world?

  17. Stan wrote: "What Godel actually implies is that no single level of intellectual endeavor can be completely verified within itself; it requires a meta-level for the perspective; similarly, the meta-level cannot be verified, except by a meta-meta-level, and so on ad infinitum."

    This is an erroneous overgeneralization of what Gödel proved. As Torkel Franzen puts it in _Gödel's Theorem: An Incomplete Guide to Its Use and Abuse_, the Gödel-Rosser theorem shows that "Any consistent formal system S within which a certain amount of elementary arithmetic can be carried out is incomplete with regard to statements of elementary arithmetic: there are statements which can neither be proved, nor disproved, in S." Attempting to generalize this result beyond statements of elementary arithmetic requires further demonstration and is not an immediate consequence of Gödel's incompleteness theorems.

    The way you've phrased it is falsified by (for example) Euclidean geometry, which is consistent and complete (in the sense of Gödel's theorem). That is, every sentence in the language of Euclidean geometry is provable or disprovable in the theory.

    Franzen's book is a good one to read for anybody inclined to draw philosophical inferences from Gödel's incompleteness theorems.

  18. My understanding is that per Franzen's book, Euclidean geometry has not been sufficiently formalized for Godel's theorem to apply. The theorem itself can be applied to any formal system that attempts completeness. i.e. it will always lose consistency first.


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