Home > methodology > Gödel’s theorems & the limits of reason

Gödel’s theorems & the limits of reason

from Asad Zaman

My article on the limits of reason was published in Express Tribune recently (Monday April 13, 2015). This essay shows that logic is limited in its ability to arrive at a definite conclusion even in the heartland of mathematics. Pluralism is required to cater for the possibility that both Euclidean and non-Euclidean geometries represent valid ways of looking at the world. The world of human affairs is far more complex. In order to study and understand societies, one must learn to deal with a multiplicity of truths. This argument, which is related to the first, has been made in my article “Tolerance and Multiple Narratives” which was published in Express Tribune earlier (March 29, 2015). These ideas form part of the background for supporting the drive for pluralism in our approaches to economic problems.

  1. April 23, 2015 at 12:26 pm

    To argue that Gödel’s theorem implies that there should be a multiplicity of approaches in economics is a bold leap of thought. It is also nonsense! There are many theories in physics and in biology. Scientists know which theory applies to which phenomenon. There is only one physics and one biology. Or, does Zaman think that it follows from Gödel’s theorem that creationism is just as valid as evolution?
    Of course, there are competing and conflicting theories at the frontiers of a science. One can also argue that economics and the other social sciences have not jet crossed even the first frontier. In my view, classical economics did provide a valid core, it was perverted by the mathematical economics of the post-WWII era.

    • Paul Schächterle
      April 23, 2015 at 1:01 pm

      The main problem in economics is not that there are certain statements that can’t be deduced from certain axioms.

      The problem in economics is that there are axioms in use that are false. We know they are false. Thus any deduction from those axioms is worthless. We know they are worthless.

      Additionally there is the problem that mainstream economists make statements that cannot be deduced from the axioms they state. That constitutes a logical error — a non-sequitur. Thus we know that those statements are worthless, too, even if the axioms were true (which they are not).

      Additionally to that, mainstream economics shuns and expulses all theoretical approaches that do not use their false and illogical axiomatic system. That makes mainstream economics de facto a non-science, i.e. a counterfactual belief system.

      When heterodox economists lament a “lack of plurality” they should actually lament the lack of reason and the anti-scientific behaviour of the mainstream.

      • Paul Schächterle
        April 23, 2015 at 1:03 pm

        That was meant as a reply to the article not as a reply to Claude Hillinger’s comment.
        I agree with Claude’s comment.

  2. Franklin Chiemeka Agukwe
    April 23, 2015 at 1:34 pm

    Absolutely.Economics needs pluralism and we should find a way to break the monopoly of neo-classical economics which is been hindered by its disciples from America and Europe whose articles are considered the best and others from outside America and Europe are not considered of worth

  3. April 23, 2015 at 1:50 pm

    There are many different systems of axioms, each of which represent valid, coherent and consistent ways to axiomatize geometry. Each system leads to a radically different geometry. Similarly, there are many different ways to organize economics affairs, like the Amish, the Cherokee, Individualistic capitalistic, communistic, socialistic, Islamic. Each different way of organizing society would have its own intrinsic laws of motion regarding economics. There is no one universal set of economic laws which applies equally to all societies and across time. This widespread misconception arises from the Deification of Science as I have argued in my paper with this title: http://ssrn.com/abstract=2260052

    • Garrett Connelly
      April 23, 2015 at 2:58 pm

      Please let me offer one proviso, Asad,

      There is one over arching set of laws guiding economic parameters and many ways to go about fitting within those laws. Each grouping of societal organizing system you have offered exists within unalterable laws of thermodynamics. Those which do not match up through a culturally determined positive accounting system balance will eventually collapse and disappear.

      My guess is individualist capitalism supported by neoliberal mainstream economic propaganda will devolve further into WW3 and something new will arise from the ashes.

    • Rhonda Kovac
      April 23, 2015 at 4:03 pm

      In addition to “different ways of organizing society”, there is as well the fact of dynamic societal change, giving us also, in a manner of speaking, a variety of ‘societies’ to organize. Since we haven’t full knowledge of the current behavioral rules, nor have we yet foreknowledge of how society will change, then we cannot say with confidence how many and which of those rules will continue to apply, or will no longer apply, in subsequent versions of what we now have.

      This calls into question the general method of ‘learning the rules of behavior’ followed by ‘applying those rules’.

      Of course, one may claim that the kinds of changes referred to here are deep ones, occurring over long periods of time. And so we don’t have to worry about a rule-set of behavior becoming obsolete before it can be utilized. But in a complex system, characterized by rapid pace, and with numerous highly interdependent parts, even subtle changes can have dramatic ramifications along shorter time-frames And so we may not be able to be sure what within any given rule-set will become no longer usable or when.

      In order to be congruent with dynamic reality, then, our method must be comparably flexible and responsive to possible system changes. This requires a different methodological framework than what our current scientific protocols allow, one accommodating a more extended process, that treats its variables more provisionally.

      • Garrett Connelly
        April 25, 2015 at 5:52 am

        Yes, and the acceleration of evolution is also geometric. When the hockey stick graph of evolution turns vertical with accelerating new combinations of information, Assad’s zen blur becomes the norm. And, Rhonda, you explain the turn to vertical as it has already happened. Thank you, now we are free to point to reality. Imagine how rich we would all be without endless war.

  4. April 23, 2015 at 2:57 pm

    The insignificance of Gödel’s theorem for economics
    Comment on ‘Gödel’s theorems and the limits of reason’

    In his Tribune article Asad Zaman writes: “Progress in science becomes possible only after we abandon the quest for logical certainty.”
    http://tribune.com.pk/story/868779/godels-theorems-the-limits-of-reason/

    True, of course, logic has limits. Yet, from this does not follow that inconsistency is acceptable. From Gödel’s theorem in particular does not follow that the pluralism of false theories is acceptable. And not by any stretch of the imagination follows that inconsistent orthodox and heterodox economic models will ever be acceptable.

    It still holds: science does not explain everything, but non-science explains nothing. Science is not logic alone and not fact alone: it is the seamless synthesis of both.

    “Research is in fact a continuous discussion of the consistency of theories: formal consistency insofar as the discussion relates to the logical cohesion of what is asserted in joint theories; material consistency insofar as the agreement of observations with theories is concerned.” (Klant, 1994, p. 31)

    Zaman’s understanding of the interaction of mathematics and physics is utterly confused. He writes
    “The axiomatic-deductive methodology of mathematics leads to logical certainty without requiring empirical confirmation — we do not assess the validity of the Pythagorean Theorem by drawing triangles and measuring their sides.” (see Tribune)

    Exactly the opposite is true.

    “One of the most famous stories about Gauss depicts him measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken for evidence that the geometry of space is non-Euclidean.” (Brown, 2011, p. 565)

    Or, with regard to Newton’s axiomatization of physics:
    “In the most fruitful applications of mathematics to the physical world, some nonmathematical axioms also enter. The Newtonian system of mathematical mechanics depends as much on the Newtonian laws of motion and gravitation as it does on the axioms of mathematics.” (Kline, 1981, p. 469)

    In pure mathematics progress has not been and will not be achieved by abandoning the quest for logical certainty.

    “In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. … Thus there will always be at least one true but unprovable statement.” (Wikipedia)
    https://en.wikipedia.org/wiki/Kurt_G%C3%B6del

    This, though, is no hindrance at all for the advancement of economics because the ‘true but unprovable’ statement can be added as an additional axiom to the initially given formal system and thus make it more comprehensive. More comprehensive, to be precise, does not mean complete. Not complete, on the other hand, does not mean, that non-scientific approaches provide better results or any acceptable results at all.

    The fundamental methodological problem of economics does not arise from Gödel’s theorem but from the sad fact that economists fail already at simple logical tasks. It can be shown, for example, that the basic Keynesian formalism is logically defective (2011). Nonetheless, economists blindly apply it for more than 70 years.* This scientific incompetence, and not the Incompleteness Theorem, is the real problem of economics.

    The manifest confusion of economists and the proto-scientific mess of contradictory and logically defective models cannot be justified with Gödel’s theorem.

    Egmont Kakarot-Handtke

    References
    Brown, K. (2011). Reflections on Relativity. Raleigh, NC: Lulu.com.
    Kakarot-Handtke, E. (2011). Why Post Keynesianism is Not Yet a Science. SSRN
    Working Paper Series, 1966438: 1–20. URL http://ssrn.com/abstract=1966438.
    Klant, J. J. (1994). The Nature of Economic Thought. Aldershot, Brookfield, VT:
    Edward Elgar.
    Kline, M. (1981). Mathematics and the Physical World. New York, NY: Dover.

    *See also http://axecorg.blogspot.de/2015/04/economics-vs-sociology.html

  5. blocke
    April 23, 2015 at 6:41 pm

    Read, RWER issue 56: Wolfgang Drechsler

    March 11, 2011 “Understanding the problems of mathematical economics: A “continental” perspective

    • April 29, 2015 at 10:10 pm

      Thank you, I will read it. See in the meantime my post
      https://rwer.wordpress.com/2015/04/17/on-dogmatism-in-economics/#comment-91932

    • April 30, 2015 at 9:17 pm

      Rubberneck’s reality
      Comment on ‘Gödel’s theorems and the limits of reason’

      In his paper ‘Understanding the problems of mathematical economics: A “continental” perspective’ Wolfgang Drechsler quotes:

      “…scientific concepts are idealizations; they are derived from experience obtained by refined experimental tools, and are precisely defined through axioms and definitions. Only through these precise definitions is it possible to connect the concepts with a mathematical scheme and to derive mathematically the infinite variety of possible phenomena in this field. But through this process of idealization and precise definition the immediate connection with reality is lost.” (Heisenberg, in Drechsler, 2011, p. 52)

      What Drechsler suggests throughout his paper is that in the scientific process reality gets lost. Note, however, that Heisenberg speaks of the immediate reality. This reality is dear to so-called social scientist and consists of what the rubberneck sees when he looks out of the window and hears when he listens to his neighbors and the talking heads.

      Atoms and quarks do not belong to this immediate reality. Neither does the economy. Nobody can see, touch, or smell the economy. So, nothing of importance is lost through the scientific process but much is gained. It can hardly be denied that Heisenberg and his colleagues reached a higher level of knowledge using the longest mathematical ladder. The same cannot be reported from the adepts of Verstehen. They are forever lost in liguisticality, that is, myth, storytelling, rhetoric and gossip.

      Economics fits Heisenberg’s characterization of the scientific process but only superficially. This has often been noticed.

      “Suffice it to say that, in my opinion, what we presently possess by way of so-called pure economic theory is objectively indistinguishable from what the physicist Richard Feynman, in an unflattering sketch of nonsense ‘science,’ called ‘cargo cult science’.” (Clower, 1994, p. 809)

      As far as Wolfgang Drechsler’s and Asad Zaman’s critique of mathematics and the scientific method is appropriate it hits the lookalike.*

      To apply the method correctly and to state ‘axioms and definitions precisely’ is the task of Constructive Heterodoxy.**

      Egmont Kakarot-Handtke

      References
      Clower, R. W. (1994). Economics as an Inductive Science. Southern Economic Journal, 60(4): 805–814.
      Drechsler, W. (2011). Understanding the Problems of Mathematical Economics: A “Continental” Perspective. real-world economics review, (69): 45–57. URL
      http://www.paecon.net/PAEReview/Drechsler56.htm/.

      * For detailed correction and refutation of Zaman’s arguments see the crossreferences
      http://axecorg.blogspot.de/2014/12/marshalls-burn-mathematics-cross.html

      ** For the new curriculum see the cross-references
      http://axecorg.blogspot.de/2015/04/new-curriculum-cross-references.html

  6. Michael Kowalik
    April 24, 2015 at 11:01 am

    Thank you Asad for introducing the essentially important Goedel’s theorems to the economics profession. Goedel’s Second Incompleteness Theorem (SIT) is, in my view, the most fundamental, being the mathematical equivalent of an earlier formulation by Alfred Tarski: the Tarski’s Semantic Truth Theorem. Both theorems state, in their own way, that the truth conditions (that which makes a statement or system true) cannot be defined within the system to which they apply. This can be translated again into legal terms: no one shall be judged in his own court, or; no Court has the jurisdiction to set the limits of its own jurisdiction. All these formulations can be traced back to the classical problem of self-reference: the liar paradox etc. It’s a shame that some commentators on this blog fail to see its relevance and arbitrarily dismiss your observations as “nonsense”. I admit that the support that SIT gives to your main claim may not be immediately apparent. Saying that, I have little doubt that a more comprehensive explanation of SIT in light of the truth conditions of any theoretic filed would ultimately convince any reasonable sceptic as to validity of your claim.

    The most interesting property of SIT (to me) is that it “precludes causal determinism, insofar as it demonstrates that no system can be simultaneously consistent and complete”. We all intuitively and performatively affirm that the world is consistent (this is the key a priori of science), and therefore the world is incomplete (the reality is perpetually emergent). A world which is incomplete is essentially indeterministic (and that includes all scientific domains). But without causal determinism science, or, for that matter, any attitude that claims universal validity loses its claim to objectivity: it may be useful but it can never be absolutely true.

    That is why “scientific rationality is capable of accommodating ostensibly contradictory theoretic domains (biology, physics, geography etc.) as long as these are associated with self-contained modes of technological application: what results is a paraconsistent, or modal, view of reality. The modal domains intersect only marginally, via complementary effects on the same object-world which is the ground of their ontic compatibility, but the laws (the meta-languages) that are thought to govern these domains are mutually inconsistent; consistency can be accomplished only in a higher-degree meta-language. The highest-degree (the unspoken) meta-language that grounds the paraconsistent modal domains in the same world includes all methodologies as its objects, but is itself not an object. The discontinuity thus created in the system of relations precludes the scientific rationality from achieving theoretic closure, and thus the familiar paradigm of approximation is invoked to evade the object-level inconsistency and the collapse of its ‘real’.”

    Given each of us is structurally precluded from apprehending a complete world, a multiplicity of perspectives is necessary for any of us to conceive of anything as being ‘real’, that is, it is necessary to maintain consistency of a common world. Validity of any discipline is tentatively granted by all ‘other’ disciplines, which are the ultimate conditions of truth for that discipline: these form the ‘world’ or ‘context’ within which any particular discipline is realised/applied. Pluralism is therefore structurally essential for the construction of any ‘truth’. This has profound consequences not only for science but also for moral theory.
    Quotations are from my forthcoming book.

  7. April 24, 2015 at 12:04 pm

    Thanks Garrett & Rhonda, for illuminating comments. The battle of methodologies (Methodenstreit) was fought in the late 19th century between the idea of historical specificity and universality. As Geoffrey Hodgson has written in “How Economics forgot History” — the issues were never really resolved, but the Universal Law approach came to dominate, and economics forgot history. There is a real paradox here which I have not been able to resolve for myself. On the one hand it seems clear that we must have some universal laws — if every event is particular and specific then no learning is possible and experience does not count for anything. On the other hand, it seems entirely clearly that this moment of time is unique — the people living, the environment, geopollitical circumstance, social structures, everything is such that there has been nothing like it in the past, and there will be nothing like it in the future. We can only GUESS at patterns in the past which might persist in the future — just like the scientists extrapolate from patterns — but there is no way to confirm or validate such guesses. That is in fact what we do in real life — guess at possiblities based on our limited experiences.
    There is another possibility which is more ZEN — suppose that every moment really is unique, full of potentials that has never been seen before and will never be seen again. Those who look for patterns will be SYSTEMATICALLY blind to such possibilities. To be able to grasp such potentials, we have to change our ways of living, to develop sensitivities to intutivey sense and reach for such potentials. Thus scientific thinking may systematically blind us to the best things life has to offer.

    • blocke
      April 25, 2015 at 6:23 am

      Every PhD in history runs into the Methodenstreit. I did in an undergraduate seminar for history majors and then in a graduate seminar, where we discussed it in the form of Hegel’s philosophy of history, where he tried to get rid of the problem by making the Idea the product of reason, and its actualization in time subjective and irrational. Aside from a lot of confused Ph.D. students, we more or less concluded that the IDEA of history could not be knowable as an Idea, but that historians were not left in an intellectual cul de sac because knowledge about the individual and collective, arose from the process of debate among historians. Each generation made sense through debate among themselves, with the proviso that the debate was free and open among them. This is the missing element in economics, today, the debate is not free and open among economists because orthodox control knowledge and the challengers are shut out. The chief problem, therefore, is constitutional within the field of economics. It is not an open club.

  8. April 24, 2015 at 5:54 pm

    Reblogged this on ihtis69.

  9. April 24, 2015 at 6:03 pm

    The limit to our thinking is such, it is impossible for any one of us to decide anything about the next individual. As Asad pointed out we can only guess to some extent, which can prove true or false and that to some extent. Some times we do such things about whom we lack any reason. Indeed Human body is a complex machine, where it is difficult to control oneself, how can we control others’.

    • Garrett Connelly
      April 25, 2015 at 6:38 am

      We are cosmic powered biology surfing big bang at life speed. How can I possible know your next move?

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