Gödel’s Theorems & The Limits Of Reason

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.

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2 comments
  1. Macrocompassion said:

    Godel’s Theorem is not about the need for pluralism, but is about how sticking implacably to the logic of any mathematical proof is not viable. Pluralism may well be desirable but not on these grounds.

  2. Michael Kowalik said:

    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:
    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. 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 field 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 performatively affirm
    that the world, the ‘real’, is consistent (this is a foundational a
    priori of science), but by the same token we affirm that the world is
    incomplete (new facts continually augment the ‘real’). A world which is
    incomplete is also 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 proven as
    universally true. In the words of Emmanuel Levinas: “Science seeks the
    condition of the datum, but only finds conditioned conditions. These
    suffice for the understanding of facts and the establishment of laws.
    They do not satisfy reason, which requires the regressive synthesis of
    the whole series of conditions back to the unconditional.” (Alterity and
    Transcendence, The Athlone Press 1999, p45)

    That is why scientific rationality is capable of accommodating
    ostensibly contradictory theoretic domains (Theory of Relativity,
    Newtonian Physics, Quantum Theory, Actor-Network Theory 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,
    all-unifying meta-language that grounds paraconsistent modal domains in
    the same world includes all methodologies, technologies, languages as
    its objects, but is itself not an object: it was said by Lacan that the
    universal ‘meta-language cannot be spoken’. The discontinuity thus
    created in the system of relations precludes 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 the world
    in its totality (the locus of perspective is a blind-spot in all
    observation), a multiplicity of complementary perspectives is necessary
    for any of us to conceive of anything as being ‘real’, that is, it is
    necessary to maintain the notion of a common world. For example, there
    could be no conception of death, and, by implication, of life, if one
    had not witnessed the death of another, and, conversely, if there were
    no others to witness my death. The existence of others who are
    of-the-same-kind is the condition of my truth. Similarly, validity of
    any discipline is tentatively granted by all ‘other’ disciplines, which
    are its ultimate conditions of truth: these form the ‘world’ or
    theoretic 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.

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