Home > Uncategorized > Gödel and the limits of mathematics

Gödel and the limits of mathematics

from Lars Syll

irrelevant model abstractions
with no bridges to real-world economies

 

Gödel’s incompleteness theorems raise important questions about the foundations of mathematics.

The most important concern is the question of how to select the specific systems of axioms that mathematics is supposed to be founded on. Gödel’s theorems irrevocably show that no matter what system is chosen, there will always have to be other axioms to prove previously unproven truths.

This, of course, ought to be of paramount interest for those mainstream economists who still adhere to the dream of constructing deductive-axiomatic economics with analytic truths that do not require empirical verification. Since Gödel showed that any complex axiomatic system is undecidable and incomplete, any such deductive-axiomatic economics will always consist of some undecidable statements. When not even being able to fulfil the dream of a complete and consistent axiomatic foundation for mathematics, it’s totally incomprehensible that some people still think that could be achieved for economics.

Separating questions of logic and empirical validity may — of course — help economists to focus on producing rigorous and elegant mathematical theorems that people like Lucas and Sargent consider “progress in economic thinking.” To most other people, not being concerned with empirical evidence and model validation is a sign of social science becoming totally useless and irrelevant. Economic theories building on known to be ridiculously artificial assumptions without an explicit relationship with the real world is a dead end. That’s probably also the reason why general equilibrium analysis today (at least outside Chicago) is considered a total waste of time. In the trade-off between relevance and rigour, priority should always be on the former when it comes to social science. The only thing followers of the Bourbaki tradition within economics — like Karl Menger, John von Neumann, Gerard Debreu, Robert Lucas, and Thomas Sargent — have given us are irrelevant model abstractions with no bridges to real-world economies. It’s difficult to find a more poignant example of intellectual resource waste in science.

  1. Meta Capitalism
    May 11, 2022 at 11:22 pm

    The hole at the bottom of math is genuinely creative thought. Creative ideas are born in a world of ambiguity, contradiction, and paradox.

    DIFFERENT WAYS OF USING THE MIND

    Mathematics has something to teach us, all of us, whether or not we like mathematics or use it very much. This lesson has to do with thinking, the way we use our minds to draw conclusions about the world around us. When most people think about mathematics they think about the logic of mathematics. They think that mathematics is characterized by a certain mode of using the mind, a mode I shall henceforth refer to as “algorithmic.” By this I mean a step-by-step, rule-based procedure for going from old truths to new ones through a process of logical reasoning. But is this really the only way that we think in mathematics? Is this the way that new mathematical truths are brought into being? Most people are not aware that there are, in fact, other ways of using the mind that are at play in mathematics. After all, where do the new ideas come from? Do they come from logic or from algorithmic processes? In mathematical research, logic is used in a most complex way, as a constraint on what is possible, as a goad to creativity, or as a kind of verification device, a way of checking whether some conjecture is valid. Nevertheless, the creativity of mathematics—the turning on of the light switch—cannot be reduced to its logical structure. (Byers, William. How Mathematicians Think (p. 5). Princeton University Press. Kindle Edition.)

    Where does mathematical creativity come from? This book will point toward a certain kind of situation that produces creative insights. This situation, which I call “ambiguity,” also provides a mechanism for acts of creativity. The “ambiguous” could be contrasted to the “deductive,” yet the two are not mutually exclusive. Strictly speaking, the “logical” should be contrasted to the “intuitive.” The ambiguous situation may contain elements of the logical and the intuitive, but it is not restricted to such elements. An ambiguous situation may even involve the contradictory, but it would be wrong to say that the ambiguous is necessarily illogical. (Byers, William. How Mathematicians Think (pp. 5-6). Princeton University Press. Kindle Edition.)

  2. Meta Capitalism
    May 12, 2022 at 12:25 pm

    1.1.2 Gödel and Turing on Rationalistic Optimism
    .
    “Rationalistic optimism” is the view that there are no mathematical questions that the human mind is incapable of settling, in principle at any rate, even if this is not so in practice (due, say, to the occurrence of the heat-death of the universe). 5 In a striking observation about the implications of his incompleteness result, Gödel said:
    .
    My incompleteness theorem makes it likely that mind is not mechanical, or else mind cannot understand its own mechanism. If my result is taken together with the rationalistic attitude which Hilbert had and which was not refuted by my results, then [we can infer] the sharp result that mind is not mechanical. This is so, because, if the mind were a machine, there would, contrary to this rationalistic attitude, exist number-theoretic questions undecidable for the human mind (Gödel in Wang 1996, 186-187)
    .
    What Gödel calls Hilbert’s “rationalistic attitude” was summed up in Hilbert’s celebrated remark that “in mathematics there is no ignorabimus” — no mathematical question that in principle the mind is incapable of settling (Hilbert 1902, 445). Gödel gave no clear indication whether, or to what extent, he himself agreed with what he called Hilbert’s “rationalistic attitude” (a point to which we shall return in section 1.3). On the other hand, Turing’s criticism (in his letter to Newman) of the “extreme Hilbertian ”view is accompanied by what seems to be a cautious endorsement of the rationalistic attitude. The “sharp result” stated by Gödel seems in effect to be that there is no single machine equivalent to the mind (at any rate, no more is justified by the reasoning that Gödel presented) — and with this Turing was in agreement, as his letter makes clear. Incompleteness, if taken together with a Hilbertian optimism, excludes the extreme Hilbertian position that the “whole formal outfit” corresponds to some one fixed machine. (Copeland et. al., 2013, 5, Computability: Turing, Gödel, Church, and Beyond, The MIT Press. Kindle Edition.)
    .
    (Copeland, Jack B. Posy Carl J. and Shagrir Oron, Eds. Computability (Copeleand et. al., ed.) [Turing, Gödel, Church, and Beyond]. Cambridge, Massachusetts: MIT Press; 2013; p. 5.)
    .
    Economics can’t be a “Euclidean” science. It reduces it to a logical axiomatic system in applied mathematics, with little bearing on real economies. As Keynes stated, we should use a more “Babylonian“ approach and aim for less universal theories and accept that there will always be binding spatio-temporal restrictions to the validity of our theories. The real economy is – to use the words of Cartwright [1999] – no “nomological machine”, but rather a “dappled” world. (Lars Pålsson Syll. On the use and misuse of theories and models in economics (Kindle Locations 737-740). WEA. Kindle Edition.)

    .
    Gödel invented a way of coding mathematical arguments into single integers with no loss of information to prove his incompleteness theorem—ironically transforming metamathematical statements into mathematics (arithmetic) to create a paradox to do so.
    .
    Gödel’s insight is the truth of the non-algorithmic nature of mathematical insight. Human mind, its ability to engage in genuinely creative thinking, is non-algorithmic. We are, in other words, as Lars notes, not “nomological machines.”

    • Meta Capitalism
      May 12, 2022 at 12:38 pm

      … as Lars notes, notes

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