Jan

1

It seems to me that Goedel's "Incompleteness Theorem" proves the limits of reason, and science. That in a system complex enough to do arithmetic, that there either exist:

1 True properties that are not provable,
2. or the system is inconsistent.

If you understand his motives and what he thought his incompleteness theory proves: Platonists were right. Not the Sophist: "Man is the measure of all things", not the rigid scientist/empiricist. Nor the Mystic.

It is said that some of Goedel's decent into madness stemmed from him not understand how others could mis-understand the implications of his proof and what was so clear to him.

Few even understand what the proof, proves, let alone the implications, and fewer still the proof.

Goedel was every bit as much the genius as Einstein according to his good friend Einstein.

Jim Sogi responds:

From my read of Luck, Logic & Whies Lies by Jörg Bewersdorff, rather than the limits of reason, Goedel marked the end of a period in the history of math and the beginning of what I might describe as the probabilistic age. Many of the advances in physics and our own market science rests on probabilistic mathematics and this is the new frontier in the same way that the mathematic fiction of a limit allowed Newton to initiate modern science. To me it is a peculiar type of math with variables representing shifting penumbras, but is what gives an advantage over those relying on linear or fixed systems.

Jason Schroeder exclaims:

Don't drag Goedel into this!

Your Bayesian Vagabond cautions over exuberance concerning Mr. G proves limits.

Moving to probabilities does not remove the problem. Probabilities are deductions taken under uncertainty. Otherwise probabilities, including the famous 0 and 1, are mental fixations aiding the proving/deducing process. Incompleteness holds that that abstract process cannot prove everything. Some things require a different tactic or strategy.

More symbols (limits and penumbras and strings of numerals) do not create more possibilities to defeat incompleteness. We all gotta work for our dinner intellectually. Take the risks and change the rules.

Mr. G proves Hilbert championed a dead-end. The scientific air at the time was using phrase  "final solution" voiced by Hilbert and his groupies. The German politicians were just being savvy by bringing the notion to the people. Showing the axiomatization, or encoding, or formalistic pretensions that the averagely clever think they automate mapping out a solutions before taking to the field.

"It should anyway be observed that Gödel's theorem is not the anti-scientist panacea… science is primarily seeking questions" not proving correctness before trying (that is called self-righteousness in another tradition).

Remember Popper's love of falsifiability ignores Goedel's work because it is not falsifiable! Goedel refutes Hilbert and Popper.

More from Girard, a mathematical logician:

It is out of question to enter into the technical arcana of Gödel's theorem, this for several reasons :

(1) This result, indeed very easy, can be perceived, like the late paintings of Claude Monet, but from a certain distance. A close look only reveals fastidious details that one perhaps does not want to know.

(2) There is no need either, since this theorem is a scientific cul-de-sac : in fact it exposes a way without exit. Since it is without exit, nothing to seek there, and it is of no use to be expert in Gödel's theorem.

…never forget Turing's contribution to computer science, a ontribution which mainly rests on a second reading of Gödel's theorem ; the fixed point of programs is nothing more than the celebrated algorithmic undecidability of the halting problem: no program is able to decide whether a program will eventually stop, and no way to pass around this prohibition. This is a simplified version of the incompleteness theorem … loses very little …

Russ Sears concludes:

Not having read "Luck, Logic & Whites Lies", but left to judge by your brief decription.

Much has been written about Goedel's proof and its implications from those that don't really understand it. Or if they do they only give the part of the story they want you to hear. This is part of the frustration Goedel had.

To quote an expert on Goedel, Rebecca Goldstein, "…the second incompleteness theorem doesn't say that the consistency of a formal system of arithmetic is unprovable by any means whatsoever. It simply says a formal system that contains arithmetic can't prove the consistency of itself. After all, the natural numbers constitute a model of the formal system of arithmetic and if a system has a model then it is consistent…In other words, when the formal system of arithmetic is endowed with the usual meaning, involving the natural numbers and their properties, the axioms and all that follow from them are true and therefore consistent. This sort of argument for consistency, however, goes outside the formal system, making an appeal to the existence of the natural numbers as a model"

The goal was "to expunge all reference to intuitions-was most particularly directed toward our intuitions of infinity: not surprisingly, finite creatures that we are, it is these intuitions that have prove themselves, from the very beginning , to be the most problematic." … "This can only be done by going outside the formal system and making an appeal to intuitions that can't themselves be formalized."

In other words, my own this time. Goedel ideas no doubt did help herald in what you call "the probabilistic age". He did so by making scientist question even the subtlest assumptions in their methods and models. But I would suggest that Heisenberg principle had much more of an effect in causing a "probabilistic age" than Goedel, if for no other reason than it came first.

The implications to reasons are that a pure Spock is not possible…that intuition must be a part of the process and hence the value of standing like a tree, running 70 miles a week in cold of December. Or, as Einstein and Goedel both did, going for long walks often together can give increase your scientific output, by giving you a chance to put it in perspective.

While this clearly has implication for a speculator, I would suggest that the bigger implication is that finite creatures that we are we should always remain humble and be open to the idea that we even as "counters" are heading down a wrong path. We do not always have an edge, despite what the numbers say. Not that "counting", reason, or science is wrong… rather we are using it wrong, that we missed something in our model. The limit to science and reason is us.


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