Godel's Lasting Legacy

Austrian logician Kurt Gödel’s incompleteness theorems showed us the limitations of mathematics within mathematics. While math is still useful for proving scientific theorems, Gödel transformed the perception of pure mathematics in a way that still makes modern mathematicians uncomfortable. Here, leading thinkers—a mathematician, a philosopher, and a physicist—wrestle, almost literally, with the implications of Gödel’s legacy.

Original program date: June 4, 2010

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[EugeneKhutoryansky]

I think this is analogous to the difference between someone expressing discomfort about the fact that he doesn't fully understand how the biological process of digestion works, and another person pointing out that this lack of understanding doesn't prevent anyone from being able to eat.

[LacerdaPT]

Well, regarding what was said in 1:40, a paper just came out where incompleteness of mathematics limits physics: "Undecidability of Spectral Gap"

[extraterrestrial]

i think wat is really incredable about godels theorem.. is in how it ties in with the ontology of the axioms.. and in what it means for the mind because something is a serious miss in our understanding of how we are able to cognitise things beyond their parameters in essence..

[erikziak]

At least since Gödel we know that we cannot be sure if we will ever know. It is hard to distill the essence into one sentence, but hopefully it made sense, at least some.

[davidroberts]

square root of -1 is a big enough mystery all on its own. The Incompleteness theorem is so much more to worry about but we're not there yet, at least I'm not. :-)

[henryg]

Nonsense! Mario Livio (the physicist) failed to understand or simply dismissed what Gregory Chaitin was talking about.
Chaitin is saying that at the heart of mathematics there is something which Hilbert wished could be solved: the foundations of mathematics on solid ground. Why is Mario Livio going on and on about how other fields can still be performed? That isn't the issue! It's about limits of understanding, mathematics, logic, and philosophical implications of the incompleteness theorem.

[jherbranson]

I think that's the point with Godel's incompleteness theorem. It doesn't make math less reliable or anything like that, it simply showed the vulnerability in formalized approaches to math. The truth of math exists outside of those systems. Nucleons are still defined by their number or protons regardless of how well we formalize our understanding of math. Goldbach's conjecture is probably obviously true. I think people want to believe that rigor and disciplined formality are always going to win out over intuition and unconventional approaches in the world, and they want to believe that formality and convention are in fact what control our world, but that simply isn't true. Formality has its place and yes it's powerful, it's just that it cannot be the end to all things always. It's just another tool in the end.

[snt_gulab]

The mathematician clearly wasn't very well equipped to combat the physicist, as he made a very illogical generalization that all physicists don't care whether there are deep fundamental flaws in quantum mechanics because the just wanted to make transistors and lasers for money... when this is far from the truth as the theoretical physicist and industrial physicist are completely different fields/industries/purposes. Just like someone who is an actuary and someone who is a pure mathematician, they may have had the same first two years in higher education, but that's it. Many physicists dedicate their life to making sense of quantum mechanics, and many mathematicians dedicate their lives to furthering what Godel proved (myself being an example for both things, here's hoping).

Very disappointed that such an embarrassingly illogical generalization came from a someone very highly educated in a STEM field. As well as how he tried to prove his logical fallacy by overtaking the person in volume/arrogance.

[lucidd.]

Any formalized or closed system of values or thinking has blind spots. Couple this with black holes, where information/ energy is lost, the physical universe is not closed but open and incomplete. Hence math cannot provide a complete description. Other significant implications as well, cause preceding effect, etc

[abhishekpratapsingh]

"Who cares about technology..."