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80-year-old problem around irrational numbers solved by new proof


It involves approximating numbers with fractions:

The duo’s solution came as a surprise to many in the field. “The general feeling was that this was not close to being solved,” says Aistleitner. “The technique of using [graphs] is something that maybe in the future will be regarded as just as important [as]—maybe more important than—the actual Duffin-Schaeffer conjecture,” says Jeffrey Vaaler, a retired professor at the University of Texas, Austin, who proved a special case of the conjecture in 1978.

It may take other experts several months to understand the full details. “The proof now is a long and complicated proof,” says Aistleitner. “It’s not sufficient just to have one striking, brilliant idea. There are many, many parts that have to be controlled.” At 44 pages of dense, technical mathematics, even leading mathematical minds need time to wrap their heads around the paper. The community, however, seems optimistic. Says Vaaler: “It’s a beautiful paper. I think it’s correct.”

Leila Sloman, “New Proof Solves 80-Year-Old Irrational Number Problem” at Scientific American

So math isn’t as cut and dried as some students fear.

See also: Some philosophical questions to keep you awake, if the prospect of partying doesn’t:

Does the size of the universe sweep us toward atheism?

Philosopher: If there is something rather than nothing, questions around God cannot be ignored Waghorn: “Firstly, that on the most plausible demarcation criterion for science, science is constitutionally unable to show theism to be a redundant hypothesis; the debate must take place at the level of metaphysics. ”

Is zero even?

Absolute zero proven mathematically impossible?

Is celeb number pi a “normal” number? Not normal. And things get worse. Surely this oddity is related in some way to the unreasonable effectiveness of mathematics.

Durston and Craig on an infinite temporal past . . .

Physicist David Snoke thinks that Christians should not use the kalaam argument for God’s existence


Must we understand “nothing” to understand physics?

Why is space three dimensions anyway? Why not six? A new theory is offered. They want to test their theory?  What a great idea! In an age of wars on falsifiability, that’s a refreshingly new/old idea. Anyway, our universe seems pretty smart and can keep us awake.

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De proof: https://arxiv.org/abs/1907.04593 kairosfocus
Of related note, Gödel’s Incompleteness Theorem is not just some interesting abstract mathematical proof, but has now been extended to physics and is also now turning out, (since Darwinism itself is based on the presupposition of reductive materialism), to be relevant in biology as well: The following article entitled 'Quantum physics problem proved unsolvable: Gödel and Turing enter quantum physics', which studied the derivation of macroscopic properties from a complete microscopic description, the researchers remark that “even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour”,,, The researchers further commented that their findings “challenge the reductionists' point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description."
Quantum physics problem proved unsolvable: Gödel and Turing enter quantum physics - December 9, 2015 Excerpt: A mathematical problem underlying fundamental questions in particle and quantum physics is provably unsolvable,,, It is the first major problem in physics for which such a fundamental limitation could be proven. The findings are important because they show that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.,,, "We knew about the possibility of problems that are undecidable in principle since the works of Turing and Gödel in the 1930s," added Co-author Professor Michael Wolf from Technical University of Munich. "So far, however, this only concerned the very abstract corners of theoretical computer science and mathematical logic. No one had seriously contemplated this as a possibility right in the heart of theoretical physics before. But our results change this picture. From a more philosophical perspective, they also challenge the reductionists' point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description." http://phys.org/news/2015-12-quantum-physics-problem-unsolvable-godel.html
And on the following episode of ID the Future, Dr. Richard Sternberg, research fellow at the Biologic Institute, speaks on his mathematical/logical work showing the difficulty of identifying genes purely with material phenomena, and that DNA doesn’t have all that’s needed to direct the development of organisms. The math, he says, is even showing gaps in the computability of what happens in the cell, which could help shed light on how machine-like organisms are or are not, how evolvable they are, and whether artificial life is possible.
Richard Sternberg on the Trail of the Immaterial Genome https://www.podomatic.com/podcasts/intelligentdesign/episodes/2019-09-11T13_37_45-07_00
Supplemental notes:
Darwinism vs Biological Form - video https://www.youtube.com/watch?v=JyNzNPgjM4w Darwinian Materialism vs. Quantum Biology – Part II - video https://www.youtube.com/watch?v=oSig2CsjKbg How Quantum Mechanics and Consciousness Correlate – video (how consciousness, quantum information theory, and molecular biology correlate – 27 minute mark) https://youtu.be/4f0hL3Nrdas?t=1634
Of related note, irrational numbers and/or fractions also played a large part in Cantor's work in trying to 'tame infinity' in mathematics. His effort to 'tame infinity' in mathematics, and the questions that his effort raised for the foundations of mathematics itself, played a large part in Godel's subsequent work formulating the Incompleteness theorem for mathematics. Here is a documentary that tells that story
BBC.Dangerous.Knowledge.1of6 https://vimeo.com/73677223
I like the way Perry Marshall's lays out the implications of Godel's incompleteness theorem
Gödel’s Incompleteness Theorem: The #1 Mathematical Discovery of the 20th Century Excerpt: Gödel’s Incompleteness Theorem says: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove.” You can draw a circle around all of the concepts in your high school geometry book. But they’re all built on Euclid’s 5 postulates which are clearly true but cannot be proven. Those 5 postulates are outside the book, outside the circle. You can draw a circle around a bicycle but the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself. Gödel proved that there are ALWAYS more things that are true than you can prove. Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions. Gödel’s Incompleteness Theorem applies not just to math, but to everything that is subject to the laws of logic. Incompleteness is true in math; it’s equally true in science or language or philosophy. And: If the universe is mathematical and logical, Incompleteness also applies to the universe.,,, Reasoning outward from a smaller circle to a larger circle is “inductive reasoning.” Examples of inductive reasoning: 1. All the men I know are mortal 2. Therefore all men are mortal 1. When I let go of objects, they fall 2. Therefore there is a law of gravity that governs falling objects Notice than when you move from the smaller circle to the larger circle, you have to make assumptions that you cannot 100% prove. Now please consider what happens when we draw the biggest circle possibly can – around the whole universe. ,,, Whatever is outside the biggest circle is boundless. By definition it is not possible to draw a circle around it. If we draw a circle around all matter, energy, space and time and apply Gödel’s theorem, then we know what is outside that circle is not matter, is not energy, is not space and is not time. It’s immaterial. Whatever is outside the biggest circle is not a system – i.e. is not an assemblage of parts. Otherwise we could draw a circle around them. The thing outside the biggest circle is indivisible. Whatever is outside the biggest circle is an uncaused cause, because you can always draw a circle around an effect. We can apply the same inductive reasoning to the Origin of Information: In the history of the universe we also see the introduction of information, some 3.5 billion years ago (Or was it longer? Was information somehow present at the beginning?). It came in the form of the Genetic code, which is symbolic and immaterial. The information appears to have come from the outside, since information is not known to be an inherent property of matter, energy, space or time All codes we know the origin of are designed by conscious beings. Therefore whatever is outside the largest circle is a conscious being. https://www.perrymarshall.com/articles/religion/godels-incompleteness-theorem/

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