“Is by necessity speculative,” which is exactly why I believe we should take our own certitude with a grain of salt, and spend a tad more time trying to appreciate our limitations. That is our Physical Reality ~ Mind divide that is ultimately unbridgeable.
Then I come across tidbits where genuine experts point out that my misgivings are grounded , (perhaps teasers is more accurate, since this article accelerates beyond my comprehension in a hurry). For instance:
On Formally Undecidable Propositions of Principia Mathematica And Related Systems
Translated by B. MELTZER Introduction by R. B. BRAITHWAITE
“Kurt Gödel’s astonishing discovery and proof, published in 1931, that even in elementary parts of arithmetic there exist propositions which cannot be proved or disproved within the system, is one of the most important contributions to logic since Aristotle. Any formal logical system which disposes of sufficient means to compass the addition and multiplication of positive integers and zero is subject to this limitation, so that one must consider this kind of incompleteness an inherent characteristic of formal mathematics as a whole, which was before this customarily considered the unequivocal intellectual discipline par excellence.”
INTRODUCTION by R. B. BRAITHWAITE
"Every system of arithmetic contains arithmetical propositions, by which is meant propositions concerned solely with relations between whole numbers, which can neither be proved nor be disproved within the system. This epoch-making discovery by Kurt Gödel a young Austrian mathematician, was announced by him to the Vienna Academy of Sciences in 1930 and was published, with a detailed proof, in a paper in the Monatshefte für Mathematik und Physik Volume 38 pp. 173-198 (Leipzig: 1931). This paper, entitled “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I” (“On formally undecidable propositions of Principia Mathematica and related systems I”), is translated in this book. Gödel intended to write a second part to the paper but this has never been published.
Gödel’s Theorem, as a simple corollary of Proposition VI (p. 57) is frequently called, proves that there are arithmetical propositions which are undecidable (i.e. neither provable nor disprovable) within their arithmetical system, and the proof proceeds by actually specifying such a proposition, namely the proposition g expressed by the formula to which “17 Gen r” refers (p. 58). g is an arithmetical proposition; but the proposition that g is undecidable within the system is not an arithmetical proposition, since it is concerned with provability within an arithmetical system, and this is a meta-arithmetical and not an arithmetical notion. Gödel’s Theorem is thus a result which belongs not to mathematics but to metamathematics, the name given by Hilbert to the study of rigorous proof in mathematics and symbolic logic.
Or to put it in simpler terms,
May 22, 2021 - Veritasium
Not everything that is true can be proven. This discovery transformed infinity, changed the course of a world war and led to the modern computer. This video is sponsored by Brilliant.
I powered through that video again. It’s got a bizarre attraction for me, and I’ve listened to it a few times, though I’ve definitely started it more often, than actually finishing. But when I’m in the mood, it’s got that magnetism of, I don’t know, perhaps driving by flashing lights around a car wreck, or perhaps a touch of intellectual porn, or something off in the mists of my mind.
In any event, listening to Gödel’s, what is it, 762 page long proof, that one plus one equals two, being described and then Gödel’s card game, and it overwhelms.
I simply listen and try to absorb the words and let my thoughts follow best they can, while other areas of my mind try bursting in. Such as Gödel’s numbers and card game of representing mathematical symbols/formulas as numbers and then manipulating the numbers to prove something about the symbols themselves, and . . .
Suddenly in a burst cinematic glory, the image of Gödel pops up as the master Gordian knot weaver. Best dang knots you ever did see. Then I wonder should I really be impressed with all this. It’s simply a Gordian knot. We know how secure those are. But then comes, the knowledge that all this underlies todays function computers, so I’m told, or at least these sorts of thoughts helped the human mind focus on building actual truly magical computers. Meaning that somewhere in all that mess, there is a there. But why does it matter. At least to me. I don’t want to be a Master of the Universe, I’m too preoccupied with being an element in the flow of Earth’s evolution.
What I believe is that much as we can do, it’s idiotic to obsess over mastering it,
and our self absorption in striving to find all the answers, in search of that ultimate answer to everything, it’s Faustian bargain, as demonstrated ad nausea.
The folly as witnessed by how we treat each other, this Earth and its creatures & varied biospheres, along with how we disregard simple honest facts in favor of hubristic doomed monument building.