godelian wrote: ↑Thu Dec 19, 2024 3:25 pm
BigMike wrote: ↑Thu Dec 19, 2024 12:21 pm
But when it comes to these “transcendent” questions, consider this: just because something transcends our current understanding doesn’t mean it’s inherently irrational or beyond the reach of reason. These answers may not satisfy the yearning for something more, but isn’t that yearning itself just another product of cause and effect? But it doesn’t mean the answers are supernatural.
If the physical universe is in any way structurally similar to the arithmetical universe, then most of its truth cannot be expressed in language, let alone, justified. Most arithmetical truth is not rational. It is instead supernatural.
http://www.sci.brooklyn.cuny.edu/~noson ... ovable.pdf
The world that Gödel described can be extended. Of all the mathematical facts, only some are expressible with language. Any mathematical fact that is provable is automatically expressible because proofs must be done in a language.
We have come a long way since Gödel. A true but unprovable statement is not some strange, rare phenomenon. In fact, the opposite is correct. A fact that is true and provable is a rare phenomenon. The collection of mathematical facts is very large and what is expressible and true is a small part of it. Furthermore, what is provable is only a small part of those.
The idea that the arithmetical universe and the physical universe are structurally similar was originally proposed by Pythagoras.
Tarski's undefinability of the truth guarantees that most arithmetical truth will forever remain beyond the reach of reason:
https://en.wikipedia.org/wiki/Tarski%27 ... ty_theorem
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic".[1]
The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.
The God-of-gaps idea is an illusion. We will not one day discover most of the truth about arithmetic. The reason why it is out of reach cannot be bridged with technology. The overwhelmingly vast majority of the truth about arithmetic is ineffable. It cannot even be expressed in language. It is transcendental, i.e. supernatural.
Ah, Godelian, you’ve plunged us straight into the deep waters of Gödel and Tarski, and it’s a fascinating place to be. As a mathematician, this is a place close to my heart. But I think you’re drawing a line where none needs to be drawn—a line between the ineffable and the supernatural, as if the limits of provability or expressibility somehow necessitate the existence of something beyond the natural order.
Let’s take a step back. Gödel’s incompleteness theorems and Tarski’s undefinability theorem do indeed reveal profound limitations in formal systems. They tell us that there will always be truths that lie beyond the reach of provability within a given framework. But this limitation isn’t evidence of the supernatural—it’s a reflection of the constraints of the systems we use to describe the world. These constraints arise from the very nature of formal languages and logical systems; they’re features of how we process and represent information, not evidence of a realm beyond nature.
Your comparison between the arithmetical universe and the physical universe is compelling, and Pythagoras would be delighted by your dedication. But let’s remember that the inability to fully capture arithmetic truth within arithmetic itself doesn’t make that truth supernatural. It makes it
structurally inaccessible. Similarly, if there are aspects of the physical universe that remain beyond our comprehension, that doesn’t mean they are inherently supernatural—it simply means our models are incomplete, bound by the tools of language, mathematics, and reasoning we have developed.
The ineffability of certain truths is not proof of transcendence in the mystical sense. It’s a recognition of the boundaries of our systems and our cognition. To label these truths as “supernatural” conflates “beyond our understanding” with “beyond nature.” But ineffable truths about arithmetic—or the universe—don’t exist outside the natural order; they exist within it, governed by the same deterministic forces and structures that shape everything we
can understand.
Here’s where this becomes important: the recognition that some truths are ineffable should inspire awe, not surrender. Gödel’s and Tarski’s results don’t point us toward a supernatural reality; they point us toward the infinite complexity of the natural one. They remind us that, while we may never fully grasp every facet of arithmetic or physics, what we
can grasp is still part of a larger, interconnected whole.
So, Godelian, instead of seeing these limitations as a gateway to the supernatural, why not see them as a testament to the endless richness of the universe? The fact that we can even
conceive of the ineffable, that we can edge toward truths we know we’ll never fully capture, is part of the beauty of being human. It’s not a sign of a transcendent realm—it’s a sign of the boundless depth of the one we’re already in.