Reddit reviews Incompleteness: The Proof and Paradox of Kurt Gödel (Great Discoveries)
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> You're talking about [1] Axioms
OP asked for Something that does not need to be postulated and axioms need to be postulated (a person who wants to study geometry has a different set of axioms than a person who wants to study number theory).
Someone could assert that, say, Modus Ponens has to be true (perhaps even that Logic is the study of those things that must be so). But on the other hand people could study derivation systems where MP doesn't hold.
I perceive that, in part, the answer to OP depends on whether you think that there is a reality of mathematical objects which axiom systems just model, or maybe you think that by writing down axioms and deriving consequences you are creating truths, etc. It seems that OP's question is inherently philosophical.
OP might enjoy Goldstein's book on Godel.
You may be interested in reading Incompleteness, and then consider how if we can convert Scripture to a formal system, we must have either an inconsistent one, or an incomplete one. The first is clearly false, but the second destroys Sola Scriptura.
Oh, and snide response: yes.
For a detailed (but ELI5 safe) exposition of the Incompleteness Theorem, I highly recommend Incompleteness, by Rebecca Goldstein.
https://www.amazon.com/Incompleteness-Proof-Paradox-G%C3%B6del-Discoveries/dp/0393327604
> My point being, the standard you're expressing is entirely incoherent.
It is not. Modern metamathematics has demonstrated conclusively that entirely self-consistent systems are impossible. All systems that do "arithmetic" will have true statements that cannot be reached from their initial foundational axioms. One consequence of this is that there can be two or more statements which are true but in contradiction.
Now, Goedel's work applies specifically to systems of formal mathematics, but they have analgous consequences in epistemics, at least epistemics that are build on system of rational argument - axiom-theorem-rinse repeat. Analagously, this means questions like "Can God create a rock so large he cannot lift it?" or "Can an omnipotent being create entities with truly free will?" are unanswerable. You cannot reach the conclusions from the premises.
In both cases, the fundamental reason is the same - they are (backward) self-referential systems. Mathematical systems and epistemological systems are built on unprovable foundational axioms/assumptions from which we derive theorems/principles. It is the building on what you built last behavior that causes the problem.
When I first came to understand Godel's proofs, it dawned on me that this was exactly the same reason these tensions between faith and reason, free will and determinism, and so on have no answers so long as they attempt to be backward-consistent.
Godel showed that this problem exists and will always exist for all nontrivial mathematical systems. Interestingly, his proof was to show that formal logic (propositional calculus) was* "complete" and thus consistent. Unfortunately, formal logic is insufficient to produce fully functional mathematical systems.
For a good, non technical, introduction to Goedel and his work, see:
http://www.amazon.com/Incompleteness-Proof-Paradox-G%C3%B6del-Discoveries/dp/0393327604
tl;dr This handwaving I am doing here, not formal proof. I am trying to motivate the shape of this discussion and have played fast and loose with technical precision...
You might be interested in Gödel's Incompleteness as it relates to the Platonic Forms (and there to God).
Insofar as anything is real, it is a reflection of God; how exactly that is may not be known to us.
Physicist here so don't pretend I don't know what science is. (Though like the ancient Pythagoreans I'm sure as soon as I discuss something that has been proven that goes against a purely scientific worldview out comes the pitchforks.) And though I love science, unlike some people here I am willing to admit to the limits of science. Science can lead to all truth in the same way that rational numbers define all numbers: it can't! and Godel proved it.
The real problem with science is that it has been mathematically proven by Godel that there are more things that are true then are provable and thus you can't ever have a scientific theory that can determine the truth or falsity of all things. As soon as you write down that theory, assuming it allows for arithmetic, Godel's incompleteness theorem immediately shows if the theory is true there will be true statements about reality that are beyond provability. Read Godel Esher Bach or Incompleteness or work through it yourself in this textbook as I have.
So like I said above, science is great in it's sphere (and in that sphere let me emphasize it is awesome!) but leads to all truth in the same way that rational numbers leads to all numbers. (And the analogy is precise since Godel used the famous diagonizational argument in his proof.) Russell and Whitehead set out to show in the early 1900s that if we could determine the axioms of reality then through logic work out everything that was true and Godel spoiled the party.
It it would be one thing if these truths were trivial things, but they are not. Some examples of true or false statements that may fall into this category of being unprovable are:
Now, at this point critics almost always tell me: but Joe, Godel's incompleteness theorem is only relative to your set of logic. (Ie... we can prove Goldbach by just adding axioms needed to do so.) Fine. But two things: (first) adding axioms to prove what you want willy nilly is not good science. (Two) You now have a new set of axioms and by Godel's theorem there is now a new uncountable set of things that are true (and non-trivial things like I listed) that are beyond proof.
Now usually comes the second critique: But Joe, this doesn't prove God exists. And this is true. But at least it has been proven God gives you a chance. It has been proven that an oracle machine is free from the problems that hold science and logic back from proving the truth of all things. At least something like God gives you a chance (whereas science falls short).
Or, like Elder Maxwell says so well: it may only be by the "lens of faith" that we can ever know the truth of all things. He maybe be right, and hence the importance to learn by study, and also by faith...