logic, mathematics, limits, rationalityI have been struggling with Goedel’s incompleteness theorem (the first one), specifically with implications it seems to have. In the realm of mathematic theorems that have been extensively studied, there is not much doubt about the validity of the proof itself. The doubtful part is what its implications are and how (or if) it relates to our life.
Here is where this becomes interesting. Since you can count, you could argue that your internal theory about the world contains Peano’s arithmetic. As a consequence (providing your theory of the world is consistent), you can never prove every single truth, even if given infinite means.
There have been arguments that this idea is flawed, since people make mistakes and their theories about the world are thus inconsistent.
Which means that either you are consistent and do not make mistakes and you will never know every truth (replace “you” with “humankind past, present and future”, same thing for these purposes), or you make mistakes and are inconsistent and thus you actually might know every truth, but sadly, thanks to inconsistency, you would also know every falsehood - it would logically be all the same to you.
It can be argued that embracing Goedel’s theorem leads to eastern philosophies that appear to reject consistency (a statement can be both true and false at the same time), and in exchange provide the potentail to gain the ultimate knowledge (which at the same time happens to be a complete lack of knowledge).
This is also seems to be reflected in Christianity, where you are told that you should abandon your logic and replace it with belief. It makes sense - logic can only get you so far.
I find it very hard to root for logic, as the logic contains within itself means of its own undoing. Thinking logically means being eternally limited with no means of escaping that jail. The only way out means to abandon logic. It seems that if we used strict logic for ruling our everyday affairs, thinking everything to its ultimate consequences, we would have to behave like pure lunatics (our theories are necessarily inconsistent, single inconsistency spoils the entire theory, making every statement both true and false at the same time, you cannot really make decisions if everything is true and false at once!). It is a great thing we do not behave logically! And apparently, religions have a grain of deep mathematical truth in them.
My main question is - what is the main flaw with the train of thought I proposed?
You’re making an unwarranted jump from undecidability (found in very strong forms in both Gödel’s incompleteness theorem and the halting problem in computer science) to rejection of consistency.
Let’s try an analogy to highlight the flaw in reasoning.
“I can’t observe a teapot buried on the far side of the moon. Therefore, I reject the idea that we can learning things from observation, and should rely entirely on divine inspiration.”
All that the incompleteness theorem tells you is that within this nice framework you’ve created where you can, starting from axioms, prove whether various things are true are false, that there are questions you can ask that you can’t answer. Some questions. The halting problem is very similar in that there are some programs whose ultimate outcome (i.e. will it complete its calculation and stop) cannot be known.
But almost nothing requires that absolutely every well formed question be answerable. So it’s interesting that there exists a class of questions whose answers cannot be known, but it’s not really worth worrying about too much given all the important questions whose answers could be known, but which we do not know yet.
Well formed propositions in a logical system are either true, false, or Godel statements. Godel statements, can be stated in the syntax of the logical system, but cannot be proven true or false by following logical steps of derivation/proof from the stated axioms of the system. As such, any further statement derived from an assumption of true/false of a Godel statement will have no relevance to statements that can be proven within the system.
So, the question is really about relevance. If we have system that is a model of some real world situation, such a system can be be fully predictive and explanatory even though it contains Godel statements. Chances are that the Godel Statements, while phrased logically, simple bear no true relevance to what the system is representing. And that is the important thing to remember- these systems, in the case of the real world, represent/model it. Statements that are purely logical may not have an analog in the represented/real system.
A glaring example of this is that mathematicians assign the status of any specific equation as a Godel Statement to very few statements. Some claim that this is because the Godel Statements, when found, are simply incorporated as new axioms for the systems. This would be because the derivations of such statements would be useful for a particular sub-field in math. Take Euclid’s Parallel Postulate as an example. This postulate was found to be unprovable from the rest of the axioms of geometry. It’s incorporation as an axiom is what allows for Euclidean geometry to be a sensible system and it derivatives include important results, such as the Pythagorean theorem. All these useful things are based on the assumptions of truth of certain statements- the axioms. And from this view all axioms are Godel statements, for if a given axiom is instead accepted as a theorem then it is unprovable from the other axioms. (as axioms are independent of each other by definition, in the sense that they cannot be derived from on another)
So in the end with logic- we must always start somewhere. We must assume the truth of certain unprovable things that are not in contradiction with each other. The question is- are the logical derivatives of these axioms useful: do they represent what we see around us? There is nothing that says what we see around us cannot be represented completely and logically. Godel Statements do not undermine the representational power and efficiency of logical systems.
This is all presented as evidence for what I think the main problem with your train of thought is: “It seems that if we used strict logic for ruling our everyday affairs, thinking everything to its ultimate consequences, we would have to behave like pure lunatics (our theories are necessarily inconsistent, single inconsistency spoils the entire theory, making every statement both true and false at the same time, you cannot really make decisions if everything is true and false at once!)” particularly the part about inconsistency ruining the entire theory- which seems to be a premise throughout your question.
Logic does not provide every answer and shouldn’t be expected to. It is only a tool. If you replace it with belief, then what belief? And how did you arrive at the conclusion to choose that belief?
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