So does Gödel's incompleteness theorem apply to scientific knowledge? I mean, as I understand it, he proved that any useful logical system contains statements that can neither be proved true nor false. But does that apply to quantum mechanics, for example? Are there self-referential statements in QM?
It applies to any formal system powerful enough to encode the properties of the natural numbers.
Quantum Mechanics, as a formal system can encode the natural numbers, and thus can form a substrate for self referential Gödel encoding. The same principles would apply to a quantum system, and any other capable of encoding natural numbers.
Well, scientific hypotheses are tested through observations or, ideally, experiments. Maybe QM could encode arbitrary statements about natural numbers. But I can't imagine how one could test them through observations or experiments.
The question wasn't about reality, but about quantum physics as a formal system, which is very much a formal mathematical system that can exist separate of hypothesis testing and whatnot. The point is that as soon as we have any finte definition of a model for reality, incompleteness kicks in.
As I understand it, rather than "finite definition of a model for reality" it's really more a matter of having a single model. No one model can cover everything but multiple models can, right?
What would be the difference between multiple models and one model containing all of them? You can keep adding on to the model, or mixing and matching models, and you will never get one that immune to go goedel. Either you have model of physics with contradictions, or you have one where there exist questions that cannot be proven true or false, in other words, undecidable.
Right. I can't imagine a single model for reality. That is, in the sense that you start the model running and it reproduces all events everywhere from the Big Bang to whatever happens eventually.
But models that generate results that can be tested, through observation and experiment? Sure. Maybe even arbitrarily integrated models.
However, there's uncertainty throughout. So models can't be deterministic. Certainly at the "ends", at the quantum level, and at the level of consciousness.
A model doesn't have to be deterministic, use finite quantities, or even be possible to execute in a simulation. Math often does not concern itself with such things. Undecidability is concerned with the set of rules themselves. Even if it is impossible to actually simulate, is there a finite set of physical laws which drive all things? If yes, and if you can construct the basic requirements for counting and making logical statements and such within that system, then you are subject to incompleteness.
That said, it's interesting to think about a universe with truly infinite rules. Each physical law could have minor exceptions caused by smaller more detailed phenomenon. Each time you would discover some new principle, it would reveal more yet unknown questions, a fractal of infinite knowledge to be refined and science to do. But I think most scientists hope for a finite set of underlying rules for reality.
I think it applies specially to scientific knowledge.
There is a connexion there between Gödel and Popper.
Except inside a formal system, you can never prove that something is true, only that one explanation is better than other in an endless pursue of better explanations.
I'm not sure there is such thing as not-scientific knowledge, by the way.
> I'm not sure there is such thing as not-scientific knowledge, by the way.
I'm fully on-board with the overwhelming, world-changing effectiveness that the scientific method provides for distilling factual, empirical knowledge and truth.
Lately, however, I've been contemplating forms of knowledge and understanding that are more difficult to assess and validate -- things that might be typically described as wisdom or keen insight. Our scientific instruments can't provide observations that let us robustly verify such knowledge, but to me it seems very evident that it exists.
Some examples: What is important to building and maintaining strong relationships? How can one prepare for and handle personal hardship? If one finds themselves in a fortunate position with excess resources, what are good ways to use those resources to help others?
Science can help us with these questions, but humans have useful knowledge to bring to bear in answering those questions that can't be yet described within the framework of science.
Differentiating by quality or truthiness is ridiculously hard in such domains, but I don't think that is a valid reason for dismissing such things altogether.
Yes, that's what I was getting at. Insight, wisdom, etc.
Even so, one can also apply the scientific method to those sorts of knowledge. One can look at performance. Quality of relationships. Success at dealing with hardship. That's part of psychology. But it hasn't received enough attention, I think.
Right, Popper. I guess that's why they were adjacent on my university reading list ;) But I don't think of that as an expression of limitations in hypotheses as formal systems. Because you don't reject an hypothesis (or not) through logic and reasoning. You do that through observation and experiment. So it's rather a fundamental limitation in the scientific method.
Anyway, I get that they're similar. But I don't see them as the same, but rather complementary.
I also get what you say about knowledge. Scientific knowledge is what you get by studying external reality. But there is also knowledge that you get through introspection.
What I hear from a collegue whose thesis is on alternative explanations for the quantum field observations, the main assertion is that there are difficulties in observing quantum particles which may simply be not true in light of newer theories with more explanatory power such as the pilot wave equation that sheds more light upon, for example, phenomenon such as quantum entanglement than just saying "it's more probable therefore it happens".
