4 ms·

didn’t Godel and Turing and Church and Heisenberg definitively answer this question in the affirmative for most of the first half of the 20th century? There ex

didn’t Godel and Turing and Church and Heisenberg definitively answer this question in the affirmative for most of the first half of the 20th century? There exist an infinite number of questions of the physical world and in math alone that one can ask, whose answers are “we cannot know within the context of these axioms”. Somewhat by definition, these questions are generally somewhat uninteresting, but there sure are a lot of them.

cdancette 9y ago
This doesn't really mean there's a limit to scientific understanding, just that there's an infinite amount of questions we can't have an answer for. But there could be an infinite number of questions we can have an answer for as well, thus there's no limit to our understanding.
- supergarfield 9y ago
  Something being infinite doesn't mean it has no limit (I guess "boundary" would be a better term). If, for instance, it were impossible to answer any question about the strong interaction, I'd clearly call that a limit to our understanding, regardless of the infinity of things we can say about black holes.
  - cdancette 9y ago
    I think we could understand why it is impossible answer this question, and this would be knowledge, and maybe tell us where to look next.
- hprotagonist 9y ago
  In what way are those infinite unanswerable questions not a bound on understanding?
mathattack 9y ago
Yes - in theory these types of questions exist. But are things like the common cold amongst these problems?
- hprotagonist 9y ago
  i would lump most questions about biological systems into this category, yes. (this is my research area) Biological systems become intractable really fast, and deterministic chaos rears it’s ugly head more than you might expect. The game isn’t “can we understand this fully”, normally. It’s more like “can we understand enough to usually be able to fix things when systems are egregiously out of whack”, and that we have a shot at.
supergarfield 9y ago
Gödel's theorem does applies to mathematical systems of axioms, and I don't think it's clear how it translates to physics. To take an extreme example, you could answer any physical question about a universe that's made up of a single particle traveling in a straight line, even though doing so requires a mathematical system complex enough for Gödel's results to apply. On the other hand, there are a number of interesting undecidable questions (in ZF, the axiom of choice and the continuum hypothesis are examples).
- naasking 9y ago
  > To take an extreme example, you could answer any physical question about a universe that's made up of a single particle traveling in a straight line, even though doing so requires a mathematical system complex enough for Gödel's results to apply... ...using current models. It's not clear them at such models are ultimately needed.
jules 9y ago
Depends on your point of view. If you something is not knowable even in principle then maybe the question was wrong, like "how many angels can dance on the head of a pin?". A better answer is perhaps that there are things that we could in principle find out, but cannot because either: 1. We can't do the experiment in practice, such as building a particle accelerator much bigger than the LHC, or factoring a 100000 digit number. 2. It's too complicated and there's no simple description in terms of macroscopic variables, such as the earth's ecosystem.
- lootsauce 9y ago
  You can explain away with reasons why something is not or will not be known. That makes it no less a limit to our understanding.
  - jules 9y ago
    Not always; the unknowable information in quantum mechanics is simply not there. It's not that there is no way to get at the information in practice, it's not even that there is no way to get at the information in theory. It's even stronger: Bell's theorem proves that the information can't be there: simply assuming that the information is there already leads to conclusions that contradict experimental facts. The idea that there ought to be information comes from erroneously applying our everyday intuition to quantum mechanics.
juicyfroot 9y ago
This. Godel's thereom frees the mind.
gbacon 9y ago
See also Rice’s theorem [0]: In computability theory, Rice's theorem states that all non-trivial, semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is non-trivial if it is neither true for every computable function, nor for no computable function. [0]: https://en.wikipedia.org/wiki/Rice's_theorem https://en.wikipedia.org/wiki/Rice's_theorem
- evincarofautumn 9y ago
  I think it’s worth noting that Rice’s theorem and the undecidability of the halting problem apply to arbitrary programs. In practice, the programs that humans write are very much non-arbitrary. Moreover, sometimes we’re happy to trade a bit of expressive power in exchange for stronger guarantees—like using a total/strongly-normalising language in which the answer to “Does this program terminate for all inputs?” is always “Yes”. In a similar vein, most real numbers are uncomputable, but we pretty much only care about the computable ones AFAIK.

cdancette 9y ago

This doesn't really mean there's a limit to scientific understanding, just that there's an infinite amount of questions we can't have an answer for. But there could be an infinite number of questions we can have an answer for as well, thus there's no limit to our understanding.

supergarfield 9y ago
Something being infinite doesn't mean it has no limit (I guess "boundary" would be a better term). If, for instance, it were impossible to answer any question about the strong interaction, I'd clearly call that a limit to our understanding, regardless of the infinity of things we can say about black holes.
- cdancette 9y ago
  I think we could understand why it is impossible answer this question, and this would be knowledge, and maybe tell us where to look next.
hprotagonist 9y ago
In what way are those infinite unanswerable questions not a bound on understanding?

mathattack 9y ago

Yes - in theory these types of questions exist. But are things like the common cold amongst these problems?

hprotagonist 9y ago
i would lump most questions about biological systems into this category, yes. (this is my research area) Biological systems become intractable really fast, and deterministic chaos rears it’s ugly head more than you might expect. The game isn’t “can we understand this fully”, normally. It’s more like “can we understand enough to usually be able to fix things when systems are egregiously out of whack”, and that we have a shot at.

supergarfield 9y ago

Gödel's theorem does applies to mathematical systems of axioms, and I don't think it's clear how it translates to physics. To take an extreme example, you could answer any physical question about a universe that's made up of a single particle traveling in a straight line, even though doing so requires a mathematical system complex enough for Gödel's results to apply. On the other hand, there are a number of interesting undecidable questions (in ZF, the axiom of choice and the continuum hypothesis are examples).

naasking 9y ago
> To take an extreme example, you could answer any physical question about a universe that's made up of a single particle traveling in a straight line, even though doing so requires a mathematical system complex enough for Gödel's results to apply... ...using current models. It's not clear them at such models are ultimately needed.

jules 9y ago

Depends on your point of view. If you something is not knowable even in principle then maybe the question was wrong, like "how many angels can dance on the head of a pin?". A better answer is perhaps that there are things that we could in principle find out, but cannot because either: 1. We can't do the experiment in practice, such as building a particle accelerator much bigger than the LHC, or factoring a 100000 digit number. 2. It's too complicated and there's no simple description in terms of macroscopic variables, such as the earth's ecosystem.

lootsauce 9y ago
You can explain away with reasons why something is not or will not be known. That makes it no less a limit to our understanding.
- jules 9y ago
  Not always; the unknowable information in quantum mechanics is simply not there. It's not that there is no way to get at the information in practice, it's not even that there is no way to get at the information in theory. It's even stronger: Bell's theorem proves that the information can't be there: simply assuming that the information is there already leads to conclusions that contradict experimental facts. The idea that there ought to be information comes from erroneously applying our everyday intuition to quantum mechanics.

juicyfroot 9y ago

This. Godel's thereom frees the mind.

gbacon 9y ago

See also Rice’s theorem [0]: In computability theory, Rice's theorem states that all non-trivial, semantic properties of programs are undecidable. A semantic property is one about the program's behavior (for instance, does the program terminate for all inputs), unlike a syntactic property (for instance, does the program contain an if-then-else statement). A property is non-trivial if it is neither true for every computable function, nor for no computable function. [0]: https://en.wikipedia.org/wiki/Rice's_theorem https://en.wikipedia.org/wiki/Rice's_theorem

evincarofautumn 9y ago
I think it’s worth noting that Rice’s theorem and the undecidability of the halting problem apply to arbitrary programs. In practice, the programs that humans write are very much non-arbitrary. Moreover, sometimes we’re happy to trade a bit of expressive power in exchange for stronger guarantees—like using a total/strongly-normalising language in which the answer to “Does this program terminate for all inputs?” is always “Yes”. In a similar vein, most real numbers are uncomputable, but we pretty much only care about the computable ones AFAIK.