Ways we can fail to answer

In what ways can we can fail to answer a question?

I mean necessarily fail: actual barriers to knowledge, rather than skill issue hurdles. But of course contingent failures are much more common: “We didn’t ask the question in the first place”, or “We didn’t have the particular insight that would have allowed for productive research”, or “We didn’t manage to remove every cognitive bias”, or “Instrumentation is really hard”, or “We are not rich enough to run this study yet”, or “We worshipped the problem”.

I also mean fail exactly; there are often excellent approximations, and we can often legitimately patch over tricky philosophical questions with our unanalysed tacit knowledge.

Conceptual problems (the question is not a question)

A malformed question or category error: the question may ask for something which doesn’t make sense (e.g. “what colour is justice?”, or “Is sigmoid jealous of ReLU?”, or “Did these things happen at the same absolute time? What are the absolute coordinates of this event?”.)
- Strong incommensurability: the question doesn’t make sense because it mixes frameworks. (e.g. “What is the Einsteinian mass of phlogiston?”, or “What’s the wavefunction of this classical field?”)
Vagueness: the question may fail to mean anything in particular. (e.g. “When exactly did you become an adult?”)
False assumptions: we can’t answer it because, while it was clear and made sense, it was wrong from the start (e.g. “Have you stopped beating your wife?” or “Is personality based on nature or nurture?”.)

Logical problems (the question has no answer)

Antinomy: it turns out that there is no answer because the question is circular or involves itself somehow (e.g. “Is this sentence false?”)
Incompleteness (“Show that Peano arithmetic is consistent with Peano arithmetic”.) and undefinability (e.g. “Is this sentence true in arithmetic?”) are about some proof answers being inaccessible inside systems powerful enough to be interesting and useful.
Mathematical independence: The question is neither provable nor disprovable with these axioms. (e.g. Is there a well-ordering of the reals? Is the continuum hypothesis true in ZFC? What is the value of BB(748) in ZFC?)
Various holes in social mathematics. Important questions like "what's the optimal voting system?", "what does the majority prefer?", "what's a democratic system where people don't have an incentive to vote strategically?", "what's the perfect design for a market?" in general don't have an answer. See e.g. here.
Anthropic self-locating effects: you can’t get at the answer because you couldn’t exist to observe it. (e.g. “What’s the probability I’m a Boltzmann brain?”, or “What’s the prior probability of observer-permitting universes?”)
Non-uniqueness is only a problem for questions which ask for the one true answer. (e.g. “What is the electromagnetic potential at this point?”)

Ontic problems (the answer is literally inaccessible)

Uncomputability: we can’t answer it because no computer or mind can. (e.g. “Is this random?”, or “Does this Diophantine equation have integer solutions?” or “What’s the shortest Python program that outputs this file?” or “Can you write a program that checks if conjectures follow from these axioms?” Or you asked about the language the problem is in.)
Computational intractability: we can’t answer it because it would take too long, including if we turned the universe into a computer. (What’s the best way to schedule these classes, avoiding all conflicts and respecting room capacities and lecturer availability? Or “I forgot my password; crack this file”.)
Inapproximability: we can’t even approximate the answer because it would still take too long. (e.g. “What’s the maximum clique size in this graph?”, “What’s a (log n)-approximation to the chromatic number?”)
Quantum indeterminacy: there’s no answer because (maybe) physics is intrinsically random. (e.g. “When will this radium atom decay?”)
- Physical constraints, speed limits and no-gos: it can’t be answered in principle because of physics. (e.g. the answer is beyond the cosmological or particle horizon, “what will happen in this galaxy 20 Gly away?” or “Can we measure this state in two bases?”, or “Make a perfectly accurate interferometer”, or “What are the microstates of this black hole?.)
- Mixing: the answer is gone; we arrived too late; the system forgot the answer. (e.g. “What was the exact microstate of the gas in this room 1 hour ago?”, “What was the original state of this thermalised system?”)
- Non-ergodicity: the answer isn’t reachable because the system doesn’t mix. (e.g. “What equilibrium will this glassy system reach?”)
- Measurement problems
  - Disturbance or decoherence: we can’t answer it because our instruments disturb the thing in question too much.
  - Complementarity: we can’t answer it because the answer was excluded by another question we asked first. (e.g. “where is this electron and how much momentum does it have?”)

Epistemic (we cannot get at the answer)

underdetermination or unidentifiability. (e.g. Is spacetime fundamentally Lorentzian? Which interpretation of quantum mechanics is true? Why do physical constants look fine-tuned? What was the ancestral DNA sequence at some past generation?)
Chaos: we can’t answer it because we can’t measure the initial conditions well enough to predict large systems. (e.g. Where exactly will this double pendulum be in 100 Lyapunov times? What’s the weather like 3 months out?)
Computational irreducibility: getting the answer is inseparable from running the system, the question has no shorter answer than a full simulation (e.g. “What will this cellular automaton look like in 10^100 steps?”)
Hidden variables: we can’t ever observe the actual variables (e.g. What are the simultaneous values of σ_x and σ_z for this electron?)
Cognitive closure: we’re not smart enough to answer it. (e.g. “What is it like to be a bat?” It kinda looks like quantum gravity could also be an example.)
Cognitive bias: we’re not rational enough to answer it (e.g. “How biased am I?”)
I suppose I should mention the original “epistemic barrier”, the putative conceptual barrier between the mind and the world.

??? (the question is logical and/or ontic and/or epistemic idk)

Observer effects
- Back action (e.g. What’s the pre-measurement spin of this electron? What are the “real” observables in this quantum system? or “Measure the full-speed time evolution for this system”.)
- Theory-ladenness:
  - Semantic theory-dependence: Our observational vocabulary already presupposes theoretical commitments that prejudge the answer (e.g. What’s the rest mass of an electron, without assuming special relativity? What’s the ‘real’ temperature of the CMB independent of blackbody theory?)
  - Perceptual theory-dependence: Our perception is shaped by theoretical expectations, so we cannot see the answer “directly” (e.g. What do electron tracks really look like? What does this fMRI show before we apply the hemodynamic response model?)
Future contingents: it doesn’t have an answer yet so we have to wait. (e.g. “What lottery numbers will win next week?”)
Hysteresis: we can’t answer it because we weren’t there at the start and the system remembers. (e.g. What’s the magnetic moment of this material at field strength H? When will this old rope snap? At what temperature will this water freeze?)
Reflexivity and strange loops: the thing in question is self-referential or causally dense, or it [changes](https://en.wikipedia.org/wiki/Demand_characteristics) when answered, and so can’t be picked apart. (e.g. “What’s the best method for finding the best method?”; “Which level of description is fundamental in this self-referential system?”, “Which part of your mind is the real you?”.) See also antinomy.

You could view some of these as not failure but just having a non-unique answer; we don’t fail to answer the question, just parametrise the observer and then answer. If non-uniqueness is a failure, it’s a happy one; it just means that you get too many answers and have the nicer problem of picking one.

You will have done well if even once in your life you fail to answer a question for these reasons. Getting so far means you have avoided hundreds of punji traps, claymores, nerve gasses, madnesses.

LESSWRONG
LW