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?”) 5
• 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?”. 1)

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.) 4

• 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?”) 3

• 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?”)2

• 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 really 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.

See also

• The great majority of unusable maths
• Against philosophy
• Aaronson
• Gelman and Yao
• The Unknowable
• The Outer Limits of Reason
• Epistemic Options in the Face of Epistemic Barriers
• Epistemic Boundedness and The Universality of Thought

Sometimes we just can't answer a question, even in principle. This is a list of the problems that make that true:

Conceptual Problems: the question is broken

A malformed question or category error is when you ask for something that doesn't make sense. Questions like "what colour is justice?" or "is sigmoid jealous of ReLU?" fail because they apply properties to things that can't have those properties. Justice isn't the sort of thing that has colour, and mathematical functions don't have emotions. Similarly, asking about "absolute time" or "absolute coordinates" presupposes a framework (Newtonian absolute space and absolute time) that doesn't correspond to physical reality.

Strong incommensurability is when a question tries to mix incompatible theoretical frameworks. For instance, you can't ask about "the Einsteinian mass of phlogiston" because phlogiston doesn't exist in any framework that includes Einsteinian physics, and you can't ask about "the wavefunction of this classical field" because classical fields don't have wavefunctions. The question assumes you can translate concepts between frameworks that are mutually exclusive.

Vagueness means the question doesn't pick out anything specific enough to answer. "When exactly did you become an adult?" has no precise answer because "adult" is a vague category: there's no one moment where you transition from non-adult to adult. The concept has fuzzy boundaries, and so you can't get a single exact answer.

False assumptions render a question unanswerable because, while the question is grammatically clear and seems to make sense, it presupposes something false. "Have you stopped beating your wife?" can't be answered yes or no if you never beat your wife in the first place. Similarly, "Is personality based on nature or nurture?" presupposes a false dichotomy, when the reality involves complex interactions between both.

Logical Problems: no answer exists

Antinomy is when a question is self-referential or circular in a way that prevents any consistent answer. "Is this sentence false?" can't be true (because then it would be false as it claims) and can't be false (because then it would be true). The structure of the question itself creates a logical impossibility.

Incompleteness and undefinability are fundamental limits which only come up in heavily formalised questions. You can't prove that Peano arithmetic is consistent using only Peano arithmetic itself, and you can't define arithmetic truth within arithmetic. For deep reasons, these kinds of question are impossible to answer when you're using any formal system powerful enough to be useful.

Mathematical independence means a question is neither provable nor disprovable from your chosen axioms. Whether the continuum hypothesis is true can't be decided within standard set theory (ZFC). The statement might be true in some models of ZFC and false in others. Similarly, certain values like the busy beaver number BB(7910) are independent of ZFC, meaning no proof can establish their value within that system.

Anthropic self-locating effects create unanswerable questions because your very existence as an observer depends on certain conditions being true. You can't determine the probability you're a Boltzmann brain (a spontaneous fluctuation that created your current conscious state) because if you were one, you'd still observe exactly what you observe now. Your existence selects for certain observations, making the underlying probability inaccessible.

Non-uniqueness is when multiple equally valid answers exist. It's only a problem when you're demanding a single correct answer. The electromagnetic potential at a point isn't uniquely determined because you can add any gradient of a scalar function without changing the physics. The question "what is the electromagnetic potential?" thus has infinitely many correct answers, not because we're ignorant, but because the quantity itself isn't uniquely defined.

Ontic Problems: the answer is physically inaccessible

Uncomputability means no computer or mind, regardless of time or memory, can provide an answer. You can't write a program that determines if an arbitrary program will halt, or if an arbitrary Diophantine equation has integer solutions. They're provably impossible to solve algorithmically. Asking "is this sequence truly random?" is uncomputable because there's no algorithm that can verify true randomness.

Computational intractability means the answer exists and is computable in principle, but would require more time than is physically available, even if you converted the entire universe into a computer. Optimal scheduling problems with all constraints, or cracking a well-encrypted file, are solvable in principle but require checking so many possibilities that they're effectively impossible. The difference from uncomputability is that these problems could be solved with enough resources; they're just practically impossible.

Inapproximability is worse: even finding an approximate answer takes too long. For some problems like finding maximum cliques in graphs or chromatic numbers, you can't even get close to the answer in reasonable time.

Quantum indeterminacy (on some interpretations of quantum mechanics) means physics is fundamentally random, so there's no answer to questions about individual quantum events. "When will this radium atom decay?" has no answer because the decay is genuinely random, not merely unpredictable due to our ignorance. The universe itself hasn't determined when it will happen until it actually happens.

Physical constraints, speed limits, and no-go theorems make certain questions unanswerable because of fundamental physical laws. Information beyond your cosmological horizon is forever inaccessible because space itself is expanding faster than light can travel. You can't measure a quantum state in two incompatible bases simultaneously, and you can't build a perfectly accurate interferometer because of quantum limits on measurement precision.

Mixing means the information you need to answer is irreversibly lost because the process erases fine details. If you want to know the exact microstate of the gas molecules in your room an hour ago, that information is gone. The system has thermalised, and the microscopic details have been scrambled into macroscopic averages that can't be reversed.

Non-ergodicity is the opposite problem: the system doesn't mix, so it can't explore all possible states and reach equilibrium. Glass is a classic example. "What equilibrium will this glassy system reach?" may have no answer because the system is trapped in a local configuration and will never reach the global equilibrium, even given infinite time.

Measurement Problems:
Disturbance or decoherence means your measuring instruments mess with what you're measuring. In quantum mechanics, any measurement strong enough to extract information collapses the quantum state. You can use weak measurements to minimise this, but you can never eliminate the disturbance entirely. The act of observation changes what you're observing.

Complementarity is when we can't answer it because the answer was excluded by another question we asked first. Measuring one property precisely makes it impossible to measure a second property precisely. If you measure an electron's position very accurately, you necessarily disturb its momentum, making it impossible to answer "where is this electron and how much momentum does it have?" The two properties can't be simultaneously known with arbitrary precision.

Epistemic Problems: we can't get at the answer

Underdetermination or unidentifiability is when multiple different theories or explanations are all consistent with the available evidence. Is spacetime fundamentally Lorentzian? Which interpretation of quantum mechanics is correct? These questions might have answers, but the evidence we can gather doesn't distinguish between the alternatives. The data underdetermines the theory.

Chaos means tiny uncertainties in initial conditions grow exponentially, making long-term prediction impossible. You can't predict where a double pendulum will be after 100 Lyapunov times (the characteristic timescale of exponential divergence) because you'd need to measure the initial conditions to impossible precision. Weather prediction fails beyond about two weeks for the same reason: accumulating uncertainties destroy predictive power. This is epistemic because there is an answer but we can never measure the initial conditions well enough.

Computational irreducibility is if there's no shortcut to the answer; you have to run the full process or simulation. For certain cellular automata or complex systems, predicting the state after many steps is just as hard as actually running the system for that many steps. There's no compressed description or formula; the answer is inseparable from the process of computation itself.

Hidden variables are quantities that affect the system but can never be directly observed. In quantum mechanics (on some interpretations), you can't simultaneously know the values of non-commuting observables like spin in different directions. The question "what are the simultaneous values of σ</i>x and σ</i>z for this electron?" is unanswerable because these variables, if they exist, are hidden from observation.

Cognitive closure is the hypothesis that humans might lack the cognitive capacity to understand certain problems, like a dog can't understand calculus. "What is it like to be a bat?" might be inaccessible because our architecture can't simulate bat consciousness. Quantum gravity might be another example: perhaps the correct theory exists but is too complex for the human mind to grasp.

Cognitive bias is patterned irrational thinking that prevents us from answering questions accurately. "How biased am I?" is nearly impossible to answer because your biases affect your assessment of your own biases.

The original "epistemic barrier" in philosophy is the supposed conceptual gap between the mind and the world. How can we know that our perceptions and thoughts correspond to reality when all we have direct access to is our own mental states?

Confusing problems: you can't tell if it's logic, physics, or epistemics