Knowledge: Limits and Potential
I’ve heard of parents who try to take their child’s questions about the world seriously. The most sincere ones become distraught. Not out of annoyance or weariness of simplistic questions, but because they realize, at bottom, much of their knowledge is memorized — knowledge by acquaintance, not by understanding — and they can’t provide real, satisfying answers to the child’s questions.
Though not yet a father, I want such answers in my quiver when I become one. Deep explanations not only of phenomena, but also of how we created such explanations, why they are good and what their limits are. Hence this essay, an extended inquiry into our epistemic limits and potential, for the future child deserving of satisfying explanations, for the reader who too seeks to answer curiosity sincerely, and for those willing to follow me into this inspired labyrinth.
This essay welds together three essays’ breadth of material into one clashing cacophony. I present three views, each with the intent of steelmanning — not only being fair but attempting to convince you — before dismantling them. A tip for reading: After each view, try to voice your agreement or objections on the typed word, so that my subsequent criticism can serve more as tests to your understanding than lectures about rarefied matters.
Now, let me walk you through a question that most every child raises. In answering it, we will be forced to map out the meta-territory of idea space and glimpse the foundations of our knowledge (or lack thereof!), plumbing to our epistemic limits and debouching back again at our infinite potential for explanation.
My imaginary child wants to know why the weather is getting colder. So she asks me,
- “Why is it colder now than before?” It is November, and winter is approaching. Winter means colder weather.
- “Why is the weather cold during the winter?” Earth’s rotation about itself is on a tilted axis relative to its orbit around the Sun. When Earth’s tilt is pointed away from the Sun, the Northern hemisphere gets less sunlight, creating winter. I sketch a picture:
- “Why does Earth orbit the Sun?” There is (kind of) a force of gravity attracting all masses. That pushes Earth towards the Sun, but since Earth is already moving, Earth continually orbits.
- “Why is gravity there?” The Sun, being a big mass, creates an imprint on spacetime that curves it, like a ball placed on a flat paper held at its edges. That curved spacetime geometry guides the Earth’s path around the Sun.
- “Why do masses create a curved spacetime?”
We’ve hit a wall with that one; no explanation for curved spacetime yet. But we still have questions to answer, for the child’s line of inquiry embeds three deeper questions. By asking us about curved spacetime, she demands an answer to the question “Why can’t you explain this claim?”
And after each “why” question, the child is not satisfied. She realizes that there is something more to explain, that each explanation given to her harbors assumptions which demand explanation. So her inquiry implies another question: “Why can’t you give me a self-contained, satisfying explanation that does not depend on any other deeper explanation?”
The Biological-Supernatural View
The first explanation forswears the universality of rationality by looking towards our origins. Our brains evolved on the savanna not to see reality as it is, but to survive long enough to rear our offspring to mating age.
Indeed, there are wavelengths of light we cannot see. There are frequencies of sound we cannot hear. There are true mathematical theorems we cannot prove. Perhaps there are thoughts we cannot think and explanations we cannot conjecture. If the world is so queer that we can’t fully explain it, it makes sense that we struggle. Any knowledge we have now is a consequence of luck, that our perceptions and logic corresponded with reality enough for us to create useful theories. But we should not take this streak of success for granted, for it can cease at any next problem, when we reach the limits of our knowledge. After all, our brains were not fashioned to accurately understand the world.
Curiously, this argument implies the existence of the supernatural. It creates an upper bound to our understanding of the world beyond which we cannot comprehend. And those things we cannot understand, even in principle, we traditionally deem ‘God’. The laws of physics or explanations not accessible to our biological brains may be nonstandard forms of the supernatural, but they are functionally equivalent to the latter. They are the domain of God. In some atheist circles, this implication is sufficient as a reductio ad absurdum, but to others, the rational justification for God only makes the idea more appealing.
I tend to be sympathetic to evolutionary appeals, so I had bought into this view at one time. However, for me, one distinction of logic unraveled the clothes on this emperor: Flawed perceptions do not imply a limit to understanding. This is a nonintuitive point, but it has backing.
We know from the Church-Turing-Deutsch Principle that the laws of physics are made up of entirely computable functions, i.e. they can be described by universal computers to an arbitrary accuracy. (This is remarkable because computable functions make up an infinitesimal subset of all mathematical relationships.) So anything downstream of the laws of physics — quasars, happy feelings or consciousness — must be similarly computable. Every phenomenon in the universe therefore can be modeled perfectly by the computer such as this one you are staring at. That is what it means to be universal. Now, our brains have different architectures from the computer you have in your hands. We don’t know our brains are universal computers, but it is only plausible that they are: Our minds could only know so much if our brains were universal.
And the question of computability is equivalent to that of understanding. Understanding something means to form a representation of that thing in our brains. For example, we understand the laws of arithmetic if we can simulate the laws in the physical circuits of our brain. That simulation forms an explanation. So our explanations can get closer to the laws of physics without bound. Accepting this, it follows that we can understand all phenomena in this universe arbitrarily well.
Justifying the lack of self-contained explanation with a limit of rationality is not plausible. Yet our explanations of the world nevertheless contain components that demand explanation. What gives?
The Foundationalist View
A series of ‘why’ questions result in one of three outcomes: a circularity, an infinite regress, or an axiom. Since the former two are logical fallacies, the goal of knowledge is to derive axioms from reality and build a corpus of knowledge on that. The foundation of our knowledge comprises axioms of which we have reason to believe are true. Only from true axioms can we build the edifice of knowledge.
Some axioms are self-evident, like those of mathematics. Others we must derive from observation, using induction, extrapolation or generalization. For instance, before Einstein proved that matter is made up of atoms, he observed the downward path of dust particles in the air. Falling tiny particles dance in an irregular way, called Brownian motion. Einstein then conceived an experiment. He placed granules in a fluid and measured how far they drift. Using the experiment, Einstein inducted the existence of atoms: He calculated the size and velocity of the particles in the air pushing the granules around. Before, Atomic Theory was mere philosophy; henceforth, it became a scientific theory.
As for the question of why spacetime is curved, that is an axiom of Einstein’s. He used deduction — thought experiments and math — to posit it, and validated it by observation. Its prediction of the trajectory of Mercury was observed to be more accurate than Newton’s in 1915. It predicted light from the Sun on its path to Earth bending, and that was confirmed in 1919. So, just like Atomic Theory, curved spacetime is supported by evidence and self-evidence, so there is no need to further explain already supported axioms.
For the child, we just have to bite the bullet and leave curved spacetime unexplained. We don’t need to demand further explanation, and indeed can’t demand an explanation about the fundamental structure of spacetime; to our best knowledge, the universe just is curved. The child has to accept that, satisfied or not. She may only be satisfied with a story for an explanation — humans crave stories — but reality is not a story. So we have to leave aside our desire for stories, for further explanation, and instead build on a solid foundation.
The whole notion of foundations in knowledge is a fallacy (and yet foundationalism is the worldview that grips the majority of today’s philosophers and thinkers). Earlier, we took axioms to be fundamental to our theories because it was the only tenable solution to a problem of logic, the infinite series of ‘why’ questions. However, if we take axioms to be not only fundamental but foundational, justified in their truth value by experimental confirmation, self-evidence etc., axioms do not solve the problem. They lead to the infinite regress. Let’s illustrate.
Foundationalists justify Einstein’s curved spacetime with observational and experimental confirmation. But no such confirmation can justify any axiom without an additional axiom being introduced. Initially, curved spacetime was a tentative hypothesis. Then, observation accorded with its predictions. To justify curved spacetime, we must introduce a criterion: If a hypothesis is supported by such and such, then it is justified. So when we ask “why curved spacetime?”, we have to justify it with that criterion. But then the criterion remains unexplained. And we can try to explain that with another criterion or idea, but then the latter remains unexplained. Ad infinitum.
To justify axioms is to plunge into a black hole of logic. Axioms that claim to be true — or even probably true — are no solution to the infinite regress; another explanation is needed to justify why the axioms are true, and another justification for that explanation, and so on.
A Theory of Truth
Justified axioms seem to be a through-and-through chimera, pinned by the infinite regress. But before the argument falls on its back, foundationalists may make claims to infallibility in some axiom, or at least assumed infallibility.
How would we access infallibility? A believer in Greek myths would regard the Pythian Oracle as infallible. Perhaps those infallible utterances by her prophetess can build a justified foundation of knowledge. Or, we can assume a stance to be infallible by necessity: Some moral philosophers assert that the worst possible misery of all sentient creatures is the worst moral situation, and even though it might not be an infallible truth, it is an idea so ingrained in human nature that it might as well be infallible. That, they say, can be a foundation to morality.
I refute the latter moral argument’s foundationalist claims in a subsequent essay. Here, I intend only to raise a counterintuitive point: Even if we have a nugget of infallible truth, it can never be a source of an infallible corpus of knowledge. For why that is, let us tell a Greek myth.
Pythia prophesied to the king of Athens, Aegeus, that “if you open the foot of the wineskin before you reach the summit of Athens, you will die in grief”. What could she mean? Aegeus thought it a command to be abstemious while journeying to the acropolis. A wise confidant interpreted differently, and accurately. “Wineskin’’ is a sexual innuendo, and “the summit of Athens” refers to when the youthful Aegeus takes the throne. The “or you will die in grief” was less metaphorical. Two decades later, a grief-stricken Aegeus jumped from a tower, into the sea now called the Aegean. This was because his son Theseus, whom he engendered before becoming king (and whose ship we will philosophically tour in a subsequent essay), accidentally returned to Athens with black sails instead of white, signaling that he was killed by the Minotaur.
The myth illustrates a deep property of knowledge. Pythia pointed to an abstract proposition with an ambiguous statement. Aegeus knew the statement, but he was none the wiser because he misunderstood what proposition it points to. In the diagram, Aegeus mistakenly points the red arrow to a different proposition. The wise bearded guy interprets the statement as intended.
We are restricted by physical statements, which are inherently riddled with ambiguity — no matter how clear or precise they are made to be. Philosopher Karl Popper put it well: “It is impossible to speak in such a way that you cannot be misunderstood: there will always be some who misunderstand you.”
So, physical statements — like Pythia’s utterances or that the worst possible misery is the worst moral outcome — cannot be true. Only the abstract propositions to which they point can be true. Yet, as physical beings, we must couch abstract propositions in physical statements. To gain knowledge, we must guess the propositions to which statements are pointing.
The only way for knowledge from a true axiom to be infallible is for that guessing process to be infallible. We can use formal logic and mathematics, but every explanation has a component of interpretation, and we can never infallibly justify why an interpretation follows from a formal system or axiom. Any justification must include a criterion for why that interpretation is true, and then that criterion must be justified by another criterion, ad infinitum. The process of seeking infallibility in knowledge is again rendered incoherent by an infinite regress.
Any axiom is fallible at the outset. But even if we claim or assume axioms to be true, our understanding of them is fallible, and therefore, any knowledge we construct on an “infallible” axiom is fallible. This violates the spirit of the foundationalist pursuit of knowledge, which seeks foundations of justified true belief, so that we can construct true knowledge.
Concerning the child’s question — why space time is curved — the foundationalists are right in a trivial sense: To the best of our knowledge, spacetime just is curved. But demand for explanation does not cease. We must answer why a curved spacetime is as yet our best knowledge. It is not an illuminating position to assume its truth because it’s a “justified, true” axiom or one “derived from observation”. It is time to start seeing knowledge as tested guesses.
In defenestrating the role of certainty or justification in knowledge, I may seem to be teetering on postmodern relativism; steering clear of the Scylla of foundationalism means navigating towards the Charybdis of relativism. There will be no need for that. Next section, I will present the theory of knowledge that I buy into, one which embraces equally fallibilism and objectivity, arbitrary guesses and critical tests.
Critical Rationalist View
Most of our knowledge about knowledge can be compressed into just two concepts: fallibilism and the growth of knowledge by conjecture and criticism.
If you followed last section, you have already grokked fallibilism. In fallibilism, we forswear justification. Instead, we propose that humans can be wrong about anything they say, think or do. We take that seriously. Any axiom of ours can turn out wrong. And even if we assume an axiom true (though they never can be infallibly true), our understanding of it can be mistaken. Nothing in knowledge is for certain. Even fallibilism.
In adopting fallibilism, we also eschew relativism. Fallibilists are necessarily realists, those who hold the obvious belief that there is an objective reality, because we posit that humans can be wrong about anything, meaning there is something to be right about. Taking the step towards fallibilism means treading towards objectivism, yet it also means sidling towards epistemic optimism: An infinite capacity to be mistaken implies an infinite capacity for progress.
But why do we propose fallibilism? We see that the growth of knowledge cannot — and does not — make use of justification in its axioms. Earlier, we saw that foundationalists characterize the growth of knowledge as a process of branching from true axioms. Critical rationalists’ conjecture and criticism make use of that process, but the axioms need not be true.
Axioms are guesses. They are hypothesized explanations to solve problems. Among countless failed hypotheses, Einstein conjectured a curved spacetime to reconcile the tension between Newtonian mechanics and Faraday’s field theory. That conjecture is regarded with confidence nowadays because it withstood subsequent criticism. But its origins were as higgledy-piggledy as all the other failed hypotheses or any other failed idea.
Now, for how I would actually respond to the child, her question of “Why is spacetime curved?”
We understand curved spacetime as an explanation. Einstein guessed that reality can be seen as a curved spacetime — he asked what if — and then tested the hypothesis by rendering it self-consistent in math, scrutinizing its implications and subjecting it to experiment. With that, he attempted to approximate reality. But like all theories, it is fallible, with no basis or justification but that it has withstood criticism and tests. The reason why I gave you curved spacetime as an explanation is that its flaws have not been ferreted out yet, and a convincing successor that explains curved spacetime has not been engendered yet.
The explanation snuffs out the infinite regress by relinquishing justification in theories and explains how we conceived the idea of a curved spacetime, in a process of conjecture and criticism.
We answered one question of a child. In doing so, we had to tour the depths of epistemology and question the foundations of our knowledge, but we came out with our explanations tested. Let us now move onwards, encountering and resolving better problems in answering children’s questions sincerely.
 Infinite regresses are fallacious arguments that explain what they purport to explain with an argument that in turn requires the same argument to explain itself. Say I want to know what black holes are. At the center of every galaxy is a supermassive bipedal black bear taking an inhale. But, as far as we know, inhaling cannot create enough suction to capture light. So, we have to posit a black hole within the bear’s maw. But according to my explanation, black holes are really bipedal black bears. Infinitely nested bears follow.
If you prefer the famous turtle example, a woman told a philosopher that he believes the Earth is suspended on the back of a colossal turtle. The philosopher asks her what that turtle stands on. She responds that it is poised on the back of another turtle. “And below that turtle?” It is turtles on down, ad infinitum.
 The existence of theorems we cannot prove was first shown by Godel’s Incompleteness Theorem, using a clever form of the infinite regress. To formulate the theorem: For any axiomatic system sufficiently powerful to represent itself, there are abstract propositions that it cannot prove to be true or false, even though those propositions are indeed true or false.
 The idea that positing a bubble of intelligible phenomena beyond which we cannot comprehend is functionally equivalent to the supernatural is made in Deutsch, David. The Beginning of Infinity.
 Deutsch, David 1985. “Quantum theory, the Church–Turing principle and the universal quantum computer.” Proceedings of the Royal Society London.
For the sticklers: How I described the Church-Turing-Deutsch principle is slightly inaccurate (it is accurate to the Church-Turing Principle). The laws of physics are only computable by universal quantum computers, not universal classical ones. And our brain is likely a classical computer (engineering problems like decoherence arising from the brain’s structure preclude it from being a quantum computer; though this point is controverted).
However, that does not create a bound on our understanding in principle, for we can interface our brain with a quantum computer, which can in turn represent the laws of physics to an arbitrary degree. This may seem wild-eyed, but it is probably not forbidden by the laws of physics. And any technology not precluded by the laws of physics is achievable, given the right knowledge.
 Rovelli, Carlo. Reality is Not What it Seems. Riverhead Books, New York, 2017. Pages 30–32.
 Paraphrased from Harari, Yuval. Twenty One Lessons for the Twenty-First Century. Spiegel & Grau, 2018.
 The following discussion makes use of ideas from the article: Deutsch, David. “Why It’s Good to Be Wrong.” 2013. Nautilus Magazine.
 Harris, Sam. The Moral Landscape. 2010, Free Press.
 Fry, Stephen. Heroes:
 This theory of truth is a modification of the Correspondence Theory of Truth, originating with Alfred Tarski, but the distinction between physical statements and abstract propositions was made by David Deutsch.
 Popper, Karl. An Unended Quest.
 Some glib philosophers think themselves to have found a paradox in fallibilism: A fallibilist cannot be fallible about fallibilism itself, for being wrong about fallibilism means being infallible. Yet one who is wrong cannot be infallible. It makes your head spin. Usually, when we encounter paradoxes, that means something is wrong in our logic. To render fallibilism philosophically self-consistent, we might have to make an extra Bertrand Russell-esque stipulation that this proposition cannot apply to itself. Or we can just note the paradox and go ahead running with the proposition.
 Note that I said in this universe. The bio-supernaturalists were mistaken because they underestimated the role of computational universality. However, there may be some places in the multiverse that we, as beings confined to the physical laws of this universe, cannot understand in principle. For example, physicist Max Tegmark speculates a “Layer 4 of the multiverse” which contains universes governed by radically different laws of physics than ours. (See Tegmark, Max. Our Mathematical Universe. 2014, Knopf.) I speculate that the laws of some of those universes cannot be modeled by our brains if they are governed by math. So, to find entities we cannot understand in principle, we had to indulge in two levels of speculation and conjure universes we could never in principle access. I hope you agree that the limit to our capacity to understand is so rarefied as to render the “evolutionary brain” argument moot, for nothing seems to bound our explanations of this universe.
 The image was generated by DALL-E, OpenAI’s text-to-image model of 12 billion parameters. OpenAI 2021 (“DALL·E: Creating Images from Text.”)