Explanations as Hard to Vary Assertions

Update: After some investigation, I found out that The Beginning of Infinity by David Deutsch contains a few minor misquotes of Popper, Turing and others. Nevertheless, it is an excellent book.


As I read through Rationality: A-Z, I kept seeing similarities to David Deutsch’s worldview. Deutsch pioneered quantum computation in the 1970s, motivated by the possibility of gaining a deeper grasp of quantum physics and as a potential way to test many-worlds.

This post is adapted from my review of The Beginning of Infinity. I read it a couple of years ago, and it is among the most formative books I have read.


We have a great deal of knowledge about the vast and unfamiliar reality that causes our observations and the elegant, universal laws governing that reality. This knowledge consists of explanations: assertions about what is out there beyond appearances and how it works. Where do explanations come from? The source of our knowledge is a process of conjectures alternating with criticism. Humans possess the capacities for creativity and rationality, enabling them to actively pursue error correction through creating, combining, altering and criticising ideas in the quest for good explanations.

Good Explanations

The role of experiment and observation is to choose among the ideas we come up with. We interpret experiences through explanatory theories, but good explanations are not obvious. Fallibilism entails not looking to authorities but acknowledging that we may always be mistaken and that no belief can ever be rationally supported or justified conclusively. Always, there remains a possible doubt as to the truth of the belief. What distinguishes science from other belief systems is that scientific beliefs are always defeasible and never final. We ought to continuously be correcting errors and updating beliefs in the quest for knowledge. We correct errors by seeking good explanations.

Good explanations in science are hard-to-vary assertions about reality. They are hard-to-vary because they provide specific details that fit together so tightly that changing them ruins the explanation. This criterion helps to eliminate bad explanations that keep adding justifications in light of refutations and counterevidence to avoid falsification. An explanation that is hard-to-vary but does not survive a critical test can be considered falsified.

We can explain what it means for a conjecture to be hard-to-vary in terms of Bayes’ theorem.

[Paraphrasing from Decoherence is Falsifiable and Testable] A good explanation offers precise assertions about reality. If there is some evidence that the assertion can’t explain, then the likelihood will be tiny. Thus, the numerator will also be tiny, and likewise the posterior probability . Updating on the near impossibility of evidence has driven the probability of the assertion down to epsilon. A theory that refuses to make itself vulnerable in this way will need to spread its probability widely by being vague. [Update: this is slightly incorrect because it doesn’t consider all possible hypotheses, please refer to Eliezer’s elucidation.]

Frank Wilczek describes hard-to-vary-ness as follows “A theory begins to be perfect if any change makes it worse.” He explains further using the Standard Model as an example of a hard-to-vary explanation:

Too many gluons! But each of the eight colour gluons is there for a purpose. Together, they fulfil complete symmetry among the color charges. Take one gluon away, or change its properties, and the structure would fall. Specifically, if you make such a change, then the theory formerly known as QCD begins to predict gibberish; some particles are produced with negative probabilities, and others with probability greater than 1. Such a perfectly rigid theory, one that doesn’t allow consistent modification, is extremely vulnerable. If any of its predictions are wrong, there’s nowhere to hide. No fudge factors or tweaks are available.

Good explanations help us achieve better map-territory convergence. They allow us to construct more accurate models of the territory. To grasp reality, we must resist the temptation to start from conclusions to bend facts to fit them. Grasping reality entails overcoming our cognitive biases, going on joyful explorations across the territory and improving our map along the journey.

Poor explanations purport to explain anything and everything. Such explanations explain nothing. Freudian psychoanalysis was equally good at coming up with an explanation for every possible thing the patient could do. Similarly, God and magic can explain anything and everything. Therefore, they offer us no explanatory power.

Some good explanations have enormous reach: they explain more than what they were initially intended to. In science, good explanations gave rise to the principle of Testability, which constrains a scientific explanation to be hard-to-vary. Still, good explanations go beyond science and apply to philosophy, politics, morality, economics, etc.

Bad Explanations

Explanations are two a penny. Good explanations are extremely hard to come by. Bad explanations are not necessarily false. They can be true but completely lacking in explanatory power.

Suppose you are watching a conjuring trick, and you are trying to explain what is happening. An example of a bad explanation would be, “Well, it is magic.” That is a bad explanation because you can apply that explanation to anything. Another example of a bad explanation is to say, “Well, the conjurer did something.” This shows that a bad explanation doesn’t necessarily have to be false but just utterly inadequate.

If we take, via analogy, the laws of physics and trying to explain things in the natural world, we could answer the questions “What is the origin of species?” and “What is the origin of adaptations in the biological world?” with “Atomic interactions cause them.” This statement is true, but it doesn’t explain. A good explanation of these phenomena is the modern variant of the Theory of Evolution.

Science and philosophy are both subsets of the quest for good explanations. Science and philosophy overlap, but Popper’s criterion of demarcation helps us avoid going down blind alleys. It states that scientific theories are in principle testable by experiment and metaphysical theories are the ones that aren’t while making no judgment about the validity of either type of theory.


Bad philosophy—a subset of bad explanations—does not only contain falsehoods, but it also disturbs our ability to search for good explanations. False philosophy is not harmful; in fact, errors are the standard state of human knowledge. We can expect to find errors everywhere, including in the theories that we most cherish as true. However, bad philosophy is harmful because it aims to cut off the progress of knowledge, coercing us to remain in the dark. It is the kind of philosophy that not only makes false claims but more dangerously says, “You mustn’t think about so and so.”

Before the Enlightenment, the Church was the authority forcefully closing off the progress of knowledge to maintain its hegemony. Today, the scientific establishment has become the new Church, oppressing creativity and imagination. Science today insists that everyone believes in the same thing and in the same way.

Empiricism can be and has been misused and misapplied throughout the history of science. Galileo’s fellow scientists pointed to empirical evidence to resist his theory. For example, when a ball was dropped from the top of a tower on a sailing ship, the ball fell at the bottom on board. This suggested to them that the Earth was stationary. However, the theory of special relativity explains this observation through inertial frames of reference. This letter from Galileo to Kepler captures his frustration:

My dear Kepler, I wish we could laugh at the extraordinary stupidity of the mob. What say you about the foremost philosophers of this University, who with the obstinacy of a stuffed snake, and despite my attempts and invitations a thousand times they have refused to look at the planets, or the moon, or my telescope?

The observation of stars with the naked eye provides another example of a parochial error in science. Generations of philosophers and scientists speculated about the reality of stars in the night sky, convinced that twinkling was a real property of stars. Modern telescopes contain automatic mechanisms that continuously change the shape of the mirror to compensate for the shimmering of the Earth’s atmosphere. Observing through such a telescope, stars do not appear to twinkle as they did to generations of observers in the past. Those observations of stars twinkling are only appearances. These appearances are certainly real aspects of our perception, but they have nothing to do with the reality of stars. Thus, we cannot be certain about our observations.

Modern cognitive science tells us that our brains reconstruct visual reality. There is no such thing as a raw experience of reality. The famous lines illusion is an illustration of visual bias.

The cognitive processes which form our experiences have been forged over many millions of years of genetic variation alternating with selection. There is no reason to believe that they have been optimised to capture reality comprehensively and accurately. We ought to acknowledge that our knowledge of reality is inherently uncertain. To see clearly, we ought to seek error correction through creatively pursuing good explanations.

Science is a human process and as such, it is unsurprising that it contains bias and dogma. It is a well-established fact that humans are far from being optimally rational agents. Science helped us move away from the tyranny of the Church, but it didn’t eliminate dogma; it merely replaced the Church with the scientific establishment. These problems don’t mean that science is bad in principle. However, the way a lot of science is done today is dogmatic, thus potentially closing off the growth of knowledge in many fields.


Explanations in science traditionally took a reductionist approach. Such an approach claims that to have a complete explanation of what is going on at the higher levels of abstraction, one must understand what is happening on lower levels. For example, to understand humans, one needs to understand their biological organs. Understanding the organs entails understanding cells, then biochemistry, physical chemistry, physics, and all the way down to fundamental physics. This quote by Douglas Hofstadter captures this prejudice beautifully:

Saying that studying the brain is limited to the study of physical entities would be like saying that literary criticisms must focus on paper and bookbinding, ink and its chemistry, page sizes, and margin widths, typefaces, and paragraph lengths, and so forth. But what about the high abstractions that are the heart of literature—plot and character, style and point of view, irony and humour, allusion and metaphor, empathy and distance, and so on? Where did these crucial essences disappear in the list of topics for literary critics?

Reductionism is a prejudice. It is historically understandable because the physical sciences developed fastest, and it so happens that some of the best explanations in physics have been bottom-up. For example, space and time, elementary particles, and so on. But it has never been the case, even within physics, let alone other sciences, that all good explanations are reductionist. For example, the Theory of Evolution has achieved immense success without dealing with atoms.

Modes of Explanations

The quest for good explanations implies that we must not have the reductionist prejudice. If we do find an explanation that is on a higher level and it is a good explanation—provides hard-to-vary assertions about reality—then it is simply irrational to reject it just because it doesn’t have the reductionist form. We have been historically taught that reductionist explanations are the kind of explanations we should pursue. Still, by deeply understanding the power of good explanations, we become more open to different modes of explanation.

It is nearly always the case that whenever someone finds a new and much deeper theory, then it is not only a better explanation, but it is also a different mode of explanation. For example, in physics, Einstein’s explanation of gravity in curved space-time was a new mode of explanation. Relativity was not merely a tweak on Newtonian gravity, e.g. instead of an inverse square law had an inverse cube law. Relativity was a different kind of explanation altogether. It explains that space and time—which Newton’s theory regards as immutable background entities—are a dynamical space-time object which bucks and weaves and explains all sorts of things apart from just the motion of planets.

Science and Humanity

Science appears to be largely a story of us fighting our way past anthropocentrism, this notion that we are at the centre of things. We are not special; we share more than half our genes with a banana. This notion is the principle of mediocrity. Deutsch believes that this is literally true but nevertheless believes that we are central to any understanding of the universe.

  • First, if you think of that chemical scum, namely us, and possibly other conscious, intelligent beings, then to study that scum fully is impossible. Unlike every other scum in the universe, this scum is creating new knowledge, and the growth of knowledge is profoundly unpredictable. As a consequence of that, to understand this scum—never mind predict—but to understand, to understand what’s happening here, entails understanding everything that is happening in the universe. The growth of knowledge is profoundly unpredictable because if we could predict it ahead of time, we can invent future inventions now, yet we cannot. To predict the future perfectly, we ought to simulate the future perfectly, and to simulate the future, we ought to simulate the universe, and the best we can do to simulate the universe is to watch the universe unfold in real-time.

  • Second, the other way around is that the reach of human knowledge and human intentions on the physical world is unlimited. We are only used to having a relatively tiny effect on this small insignificant planet and the rest of the universe to be completely beyond our ken, but that is just a parochial misconception. We know that there are no limits (by the universality of computation) on how much we can affect the universe if we choose to.

We, and all other conscious, intelligent beings, are completely central to any understanding of the universe.

Remark: It is important to define what Deutsch means by computation. Computation—within any laws of physics—is the instantiation of abstract objects and their relationships using physical objects and their motions and interactions. Computation is universal because we can create a computer within the universe that can simulate any physical process.

The goal of science isn’t to deem humanity worthless. Experience plays an important role in science. Our knowledge is theory-laden, meaning there is no such thing as the raw comprehensive, accurate experience of reality—all our experience of the world comes through layers of conscious and unconscious interpretation. The role of human experience in science is to guess new conjectures and choose between conjectures that have already been guessed. That is what learning from experience is about.


All progress comes from the quest for good explanations—hard-to-vary assertions about reality. There isn’t an authoritative source of knowledge. Still, we can use this process of seeking good explanations through conjectures alternating with criticism to grind out knowledge about reality that is sufficiently reliable for us to treat as provisionally true and act upon.