## How do we know anything?

#### August 2022

Consider the following mathematical idea:

• A function $f(x)$ is smooth and continuous (i.e., it is differentiable everywhere)
• At some point $x_1$ the slope of $f$ is negative, $f’(x_1) < 0$
• At some other point $x_2 > x_1$ the slope is positive, $f’(x_2) > 0$
• Therefore by the intermediate value theorem there must be a minimum where $f’(x) = 0$ somewhere between $x_1 < x < x_2$

This strikes us as obviously true and so we accept it, at least at the undergraduate level, as “proof” of the existence of the minimum.

But really this is a feeling. Where does it come from? Some process in our brains is telling us to see something here, but when you try and pin it down our initial sense of clarity dissolves into a feeling of slipperiness. Mathematics has a long history of producing results that seem obviously true but which turn out later to be at least incomplete. More simply, everyone has had the experience of being struck by an idea clearly being right only to realize later something important was missed.

Thinking about this, there is a fork in the road. On one side are interesting questions of how the brain produces the overall feeling of correctness and its reliability. On the other are the even deeper questions about, if this feeling is usually correct, what kinds of physics support the brain doing this in the first place. I’m not particularly concerned with the philosophy — formal epistemology etc — as I’m less concerned with what it all means and more practically with how is this possible.

“Common sense” is better thought of as “basic reasoning”1: a collection of fast heuristics that work well for regular life but may not generalize very well to novel or exotic situations. There are a bunch of these, let’s say, “core cognitive approximations” that underlie these fast basic reasoning capabilities, including, to just pick three:

• the feeling of a thing being “true”
• the recognition of causation
• the feeling of two things being “similar,” for example across metaphors

All of our brains evaluate these things, and we usually assume that they do so in roughly similar ways. This is critical for having a shared foundation of reason and the application of logical thinking. Two brains with different senses of truth will have different senses of reason, and they will find it difficult to share their thinking.

On the other hand, as products of evolution, we developed these feelings because they conferred a survival advantage. The fact that this works over a distributed system of independent agents tells us something deceptively profound: there is, at the deepest levels, an absolute arrow of truth, some real sense in which some things can be “correct” and others can be “wrong.” We can’t arbitrarily make up conflicting math and call it equally valid (and believe it) without compromising this differential survival advantage. But also as products of evolution, all of these learned functions are approximations; they break down in unpredictable ways on the margins, and may be completely unhelpful for truly out-of-distribution observations. I suspect this is why words alone are wholly insufficient for really understanding quantum mechanics, for example.

Returning to the question of how we might verify our internal feeling of correctness, we can take this same line of reasoning back in the direction from which we came: we might believe that a belief is truly correct if we can use it to improve our relative competitiveness. It is probably impossible, in the sense that it is ill-defined, to nail down fundamental truth except by reference to a relative effect in this way.2 Until a suspected conclusion has been applied, the best we can do is limited by these neural approximations. In many cases this means that we don’t (can’t?) verify an idea directly, but we can infer its correctness because it is a component of a longer reasoning chain that produces a useful result at the end.

For example, calculus is abstract mathematical tool which turns out to be very useful for improving our competitiveness, i.e., developing testable theories of physics which we can use practically. A civilization that possess math at the level of calculus or beyond would possess a huge advantage relative to one that doesn’t. Calculus has important internal “proofs,” but at the end of the day these need this kind of relational support grounded in differential survival advantage to exit circular reasoning.

This also hints at an answer to whether math is “discovered” or “invented”: if a given concept can be correlated to the physical world in some useful way that produces a competitive advantage, if even at a great distance3, there is a strong argument for it being discovered, whereas if not and it remains a wholly abstract concept disconnected from the graph of ideas correlatable to the physical world, it might be said to have been invented. Mathematicians especially require very well tuned cognitive approximations, as they have to rely on these feelings of correctness while wandering in the desert looking for insights.

Of course, the approximative feelings underlying language are useful in that they facilitate fluent reasoning. If we needed to stop to carefully formalize every step of our thought process, it would be very difficult to do anything. It is important to put care into your metacognitive processes validating how good these feelings are and especially preventing them from corruption by the propaganda we encounter throughout life.

The feeling of intuitive correctness is often suspect but in day-to-day life language is what we rely on to think. The earlier example of quantum mechanics is instructive: language is completely useless for initially wrapping your head about the theory since it is simply too far out of distribution relative to the environment that produced the concept spaces which underlie the language we developed in childhood. Even to this day professionals actively debate the correct “interpretation” — what it really means — while others have simply resigned themselves to shut up and calculate. But this is a problem with language, not the physics!

The key idea here is that this limitation is present in all of our understanding of the world, and so while language spans a lot of our useful life experience, it is still an approximation, and we should be weary about both venturing out-of-distribution (developing new theory) and talking past each other endlessly (organizing groups of humans). We would do well to keep a voice in the back of our heads warning us of the approaching edge of the linguistic usefulness, so to speak, and the importance of not simply powering through on the feeling of sense alone, no matter how obvious and inescapable it may appear at the time.

1. Cognitive scientists sometimes call this “heuristic” processing as opposed to “analytic” processing. See e.g., Heit (2015), or Kahneman & Frederick (2005)

2. This is slightly different from epistemic relativism, since I’m not saying that truthfulness is perspectival — constraining to a consensus truth might be possible — but only that the epistemic status is only discoverable by reference to a relation. Also, one might argue this is related to the need for prospective intervention to establish causality in the scientific method. But causality is a very tricky and difficult topic in its own right and so I think best to avoid it for now here.

3. By “at a great distance” I mean an extended reasoning chain of abstract concepts. For example, the geometric idea of a fiber bundle is very abstract, but through a long chain of dependencies differential geometry turns out to be critical for classical electromagnetism (also an abstract idea), which can be used to make electronics. It is this powerful explanatory power that works in both directions that allows us to do better than merely the cognitive feeling of “proof.”