Math, Science, and Logic Are Predictive Models

Epistemic status: content summarized and synthesized (0-1 steps of reasoning) from the Sequences by Eliezer Yudkowsky.

Unreasonable Effectiveness

I’ve heard several people – professors, peers, and folks on the internet – express surprise at the “unreasonable effectiveness” of mathematics in describing the physical universe. In other words, they claim to be surprised that mathematical laws are able to describe the workings of the universe so precisely. From a certain point of view, this does seem surprising. If mathematics is entirely a priori, abstract, and not tied to physical reality, why should it be able to predict the behavior of physical quantities so well? Surprise is a sign that your map does not match the territory. Let’s try to answer this question, so that we can explain this surprise and update our map.

We can roughly break down the ways our model could be wrong into two possibilities – either the reasoning that leads us to be surprised is wrong, assumption that math is not tied to physical reality is wrong. To me, the reasoning that leads to the surprise seems pretty ironclad. If math is not tied to physical reality, the only way it could do this well is by pure chance, which would be highly unlikely and thus highly surprising. This is not to say that breaking down the possible problems this way or saying that “the reasoning seems pretty ironclad” is a valid way to prove that the assumption must be the problem, I’m just outlining some heuristic reasoning to suggest what the most fruitful line of further thought might be.

Is Mathematics A Priori?

So let’s challenge the assumption that mathematics is actually a priori, and not tied to physical reality. If mathematics is deduced a priori, that means it should be self-evident: no evidence would be necessary to prove it, and no evidence could possibly disprove it. Take the simple mathematical statement 2 + 2 = 4. Can we imagine evidence that could disprove it? Well, I don’t know of any way to say this better than Eliezer Yudkowsky did, so I’ll just rip this thought experiment from the Sequences wholesale:

Suppose I got up one morning, and took out two earplugs, and set them down next to two other earplugs on my nighttable, and noticed that there were now three earplugs, without any earplugs having appeared or disappeared—in contrast to my stored memory that 2 + 2 was supposed to equal 4. Moreover, when I visualized the process in my own mind, it seemed that making xx and xx come out to xxxx required an extra x to appear from nowhere, and was, moreover, inconsistent with other arithmetic I visualized, since subtracting xx from xxx left xx, but subtracting xx from xxxx left xxx. This would conflict with my stored memory that 3 – 2 = 1, but memory would be absurd in the face of physical and mental confirmation that xxx – xx = xx.

I would also check a pocket calculator, Google, and perhaps my copy of 1984 where Winston writes that “Freedom is the freedom to say two plus two equals three.” All of these would naturally show that the rest of the world agreed with my current visualization, and disagreed with my memory, that 2 + 2 = 3.

How could I possibly have ever been so deluded as to believe that 2 + 2 = 4? Two explanations would come to mind: First, a neurological fault (possibly caused by a sneeze) had made all the additive sums in my stored memory go up by one. Second, someone was messing with me, by hypnosis or by my being a computer simulation. In the second case, I would think it more likely that they had messed with my arithmetic recall than that 2 + 2 actually equalled 4. Neither of these plausible-sounding explanations would prevent me from noticing that I was very, very, very confused.

What would convince me that 2 + 2 = 3, in other words, is exactly the same kind of evidence that currently convinces me that 2 + 2 = 4: The evidential crossfire of physical observation, mental visualization, and social agreement.

https://www.lesswrong.com/posts/6FmqiAgS8h4EJm86s/how-to-convince-me-that-2-2-3

The argument that this thought experiment is supposed to make is that mathematics is not actually derived entirely a priori, independently of physical reality. If I lived in a universe where the rules were different, it seems quite likely that the mathematics I would formulate would be different. Why would I insist that two plus two really equals four when all the evidence around me says otherwise? The claim that 2 + 2 = 4 as a matter of self-evident common sense is actually dependent on the vast amount of inductive evidence we all have, from heaps experience of seeing two things and two things combine to make four things.

Perfectly Reasonable Effectiveness

So then what is mathematics, if it’s not a set of truths about the nature of reality that we’ve deduced a priori? Well, at the most fundamental level of formalism, math is built on a set of rules about how to manipulate symbols. The Peano Axioms, for example, are a set of symbol-manipulation rules used to describe arithmetic with the natural numbers. The system of character strings that the axioms describe matches how indivisible objects seem to behave in real life. In other words, if I map the “=” symbol from Peano arithmetic to the real-life experience of two groups containing the same number of objects, and I map the “+” symbol to the real-life operation of combining groups, then the Peano arithmetic system will behave just like my real-life system of groups of objects. Crucially, this lets me predict real-life behavior just by manipulating symbols on paper according to certain rules. And it’s not just Peano arithmetic that works this way. All of modern mathematics is formalized as a set of rules about manipulating symbols, which describe a system of character strings with properties that map to the properties of quantities in real life.

This gives us the answer to our original question. Math is a set of rules we’ve designed to have properties that match the properties we think reality has. If we were to observe two and two adding up to three somehow, the correct response would not be to throw out the observation because it’s “mathematically impossible”. The correct response would be to throw out that part of math because we now know it does not actually match reality. Reality is not bound by the absolute rules of mathematics – mathematics is bound by the rules we think are absolute in reality. This makes it completely unsurprising that math does a really good job of predicting the behavior of physical quantities – that’s what we designed it to do.

Other Things That Are Predictive Models

The same goes for science: this is a bit more obvious than for math, because it’s easier to imagine amending a scientific principle in the face of conflicting evidence. Scientists do it all the time. The point is that science is fundamentally the same sort of activity as math – tweaking and fine-tuning a model until its properties match the ones we observe in reality. Maxwell’s equations describe a symbol-manipulation system that is just ordinary math plus some extra symbols and rules – they are extra axioms in the model. This model describes a part of reality (how electromagnetism behaves) just like Peano arithmetic describes part of reality (how indivisible items behave).

The same goes for logic: this is a bit less obvious than for math, because it’s harder to imagine amending a logical principle in the face of conflicting evidence. But just like a sufficient heap of evidence could convince Yudkowsky that 2 + 2 = 3, another heap of evidence could convince me that T & F = T. The point is that logic is fundamentally the same sort of activity as math. The “&” operator describes a part of reality whose properties we are very confident about, but it’s still just a model to describe a part of reality.

Now, I couldn’t exactly include these in the title of the article, but there are other things which are also secretly predictive model-building. Computer programming, that pebble-bucket parable from the Sequences, and epistemic rationality, to name a few. They’re all fundamentally the same kind of activity – predictive modelling, the activity that underpins most (all?) of our knowledge about the world.