Empirical vs. Mathematical Joints of Nature
We (Alex and Elizabeth) are thinking about doing this project where we figure out how paradigm formation happened in chaos theory. Alex has also been thinking about paradigm formation for agent foundations (which people often talk about as being pre-paradigmatic). These are some thoughts on what paradigm formation means.
There’s an issue which is something like, agent foundations is more like math than like science, and so it’s unclear to me exactly what it means to have a paradigm. And I think this is similar to chaos theory, so it might be useful to talk it through and compare them.
On one hand, you could say that a paradigm is just a set of methods that people generally agree successfully solve problems. And in that sense, domains of math could totally have paradigms.
But in the sciences, there’s another standard of whether a problem was solved, which is that reality was successfully predicted. Domains of math don’t exactly try to “predict” phenomena, and so whether some problems were solved feels more subjective.
And I feel like both chaos theory and agent foundations are more like fields where people will just generally agree whether or not their confusion was resolved by certain frameworks, rather than fields where empirical phenomena are predicted.
One definition I have been using of a paradigm is that it is a single solution to problems people believed were disparate. Newtonian physics predicted the rate of a falling apple and the orbit of the moon. Plate tectonics explained magnetic striping on the ocean floor, mountains, and the Wallace line (and let you use information about one to make predictions about the other that were born out).
Ah, interesting. That sounds related to something I’ve been thinking about which I might call paradigm shifts vs paradigm formation. I think Kuhn mostly talks about shifts, where there exists a previous strong paradigm. But if a field has no paradigm at all, then the formation of the first one might look different from a shift. Showing that multiple problems have a single solution sounds more like what paradigm formation might look like.
I’m excited about chaos theory because it seems like it might be doing the same thing as the theory of plate tectonics- combining weather systems, eye movement, and a dripping faucet under a single set of equations, and letting insights from one inform our understanding of the others.
There’s maybe a pure/recreational version of chaos theory that’s math only, but there’s also chaos field that is supposed to spread insights useful across domains.[1]
I think it’s reasonable to call that shift vs. formation, although I’m not sure it will break down cleanly. My sense is biology and geology have single unifying paradigms that don’t explain everything but are clearly the foundation of a field. Whereas physics have multiple paradigms that exist in parallel (e.g. classical mechanisms, relativity, and quantum mechanics). I’m not sure where chemistry fits in- probably more like biology and geology in that nothing in in chemistry makes sense except in light of atoms and molecules, but I haven’t thought about it that much.
I have no idea how this compares to agent foundations, which I’m basically ignorant of.
[1]One reason I chose chaos theory in particular to study is that it’s still at the stage where this is considered cross-domain pollination and not “well yeah, they’re both chaotic systems”, the same way magnetic stripes and mountains are just obviously both geography. It is easier to study the process of paradigm coalescence before it has finished. But of course this risks the possibility that Chaos the field is just not actually carving reality at the joints.
I see. My impression of chaos theory is more like… people noticed that some systems seemed to be kinda predictable but kinda not. And they were confused about what was up with that, especially given that the systems were sometimes very simple equations. Those equations do match some empirical systems. But figuring out the concept of chaos doesn’t exactly let you make better predictions; instead it lets you understand which systems can be expected to be predictable at all. So for example, we don’t even bother trying to make weather forecasts for a year out.
But I’m not very confident about this (having not yet studied the history of chaos theory). Maybe it lets you get some better statistical predictions, or something.
The part of chaos I find interesting is the part that specifies underlying patterns in spite of inability to predict exact state at a given time, such as Lorenz systems. I’m not yet confident these are genuinely useful and not Big Chaos PR stunts, but if they are, and if learning from one application is transferable to another, that’s the kind of thing I would find meaningful.
What types of underlying patterns are you thinking of for the Lorenz system?
I believe this has improved weather forecasting.
I think people had the equations for the Lorenz system pretty early on, and the “problem” of chaos was that the behavior the system just looked really weird, even though we could calculate it all out. If weather forecasting was improved not just from running the equations, but from some further deconfusion that the concept of chaos helped with, I’d be super interested to know how.
Yeah that’s a great question we should put on our list.
Going back to
And I feel like both chaos theory and agent foundations are more like fields where people will just generally agree whether or not their confusion was resolved by certain frameworks, rather than fields where empirical phenomena are predicted.
could you say more about this? Does reducing confusion without predicting phenomena do anything useful?
Is agent foundations actually at the point where it’s reducing anyone’s confusion?
Does reducing confusion without predicting phenomena do anything useful?
I’m sure it often does, although humans can notoriously feel deconfused even when their beliefs make no sense. I think that deconfusion mostly makes research go faster, because it means your ideas are better.
Is agent foundations actually at the point where it’s reducing anyone’s confusion?
Oh, no. I just mean that if agent foundations attains a paradigm, I don’t think it will be because a bunch of empirical data is explained and/or predicted, I think it will be because someone published a framework where everyone reads it and goes, “Oh yeah, they nailed it. Let’s go with this.” This is largely what happened with Turing machines and Shannon’s information theory.
Can you say more about one of those examples?
Sure. In the early 20th century, mathematicians were trying to figure out what they all intuitively meant by an “effective method”. They all agreed that there was some class of thing that was a calculation you could actually sit down and do (versus other mathematical objects which could be described not but not actually written out somehow). People spent a while proposing different formal definitions, which I think were usually classes of functions. Eventually, Turing came up with a way to formally model “sit down and follow an algorithm”, which was his description of what we now call Turing machines. When other mathematicians read this, I think they found it pretty compelling. And after a few more years, people proved that all the other proposed definitions were either equivalent, or subsets of Turing machines. And so everyone felt like they had gotten deconfused on what an effective method was.
I think this genuinely helped humanity build better computers, et cetera. But no one was doing it to predict an empirical phenomenon.
Shannon’s formal definition of information was similar. He described it in a convincing way, including giving a derivation from intuitively satisfying axioms. He then proved some theorems that were clearly going to be super useful for building communications systems.
Everyone read his paper and was immediately deconfused about information, and then moved forward using it as the paradigm. But I also don’t think Shannon or others were doing science in the sense of predicting a phenomenon.
And I think agent foundations is similar. There’s a conceptual category that I already have (analogous to “effective method”, or “information”) and I’m trying to figure out what its formal characterization is.
Let me see if I understand:
The best answer to “how fast does an apple fall?” was always going to be described by gravity, with bonus points for air resistance. But the best answer to “what is an effective method?” didn’t have to be “calculable by a Turing machine”? Except there were multiple attempts to describe calcuability that turned out to be isomorphic to turing machines, so maybe turning/lambda calculus/etc are carving reality at some joint?
More like, “how fast does an apple fall?” is directly checkable against reality at all. Whereas the definition of “effective method” is… somehow instrumentally useful to humans, even though it doesn’t exist in the territory.
Got it. Yeah I do think chaos theory is going to be more like that than like plate tectonics.
[edit: pinned to profile]
The claim that an “effective method” is in the map and not the terrain feels deeply suspect to me. Separating map from terrain feels like a confusion. Like, when I’m doing math, I still exist, and so does my writing implement. When I say some x “exists”, in a more terrain-oriented statement, I could instead say it “could exist”. “there could exist some x which I would say exists”. for example, I could say that any integer can exist. I’m using a physical “exists” here, so I have to prefix it with “could”. it’s also conceivable that the thing existed before I write it, if some platonic idealism is true, and it might be. But it seems like the only reason we get to talk about that is empirical mathematical evidence, where a process such as a person having thoughts and writing them happens. Turing machines similarly seem like a model of a thing that happens in reality. It’s weirder to talk about it in the language of empiricism because of the loopiness of definitions of math that are forcibly cast into being physicalist, but I don’t think it’s obviously invalid. I do see how there’s some property of turing machines, chaos theory, arithmetic, and linear algebra that is not shared by plate tectonics, newtonian gravity, relativity, qft, etc. but all of them are models of something we see, aren’t they?