Finding pi and G in Mathland
This post is a Cunningham’s law draft, c. 75% finished. Consider a) waiting until this notice has disappeared to read a more coherent post, or b) criticizing it with a focus on what would be right, not just what is wrong.
Science explains physical phenomena through mathematical theories. If an explanation is true,[1] the physical phenomena form a model[2] of the mathematical theory.
Because mathematicians explore theories independently of their connection to our universe, it creates the false impression that the only direction relevant for science (and thus for the real world) goes from physical phenomena to theories: one finds the mathematical theories that describe our universe by abstracting from physical phenomena. Mathematicians might occasionally find theories that are relevant for not yet discovered / understood physical phenomena, but the relevant link (according to this wrong view) goes from those physical phenomena (once understood) to the mathematical theory.
However, not all mathematical theories are equal. In “Mathland”, each of them sits next to (usually infinitely many) other similar but (potentially infinitesimally) different theories. A question that can be asked is: inside Mathland, how visible is a theory?
Let’s start with a simpler case: how visible is pi? That depends on where in Mathland we are. If we are in the regions with all the possibles series of the form
with integer, real or complex a, b, c, we might not be able to find the values −1, 2 and 1 that together produce
But of course
What about the gravitational constant, G=6.6743×10−11 m3⋅kg−1⋅s−2? We can imagine the region of Mathland that contains all the different version of Newtonian mechanics, each with a different real value of G. Different things are possible inside each version, but there is no way to find the one with “our” value of G, since it is a fundamental physical constant of our universe.[4]
I’ll call the “points” of Mathland that are “visible” Schelling math; the rest, mundane math.[5]
Most of science is about finding which point of mundane math describes our universe. Some engineering consists in taking some small piece of Schelling math and trying to reach it from within the mundane math the describes the physical system in question.[6]
And sometimes, very rarely, something entirely different happens: a consequential piece of Schelling math is found, and if the path to it can be found inside our universe, a new domain of physical possibilities opens up. Computability theory is the clearest and possibly only example of this happening.
Coming soon: the debate between Agent Foundations vs. realism about rationality / prosaic alignment is crucially about whether the theory of agents in our universe is mundane math or Schelling math.
- ^
Or rather: “to the extent that the explanation is true”.
- ^
Confusingly enough, the word model is also used to mean a mathematical theory created as an abstraction from physical phenomena. In this post the word is always used in the model-theoretical sense.
- ^
Actually I don’t know if this is the case: the series might have peculiar convergence properties setting it apart from its neighbors. If that were the case, the example is wrong and the post needs another example.
- ^
The reverse isn’t true: Some scientific theories take constants from physical reality that might actually be a consequence of deeper stuff we ignore.
- ^
As a shorthand for Schelling/mundane mathematical theories. I might change the name later, because this one implies game-theoretical relevance that is lacking here.
- ^
The examples from the hyperpolation paper seem to me to fall under the same pattern: realizing that you are in a region of mundane math which has a very close Schelling math neighbor.
It seems to me that math and physics are fundamentally different. Our understanding of physics doesn’t rule out the existence of parallel worlds with a different value of G or even the possibility that the true value of G differs from what we believe by, say, 1E-100 N/kg^2*m^2, but it does fully exclude parallel worlds with a different value of π.
Unlike dimensionless constants like the fine structure constant, which is a pure number, G is a “dimensionful” constant which includes physical units. So it doesn’t even have a fixed numerical value, its value depends on the units you use.
Very good point, thanks. I’ll use your example to avoid that problem. Though I feel that we miss the intuitive aspect that the constant is part of a theory with a relatively small number of axioms.
I find the question interesting, though, of how to characterize the value of G in the mathland region where all the variations of Newtonian mechanics with different values of G are. Each variations allows different configurations, and it seems that it should be possible to point at one of them in some way, though maybe I’m wrong.
Thanks for your thoughts. I’m not sure I understand what you mean, though:
> It seems to me that math and physics are fundamentally different.
> parallel worlds with a different value of G
My post is trying to make both points, among other things. Did I fail to convey them?
> [physics] does fully exclude parallel worlds with a different value of π.
What makes you think that it is physics that is excluding this possibility?
the question asked here is “how natural is plane geometry?”, and the answer is not very clear to me. for example, our notion of distance seems physically motivated, and may be contingent on the substrate. i have no way of knowing whether our substrate is ‘common’ or ‘unusual’ (or even what those words could mean in this context), so locating pi through geometry may strike “most” others as bizarre.
imo the simplest definition of pi is via differential equations. but again we have to ask whether calculus is particularly widespread, or is contingent on the (perhaps distinctive) property that newtonian mechanics is a reasonable approximation at our scale. would intelligent agents living in some (more-obviously-) discrete-land have reason to invent “continuous rate of change”? perhaps not.
(and, perhaps, intelligence requires sufficient complexity that any such agent is necessarily buffered from the underlying substrate, in such a way that conservation laws and continuous approximations are always relevant. i have no way of knowing.)
as well, i find that this essay neglects entirely the psychological/aesthetic aspect of mathematics. mathematicians—as much as various adages would disagree—are not machines that produce theorems by rote. rather math-as-practiced is a consequence of human minds and proclivities, as much as any other activity. why do we find some math simple, or compelling? surely this is a consequence of the human architecture, as much as it is a fact about the mathematical landscape.
not to mention historical contingencies: the history of geometry starts with architecture and masonry. trigonometry was important for maritime navigation. a culture less warlike and conquesting may be impossible across the multiverse… but we may wonder nonetheless what math they would first find, and how that would affect their choices of constants, or the ways they express their physical theories.
Thanks for your thoughts.
The reference to what is “natural” or “widespread”, to other intelligent agents, and the psychology and history of mathematics, are alien to the point that I’m making here. My point is that it is possible to discuss math without mentioning all these things; if this assumption is wrong, my post cannot be salvaged.
Regarding your example: Plane geometry is indeed only one among many, but the other geometries are distinct enough, and discrete enough, that plane geometry is a clear object in mathland, as is pi by having a special role in it.
i suppose i misread your essay, then. i took
to mean that the project was to ask what objects are “easy” to discuss. my point is that what is visible depends on where you are standing.
of course (putting aside translation/bootstrapping issues) it is possible to discuss mathematical objects without reference to a specific world in which they were discovered.
i guess i still don’t know what you mean. mind clarifying?
This was indeed my goal, but restricting the possible “wheres” to those that don’t mention facts about our universe (which includes facts about humans, like their psychology or history).
I’m working on a clearer version of my post which hopefully clarifies a bit what I mean.