Herewith some trains of thought leading through various more or less obscure corners of philosophy, theology, set theory, analytic topology, and/or differential geometry. The math parts may well be wrong, as I’ll be mostly just making it up. (Unlike the philosophy and theology, ha ha ha!) This is the kind of thing I think about on the subway sometimes. When not mindlessly scrolling social media.
In the comments section of a YouTube video titled “Evolution Doesn’t Explain Morality”, I was for some reason talking about objective truth, and whether basing your moral system on a God of some sort gives you an objective source of truth more than basing your moral system on anything else does. (It doesn’t.)
And unrelatedly I stumbled across something about the Axiom of Infinity, which is an optional axiom in good old Zermelo–Fraenkel set theory, saying that there is at least one infinite set.
If you include the Axiom of Infinity (and the Axiom of Choice) among your axioms, you can prove all the usual stuff of ZFC, which is to say, all the usual stuff. If you leave it out, as that article there says, not only do you lose any theorems about infinite sets, but you also can’t prove a bunch of theorems about finite sets, because the proofs go through infinite sets on the way to the conclusion (basically).
And then there are all of the non-Euclidean geometries, where you put in an axiom saying that instead of there being exactly one line parallel to a given line and a given point not on that line, there are no such lines, or there are lots and lots of them, and then you get geometries in which the angles of a triangle add up to more or less than a straight angle, and other fun stuff.
So it’s objectively true (let’s say) that in Euclidean geometry the angles of a triangle add up to a straight angle, and it’s objectively true that in ZFC with the Axiom of Infinity you can prove certain things; but it would be silly to say that the parallel postulate, or the axiom of infinity, or the axiom of choice for that matter (celebrate the Banach-Tarski Paradox!) is objectively true simpliciter. They are axioms; everyone chooses which ones to work with at any time.
Back in the arena of theology, it may be objectively true that within a certain reading of Catholicism, things X and Y are objectively true; and within a certain reading of Hedonic Rule-Utilitarianism, things R and S are objectively true. But there is no (objective) argument that that reading of Catholicism, or that reading of Hedonic Rule-Utilitarianism, is objectively true simpliciter. They are moral systems; everyone chooses which one to work within.
And that got me thinking about just how few axioms you can have, and still do interesting things. Say that all you assume you have are a bunch of points (which we could call “fnorbs” except that we want to lean a little bit on pre-existing ideas because we are lazy), which themselves have no particular properties, and you have a “distance” function (which we could call “doppity” except that etc) d that takes two points and returns a real number. We can call d(x,y) for any points x and y “the distance between x and y“. Let’s assume, I dunno, that for all points x and y, d(x,y) is greater than or equal to zero, and that d(x,y) = d(y,x).
Can we get anything at all out of this tiny set of axioms? If we think for a moment about points being points in some usual Euclidean N-space, and distances being the usual Euclidean distance, then we can think about what something like a “line” might be for instance.
If we have two points x and y, we could define say “the line segment between x and y” as all points z (if indeed there are any) for which d(x,z) + d(z,y) = d(x,y). And that’s sort of fun! Note that so far we don’t know if there are any points on that line segment; there might not be.
A (long) tangent! We had to say that d(x,y) = d(y,x) because our notation strongly suggests that the function d applies to ordered pairs of points: a first argument one and a second argument one. We could instead define d to operate on sets of points, defined only for sets of cardinality two (i.e. having two elements). Since sets aren’t ordered, “{x,y}” is the same set as “{y,x}”, and d({x,y}) is the same as d({y,x}) without our having to say so as an axiom.
This brings us, interestingly, to another axiom. It would be quite natural to write d(x,x)=0, that is that each point is at distance zero from itself. (And in fact if one adds that axiom, one has got something that’s often called a “semi-metric space“, about which there’s a certain amount of theory.)
But what happens to that if we’re using the set form of the distance function? We’d have to talk about the set {x,x}, and it would have to have cardinality two. Is the set {x,x} any different from the set {x}, and does(n’t) it actually have cardinality one, not two?
A good question! One simple way to model a set of points is as a function s from points to Boolean values (true and false), where if s(x) = true, we say that x is “in the set s“, and otherwise we say that it “isn’t”.
(If the set of points is enumerable, or even just countable, which is to say if we can in principle try s on each and all of them, then each set is well-defined; otherwise we may have some sets where we can’t say even in principle which-all points are in it, or what its cardinality is.)
If we model sets that way, then {x,x} is just a weird say of writing {x}, which is just a short way of writing the set defined by “s(z) is true if z=x, and false otherwise”. And that set has (I think uncontroversially) just one element, and therefore the distance between a point and itself is not zero, but is undefined (just like, say, the distance between three points is).
Which I thought was kind of neat, although I can imagine that it might make it harder to do proofs, since you have to consider x=y as a special case all the time, before you can talk about d({x,y}) even having a value.
Then there are “multisets”, which are like sets (in that they are unordered), but which can contain more than one of an element. These are represented by a function (oh I dunno) t, from points to integers, where t(x) tells us how many times x occurs in the (multi)set t. And the cardinality of the set has what I hope is the obvious definition, as the sum over all x of t(x).
If we define our distance function as being defined for any multiset of points of cardinality two, then we can write d({x,x}) just fine, and therefore we can specify that d({x,x}) = 0 for any point x.
(Beyond multisets there are for instance fuzzy sets, where t isn’t integer-valued but real-valued, so an element can belong to the set zero times, or one, or 0.5, or the square root of two, and well, cardinality… But we aren’t using those tonight.)
The next fun question is whether there are any cases where d({x,y}) = 0 and x != y. According to my poking around on the interwebs a bit, it appears that one can take either answer to this as an axiom, and get a different bunch of semi-metric spaces either way. When I asked Bard Advanced or some AI like that about this at work (more perhaps on that below) it called “d({x,y}) = 0 if and only if x = y” something like “the identity of indiscernibles”, which appears to be a thing. And the opposite axiom, that there exists at least one pair of points x and y such that ({x,y}) = 0 and x != y, apparently makes the space into a “pseudo-metric space”, which is another Fun Word To Use Daily.
End of tangent I think. :) What else can we maybe define in our minimalist semi-metric space? We defined a line segment, is there a fun way to define a line? I was thinking about a way to define it as the union of three sets (the segment between x and y, the part beyond x, and the part beyond y), and that probably works, but then I ran across a simpler one: given any two points x and y (for which d({x,y})>0, just to be safe), we can identify all of those points z (if any) where d({z,x}) = d({z,y}) as “the line smoofled by x and y“, or in order to use our pre-existing Euclidean intuitions as “the line which is the perpendicular bisector of the segment between x and y“.
And I thought that was fun. :)
Note that (haha, “Note that”) the “line” up there isn’t really a line in the Euclidean sense necessarily at all. In fact if we imagine our little semi-metric space to be embedded in a three dimensional Euclidean space, then the locus of points equidistant from two given points is actually a plane. So we might want to call the generalized d({z,x}) = d({z,y}) as a “subspace” rather than a “line”. We might be able to figure out how many “dimensions” a semi-metric space “has” by looking at how many times we can take a subspace of a subspace of a… before we have zero points left. Or something.
If we go off and read about semi-metric spaces (and also metric spaces, which is what you get roughly if you include the identity of indiscernibles and also the Triangle Inequality d(x, z) ≤ d(x, y) + d(y, z)), one of the things that people like to talk about is spaces where there are lots and lots and lots of points, and they are really really really close together.
Like, a point x is an accumulation point iff (and I think I have this right) for any distance D however teeny, there is a point y where d({x,y})>0 and d({x,y})<D. That is, you can move as close to x as you like, and there will still be another point that’s closer to x than you are. This requires, clearly, a whole bunch of points, and some of them have to be really close together.
Somewhat relatedly, people also talk about open balls (which makes me chuckle because I am twelve), although usually in metric spaces (with the triangle inequality) for some reason I’m not sure. The open ball at x with radius e is all of the points z (if any) where d({x,z})<e. Nice and simple! With only the properties we have so far, it’s not clear that the open ball is “open” or a “ball” in the intuitive senses.
If every point is an accumulation point, then roughly there are as many points as you could want everywhere, and now we can imagine that an open ball is actually a cute fuzzy sphere (or N-sphere since we don’t know about dimensionality), open in the sense that no point is quite out there at distance equals e, and a sphere in that… it’s spherical.
Actually I don’t think I’ve put enough properties into the space yet; there might not be any points even at distance e/2 from the center. We need something like for any point x and distance D, there is a point closer to x than D, and also a point farther from x than D. And they’re all accumulation points. That seems nice.
I can’t find a term for a (semi-)metric space where all points are accumulation points; I’ve posted on r/learnmath about it (because I am too shy to post it to r/math); we’ll see if I get deleted for not following some sub protocol or something. :) I’m not positive about the one where there are points further away, either; that might be “unbounded” or something.
Once you have open balls (heh heh) you can say things like “the open balls of the space can serve as a base for a topology, where the open sets are all possible unions of open balls,” which is where my mind sort of starts to slide off, which is why I ended up not majoring in Math.
Fun up to there, though! :) Which is why I ended up majoring in Philosophy…
The AI connection here is that I discussed various aspects of all this with Bard (which may or may not be Gemini who knows) and ChatGPT (4.0?), and was very much struck by how plausible and confident and often correct they were even on this rather niche topic, and equally struck by how they would sometimes be just randomly wrong about something, and then be all apologetic and try again if I pointed it out.
This makes it rather unnerving to try to learn anything from them, because who knows if you’re learning some wrong stuff that’s wrong subtly enough that you don’t notice.
And I think I won’t try the WordPress AI Assistant feedback on this post. :) That joke has just about run its course.
ceoln 