Quote Originally Posted by TheGreatCthulhu
if I tied (1), I’d get some ratio-like expressions that wouldn’t really tell me much more that what I had without them?
Right, I suppose you known how to construct the set of rational numbers from the set of natural numbers ? you can do something similar, but with some technical difficulties. Anyway, as you said, the resulting "ratios" would be useless in modeling the concept of probability.

>> If I tried (2), someone might argue that this is only one of possible results, since there's really no "correct" probability distribution function.

Yes that's totally right, but...

>> ...this notion that there is some natural probability related to each outcome (that a result of a die roll, or of a point selection has some inherent probability), but this was wrong.

the key is what do you mean by "natural". Actually, modern mathematics has a well defined and sophisticated meaning of that word ( that comes from category theory ) capable of modeling a very general concept of structure.

Anyway, intuitively, given an object and its structure you can sometimes specify a set of natural properties or naturally related objects. These might include a probability structure.

For example, every scientist will understand you if you say "consider a uniform probability density over the interval [0,1]" or "over a sphere" or "over a finite set" or "over a thorus" or over "a projective space", "a flag manifold", "a Moebius strip" or even over the set of "right continuous left bounded paths"... but every scientist will NOT understand you if you say "consider a uniform probability density over the set of reals", or over "the set of triangles", "the set of continuous paths", "the set of chords of a circle", "the set of natural numbers"...

what does the former geometrical objects have in common that the latter have not ?

it's not cardinality: excluding finite sets, common cardinalities appear in both situations ...
it's not an intuitive notion of "boundedness": like, bounded interval vs the whole real line; but the set of chords of a circle is bounded ( or better, compact would be the right term ) ...

Most geometrical objects in both lists ( and every object in the first ) specify (maybe implicitly, sometimes it depends on the context in which they appear) a group of symmetries ("compatible" with their structure, this might imply continuity, measurability, differentiability, or whatever): a finite set has its permutation group, a sphere its group of rotations, the real line its group of translations, a bounded interval its semigroup of local translations, a 2-dimensional projective space its group of Moebius transformation and so on...

Now, each symmetry carries a notion of "uniformity", in the sense that "uniform things" will be somehow compatible with and invariant with respect to that symmetry.

Finally, using nuzzle's words, this is the heart of the problem: Does every geometrical object admit a probability structure that is compatible with and invariant with respect to its symmetry group ?

The answer is NO.

And this is why we cannot get an unanimous/unique answer to the question "what's the probability that of a triangle being acute?" : because we consider rotated\scaled\translated triangles naturally equivalent AND because the set of triangles does not admit a probability measure invariant with respect to its group of "natural" transformations.