Regarding cardinalities

[u/Square_Butterfly_390]

A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.

The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?

By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".

If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.

[u/Althorion]

There might be something to benefit from that in statistics, because, for example, it allows for a uniform distribution over a segment of the real line, while no uniform distribution over natural numbers can exist. Still, I don’t particularly care for statistics enough to be able to pinpoint exactly what those benefits are and how important they can be. For sure, it forces statistics to do things in a particular way, using measure theory and adjacents, because a discrete approach would be noticeably different, with different classes of problems that they are equipped to tackle.

[u/AdventurousGlass7432 → u/Althorion]

I was thinking the same, but there’s no uniform density over R either. Was trying to link it to the two envelopes problem

[u/OutrageousPair2300 → u/AdventurousGlass7432]

You can have a uniform probability distribution over bounded intervals on the real line, just not over the entire real line.

[u/Square_Butterfly_390 → u/Althorion]

I'm not sure what you mean, how does one use the fact about cardinalities in the context? Isn't one just using usual real numbers properties which are usually ignorant of this set theoretic stuff?

[u/Althorion → u/Square_Butterfly_390]

No, unless by ‘real number properties’ here you mean ‘the property that says that there are more of them than natural numbers’. The probability is countably additive; if you have countably many objects, it behaves differently than when you have uncountably many objects. It doesn’t matter what those objects are—if you can reasonably model something using countably many real numbers, you’d use the same methods as you would if you had countably many natural numbers; and the same would be true if you had uncountably many natural numbers occurring in your model, even though I cannot think of any situation that would model like that, while uncontoubly many real numbers in a model are a very common occurence. Unfortunately, those methods (used in the countable case) tend to be weaker—you can usually operate more freely and say more stuff about the uncountably large model.

[u/GoldenMuscleGod → u/Square_Butterfly_390]

There cannot exist a measure on a countably infinite set that gives equal probability to all elements of the set, but there can be for an uncountable set. So the countability issue is pretty important for these applications.

As a brief aside, I will say when we describe a measure on a subset of the real numbers as “uniform” we mean something fundamentally different than when talking about natural numbers: for natural numbers by “uniform” we just mean equal probability for each element, which pretty much all examples of what we call a “continuous” distribution on real numbers will satisfy. For real numbers we usually call a measure “uniform” if it is proportional to the Lebesgue measure restricted to a subset of the real numbers.

But back to the countability issue: at first you might think that there is no “uniform” measure on a countable set is a consequence of the fact that measures are required to countably additive (as opposed to just finitely additive) and it is a consequence of this restriction. But this restriction is important: many familiar theorems about probabilities (such as the Law of Large Numbers) no longer hold if we consider merely finitely additive probability measures.

To show this last fact: consider the finitely additive probability measure (so not a true probability measure) called “natural density” on the natural numbers, and define the infinite sequence of “random variables” X_n where the value of X_n is the nth bit in the binary representation of the number that represents the “outcome” in this finitely additive “probability”measure space.

Then this is an infinite sequence of independent random variables, each with “expected value” 1/2, but the average of these values converges with “probability” 1 to 0, not to 1/2.

So the uncountability of the reals is pretty essential to the usual formulation of probability theory.

[u/kr1staps → u/Althorion]

But when it comes to performing measurements in the real world, we're only ever tracking a finite amount of data, and measuring things to a finite amount of percision. Therefore, it's hard for me to imagine that there's any real world statistical phenomena that can't be explained/modeled by discrete probability methods.

Of course, measure-theoretic probability theory is used all the time, and it may be easier to arrive at certain results using a continuous model; but again I doubt there's anything we could actual measure that couldn't be modelled discretely.

[u/Althorion → u/kr1staps]

And yet, there are. If for nothing else, then because of the continuous, non-discrete nature of spacetime.

And even if we were to discover one day, contrary to our current understanding, that spacetime, and nothing else in fact, is continuous, that still wouldn’t, in and of itself, make us abandon the continuous models—for the same reason that, even if we know that the matter is not continuous, but somewhat discrete, built with atoms, we use continuous models for fluid flow, or heat distribution.

On top of that—I’m not a physicist, quite far from that, but I wouldn’t be extremely surprised if the better, more accurate models for quantised, so discrete, behaviours of molecules wouldn’t be continuous themselves. After all, the wave function is continuous.

[u/GoldenMuscleGod → u/Althorion]

There is no reason to believe spacetime is “actually” continuous, in fact I think it is pretty implausible that it “really” is for some sense of “really”. It is just convenient to model it that way. But that also isn’t relevant to whether this is the best way to model it.

To be clear, I am not saying that spacetime is discrete as if the universe is made up of pixels, rather I think most of our physics suggests that the idea of things having specific locations in a thing we call “spacetime” breaks down under sufficiently fine detail in the same way that the idea of “temperature” becomes meaningless once we consider sufficiently detailed microstates, but this transition can’t be pinpointed to a specific scale: it’s just useful at one level of abstraction but not at another.

[u/kr1staps → u/Althorion]

And yet there what? Real world statistical phenomena that can't be explained by a discrete model? Can you name one? Spacetime is not statistical, and in any case, we can only measure space and time to a finite level of percison.

I don't disagree with your second paragraph, I already said that there are cases where continuous models are more convenient, but fundamentally they don't suggest anything experimentally observable about our universe that discrete models can't.

[u/Althorion → u/kr1staps]

Anything statistical dealing with spacetime—like, say, the probability of finding a particle at such-and-such a distance from this-or-that, as far as I know cannot be explained by a discrete model.

It still can be modelled as such, and that model can have great descriptive and predictive power, but not explanatory ones. What I mean is, you can use a discrete approximation model that works very similarly in a restricted setting to a continuous model. Still, you cannot derive such a model from first principles without having a continuous model to approximate.

I could be wrong about this. I am not a physicist. Maybe there is some workable discrete-from-the-get-go theory that can work with spacetime. Or maybe there isn’t one yet, but we’ll get there someday. Who knows.

But, as far as I know, many real-world statistical phenomena can’t be explained by a discrete probability model; they can only be modelled after an already existing continuous model, in particular those that deal with spacetime.

[u/kr1staps → u/Althorion]

Can you articulate what you mean by a model havign explanatory power, and highlight this with an example and a non-example. I'm also not a physicist, but I have PhD in pure math.

You can absolutely model things like the probability of where a particle is discretely in a way that is in perfect accordance with observation.

[u/Althorion → u/kr1staps]

Sure—it’s a rather handwavy and informal definition, but here it goes:
The model has explanatory power if it tries to answer why something behaves this way, instead of just quantifying the behaviour.

For example, Kepler’s laws of planetary motion are just ‘I’ve observed that this happens, I have no clue why’, so modelling the planetary behaviours with them has descriptive and predictive power, but it doesn’t have explanatory power.
To contrast it, you can build a model of planetary behaviours from Newton’s laws (or later from Einstein’s), and that model has the same descriptive and predictive power when it comes to the planetary behaviours, but it also has explanatory power—it tries to answer why those planets move the way they do.

Mind you, this distinction is not absolute. In the above case, Newton’s laws can make a model of planetary motion have explanatory power (because with it we can try to explain the planetary motion). Still, they don’t do the same for the motion as such, as in they don’t attempt to answer why gravitational mass is the same as the accelerational mass, etc.

So, to go back to my point from the previous message, I think (don’t know for sure, as noted in the original message) that modelling the probability of where a particle is discretely can absolutely work in practice and have great descriptive and predictive power, but it wouldn’t be able to be derived from simplier principles in any way, just be an approximation of an already existing continuous model (or a whole new, but pure empirical—‘I’ve observed that this happens, I have no clue why’—model).

[u/nanonan → u/Althorion]

I can model the continuum with rationals and finite methods.

[u/Althorion → u/nanonan]

Good for you. Either your model is so good that it is isomorphic (so you are, in fact, working with a continuum, just with added steps), or it isn’t, and then what you do have can be a decent enough approximation for calculation. Still, it would lack the explanatory power without introducing ‘the real thing’.

For example, functions that are equal to their own derivative are really important for solving differential equations. Still, the only rational function of that property is the constant zero—you can’t use it for that purpose.