Finite number divided by infinity

[A.T.]

What if all natural numbers are equally likely?

By this I take you to mean there is a valid (satisfies the Kolmogorov axioms) ... We can show that this is impossible.

What does it tell us, that we cannot assign a probability in this case, if we assume these axioms? Seems like a limitation of the theory build on these axioms.

 

[Matterwave]

What does it tell us, that we cannot assign a probability in this case, if we assume these axioms? Seems like a limitation of the theory build on these axioms.

First I would like to ask what is the purpose for which you would want a uniform distribution over all of $\mathbb{N}$? Or is it "just because"?

Kolmogorov formalized probability via his axioms in the 1930s and a vast literature of probability theory and things that depend on probability theory has been built on top. It also provides the axiomatic foundation for things prior to it like the law of large numbers and the central limit theorem.

You may formalize probability your own way, but before we embark on such an arduous journey, we might want to know where the intended destination is.

This path might have other travellers on it and I just don't know them. :)

 

[A.T.]

First I would like to ask what is the purpose for which you would want a uniform distribution over all of $\mathbb{N}$? Or is it "just because"?

I thought it's only real math, when it doesn't have an application.

Kolmogorov formalized probability via his axioms in the 1930s

100 years is not much, compared to how long it took mathematicians to do something about not having an answer for $1-2$ or $\sqrt{-1}$.

 

[Dale]

Not in QM. Only in classical statistical mechanics.

https://en.wikipedia.org/wiki/Born_rule

In QM, the sample space is determined by the measurement apparatus.

I am talking about the sample space of the probability distribution from the Born rule. There is a separate sample space for the measurement apparatus which is conditional on the first.

You claim you can get an arbitrary real number from identifying the position of a particle

What I claim is that physical positions are isomorphic with the reals. I.e. they are a continuum.

The wave-function is defined everywhere

And “everywhere” in this meaning is a continuum.

It's a question of the continuum being absolutely necessary, with no possibility of a theory that avoids it.

I am not making that claim.

But I do take the continuum model seriously because it is experimentally validated, and to my knowledge discrete models are not as well-supported experimentally.

It's not a question of the continuum being a working model.

So you do think that the continuum is a working model? Your arguments seem to indicate a stronger claim to me. Your arguments seem to indicate that you believe the continuum is not a working model.

Perhaps you are making a weaker claim than I thought, as I am making a weaker claim than you think.

 

[Dale]

What does it tell us, that we cannot assign a probability in this case, if we assume these axioms? Seems like a limitation of the theory build on these axioms.

It is a limitation, but sometimes such limitations are actually a good thing.

I have occasionally wondered if it would be worthwhile to define a probability measure on the hyperreals rather than on the reals. You could get rid of that specific limitation by doing so, but I am not sure that it would be a good thing to do. I am not aware of any literature on that topic, so I don't know what problems would be introduced by "fixing" that limitation.

 

[Matterwave]

I thought it's only real math, when it doesn't have an application.

Well, you could also give a mathematical application!

"I just want this to be" is also ok, but you would be the best person to build the theory in that case~

100 years is not much, compared to how long it took mathematicians to do something about not having an answer for $1-2$ or $\sqrt{-1}$.

There are other formalisms / axiomatizations of probability. They might give rigor to "a uniform distribution over the natural numbers". I'm only familiar with Kolmogorov's (it's the "main" one, like ZFC).

 

[pbuk]

Seems like a limitation of the theory build on these axioms.

Only in the sense that the fact that we cannot assign a value to $\frac10$ is a "limitation" of algebraic field theory. But the non-existence of a multiplicative inverse of 0 is necessary to provide consistency to $\mathbb R$ and $\mathbb C$.

What does it tell us, that we cannot assign a probability in this case, if we assume these axioms?

It tells us that the word "probability", as we have defined it, does not make sense in this context. In this case, it does not make sense because the phrase "uniform distribution over an infinite number of discrete events" does not make sense in the context of the Kolmogorov axioms (specifically it is not consistent with σ-additivity).

Although I am not familiar with the detail, work has been done on alternatives to σ-additivity, specifically "finitely additive probability" - see e.g. https://www.ma.imperial.ac.uk/~bin06/Papers/favcarev.pdf - which has I believe led to useful results.

 

[A.T.]

I don't know what problems would be introduced by "fixing" that limitation.

Yes, usually new problems arise. Just like introducing negative numbers, to cover the results of all subtractions, introduced the problem with roots of those negative numbers.

 

[PeroK]

What I claim is that physical positions are isomorphic with the reals. I.e. they are a continuum.

Here you make the claim that physical positions are isomorphic to the reals.

I am not making that claim.

And now you deny that claim.

You can't have it both ways. You can't say that a particle can be at any one of an uncountable number of positions - and thereby have an event of probability zero that actually happens. And, then concede that the set of positions could be finite or countable. If positions are uncountable, then they cannot then later to be found to be finite.

If you claim there are an infinite number of positions in spacetime, then there cannot possibly be only a finite number.

If you can really do what you claim, then that rules out finite or countable spacetime models.

Perhaps you are making a weaker claim than I thought, as I am making a weaker claim than you think.

I claim that we do not know whether spacetime is finite, countable or uncountable. I don't know why that seems to be so difficult to grasp. It might be continuous; it might be countable; it might be finite.

Note that in this thread, I seem to be the only one not claiming to be able to do things that have zero probability.

 

[pbuk]

Note that in this thread, I seem to be the only one not claiming to be able to do things that have zero probability.

I don't think that is true. Can you point to such a claim?

 

[PeroK]

I don't think that is true. Can you point to such a claim?

You're right. I let my frustration get the better of me.

 

[Dale]

Here you make the claim that physical positions are isomorphic to the reals.

And now you deny that claim.

You can't have it both ways

Sure I can. The first statement is my assessment of our best current models and evidence. But as a scientist I am always open to new evidence and models. So while I do claim that space is isomorphic to the reals I will not claim that there is “no possibility of a theory that avoids it”.

You can't say that a particle can be at any one of an uncountable number of positions - and thereby have an event of probability zero that actually happens. And, then concede that the set of positions could be finite or countable.

Why not? Must I pre-reject all future models and evidence before I can make any claims today? That doesn’t sound very scientific.

We have good models supported by good evidence that I take seriously. But I will not close my mind to future models and evidence.

I claim that we do not know whether spacetime is finite, countable or uncountable. I don't know why that seems to be so difficult to grasp. It might be continuous; it might be countable; it might be finite.

OK. Do you have any existing models to support the countable or finite ideas, and are those models supported by existing evidence?

 

[PeroK]

Sure I can. The first statement is my assessment of our best current models and evidence. But as a scientist I am always open to new evidence and models. So while I do claim that space is isomorphic to the reals I will not claim that there is “no possibility of a theory that avoids it”.

Why not? Must I pre-reject all future models and evidence before I can make any claims today? That doesn’t sound very scientific.

We have good models supported by good evidence that I take seriously. But I will not close my mind to future models and evidence.

Perhaps we have very different views of what a mathematical model is. It doesn't make sense to me to treat the best current model as definitive and accept all the implications of it. For example, the best current model is that the universe is spatially infinite. But, that doesn't mean we must accept all the implications of an infinite universe. Although, that does seem to be an alternative view among many scientists - who will claim definitely that something is the case, when in fact it's dependent on the current model and may not be the case at all. E.g. that there are infinitely many duplicates of planet Earth out there.

And, in this case, that you can sample from an uncountable sample space by localising a particle to a single point. That's an implication of classical statistical mechanics. But, to say you can really do it, implies (IMO) that the theory is definitive in this respect. (We'll have to agree to disagree about whether QM allows a point particle in any case.)

OK. Do you have any existing models to support the countable or finite ideas, and are those models supported by existing evidence?

There are several ideas, especially in quantum gravity, that would replace the continuum by a discrete spacetime. E.g.

https://en.wikipedia.org/wiki/Causal_sets

From a mathematical point of view, there are ideas to replace the Real numbers with the set of computable numbers:

https://en.wikipedia.org/wiki/Constructive_analysis

Let's say, for the sake or argument, that we could develop the required mathematics from this alternative to standard Real analysis. Then we have a strange situation where you believe you can generate the Real numbers from experiments in spacetime, whereas, the Real numbers are not essential to do physics.

Finally, on this point, if we take up your suggestion to use the Hyperreals and use non-standard analysis, then does that imply that spacetime must have an infinitesimal structure?

My view is that I would not demand that spacetime has all the properties of the mathematical model that describes it. In particular, I would not demand that spacetime can in fact produce the set of Real numbers. Let alone the Hyperreals.

 

[SSequence]

From a mathematical point of view, there are ideas to replace the Real numbers with the set of computable numbers:

https://en.wikipedia.org/wiki/Constructive_analysis

I have nothing to add to specific discussion, but I would briefly mention a further comment on what you just mentioned. To what extent what can or can't do mathematics [for example, to what extent some of the well-known/standard theorems have to be changed or even abandoned] in weaker axiomatic systems is studied in reverse mathematics. It is known that one can do a quite significant amount of real analysis in a system called $ACA_0$ [it is a fairly weak system]. Slightly stronger systems have been suggested to manage even more.

 

[Dale]

There are several ideas, especially in quantum gravity, that would replace the continuum by a discrete spacetime. E.g.

https://en.wikipedia.org/wiki/Causal_sets

Interesting. Although the statement "An ongoing task is to develop the correct dynamics for causal sets" sounds like it doesn't yet make any experimental predictions. So that sounds like more of a promising idea than an actual discrete model of physics. Am I misunderstanding the situation there?

From a mathematical point of view, there are ideas to replace the Real numbers with the set of computable numbers:

https://en.wikipedia.org/wiki/Constructive_analysis

Can you do derivatives and differential equations with this constructive analysis?

Perhaps we have very different views of what a mathematical model is. It doesn't make sense to me to treat the best current model as definitive and accept all the implications of it. For example, the best current model is that the universe is spatially infinite. But, that doesn't mean we must accept all the implications of an infinite universe.

It does mean that accepting such implications is a reasonable thing to do, and that such implications deserve careful consideration before rejection.

By careful consideration I mean that if you are going to reject an implication of an experimentally validated model then, IMO, you should have more than just a dislike of the implication or a hope for something different in the future. IMO, you should have a working alternative model that is consistent with the existing experimental data which does not share that implication you wish to reject.

Let's say, for the sake or argument, that we could develop the required mathematics from this alternative to standard Real analysis. Then we have a strange situation where you believe you can generate the Real numbers from experiments in spacetime, whereas, the Real numbers are not essential to do physics.

That wouldn't be such a strange situation because I would change my belief. There are three main possibilities here:

1) Only the current model matches the current data -> I accept the implications of the model
2) The current model matches the current data, and a new model agrees with the current model for all current data and all future data -> I accept the implications of both models
3) The current model matches the current data, and a new model agrees with the current model for current data but predicts different future data -> I accept only implications that both models share

Currently we seem to be in situation 1) regarding the continuity of spacetime.

 

[PeroK]

Can you do derivatives and differential equations with this constructive analysis?

I don't know. I think calculus is ultimately much easier using the Real numbers! Can you do physics without the Real numbers? I don't know. This question came up a few years ago. I think our friend from Vienna has views on this.

Instead, an interesting digression is to consider whether non-measurable sets exist in nature. If you can produce Real numbers from spacetime, then you can ask whether spacetime has non-measurable subsets. To construct a non-measurable subset of $\mathbb R$ requires the axiom of choice. An alternative to the axiom of choice is to assume that all subsets of $\mathbb R$ are (Borel) measurable. Does nature entail the axiom of choice or not? This question also came up several years ago.

This is my question: if you are getting these sets directly from nature (Isomorphic to their mathematical equivalents), then how do you deal with the set-theoretic subtleties within the foundations of mathematics? Does nature determine a precise flavour of axiomatic mathematics?

This is probably getting into the philosophy of physics now. My solution is to cut the Gordian knot and deny that anything beyond rational (or computable) numbers is implied by nature. In particular, the Real numbers (with or without non-measurable subsets) exist only as a mathematical abstraction. That said, I'm not pushing this. It could be that nature is actually the real deal, as it were, and the ZF(-C) axioms are encoded in nature. And that the Real numbers are embedded in spacetime.

 

[Dale]

Can you do physics without the Real numbers? I don't know.

Until we do know, then I think that it is best to take the model seriously. We know that current evidence is consistent with continuity. We do not know (yet) that current evidence is consistent without continuity.

 

[Rodney182]

First, a wording correction -- Subset. You take subsets of sets. You don't "take a set of a set", you "take a subset of a set".

Second, right, Adams is (roughly, with some translation) making the if-then statement:

If you take the subset of an infinite set you end up with a finite set.

Which is my translation of:


Notice the "must be".

This argument from Adams is invalid because you could still get an infinite set back (my example with odd numbers). Of course, you could get a finite set back, e.g. if you take the subset as Dale suggested, or the subset $\{1,2,3\}$ (notice no $...$ this time).


In my analogy, it would be because the set of odd numbers is infinite.


Just the difference between a statement that could be true vs deducing the truth of a statement from an argument.

You're absolutely right. I checked this later with another friend of mine who has great mathematical reasoning. Thank you for the patient explanations on the thread so far.

 

[Rodney182]

You're trying to make sense out of whimsy. The author is having fun with you!

From a google search:

"Radio origin: March 1978 (broadcast as a BBC radio comedy series before it became a book)."

Emphasis on comedy added.

Plus, if you want to get serious, and I can't fault you for that, our understanding of cosmology has progressed since 1978. The descriptions of the universe in 1978 compared to 2026 are entirely different.

I'm having fun with the author. Some of my interest in physics comes a lot from wanting to see if things I see in sci-fi movies and books could actually be realized. I know it's not supposed to be used for study, and that Douglas Adam's isn't a mathematician or real physicist.

Same with Red Dwarf. The part where they play pool with the planets was funny. I'd love to understand the astrophysics of behind why we can't, or in what way we could, like what would have to happen or change.