Undergrad How Can Improbability and Infinitesimal Probabilities Exist in Real Life Events?

[PeroK]

Impossible physically or impossible conceptually? In the real world, it is not possible to have an absolutely fair coin, so should we stop talking about the probabilities of a fair coin?

No. But we have to be careful what we conclude. A real coin can be associated with a fair coin in a number of contexts. This is part of the mathematical modelling process.

 

[PeroK]

@FactChecker Let me boil down our debate as follows. First, I'm going to say:

Let $x_0 \in [0,1]$.

I've chosen one arbitrary real number.

You believe I have done something impossible. I don't believe I have done something impossible; I believe I've done something mathematical.

And, if by doing mathematics we are all doing the impossible all the time and that is part of your definition of impossible, then I guess there's no argument.

 

[PeroK]

Please read my post carefully!

I said, that we can show that there exists a sample space where this probability is 1 and NOT 0!

The proof relies on limits at infinity, so my previous example is realistically viable but if you want to argue trivialities with me, I might as well leave this conversation...

Limits are, quite explicitly, abstract mathematical constructions. And, the history of mathematics shows how important it is to have a rigorous mathematical definition. You can't mix up limits with real physical processes.

 

[Quasimodo]

Limits are, quite explicitly, abstract mathematical constructions. And, the history of mathematics shows how important it is to have a rigorous mathematical definition. You can't mix up limits with real physical processes.

Let k be the size of a sample probability space, how big we will soon find out.

Let a true random number generator produce a string of digits size n, each digit takes 1 second to be generated, etc., how big this n would be, we'll find out soon.

Is this ok with you? Shall we proceed?

 

[PeroK]

Let k be the size of a sample probability space, how big we will soon find out.

Let a true random number generator produce a string of digits size n, each digit takes 1 second to be generated, etc., how big this n would be, we'll find out soon.

Is this ok with you? Shall we proceed?

Don't let me stop you!

 

[FactChecker]

No. But we have to be careful what we conclude. A real coin can be associated with a fair coin in a number of contexts. This is part of the mathematical modelling process.

I'll buy that. I think that I understand where each of our positions is appropriate -- the physical versus the conceptual (including physically impossible). With that in mind, I see your point and will look at "computable" some more. Even if I allow myself to include the physically impossible, I still have the problem of distinguishing the mathematical concepts of "nearly impossible" from "logically impossible". That is what I was trying to address at the beginning.

 

[sysprog]

I'll buy that. I think that I understand where each of our positions is appropriate -- the physical versus the conceptual (including physically impossible). With that in mind, I see your point and will look at "computable" some more. Even if I allow myself to include the physically impossible, I still have the problem of distinguishing the mathematical concepts of "nearly impossible" from "logically impossible". That is what I was trying to address at the beginning.

Some good insights on those concerns and on related matters can be found here: https://terrytao.wordpress.com/2015/09/29/275a-notes-0-foundations-of-probability-theory/

 

[Twodogs]

And events with a probability of exactly one, what then? In a deterministic universe the state of the universe at one moment follows directly from the state of the universe at any prior time with a probability of exactly one. Anything less and the wheels come off. Is that the case?

 

[FactChecker]

No. One must distinguish between a logical certainty and a probability of one. They are not the same.
Suppose a number is selected randomly on the line segment [0,1]. The probability that the number is irrational is 1 because the subset of irrational numbers has a probability measure of 1. The rational numbers are countable and the rational subset has a probability measure of 0. If the number selected turns out to be rational, the consequences are not that "the wheels come off".

 

[sysprog]

And events with a probability of exactly one, what then? In a deterministic universe the state of the universe at one moment follows directly from the state of the universe at any prior time with a probability of exactly one. Anything less and the wheels come off. Is that the case?

From the link I posted in #67:

By default, mathematical reasoning is understood to take place in a deterministic mathematical universe. In such a universe, any given mathematical statement

(that is to say, a sentence with no free variables) is either true or false, with no intermediate truth value available. Similarly, any deterministic variable

can take on only one specific value at a time.

However, for a variety of reasons, both within pure mathematics and in the applications of mathematics to other disciplines, it is often desirable to have a rigorous mathematical framework in which one can discuss non-deterministic statements and variables – that is to say, statements which are not always true or always false, but in some intermediate state, or variables that do not take one particular value or another with definite certainty, but are again in some intermediate state. In probability theory, which is by far the most widely adopted mathematical framework to formally capture the concept of non-determinism, non-deterministic statements are referred to as events, and non-deterministic variables are referred to as random variables. In the standard foundations of probability theory, as laid out by Kolmogorov, we can then model these events and random variables by introducing a sample space (which will be given the structure of a probability space) to capture all the ambient sources of randomness; events are then modeled as measurable subsets of this sample space, and random variables are modeled as measurable functions on this sample space.

Prof. Tao explains difficult things with clarity, but he can't thereby make them not difficult.

 

[PeroK]

No. One must distinguish between a logical certainty and a probability of one. They are not the same.
Suppose a number is selected randomly on the line segment [0,1]. The probability that the number is irrational is 1 because the subset of irrational numbers has a probability measure of 1. The rational numbers are countable and the rational subset has a probability measure of 0. If the number selected turns out to be rational, the consequences are not that "the wheels come off".

For example, if I select $1/\pi$, which is irrational, that is okay. But, if I select $0.5$, which is rational, then that is also okay. Hmm?

How would you tell whether the number you selected "turned out to be" rational?

 

[PeroK]

No. One must distinguish between a logical certainty and a probability of one. They are not the same.
Suppose a number is selected randomly on the line segment [0,1]. The probability that the number is irrational is 1 because the subset of irrational numbers has a probability measure of 1. The rational numbers are countable and the rational subset has a probability measure of 0. If the number selected turns out to be rational, the consequences are not that "the wheels come off".

I found something here. The reply from Kevin Carlson.

https://math.stackexchange.com/ques...andom-natural-number-and-a-random-real-number

He concludes that there is no way to pick a random real in the way that you describe, for example. I agree with this. As soon as you say "select a real number at random" you are no longer talking about something that makes mathematical sense.

 

[Quasimodo]

How many times you are picking out a number from [0,1] is what matters! The size of the sample space, There the probability becomes 1, even for 1 trial is only 0.
But this is an unfriendly forum. I'll leave you to sort it out yourselves...

 

[FactChecker]

I found something here. The reply from Kevin Carlson.

https://math.stackexchange.com/ques...andom-natural-number-and-a-random-real-number

He concludes that there is no way to pick a random real in the way that you describe, for example. I agree with this. As soon as you say "select a real number at random" you are no longer talking about something that makes mathematical sense.

It is easy to define ways to make a random selection of a point on a line segment. The only problem is in recording the selection with perfect accuracy, but that is another subject.

 

[PeroK]

You can select from a countable subset of real numbers. To do that, you choose a whole number $n$ and map that to $r_n$ where $\{r_n \}$ is a countable subset of real numbers. Trivially, for example, you can choose from $\frac{1}{\pi}, \frac{1}{2\pi}, \dots$.

Moreover, you start by doing what I claim is impossible: selecting lengths from the set of (all) real numbers! You can't do that either. This process is not well defined:

Suppose one has two line segments of different lengths, randomly chosen.

 

[FactChecker]

You can select from a countable subset of real numbers. To do that, you choose a whole number $n$ and map that to $r_n$ where $\{r_n \}$ is a countable subset of real numbers. Trivially, for example, you can choose from $\frac{1}{\pi}, \frac{1}{2\pi}, \dots$.

Moreover, you start by doing what I claim is impossible: selecting lengths from the set of (all) real numbers! You can't do that either.

I can let nature provide segments of different lengths and just select a line segment. Nature has no predisposition to any particular unit system. When I independently provide the units to determine the length, there is a probability of 1 that the length will be irrational.

 

[PeroK]

I can let nature provide segments of different lengths and just select a line segment. Nature has no predisposition to any particular unit system. When I independently provide the units to determine the length, there is a probability of 1 that the length will be irrational.

If I ask you for a random real number, then your answer might be: the width of your desk in $cm$? Something like that?

But, beyond a certain accuracy the width of your desk is not well-defined. Where exactly does it start and end to a scale less than an atom? And, if the atoms are moving, then the width is changing with time. Not to mention any QM uncertainties.

PS I would say that measurements are one way to generate random numbers. But, all measurement outcomes must be one of a finite (possibly countable) set of numbers. You can't have an uncountable number of possible outcomes from a measurement.

 

[FactChecker]

If I ask you for a random real number, then your answer might be: the width of your desk in $cm$? Something like that?

But, beyond a certain accuracy the width of your desk is not well-defined. Where exactly does it start and end to a scale less than an atom? And, if the atoms are moving, then the width is changing with time. Not to mention any QM uncertainties.

I am leaving the problem of accuracy as a separate subject and assuming infinite accuracy in this "thought experiment". Regardless of the uncertainties and the method used, as long as the units are human-defined, independent of the line segment selected, the result will be irrational with a probability of 1. A naturally occurring length does not have any predisposition to the rational numbers in any independently-defined, human-created unit system. The probability that the length is irrational is 1. If you are claiming that there is a predisposition to rational lengths in a unit system that it has no knowledge of, then you need to prove that.

 

[PeroK]

I am leaving the problem of accuracy as a separate subject and assuming infinite accuracy in this "thought experiment". Regardless of the uncertainties and the method used, as long as the units are human-defined, independent of the line segment selected, the result will be irrational with a probability of 1. A naturally occurring length does not have any predisposition to the rational numbers in any independently-defined, human-created unit system. The probability that the length is irrational is 1. If you are claiming that there is a predisposition to rational lengths in a unit system that it has no knowledge of, then you need to prove that.

Okay, I'll accept that as a hypothesis.

But, the number of lengths we can define is countable. We can start with your desk, your piano etc. Even if the universe is infinite, there are only a countable number of atoms, so only a countable number of things that can ever exist and have a length.

Now, we list these objects that (hypothetically could ever exist in this universe) and their lengths are $L_1, L_2, \dots$.

And now, you are selecting your real number from this countable subset of the real numbers.

If you appeal to nature, you do not have an uncountable number of lengths to choose from.

 

[FactChecker]

Suppose I define a line segment, with no knowledge of the units of measurement. Suppose that another person independently defines the units of length measurement. Although we must eventually give up on getting a completely accurate length determination, the fact remains that the length in those units is irrational with a probability of 1. That is forced by the enormously larger quantity of irrational numbers. There is NO positive probability that the length is rational.

 

[PeroK]

Suppose I define a line segment, with no knowledge of the units of measurement. Suppose that another person independently defines the units of length measurement. Although we must eventually give up on getting a completely accurate length determination, the fact remains that the length in those units is irrational with a probability of 1. That is forced by the enormously larger quantity of irrational numbers. There is NO positive probability that the length is rational.

First, a line segment is a mathematical object. It cannot be physically measured.

Second, I'm not sure it's well defined to talk about "the probability that the length of an object is irrational". That depends on how you define length; which is a physical process. One process may define length one way and another process a different way. There's no mathematical axiom to define the length of an object in one specific way.

Third, the real numbers are - whatever you say - a difficult, abstract mathematical construction. You can't define or study the real numbers by an appeal to nature. You cannot prove the Archimedian property, say, by an appeal to nature and a thought experiment about length measurments. You cannot prove anything about the real numbers by an appeal to nature.

Fourth, that the rational numbers have measure zero is an entirely mathematical statement. You cannot by mathematics alone prove that every length is an irrational number (in any system of units). There is always at least one system of units where a given length is precisely $1$ unit.

 

[FactChecker]

You seem to imply that objects in nature are all in a specific set of precise positions that can only be specified using the rational numbers. IMHO, that is wrong in general and especially wrong at the quantum level.

I do not consider the irrational numbers to be any stranger than the rational numbers. They are just a different sequence of digits, but far less restricted and more numerous. The fact that humans want to define numbers using a finite series of arithmetic operations reflects a lack of imagination on the part of humans.

PS. I think this discussion has gotten too philisophical for me. I will leave further comments up to smarter (and wiser) people than myself.

 

[PeroK]

You seem to imply that objects in nature are all in a specific set of precise positions that can only be specified using the rational numbers.

I've no idea where you think I've said that. I said there are only a countable number of objects in nature.

I do not consider the irrational numbers to be any stranger than the rational numbers.

"Strange" is not a precise term. But, for example, most real numbers (all but a countable subset) are not computable.

The issue of non-computability is the real issue here. Not rationality/irrationality. And countability.

The fact that humans want to define numbers using a finite series of arithmetic operations reflects a lack of imagination on the part of humans.

Given that the real numbers are a human mathematical construction, I can't see the point of this statement.

It's been an interesting debate, but I'm not sure there is any point in taking it further.

 

[Twodogs]

"By default, mathematical reasoning is understood to take place in a deterministic mathematical universe. In such a universe, any given mathematical statement (that is to say, a sentence with no free variables) is either true or false, with no intermediate truth value available. Similarly, any deterministic variable can take on only one specific value at a time."

This makes sense, but there is always the question of whether the attribute of being deterministic is a projection of our mathematics upon a physical universe that is not so rigidly governed. The word determine connotes limitation, a meaning that can be traced back to the PIE root meaning "peg, post; boundary marker." Perhaps, in the physical universe, limitation has its limits.

As to making a random choice, there has been much discussion about whether this is possible in a deterministic universe. Can one actually build a device that makes a random choice without reference to a truly "random" physical process?

 

[sysprog]

Suppose I define a line segment, with no knowledge of the units of measurement. Suppose that another person independently defines the units of length measurement. Although we must eventually give up on getting a completely accurate length determination, the fact remains that the length in those units is irrational with a probability of 1. That is forced by the enormously larger quantity of irrational numbers. There is NO positive probability that the length is rational.

Saying that the probability that a number chosen at random will be irrational is one is logically equivalent to saying that the probability that it will be rational is zero. I reject this expedient as incorrect language. Only the impossible has probability zero, and only the absolutely certain has probability one. The probability that the number will be irrational is actually one minus the possibility or probability that it will be rational. Because it is not impossible that the number could be rational, the probability that it will be rational is not zero.

 

[PeroK]

Saying that the probability that a number chosen at random will be irrational is one is logically equivalent to saying that the probability that it will be rational is zero. I reject this expedient as incorrect language. Only the impossible has probability zero, and only the absolutely certain has probability one. The probability that the number will be irrational is actually one minus the possibility or probability that it will be rational. Because it is not impossible that the number could be rational, the probability that it will be rational is not zero.

The root of the problem is the incorrect application of mathematics (probability theory) to number selection (algortithms and computability). In abstract measure-probability theory:

If $X$ is a random variable distributed uniformly on $[0,1]$, then the following are true:

$\forall x: p(X = x) = 0$

$p(x \in \mathbb Q) = 0$

$p(x \notin \mathbb Q) = 1$

Note that, as pointed out above, abstract probability theory says nothing about "possible" or "impossible". That's an interpretation of the mathematics.

The real problem comes when one invokes this to say something like "if you choose a number in $[0,1]$, the probability it is rational is zero." Some people go further and say "whatever number you choose, the impossible has happened." If you look online, you will see this nonsensical assertion in many places.

This I believe to be misguided and a mis-application of abstract probability theory.

There is no algorithm that can select from more than a countable predefined set of numbers. Not least because the set of real numbers that you can even describe (the "computable" numbers) is countable.

This is why the impossible (something with a probability of $0$) can't "happen". There's no real-world algortithm or process to which the mathematics applies.

PS The probability that a number chosen at random is rational depends on the algorithm with which you choose the number. You can assign a probability to $p(x \in \mathbb Q)$ based on an analysis of your algortithm. It will be zero only if your algorithm is incapable of selecting any rational number! But not otherwise.

PPS all mathematics is physically "impossible" if you think about it. Something like "let $f(x) = \sin x$" conjures a physically impossible infinite sine function. Similarly, and rather prosaically actually, "let $X$ be a random variable uniformly distributed on $[0, 1]$" conjures a physical impossibility. It's just that in this case some people confuse themselves by imagining that what you can do mathematically with numbers you must be able to do physically with numbers.

 

[sysprog]

The root of the problem is the incorrect application of mathematics (probability theory) to number selection (algortithms and computability). In abstract measure-probability theory:

If $X$ is a random variable distributed uniformly on $[0, 1]$, then the following are true:

$\forall x \ p(X =x) = 0$

$p( x \in \mathbb Q) = 0$

$p( x \notin \mathbb Q) = 1$

Note that, as pointed out above, abstract probability theory says nothing about "possible" or "impossible". That's an interpretation of the mathematics.

The real problem comes when one invokes this to say something like "if you choose a number in $[0,1]$, the probability it is rational is zero." Some people go further and say "whatever number you choose, the impossible has happened." If you look online, you will see this nonsensical assertion in many places.

This I believe to be misguided and a mis-application of abstract probability theory.

There is no algorithm that can select from more than a countable predefined set of numbers. Not least because the set of real numbers that you can even describe (the "computable" numbers) is countable.

This is why the impossible (something with a probability of $0$) can't "happen". There's no real-world algortithm or process to which the mathematics applies.

PS The probability that a number chosen at random is rational depends on the algorithm with which you choose the number. You can assign a probability to $p(x \in \mathbb Q)$ based on an analysis of your algortithm. It will be zero only if your algorithm is incapable of selecting any rational number! But not otherwise.

A problem with that analysis is that: $\exists x \ (X =x)$ is asserted implicitly by the postulate that a number is selected, and that is provably inconsistent with $\forall x \ p(X =x) = 0$.

 

[PeroK]

A problem with that analysis is that: $\exists x \ (X =x)$ is asserted implicitly by the postulate that a number is selected, and that is provably inconsistent with $\forall x \ p(X =x) = 0$.

That's mathematical "selection". That's not something that you can "really do". If you reserve "possible/impossible" to describe something you can actually do, then the problem disappears.

You must see $p(X = x) = 0$ or $\mu(\mathbb Q) = 0$ as mathematical statements.

You can't argue on physical grounds that $\mu(\mathbb Q) \ne 0$ because "rational numbers exist". The measure is well-defined and the measure the rational numbers is 0.

Likewise, you mustn't take this to say "rational numbers are impossible to find".

 

[sysprog]

Again, I claim that saying that something which is obviously not impossible has a zero probability is a pet inconsistency in the use of language. It's logically provable that the impossible, and only the impossible, has zero probability. I disagree with LeBesgue's use of zero for the measure of the rationals. I regard it as a misuse of 'zero' and consequently as incorrect language. I would assign measure zero only to the empty set. I have no problem with assigning the rationals a measure $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0));$$ i.e. the measure is no less than some minimally positive number and therefore is non-zero.

 

[WWGD]

Again, I claim that saying that something which is obviously not impossible has a zero probability is a pet inconsistency in the use of language. It's logically provable that the impossible, and only the impossible, has zero probability. I disagree with LeBesgue's use of zero for the measure of the rationals. I regard it as a misuse of 'zero' and consequently as incorrect language. I would assign measure zero only to the empty set. I have no problem with assigning the rationals a measure $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0));$$ i.e. the measure is no less than some minimally positive number and therefore is non-zero.

But there is no "Minimally-positive" Standard Real number.