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

[PeroK]

Is there a center of the range? Does nature have a prior preference that the center of the range must be rational or in some countable set? What would you say "selected" the center location?

PS Another good example is a clock. Let's assume theoretically that time is continuous. But, every clock works by some mechanism that counts things, essentially. A caesium atomic clock, for example. Any time measurement can only be made with a finite number of these units. You can try to exploit the continuity of time to get a sample from the real line, say. But, all you can get from a measurement is $1, 2, 3 \dots$ units, where each unit is a tick or cycle of your clock.

 

[WWGD]

@PeroK: Similarly, zero is a purely abstract concepts. In nature, within a finite volume, I can find zero of something, but I won't find any space with actually zero content. I'm not arguing that we can as a practical matter select arbitrary real numbers. Practically, we can perfectly select only integer numbers of things.

If I encounter a mathematician who self-delightedly shocks the sensibilities of a non-mathematician by saying that the measure, which most people in ordinary language take to mean size, or measurable size, of the infinite number of integers is the same as that of the empty set, i.e. zero, my ire will be aroused at the bullying, and I will patiently explain to both parties that no, the size of the integers is greater than the size of the empty set, but so small compared to that of the irrationals, that it's treated as the same by mathematicians.

My concern is that the term 'zero' had a well-established meaning before people started using it inconsistently, and that it is not necessary to use the term inconsistently, although there is clearly much benefit to be had from in a limited way doing so. I think that the benefit can be achieved without the inconsistency, simply by modifying the symbols and terminology used in the descriptions, to reflect when we are using which meaning.

As soon as you call both of two unequal things simply zero, you have at best discarded information, which information, although it may not be useful for your purposes, remains part of the actual truth.

Euclid's first postulate or axiom (Elements, Book I, Common Notions) says:

1. Things which equal the same thing also equal one another.​

​

I think it is irresponsible of LeBesgue to say that the empty set has exactly the same measure as the set of integers has, viz. zero, without first explicitly confronting the fact that saying so violates what Euclid said, i.e., violates something which is amply confirmed by everyday experience, and denies the validity of the abstract distinction between zero and infinitesimally more than zero.

I don't know if you're referring to my post, but I addressed your point for how to do away with what you claim is an inconsistency and you never addressed mine and now you rail against mathematicians engaging in deceptive usage of the term zero. It would have been nice if you had chosen to explain why you believed my attempted solution does not work instead of continuing to rail against an inconsistency you believe exists in the use of the term 0 or, better, suggested a solution, after many pointed out that probability theory within the Standard Reals or uncountably-infinite sets do not allow for assignment of nonzero probability to singletons. Maybe your ire should be directed too to the interested outsider who does not bother to address counters to their claims.

 

[WWGD]

No, but to add to my comment above, I think QM does actually resolve this issue, as far as it is possible to resolve it. If we take a single particle in place of your dart, then a measurement of an exact position would be a physically unrealisable state for the particle. Instead, according to the UP (Uncertainty Principle), you must have a range for position and a range for momentum. And, perhaps more important, there is no sense in which at a given time $t$ the particle really had a well-defined position $x(t)$. Nature does not, in fact, pick out a real-valued position every instant for every particle. Instead, positions are only meaningful in terms of position measurements.

Now, you could argue that this may not be correct. But, you cannot simply assume this is wrong. In short, one cannot appeal to nature to select a real number. In QM nature makes no such selection. Instead, nature allows a position within the range of your measuring apparatus. You always get a range; and, in QM, there is no sense in which the particle really is at a specific, exact location within the range.

In those terms, it is not just practically but theoretically impossible to select an arbitrary real number. That is, therefore, purely a mathematical process.

I am not sure but I believe a 3-pendulum system may be able to generate genuinely-random output.

 

[PeroK]

I am not sure but I believe a 3-pendulum system may be able to generate genuinely-random output.

Yes, but with a level of precision defined by your measurement apparatus. And, although classically you can theoretically define and measure the COM of a pendulum to any accuracy, you can't once you are down to the quantum level.

 

[sysprog]

I don't know if you're referring to my post, but I addressed your point for how to do away with what you claim is an inconsistency and you never addressed mine and now you rail against mathematicians engaging in deceptive usage of the term zero. It would have been nice if you had chosen to explain why you believed my attempted solution does not work instead of continuing to rail against an inconsistency you believe exists in the use of the term 0 or, better, suggested a solution, after many pointed out that probability theory within the Standard Reals or uncountably-infinite sets do not allow for assignment of nonzero probability to singletons. Maybe your ire should be directed too to the interested outsider who does not bother to address counters to their claims.

I acknowledge your point about the consequences of eliminating the dual of the rationals from the sample space, but I think that the same people who say that the chance of choosing a rational from the reals is equal to zero would also say that the chance of choosing a member of any finite subset of the rationals from the rationals is zero, and I would raise the same objection.

Using the infinitesimal when you're summing it and then treating it as zero when it's individuated is not a problem for me; it's saying that it's exactly equal to zero that I regard as inconsistent and as incorrect use of language.

I did propose alternatives which I think would satisfy the mathematical exigencies without doing injury to the language, e.g. saying "is treated as zero" (i.e., is treated as if it were equal to zero) instead of saying "is equal to zero".

 

[FactChecker]

I think that the same people who say that the chance of choosing a rational from the reals is equal to zero would also say that the chance of choosing a member of any finite subset of the rationals from the rationals is zero,

My understanding is that an important part of their idea is that there can only be a uniform distribution on a finite set. Any infinite set of points must have a non-uniform probability distribution that sums to 1. That allows positive probabilities on any subset.

 

[sysprog]

My understanding is that an important part of their idea is that there can only be a uniform distribution on a finite set. Any infinite set of points must have a non-uniform probability distribution that sums to 1. That allows positive probabilities on any subset.

You're right -- the countably infinite is a different beast -- thanks for the gentle correction -- I'm still not sure about how to assign probabilities to finite subsets of the countably infinite, so I'm reading up on that.

 

[FactChecker]

You can assign probabilities to every element of a countable set. Here is one example.
P(x1)=1/2; P(x2)=1/4; ..., P(xi)=1/2^i, ...
The total of all the probabilities is 1. The probability of any subset is simply the sum of the probabilities of all the elements in the subset.

 

[WWGD]

You can assign probabilities to every element of a countable set. Here is one example.
P(x1)=1/2; P(x2)=1/4; ..., P(xi)=1/2^i, ...
The total of all the probabilities is 1. The probability of any subset is simply the sum of the probabilities of all the elements in the subset.

Or take any series $\{x_n\}$converging to $L$ and assign probability $x_n/L$ to each point.

 

[sysprog]

You can assign probabilities to every element of a countable set. Here is one example.
P(x1)=1/2; P(x2)=1/4; ..., P(xi)=1/2^i, ...
The total of all the probabilities is 1. The probability of any subset is simply the sum of the probabilities of all the elements in the subset.

That's not what I meant by 'assign' -- even with a non-uniform distribution, there is no reason that I can see for supposing each number to be half as likely as its immediate predecessor.

 

[FactChecker]

That's not what I meant by 'assign' -- even with a non-uniform distribution, there is no reason that I can see for supposing each number to be half as likely as its immediate predecessor.

Correct. It is just one example. If the probabilities of the entire countably infinite set are anything that sum to 1, then you can simply add the probabilities of the elements of any subset.

 

[sysprog]

Please suppose that we know they sum to 1, because we know that some number will be chosen, but we don't how they sum to 1, i.e., we don't know how they are distributed -- how would we then sum the probabilities of the finite subset?

 

[FactChecker]

Please suppose that we know they sum to 1, because we know that some number will be chosen, but we don't how they sum to 1, i.e., we don't know how they are distributed -- how would we then sum the probabilities of the finite subset?

I don't think it would be possible.

 

[PeroK]

That's not what I meant by 'assign' -- even with a non-uniform distribution, there is no reason that I can see for supposing each number to be half as likely as its immediate predecessor.

The simplest way to do this is to toss a fair coin. The chosen number is the number of tosses to get the first head.

 

[PeroK]

My understanding is that an important part of their idea is that there can only be a uniform distribution on a finite set. Any infinite set of points must have a non-uniform probability distribution that sums to 1. That allows positive probabilities on any subset.

The critical point is that with a countable set you have probabilities and with an uncountable set you have a probability density function.

You cannot have a uniform distribution on a countable set, therefore, as you cannot have an infinite sum of equal probabilties.

But, you can have a uniform distribution on an uncountable set, as you integrate a constant function over a finite range.

I guess another take on my point is that one can physically sample countable probabilities (either via nature or by computer algorithm); but, one cannot sample a probability density function - except by a countable set of intervals.

In QM, for example, the modulus squared of the wave function, $|\Psi(x)|^2$ is a probability density function. And, $|\Psi(x)|^2\Delta x$ is (approx) the probability of finding the particle in a small interval of width $\Delta x$. You cannot say that the probability of finding the particle at $x$ is $0$.

 

[sysprog]

The simplest way to do this is to toss a fair coin. The chosen number is the number of tosses to get the first head.

That procedure would introduce a drastic bias in favor of low numbers.

 

[PeroK]

That procedure would introduce a drastic bias in favor of low numbers.

What are we trying to do?

 

[FactChecker]

In QM, for example, the modulus squared of the wave function, $|\Psi(x)|^2$ is a probability density function. And, $|\Psi(x)|^2\Delta x$ is (approx) the probability of finding the particle in a small interval of width $\Delta x$. You cannot say that the probability of finding the particle at $x$ is $0$.

Even in the limit? I agree with everything you say but I would be inclined to say that the probability at a single point is zero in the limit since you can make the neighborhood of the point as small as you like.

 

[PeroK]

Even in the limit? I agree with everything you say but I would be inclined to say that the probability at a single point is zero in the limit since you can make the neighborhood of the point as small as you like.

Again, that is mathematics. You can't take a limit in reality! The limit is a mathematical idea. You do one measurement, or a finite sequence of measurements. Even if you can have the resolution as small as you like, it's still finite. And the wavefunction tells you the probability of finding the particle in the region defined by your experiment. No experiment defines a single point.

Let me give you an example. This is from SR. Experiment #1 you accelerate a massive particle to $c- 1/2$. Experiment #n, you accelerate the particle to $c - 1/2^n$. That defines a sequence of experiments.

In the limit, of course, the particle is accelerated to $c$. Therefore, it is possible to accelerate a massive particle to $c$. You just take the limit of these experiments.

In general, you cannot take a limit and assume you have a valid physical statement.

 

[PeroK]

@FactChecker I just realized there is an interesting parallel here. As you accelerate a massive particle its energy becomes unbounded as its speed tends to $c$.

In QM, for a particle trapped in an infinite well. I.e. confined to a small space, the ground state energy is:

$E_1 = \frac{\pi^2\hbar^2}{2ma^2}$

Where $a$ is the width of the well. Hence, as you try to confine a particle to a smaller and smaller region, it's energy also becomes unbounded.

 

[FactChecker]

I'm not sure that we are talking about exactly the same thing. I am imagining a mixture of physical and mathematical to "select" a point. Suppose I have a probability distribution from QM. Instead of trying to restrict the location of the particle, can't I say that the mean is a "selected" single point? Even though I would not be able to record the mean to its full acuracy, it seems to me that it does exist as a number and could be any real number that depends on my selection of a coordinate system and units.

 

[PeroK]

I'm not sure that we are talking about exactly the same thing. I am imagining a mixture of physical and mathematical to "select" a point. Suppose I have a probability distribution from QM. Instead of trying to restrict the location of the particle, can't I say that the mean is a "selected" single point? Even though I would not be able to record the mean to its full acuracy, it seems to me that it does exist as a number and could be any real number that depends on my selection of a coordinate system and units.

Okay, but how do you select the coordinate system? If we stick with the particle in a well. The expected value of a position measurement (for any energy eigenstate) is the middle of the well.

At this point, all we have is the function $\sqrt{\frac 2 a}\sin(\frac{n\pi x}{a})$.

You can do any change of coordinates on that: $x' = x + x_0$. How do you select $x_0$?

We only invoked a physical system to try to get a natural selection for $x_0$. So, we are back where we started.

 

[FactChecker]

You can do any change of coordinates on that: $x' = x + x_0$. How do you select $x_0$?

Independent and prior to the experiment.

We only invoked a physical system to try to get a natural selection for $x_0$. So, we are back where we started.

Not in those words. We are invoking a physical system to get a location or a range, not a coordinate number. When the mean of the range is determined, it can be determined where that is on the independently-defined coordinate system. So there can be no preference for the physical process to "select" a mean whose coordinate is in any given countable set.

 

[PeroK]

Independent and prior to the experiment.

What's the point of the experiment if you already have your random real $x_0$? You've already done what you wanted to do.

 

[sysprog]

What are we trying to do?

I understand that by virtue of countable additivity we can't get a completely unbiased (uniform) distribution -- the assumption that such a distribution exists leads to contradiction:

Let $X$ be a random variable which assumes values in a countable infinite set $Q$. We can prove there is no uniform distribution on $Q$.

Assume there exists such a uniform distribution, that is, there exists $a≥0$ such that $P(X=q)=a$ for every $q∈Q$.

Observe that, since $Q$ is countable, by countable additivity of $P$,

$1=P(X∈Q)=∑_{q∈Q}P (X=q)=∑_{q∈Q}a$

Observe that if $a=0, ∑_{q∈Q}a=0$. Similarly, if $a>0, ∑_{q∈Q}a=∞$. Contradiction.

Accordingly, I'm doing some reading on distributions of probabilities of finite subsets over countably infinite sets, and seeing how the definitions of the subsets may influence the distributions -- you guys (in this thread you, @FactChecker, @WWGD, and @Stephen Tashi) have got me reviewing familiar territory, along with breaking new-to-me ground.

 

[Stephen Tashi]

however, it seems to me that it's not germane to whether it's reasonable to hold, as I do, that e.g. calling a number both positive and zero, or both zero and non-zero, is inconsistent, and is therefore an incorrect use of language.

However, it isn't mathematics that is calling a number both positive and zero. It is you that is doing that. You add your own interpretation to mathematical statements about probability and limits and conclude (with any mathematics to support your conclusion) that a number which mathematics evaluates as zero must be positive. This is not a problem for the self-consistency of mathematics. It's a problem for the self-consistency between mathematics and your own definitions.

There is a Platonic philosophy of mathematics that holds that mathematical concepts exist independently of any attempts to define them. It's a useful and common way of thinking about math. For example, we often think of "zero" as having the common language meaning of "nothing". However, the Platonic approach to actually proving anything in mathematics fails because there can't be a consensus about the validity of a proof based on various personal opinions about the things being discussed. The effective way to do mathematics is to make assumptions and definitions explicit.

As to the opinions of Euclid and Dr. Johnson (or even Newton), ancient discussions of mathematics don't set the standards for definitions in contemporary math.

One theme of this thread is that mathematical probability theory says nothing about the concepts of "possibility" and "impossibility". Only applications of probability theory consider such concepts.

To that theme, we can add the analogous theme that the theory of real numbers doesn't define "zero" to be "nothing". It defines "zero" to be the additive identity. The interpretation of "zero" as "nothing" or "do nothing" is a useful application of mathematics. However, the applications of "zero" aren't the mathematical definition of "zero".

The contemporary scheme of mathematical education is (correctly, I think) a hybrid of the formal and Platonic approaches. It's easier to let young students think of "zero" as "nothing" than to have them think about it as the additive identity. This leads them to think that the definition of zero is a theorem. The think that $a + 0 = a$ is a consequence of the fact that zero is nothing rather than a definition of what zero is.

Likewise, introductory texts on probability introduce the concept that random variables have "realizations". All of statistics is an application of probability theory.

A consequence of our approach to education is that as students began to study advanced mathematics, they face the task of un-learning their personal definitions of mathematical concepts and replacing them with the formal definitions.

 

[sysprog]

@Stephen Tashi, I disagree with your contention to the effect that I am responsible for the inconsistency of asserting a number to be both positive and zero. If each of the summands in an integration is zero when individuated, yet the integration results in a non-zero positive sum, then none of those individual instances of zero is really the additive identity, because by definition adding the additive identity results in no change to the sum, i.e., each new sum is identical to its predecessor if each of the infinitesimal individual summands in an infinite series of summations is really zero. I think it's more accurate linguistically to say that each of them is treated as zero when individuated.

 

[Stephen Tashi]

@Stephen Tashi, I disagree with your contention to the effect that I am responsible for the inconsistency of asserting a number to be both positive and zero. If each of the summands in an integration is zero when individuated, yet the integration results in a non-zero positive sum, then none of those individual instances of zero is really the additive identity,

You are making a personal interpretation of the mathematical concept of "integration". A Riemann integration isn't a sum, it is a limit of a sum. If you want to prove a mathematical statement describes both zero and also a positive number , you need to observe the formal mathematical definitions that are involved. If we add our own interpretations, it isn't mathematics that is the cause of inconsistencies.