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

[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.