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

[sysprog]

I think any definition set which requires admission of such absurdities as the notion that the set of rational numbers has 'zero content' is faulty.

The cardinality of any non-empty set is strictly greater than that of the empty set. The cardinality of the rationals is strictly greater than that of any finite set. The cardinality of the reals is strictly greater than that of the rationals. The cardinality of the power set of the reals is strictly greater than that of the reals.

None of those remarks about cardinalities is linguistically self-inconsistent or inter-inconsistent. Please recall that my objection is to the misuse of language; not to the mathematical insights.

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

That's an informal description of what was intended by the reference to $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0)).$$Another way to describe that is that it is a positive number $x$ such that any other positive number is either equal to $x$ or greater than $x$. I can't say the value of $x$ but I can indicate that it has that contemplated property and let that suffice because I can't do better. To call it zero would be inconsistent with calling it positive. Once you say a number is positive you can't consistently with that statement also say it is zero.

 

[WWGD]

I think any definition set which requires admission of such absurdities as the notion that the set of rational numbers has 'zero content' is faulty.

The cardinality of any non-empty set is strictly greater than that of the empty set. The cardinality of the rationals is strictly greater than that of any finite set. The cardinality of the reals is strictly greater than that of the rationals. The cardinality of the power set of the reals is strictly greater than that of the reals.

None of those remarks about cardinalities is linguistically self-inconsistent or inter-inconsistent. Please recall that my objection is to the misuse of language; not to the mathematical insights.

That's an informal description of what was intended by the reference to $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0)).$$Another way to describe that is that it is a positive number $x$ such that any other positive number is either equal to $x$ or greater than $x$. I can't say the value of $x$ but I can indicate that it has that contemplated property and let that suffice because I can't do better. To call it zero would be inconsistent with calling it positive. Once you say a number is positive you can't consistently with that statement also say it is zero.

It's the best we have thus far. Our probability theory on subsets of the Reals does not have enough resolution to distinguish impossible events outside of the sample space and sets with countably-many elements. How do we improve on this? I am not sure.

 

[sysprog]

It's the best we have thus far. Our probability theory on subsets of the Reals does not have enough resolution to distinguish impossible events outside of the sample space and sets with countably-many elements. How do we improve on this? I am not sure.

You just ably made the distinction in the very act of denying the ability to do so.

 

[WWGD]

You just ably made the distinction in the very act of denying the ability to do so.

I am not saying there is no distinction, just that our present Mathematical models don't allow for an effective way of making it. Edit: to the best of my knowledge.

 

[Stephen Tashi]

It's logically provable that the impossible, and only the impossible, has zero probability.

What is the proof? Logic can prove nothing by itself without assumptions or definitions.

I disagree with LeBesgue's use of zero for the measure of the rationals.

That's a statement of your personal preference. If you can propose a different probability measure then this can be discussed in the context of mathematical probability theory.

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.

The fact a definition is made doesn't prove the thing defined actually exists. (It also doesn't prove the thing defined is unique - even if it does exist.) Further, defining a probability measure for a certain type of subset of [0,1] doesn't completely define the measure. It must be defined for all subsets of some sigma algebra of sets. As someone suggested in an earlier post, your might be able to implement the concept of a "minimally positive number" by extending the real number system as in done in non-standard analysis https://en.wikipedia.org/wiki/Non-standard_analysis. Perhaps somebody has already worked this out.

 

[FactChecker]

Probabilities are not necessarily tied to the human ability to devise a finite, terminating selection process. There is much that happens and exists in nature that has probabilities with no human involvement and no known finite "selection" process.
Many comments in this thread are attempting to discard a great deal of standard probability theory that is based on measure theory. That would require a lot of work and would greatly increase the complexity of the theory. I am not sure that anyone here has identified a single benefit of that approach.

 

[sysprog]

What is the proof? Logic can prove nothing by itself without assumptions or definitions.

If a definition of a term can be shown to be inconsistent with another definition of the same term, that is in my view adequate proof of invalidity of at least one of the definitions.

That's a statement of your personal preference. If you can propose a different probability measure then this can be discussed in the context of mathematical probability theory.

I just did. This is a proposed non-zero definition whereby $x$ is the measure of the rationals: $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0)).$$ I understand that $x$ would thereby be small enough that it could be treated as zero, but it wouldn't thereby be asserted to actually be equal to zero.

The fact a definition is made doesn't prove the thing defined actually exists.

An inconsistent pair of definitions proves that at least one or the other does not actually exist.

(It also doesn't prove the thing defined is unique - even if it does exist.)

That's not something I'm quibbling about.

Further, defining a probability measure for a certain type of subset of [0,1] doesn't completely define the measure.

I'm objecting to inconsistency; not offering completeness.

It must be defined for all subsets of some sigma algebra of sets.

In my view, resorting to inconsistent definitions of zero to achieve this, while it is certainly convenient, is incorrect use of language, and therefore objectionable.

As someone suggested in an earlier post, your might be able to implement the concept of a "minimally positive number" by extending the real number system as in done in non-standard analysis https://en.wikipedia.org/wiki/Non-standard_analysis. Perhaps somebody has already worked this out.

I don't think that eliminating inconsistency in the use of the term 'zero' requires extending the reals beyond whatever is entailed by inclusion of the infinitesimal within the standard. It may require use of a different symbol, such as $0^+$, and a corresponding definition and set of rules, that allows an infinitesimal to be treated as zero without it being asserted to actually be zero.

 

[sysprog]

Probabilities are not necessarily tied to the human ability to devise a finite, terminating selection process. There is much that happens and exists in nature that has probabilities with no human involvement and no known finite "selection" process.
Many comments in this thread are attempting to discard a great deal of standard probability theory that is based on measure theory. That would require a lot of work and would greatly increase the complexity of the theory. I am not sure that anyone here has identified a single benefit of that approach.

I'm merely trying to object steadfastly to the complacent use of inconsistent definitions for zero.

 

[WWGD]

I'm merely trying to object steadfastly to the complacent use of inconsistent definitions for zero.

Please point out the inconsistency that follows from using zero as you mention. I don't see it.

 

[sysprog]

Please point out the inconsistency that follows from using zero as you mention. I don't see it.

Only the impossible actually has probability zero. To say of an event that it is possible for it to occur is to say that its probability of occurring, however small, is non-zero. Saying that if a positive quantity is so small that we can't measure it then it is equal to zero, is saying that the quantity is at once positive and therefore non-zero and also equal to zero and therefore non-positive. Nothing can be both zero and positive because the definition of positivity is that the referent is strictly greater than zero and therefore strictly not equal to zero.

 

[WWGD]

Only the impossible actually has probability zero. To say of an event that it is possible for it to occur is to say that its probability of occurring, however small, is non-zero. Saying that if a positive quantity is so small that we can't measure it then it is equal to zero, is saying that the quantity is at once positive and therefore non-zero and also equal to zero and therefore non-positive. Nothing can be both zero and positive because the definition of positivity is that the referent is strictly greater than zero and therefore strictly not equal to zero.

It seems you can just state or describe some elements, here countable subsets as being in the sample space yet with measure zero and others as not being in the sample space and avoid any confusion.

 

[sysprog]

It seems you can just state or describe some elements, here countable subsets as being in the sample space yet with measure zero and others as not being in the sample space and avoid any confusion.

I'm not trying to assert that the misuse of language to which I refer entails the existence of any confusion on the parts of those who so misuse language. I'm merely asserting that it's incorrect. Part of the definition of 'measure zero' is 'having zero content'. Saying that the set of rationals has 'zero content' is saying something that is patently false. Among sets and their subsets, only the empty set has zero content. That's what 'empty' means. Non-empty sets are non-empty because they have more than zero content. Again, I'm advocating for consistent use of language; not trying to cast aspersions on anyone's mathematical insights.

 

[PeroK]

Only the impossible actually has probability zero.

That depends on how you define "impossible". "Impossible", in mathematics usually has nothing to do with probabilities. Usually it means that some set is the empty set:

It's impossible to find a real solution to the equation $x^2 + 1 = 0$ is an informal way of saying: $\{ x \in \mathbb R : x^2 + 1 = 0 \} = \emptyset$.

It doesn't mean:

If you choose a real number, $x$, on a uniform distribution of all real numbers, then the probability that $x^2 + 1 = 0$ is zero.

Also, $\mu(\emptyset) = 0$, which means that the empty set has "probability" zero. But, that doesn't mean that the empty set is "impossible".

Your whole argument is based on a confusion of terminology.

 

[sysprog]

That depends on how you define "impossible". "Impossible", in mathematics usually has nothing to do with probabilities. Usually it means that some set is the empty set:

It's impossible to find a real solution to the equation $x^2 + 1 = 0$ is an informal way of saying: $\{ x \in \mathbb R : x^2 + 1 = 0 \} = \emptyset$.

It doesn't mean:

If you choose a real number, $x$, on a uniform distribution of all real numbers, then the probability that $x^2 + 1 = 0$ is zero.

It means that, too, because that too is entailed by the premises.

Also, $\mu(\emptyset) = 0$, which means that the empty set has "probability" zero. But, that doesn't mean that the empty set is "impossible".

I wouldn't say that the empty set is impossible, but I would say that the set of possible impossibilities is empty.

Your whole argument is based on a confusion of terminology.

I'm not the one who is confusing the terminology. It's inconsistent usage that confuses the terminology.

If you choose a real number, $x$, on a uniform distribution of all real numbers, then the probability that $x^2 + 1 = 0$ is zero.

That's fine. You have to resort to complex numbers to satisfy that equation. What I'm objecting to is, e.g., given a choice of a real number $x$ as you postulated, the assertion that the probability that $x - 1 = 0$ is zero. It's possible that $x=1$ because the specified conditions don't rule it out; wherefore, it has probability greater than zero.

 

[PeroK]

What I'm objecting to is, e.g., given a choice of a real number $x$ as you postulated, the assertion that the probability that $x - 1 = 0$ is zero. It's possible that $x=1$ because the specified conditions don't rule it out; wherefore, it has probability greater than zero.

Whether $p(1) = 0$ or not depends on the distribution. If the distribution is uniform on $[0,1]$, then $p(1) = 0$. I'm sure you know the argument.

This is all mathematics. There is no sense in which we are dealing with "possible" or "impossible" events. If you define these terms mathematically, then they have the properties they have through their definition. They do not have properties based on the English-language definition of the word used. If you define an "impossible" set as one having measure zero, then that is your definition. You can't invoke an English-language meaning of the word to override your mathematics.

Here is an example of where you are going wrong. One could argue that all numbers are "rational" because they all obey logic. One could argue that an "irrational" number is a contradiction. But, that argument confuses "irrational" as an English word; and "irrational" as a well-defined mathematical term.

You are likewise confusing "impossible" as an English word with a defintion inside probability theory. In one sense it's worse because actually "impossible" has no meaning inside probability theory, except as an informal term for a set of measure zero.

Another example:

All functions and matrices are "invertible" because you can write them upside down.

 

[PeroK]

@sysprog if you then ask: what happens if I really choose a number from $[0,1]$ using a uniform distribution. Then my answer is that is physically impossible. The mathematics does not directly translate to a real-world selection process. To select a number in reality, you must have an algortithm. This introduces two constraints:

1) You can only choose from a predetermined countable set of numbers.
2) All numbers must be computable, which is countable subset of the reals.

Ultimately, this is just an example of mathematical processes being abstract and not necessarily something you can directly do: draw a circle, generate an infinite sine function, select a real number uniformly from an uncountable set. These are things you can only approximate in the real world.

 

[Stephen Tashi]

Part of the definition of 'measure zero' is 'having zero content'.

If you state that precisely, I think it is a theorem, not part of a definition. A "measure" function is also a "content" function. So a set with zero measure , as measured by a measure $\mu$ also has zero content as measured by $\mu$.

However, there is no mathematical formulation of the notions of "possible" and "impossible" within the theory of functions that are contents (in the technical sense of the term "content" https://en.wikipedia.org/wiki/Content_(measure_theory) ). So nothing mathematical can be proven or contradicted about the common language notions of "possible" and "impossible" within the framework of that theory.

You seem to be objecting to inconsistencies in common language notions of "zero" and "content" - as well as "possible" and "impossible".

From the point of view of those common language words, unmeasureable sets ( https://en.wikipedia.org/wiki/Vitali_set ) are also a problem for thinking in common language.

Although reasoning with the common language meanings of words may lead to ideas about mathematical structures, reasoning in that manner is not precise enough to be mathematical reasoning.

 

[sysprog]

All of that seems accurate to me, @PeroK and @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. I understand that Mathematics has its own argot; however, I think it's reasonable to insist that mathematicians make the effort, when attempting to convey ideas in English, to avoid roiling the waters with inconsistent usages.

 

[WWGD]

All of that seems accurate to me, @PeroK and @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.

At least in this case you can do away with this problem by stating Rationals are part of the sample space while numbers in the complement of $[0,1]$ on the Real line are not, or the equivalent open set definition I gave. I proposed this before. Edit:I don't know if a similar distinction can be made in each case we have uncountably-many outcomes but makes a distinction between impossible and extremely unlikely , addressing your opposition to the use of probability 0 for both cases. But you do bring up a valid point, one worth addressing.

 

[FactChecker]

I suppose that one could always call it "infinitely small magnitude", but that is a lot of words, even if more correct. For convenience, I will continue to call it "zero". If there is an objection to limits of infinite processes, then that would eliminate all of calculous and beyond.

 

[WWGD]

As I see it it comes down to or is parallel to ,

the fact that a sum with uncountable support will not converge within the Reals.

 

[sysprog]

I suppose that one could always call it "infinitely small magnitude", but that is a lot of words, even if more correct. For convenience, I will continue to call it "zero". If there is an objection to limits of infinite processes, then that would eliminate all of calculous and beyond.

From Johnsons Dictionary:

infinitesimal: infinitely divided​

​

I don't object to the concept of limits, and I think the application of a version of that concept can be used to placate Xeno when he watches Achilles pass the tortoise; similarly, it can be used to appease people like me -- saying that a number tends asymptotically toward zero and so may be treated as zero or taken to be zero is different from baldly asserting that it is actually zero, even while asserting simultaneously that it is non-zero.

 

[PeroK]

From Johnsons Dictionary:

infinitesimal: infinitely divided​

​

I don't object to the concept of limits, and I think the application of a version of that concept can be used to placate Xeno when he watches Achilles pass the tortoise; similarly, it can be used to appease people like me -- saying that a number tends asymptotically toward zero and so may be treated as zero or taken to be zero is different from baldly asserting that it is actually zero, even while asserting simultaneously that it is non-zero.

There is no such thing as a number tending asymptotically to zero. A real number is either zero or it is not. A number is not a process.

Take the following result from elementary real analysis. Assume $x \ge 0$.

If $\forall \ \epsilon > 0, \ x < \epsilon$, then we have that $x = 0$.

$x$ here is not some "different" type of $0$ or some "asymptotic" number or some "infinitesimal". In this case, the number $x$ with these properties is simply $0$.

If you don't understand that, the you need to learn more real analysis.

 

[sysprog]

Assume $x≥0$.​

​

If $∀ ϵ>0, x<ϵ$, then we have that $x=0$.​

This says simply that the only non-negative number less than the infinitesimal is zero. I don't dispute that.

The results of an infinite series of calculations can tend asymptotically to zero or to any other number. I'm using the term 'asymptotically' to mean 'approaching arbitrarily closely', as Hardy and Wright did on page 7 of:

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.​

If we imagine a dart board with unit circle area, covered by the consecutive reals in all their density, and a dart with an infinitesimally sharp point: some will say that if we throw the dart at the board we will never hit a rational because their size on the board is zero; others will say that if we keep throwing the dart forever, we will eventually hit a rational number, because there are rationals in the target area; I say that only if we remove a number from the board can we be absolutely certain that we will never hit that number anywhere on the board. As long as something is possible, given a infinite number of tries, we cannot be certain that it will not eventually happen. Only the impossible can never happen.

 

[PeroK]

If we imagine a dart board with unit circle area, covered by the consecutive reals in all their density, and a dart with an infinitesimally sharp point. Some will say that if we throw the dart at the board we will never hit a rational because their size on the board is zero. Others will say that if we keep throwing the dart forever, we will eventually hit a rational number, because there are rationals in the target area. I say that only if we remove a number from the board can we be absolutely certain that we will never hit that number anywhere on the board. As long as something is possible, given a infinite number of tries, we cannot be certain that it will not eventually happen. Only the impossible can never happen.

This is a purely mathematically construction. There is no such thing as a "unit circle", infinitesimally sharp dart or an infinite number of tries. These things are all in the realm of abstract mathematics; not something you can actually do. In fact, you say yourself "imagine" these things. You can't physically construct them.

Eventually, if nothing else, you have QM to deal with.

This is the same confusion between a mathematical system and something you can actually do.

If you want your dart board, I want a real infinite sine function for the purpose of a physical experiment. Or, better, I want the Weierstrass function, which is continuous everywhere and differentiable nowhere. I want you to draw me the Weierstrass function and then conduct a series of physical experiments using it. These things do not exist in reality and cannot be done.

 

[sysprog]

This is a purely mathematically construction. There is no such thing as a "unit circle", infinitesimally sharp dart or an infinite number of tries. These things are all in the realm of abstract mathematics; not something you can actually do. In fact, you say yourself "imagine" these things. You can't physically construct them.

Eventually, if nothing else, you have QM to deal with.

This is the same confusion between a mathematical system and something you can actually do.

If you want your dart board, I want a real infinite sine function for the purpose of a physical experiment. Or, better, I want the Weierstrass function, which is continuous everywhere and differentiable nowhere. I want you to draw me the Weierstrass function and then conduct a series of physical experiments using it. These things do not exist in reality and cannot be done.

@PeroK does the K in your screen name stand for Kronecker?

 

[PeroK]

@PeroK does the K in your screen name stand for Kronecker?

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.

 

[FactChecker]

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.

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?

 

[sysprog]

@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.

 

[PeroK]

Is there a center of the range? Is there a prior preference that the center of the range must be rational or in some countable set?

The range is defined by your measurement apparatus. One critical argument from the history of QM was that the aperture on a microscope was the defining factor in how accurately you could measure the position of a particle. Crudely, the microscope is picking up light from anywhere inside a cone. What you measure is light that reflected off the particle and entered the microscope. That gives you a position measurement for the particle within some range.

In this simple case, you either see the particle or you don't. There are only two outcomes. Rational numbers and countable sets don't enter into it.

Alternatively, you could detect a particle on a screen. The screen would have a finite number of sensitive cells and the apparatus would record which cell was impacted by a particle. Again, you have a finite number of outcomes.

Every experiment that has ever been conducted can only have produced one of a finite number of results - you can more or less squeeze that to being countable if things are open-ended. Finitely many lines on a ruler, finitely many cells on a screen, finitely many microscopes. Finitely many readings on a voltmeter.

You can do calculations on these outcomes, but there are only ever finitely many. This is one difference between experimental physics and mathematics.