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

[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.

 

[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.