# B What's the meaning of "random" in Mathematics?

#### fbs7

Physics, Economists, Biologists, Astronomers and my brother all love the word "Random", as that allows allows them to get out of clockwork processes and allow for variations due to unknowns or whatever else.

But, how does a Mathematician reconcile itself with the idea of random? There's no axiom for "choice", no function for "random value", no explanation of what "chance" is.

Meanwhile I heard that someone spent 500 pages of logic to prove that 1+1 = 2 (or something like that), so how is it possible that mathematicians and logicians spend all that trouble to prove some really basic stuff, while at the same time just accept theories around probabilities and random numbers without (as least from my untrained point of view) an axiomatic foundation for choice?

#### fresh_42

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Physics, Economists, Biologists, Astronomers and my brother all love the word "Random", as that allows allows them to get out of clockwork processes and allow for variations due to unknowns or whatever else.

But, how does a Mathematician reconcile itself with the idea of random? There's no axiom for "choice", no function for "random value", no explanation of what "chance" is.

Meanwhile I heard that someone spent 500 pages of logic to prove that 1+1 = 2 (or something like that), so how is it possible that mathematicians and logicians spend all that trouble to prove some really basic stuff ...
As a programmer you should know that $1+1=0$, so basic stuff is quite relative here.
... while at the same time just accept theories around probabilities and random numbers without (as least from my untrained point of view) an axiomatic foundation for choice?
Random are the possible values of a measurable function on a probability space $(\Omega,\Sigma,P)$ to a measure space $(\Omega',\Sigma')$.

Axiomatic enough?

#### fbs7

As a programmer you should know that $1+1=0$, so basic stuff is quite relative here.

Random are the possible values of a measurable function on a probability space $(\Omega,\Sigma,P)$ to a measure space $(\Omega',\Sigma')$.

Axiomatic enough?

Well, f(x) = 1/(1+x) has possible values on [0,1] on a space defined by [0,∞]... how is that different than random?

Also, if that "measurable function" is a random value, then can someone write a formula for a truly random function (as opposed to pseudo-random)? How would it define a value, if the value is random?

I know one can write a formula to express probability, and that's fine. Say, probably = 1/(2+x) or something like that. But does that doesn't define a random value, it just gives an average, over a large enough number of samples, for the sampled values to fall within a range.

Like, probability of a duck to be male, if you count 1 million ducks, is I guess 50%. But then the "choice" that the duck is male or female is not random, it's defined by a well-known biological process.

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#### fresh_42

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Well, f(x) = 1/(1+x) has possible values on [0,1] on a space defined by [0,∞]... how is that different than random?
You need a bit more than just the intervals: sigma algebras (the measures) on $[0,1]$ and $[0,\infty)$, plus a probability measure to get the usual properties on $[0,1]$: $P:\Sigma \longrightarrow [0,1], P(\emptyset)=0, P(\Omega)=1, P(\dot{\cup} A_i)=\sum P(A_i)$.
Also, if that "measurable function" is a random value, then what is the function?
The function isn't random, it's a random variable $X\, : \,(\Omega, \Sigma ,P) \longrightarrow (\Omega',\Sigma')\,.$
How would it define a value, if the value is random?
$P$ by relating a certain real value to each element of $\Sigma \subseteq \mathbb{P}(\Omega)$. The values themselves aren't random, they are determined. They represent a probability, and this is fixed by the function, aka random variable. Randomness is just the common word for probability. It is a confusing word, as the probability value itself is not random.

You can handle the subject as ordinary real analysis. Randomness is it's interpretation, not the method.

#### fbs7

Oh... randomness is interpretation...

So a "random" variable is really just another variable, just like "time" is just a variable without anything different than say a "mass" variable. Our interpretation of "time" makes it special, but that meaning of something flowing over.. whatever... is not mathematical, it's just human interpretation.

Likewise, the "random" nature of a variable is just human interpretation, right?

Does that mean that, because "random" is a semantic thing that we use to interpret the variable, that "random" is not really a mathematical concept (any more than "time" is)? After all, a variable is just a variable, and mathematically nothing makes them any different from one another?

#### fresh_42

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Well, I would agree here. Of course the setup is made for probabilistic calculations and to grasp random processes. But the machinery is real analysis, although on special domains and with certain restrictions. The art of all is, and this cannot be solved by the machinery, to determine which specific setup describes a given random outcome. You can calculate your motions relativistic or classic, the equations cannot decide for you which one to use. I think the separation of a given experiment from the calculus necessary to describe it is actually an achievement which wasn't obvious from the start. It helps to avoid confusion and to encapsulate the discussion. And of course we won't need to talk about sigma algebras, if - as at school - some colored balls are selected from some pots. But in the end it led to a false understanding of randomness by equaling it with uncertainty. It's a bit as in physics, where old metaphors from a century ago are still in the heads of people.

#### fbs7

Once again thank you so much for the kind and patient explanations to this old bloke here.

I see that many people here cherish teaching and explaining as the highest form of discourse, and for my part I try to understand as much as I can, to not waste even a word from such wonderful mentoring -- for which you have my dear thanks!

I guess that many people here are teachers in their areas, is that a reasonable assumption?

#### fresh_42

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I guess that many people here are teachers in their areas, is that a reasonable assumption?
We have all kind of members, including teachers and current, future or former professors. Many others just have studied the fields they answer in. So yes, the general level of competence is significantly higher than on other internet forums. That does not mean that people wouldn't mistakes, at least they happen to me, usually due to bad reading, but those are normally quickly corrected. Me, too, finds it refreshing to read about things I have no expertise in, but can be sure it's on a scientific level.

#### Mark44

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As a programmer you should know that 1+1=0, so basic stuff is quite relative here.
In a one-bit adder, this is true, but only because we've lost the carry digit. In any other context, 1 + 1 = 2, in any base higher than 2, and 1 + 1 = 10, in base-2.

#### fresh_42

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In a one-bit adder, this is true, but only because we've lost the carry digit. In any other context, 1 + 1 = 2, in any base higher than 2, and 1 + 1 = 10, in base-2.
I thought of a Boolean variable. In many languages "IF X = true" can also be written "IF X = 1". In any case, I wanted to stress that even $1+1=2$ isn't automatically a given truth.

#### Mark44

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I thought of a Boolean variable. In many languages "IF X = true" can also be written "IF X = 1". In any case, I wanted to stress that even $1+1=2$ isn't automatically a given truth.
But strictly speaking, the operation on Boolean variables that corresponds to addition is OR, in which case true OR true = true. Using 1 for true, we have 1 + 1 = 1. If we take this further to include AND, we would have true AND true = true, as well. In neither case do we have 1 + 1 = 0.

#### fresh_42

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You're right. My only excuse is: far too many COBOL and RPG switches ...

#### FactChecker

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Oh... randomness is interpretation...

So a "random" variable is really just another variable, just like "time" is just a variable without anything different than say a "mass" variable. Our interpretation of "time" makes it special, but that meaning of something flowing over.. whatever... is not mathematical, it's just human interpretation.
I wouldn't say that. Whether a variable is truely random or is determined by unknown factors is a physics concern, not a mathematical concern. Mathematically, once a variable is assumed to be random, it is treated rigorously as such. It has a probability distribution, etc. The subject is well established and there are many good texts on probability theory.

#### fbs7

I wouldn't say that. Whether a variable is truely random or is determined by unknown factors is a physics concern, not a mathematical concern. Mathematically, once a variable is assumed to be random, it is treated rigorously as such. It has a probability distribution, etc. The subject is well established and there are many good texts on probability theory.
Hmm... my mind screws were more in place with the idea that "random" is an interpretation thing.... :-(

If I say x ∈ X, how do I know if this is a random variable or not? What makes this set X different than all other sets that can be defined in mathematics, that makes it a "set of random values" (or whatever the proper terminology for that), if not human semantic that attaches the word "random" to X?

#### FactChecker

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My point is that mathematics does not know or care why something has been identified as random, as long as it has been and a probability distribution of its values is defined. It may be "random" due to a human decision to rely on probabilities rather than attempt to figure out all the necessary physics to make it deterministic. We treat a coin toss as random because the idea of making it deterministic is inconcievable. There are innumerable examples like that. The location of raindrops on a square foot of ground would be random for all practical purposes.

#### StoneTemplePython

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As is often the case, I look to Feller volume 1 for inspiration.

Feller said:
A function defined on a sample space is called a random variable... The term random variable is somewhat confusing; random function would be more appropriate (the independent variable being a point in the sample space, that is, the outcome of an experiment).
Oh... randomness is interpretation...
So a "random" variable is really just another variable, just like "time" is just a variable without anything different than say a "mass" variable...
Likewise, the "random" nature of a variable is just human interpretation, right?
The issue is: I don't think there is a satisfying B level answer to this thread.

#### Mark44

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You're right. My only excuse is: far too many COBOL and RPG switches ...
There is one Boolean operation that works like what you described: XOR. true XOR true = false. In symbols, 1 ⊕ 1 = 0.

#### fresh_42

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So a "random" variable is really just another variable, just like "time" is just a variable without anything different than say a "mass" variable.
I wouldn't say that.
I do agree. My point of view is
In general, I think the connection between probability theory and the measure theory is typically underemphasised in introductory courses on probability (at least for non-mathematicians). Also, just for OP's reference: https://en.wikipedia.org/wiki/Measure_(mathematics)
Once we setup the mathematical framework, we are in the middle of measure theory and the word random is obsolete. Distributions and other probability specific quantities are merely more properties and well defined functions.

To transport the term randomness from the experiment into the math does in my opinion more harm than good. It is neither necessary nor does it provide additional insights. Randomness is coded by the choice of specific measure spaces, a probability measure, resp. a distribution function. Once we arrived there, it will only be used to interpret the results in terms of the experiment again, but cannot affect calculations itself. Within mathematics, randomness, which we discuss here to be axiomatically defined, is in my opinion simply a synonym for a specific and deterministic calculus. Thus randomness is left behind as a property of the experiment only, and in this regard as a property of a variable same as time or mass would be.

#### FactChecker

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I do agree. My point of view is

Once we setup the mathematical framework, we are in the middle of measure theory and the word random is obsolete. Distributions and other probability specific quantities are merely more properties and well defined functions.
I see your point. But general measure theory does not include the requirement that the total measure remain 1. That is the essential property that allows one to interpret it as a probability of a random variable. The scaling of Bayes' rule is to retain it as a probability.

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