Jump to content
Objectivism Online Forum

Randomness in formal Mathematical functions?

Rate this topic


Recommended Posts

Is there a proper way to define randomness in a formal mathematical function?  I'm thinking of generating a sequence of numbers yi from an algorithm using randomness i.e. random xi and the number i.

 

Can that be defined as a function as long as the random numbers xi (domain) map to only one (via the function) value yi?

 

 

 

 

 

 

 

Link to comment
Share on other sites

I'm wondering if it mathematically "acceptable" to mathematicians to define a function based on randomness, eg  Yi(x) = Random (Xi | 0<Xi<1) * i

No, that is not a function. Simply saying: well, there's this set A of things, and then there's this set B of things, and each element in A maps to one in B, does not amount to specifying a function.

You'd have to also say which element each element in A maps to. No matter how complex a function is, you can determine what it maps each element to. Whey you're saying "to a random element", you haven't told me which it is. In fact you told me that it's not determined which it is, it could by any of them.

So no, you can't define randomness with a function, that's a contradiction in terms.

Edited by Nicky
Link to comment
Share on other sites

What you want is, a random member of the set of all functions (from some domain to some range). It maps each value in the domain to a random value in the range.

Say the domain and range are both { 0, 1 }.

Then the set of functions is:

  • 0 -> 0, 1 -> 0
  • 0 -> 0, 1 -> 1
  • 0 -> 1, 1 -> 0
  • 0 -> 1, 1 -> 1

A random function in that context is a random one of those four.

 

As Nicky pointed out, you can't say that randomness is within a function. But you can randomly choose your function, which is then deterministic.

Link to comment
Share on other sites

Can that be defined as a function as long as the random numbers xi (domain) map to only one (via the function) value yi?

You could informally say that you have the set of all rational numbers then a choice function selects an arbitrary number from the set. The concept you're looking for is arbitrary. Or perhaps you want the axiom of choice - there is always a way to pick a number from a non-empty set. By the axiom of choice, completely arbitrary choices are acceptable.

 

Take f(x) = 2x. For any arbitrary value x, the output is always twice x. If you mean a specific output, well, just say f(x) = 2. For any arbitrary value x, the output is always 2.

 

If you're asking if you can make a function to pick random numbers, that doesn't make sense, like Nicky suggests. As soon as you work out a method (define a choice function beyond just arbitrary), the number is neither an arbitrary selection nor a random selection.

Link to comment
Share on other sites

What about building a set of ordered pairs

 

{xi, yi}

 

eg

 

{{x1, y1}, {x2, y2}, {x3, y3}, .... {xn, yn}...}

 

this set, can it be called a function from domain (xs) to range (ys) as long as (as you can see by definition) each xi maps to only one yi  

 

Now I'm contemplating generation of this fixed set using an algorithm, which involves randomness (axiom of choice I suppose).  Can that set of ordered pairs be formally accepted as a function?

Edited by StrictlyLogical
Link to comment
Share on other sites

What about building a set of ordered pairs

 

{xi, yi}

 

eg

 

{{x1, y1}, {x2, y2}, {x3, y3}, .... {xn, yn}...}

 

this set, can it be called a function from domain (xs) to range (ys) as long as (as you can see by definition) each xi maps to only one yi  

 

Now I'm contemplating generation of this fixed set using an algorithm, which involves randomness (axiom of choice I suppose).  Can that set of ordered pairs be formally accepted as a function?

Sure. I don't know if picking the numbers yourself will make them random though. It's not like there's a mechanism in our heads that allows us to press a switch and start generating random sequences. More than likely you'll just follow a pattern.

Even if you pick by chance, chance does not equal random. You could stumble on the sequence 1,2,3,...,10 by chance too, for instance. That sequence doesn't fit any definition of randomness.

Link to comment
Share on other sites

{{x1, y1}, {x2, y2}, {x3, y3}, .... {xn, yn}...}

 

this set, can it be called a function from domain (xs) to range (ys) as long as (as you can see by definition) each xi maps to only one yi

Any set of ordered pairs can represent a function (as long as there are no duplicated xi), but you are missing a few parameters. What are xi and yi chosen from? How many pairs are there? Do you really want to be randomly generating functions with "holes" in them where no xi was generated?

 

One example of random generation of functions is noise in digital audio. The domain is a set of times, and the range is -amplitude to amplitude.

 

 

You could stumble on the sequence 1,2,3,...,10 by chance too, for instance. That sequence doesn't fit any definition of randomness.

No sequence is inherently random. But a perfect 1-10 can be just as random in origin as [1,7,6,3...] even if it doesn't seem as "random".

Link to comment
Share on other sites

Imagine building sets defined as Sj

 

S1 = {{{x1, y1}}

S2 = {{x1, y1}, {x2, y2}}

.

.

.

 

Sj = {{x1, y1}, {x2, y2}, ... {xj, yj}}

 

Where xk = Random Real Number in [0,1] such that it does not equal any of x0, x1, ... xk-1. (do I need the axiom of choice for this to be a valid definition of xk?)

 

We also set Yi = i.

 

So really Y1 = 1, Y2 = 2, y3 = 3 etc.

Edited by StrictlyLogical
Link to comment
Share on other sites

No sequence is inherently random. But a perfect 1-10 can be just as random in origin as [1,7,6,3...] even if it doesn't seem as "random".

What definition are you using for random?

While I was mainly thinking of Kolmogorov's definition (as per wikipedia, it reads: "a string is random if it has no description shorter than itself via a universal Turing machine) when I wrote it, I'm confident that my statement (that 1,2,3,...,10 is not a random sequence) would be true as per the definition of any mathematician on the face of the Earth.

Edited by Nicky
Link to comment
Share on other sites

If you get that, you know what I mean by 'random'. No need to turn it into a debate.

I know what you mean, I just don't know why you're saying it. It has nothing to do with my post, which stated that 1,2,3,...,10 is not a random sequence.

You might as well have said: "I have an apple and an orange in a box, and I reached in and picked out the apple at random. Therefor anyone who says that apples don't fit the definition of a random sequence in Mathematics are wrong. Because I picked it at random, so it's a random apple.".

No, it's not. How you picked it has nothing to do with anything. Randomness in Mathematics is defined in terms of complexity. 1,2,3, ..., 10 is not that complex. Meanwhile, 3,5,2,8,9,4,7,6,10,1, is. That's why I could use the dots to shorten the description of the first, but not the second.

Edited by Nicky
Link to comment
Share on other sites

This topic is akin to a discussion I had with my nephew (who is studying how computers work in a much deeper sense than I've ever encountered) regarding a computer's ability to generate a "random number". The key lies in the algorithm. No matter what inputs selected, and put into a formula, the same inputs would generate the same result. The inputs are selected such that the likelihood of being the same between any two machines at any one time is unlikely, the result of the formula simply giving the impression or illusion of a "random number".

 

While I enjoy aspects of mathematics, the idea of a formula, based on number and operations (as I grasp them), generating a truly random number thus runs contrary to my intuitions on the matter.  

 

That aside, the precious little I know about the "axiom of choice", I might ask, could an 'axiom of choice' set be comprised of random numbers? If so, how might it be delineated? And if delineated, could it be demonstrated as random? After all, once you have an 'ordered' set of inputs, if you will, by what criteria could it be categorized as "random"? Is this thought extension becoming circular?

Link to comment
Share on other sites

... the result of the formula simply giving the impression or illusion of a "random number".

"Pseudorandom". For specialized uses, where it really matters, one can get hardware that will do things like sample the "noise" from some physical phenomena, and use that.

Link to comment
Share on other sites

Interesting. Yet simply augment the number of variables used to generate the inputs. Include such things as, longitude/latitude of the location of the computer, the time, current outside as well as inside temperature, the current wind-speed and direction in some form of degree capture, . . . add the number of planes in the sky, if an automobile (or number of them) is passing by on the street at the moment. . .

 

Could such a variety of inputs qualify as an "axiom of choice" set?

Link to comment
Share on other sites

That aside, the precious little I know about the "axiom of choice", I might ask, could an 'axiom of choice' set be comprised of random numbers? If so, how might it be delineated? And if delineated, could it be demonstrated as random? After all, once you have an 'ordered' set of inputs, if you will, by what criteria could it be categorized as "random"? Is this thought extension becoming circular?

 

If you already have a number written down, it is not random to you. But if I know that you are about to give me a number in the range 0-1 and I have no way of knowing anything about it except for that (harder than it sounds), then it is random to me.

 

We can extend this to the "axiom of choice" by taking this set, the reals 0-1, and putting two of it into a set.

Then we have our set of sets, { 0-1, 0-1 }, and can produce one element of each; for example, (0.5, 0.4). That would be a random pair of reals 0-1.

Now say we want a function whose domain is 3 and 7. Let the pair's first member be f(3) and the second be f(7). That would be a randomly chosen function.

 

Note that every time we chose something randomly we do so from some set; in this case, the set of functions whose domain is { 3, 7 } and whose range is 0-1.

Link to comment
Share on other sites

The difference as I see it, rowsdower, the number I have written down is already part of my "axiom of choice" set.

 

By you querying myself, softwareNerd, Nicky, StrictlyLogical, and Eiuol; or are you simply extending the number of variables used to create the "axiom of choice".

 

Numbers spawned by our volitional consciousness is random in my book. Numbers spawned by temperature, cars passing on the street, and the current dollar volume generated by wall street in the last X # of seconds are products of deterministic processes conjoined with volitional consciousness'. In a computational sense. Are you going to include the last number pressed on the number pad, or the top row, or some other combination of key-presses. Answering this question puts you at dealing consciousness as a process of identification vs. existence as a process of identity. Ironically, all I can say at this point is "Choose."

Edited by dream_weaver
Link to comment
Share on other sites

Perhaps we should agree on the definition of randomness, and then continue talking about it. Otherwise, I doubt we're talking about the same thing. Random does not equal "chosen".

By the definition of randomness I posted, the axiom of choice doesn't mean you can select random numbers. It just means you can select from a set of indistinguishable elements (I think, I'm not THAT familiar with it). For instance, just because you have an infinite amount of socks that all look the same, doesn't mean you can't select a set of them.

On the other hand, numbers aren't indistinguishable, so you don't really need the AC to accept the possibility of selecting some of them. I don't understand why the AC would allow you to select "randomly" (meaning there's an equal probability of you selecting any numbers). Where does the axiom of choice guarantee that a human being, selecting from an infinite set of numbers, has the same probability of selecting 7 or 108714801.12489048(293)? For one, that's obviously not true. Second, even if it could be true, it doesn't seem like it would be a consequence of the AC.

Link to comment
Share on other sites

When I say "random", I mean "uniformly random" such that the probability is a constant within the range you are choosing from.

 

Now I am about to spawn an integer 1-10 from my volitional consciousness.

It's gonna be 3.

Now that I've told you this, it's not random; it has a higher chance of being 3 than being 7. (Do you trust me more than 10%?)

 

Oh, and: "3".

Link to comment
Share on other sites

So there is nothing equivalent (in formal mathematics) to selecting arbitrarily one value of a range or a set ?

 

 

What about a standard definition like this

 

Let Xi be an element of [0,1]

 

("Be/is an element of" usually has the membership symbol which I do not know how to show here)

 

If no mention is made of the particular Xi, is it simply arbitrary?  I.e. we should not assume pattern or absence thereof, it simply is a element of the set which here is an interval.

Edited by StrictlyLogical
Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Reply to this topic...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
  • Recently Browsing   0 members

    • No registered users viewing this page.
×
×
  • Create New...