Randomness and discreteness

[picnic]

For what it's worth, as an undergraduate I couldn't fit most of the courses I wanted into my schedule (I was overloading courses every semester to graduate in 3 rather than for years and had already made the stupid decision to add a major in ancient Greek & Latin and a minor, so when I discovered mathematics was awesome I was already going to have something like two full semester's worth of elective credits that didn't count towards graduation). I had never taken trig, precalculus, and didn't even know what calculus was, but a course in statistics and symbolic logic had made me think there might be aspects of math I would enjoy. As I had liked statistics, I figured reading more advanced sources on statistics would be a perfect place to start. I went to the library and got some books.
I became familiar with vectors before I knew what calculus was even about, and quickly encountered terms that were not explained because I was expected to know them. So, for example, when I first read the term "2nd derivative", not having a clue what that meant, I looked it up and found that it meant taking the derivative of the 1st derivative. Of course, I had no clue what the "1st derivative" was.
Basically, I started learning mathematics at one level, proceeded backwards until I reached the level I could understand, and then retrace my steps.
Worse still, I discovered that the standard mathematics curriculum for pre-college and undergraduate students is awful. Take calculus:
...

That's interesting that you didn't know you liked math and didn't apparently take much math until you were in college. Then you immersed yourself entirely for life. Love at first sight I guess.

I have always been artistically-inclined, so I needed to be able to visualize problems or simpler analogues of problems. I was able to finish the first two years of math in one year at college, but I started taking the 3rd year of math in my sophomore year and that was where I fizzled. It was called AMa 95. I think I got an A in the class, but I had no idea what we were doing most of the time. I remember "complex contour integration", "branch cuts", blah, blah, blah. It was just monkey-see-monkey-do for me. I was also struggling with depression, so that was a hard year. I gave up my hope of being a physicist and switched to computer science so that I could just get the hell out of that place and get a job. I guess they still teach the class under a different number. I imagine it is just as awful as it ever was. I can still remember sitting all weekend long working on some problem for page after page only to discover that I made a sign error somewhere near the beginning. LOL That really took the fun out of science and math.

 

[Robert.Evans]

To me, randomness is very important, because randomness appears in quantum mechanics, and that has some implications about reality and religion IMO.

What is randomness? It seems to me that randomness is easy to define when there are discrete possibilities, and maybe it is difficult or impossible to define when there are not discrete possibilities?

Imagine flipping a coin 10 times. Imagine reality is every possible sequence of 10 coin tosses (kind of like a multiverse). If our observation is limited to only ONE possible sequence, how do we know if this sequence is a random variable? It seems to me that we can't be certain, but we can make a good guess. We know that only one possible sequence is all heads, but many possible sequences are 50% heads.

If the possibilities are discrete, then the multiverse of all possibilities has a finite number of threads of time. When the possibilities are not discrete, then the number becomes infinite. It seems to me that everything works better if reality is discrete.

I hope I explained my thoughts. I know random variables can be continuous, but I think they become less meaningful. It is harder to think about a multiverse of all possible threads of time, and then we are left with a mystery of why one observation happens instead of all the other observations.

Any thoughts?

What if there is no such thing as randomness

 

[Rick O'Shez]

What if there is no such thing as randomness

The national lottery machine wouldn't work.

 

[picnic]

What if there is no such thing as randomness

That is how it seems to me. I guess it all depends on what you mean by "randomness".
- past events are called "random" if we can't explain them fully
- uncertain future events are called "random" if we can't predict them fully
- apparently there are mathematical definitions of "random" for specific applications
- entropy and quantum mechanics have "degrees of freedom", "uncertainty", "probability waves"

It seems to me that entropy and quantum mechanics is where "random" might have objective meaning. I know that @LegionOnomaMoi mentioned that "random" isn't the correct word to use in quantum mechanics. I suppose that depends again on what we mean by "randomness".

 

[Robert.Evans]

That is how it seems to me. I guess it all depends on what you mean by "randomness".
- past events are called "random" if we can't explain them fully
- uncertain future events are called "random" if we can't predict them fully
- apparently there are mathematical definitions of "random" for specific applications
- entropy and quantum mechanics have "degrees of freedom", "uncertainty", "probability waves"

It seems to me that entropy and quantum mechanics is where "random" might have objective meaning. I know that @LegionOnomaMoi mentioned that "random" isn't the correct word to use in quantum mechanics. I suppose that depends again on what we mean by "randomness".

It's certainly a word that is used to explain things, and we have to have such things to understand what we see. But we also use luck and chance and we might say they don't exist either. I suppose if you can't prove something else is there though you are stuck with it/ It is an explanation that works, like Newton's laws at one time, till they realised there was more to it than that.
Personally I think everything follows something that has happened before, stimulus etc, but still nevertheless a term I would use.

 

[picnic]

If you wish to consider a multiverse of branching universes independently of quantum physics, this is certainly possible, but one must ask "what for?" We can imagine that there are exactly ten parallel universes, or that branch universes are formed in triplets every 365 days, etc. But we have no reason to do this. And if such a proposal is intended to be a thought-experiment or conceptual exercise, I confess I don't really understand the merit.

The branches would represent every possible observation of a particle in quantum mechanics. Imagine the collapse of a probability wave into an observation. Instead of a continuous probability wave, imagine a finite number of "possibility points" that are denser in areas where the probability wave has a crest. Imagine time is discrete like that also, so that observations are only possible at particular points in time. With this model, it seems that a finite multiverse might exist representing all the possible branches of these observations.

A universe at a moment in time is like a node on this finite tree. A universe has conservation laws, but maybe there are similar laws that restrict the entire tree of universes. Maybe some of the weird things about entropy and quantum mechanics would seem less weird when viewed from this decision tree perspective. Maybe we could make some predictions from this perspective that could actually be tested from our own limited perspective.

I don't know if this approach would work with continuous possibilities. It is harder for me to imagine it visually and a computer would probably have more trouble simulating this approach.

I just have this intuitive feeling that this might be a useful way of thinking about reality. Also I saw something similar in a vision or dream where I felt that I was observing this reality from the outside. It's like I forgot what I saw except for vague shadows. (I know that sounds silly, but this is a religious forum LOL)

It isn't intuitive. But then neither is probability, even when it concerns random (finite) trees. I wish I could hand you:
Random Trees: Interplay Between Combinatorics and Probability

Probably that book is too advanced for me, but I would like to learn more about statistics. I enjoyed the one statistics class I took in college. Statistics is the essence of science, and it is really tricky to do it correctly (IMO). The math is not so hard like differential equations, but it requires careful reasoning. I recently read this overview book on statistics that made the topic interesting ( http://www.goodreads.com/book/show/5115533-statistics )

The "a very short introduction" books are about my level at this stage in my life

 

[Etritonakin]

Why think about places to which you can never go?

 

[LegionOnomaMoi]

The branches would represent every possible observation of a particle in quantum mechanics.

In that case they can't be discrete.

Instead of a continuous probability wave, imagine a finite number of "possibility points" that are denser in areas where the probability wave has a crest.

Probabilities are the mod square of the wave function. They aren't really different from probability distributions more generally, except for how they are derived. Also, discrete sets can be infinitely dense (consider the rational numbers).

Imagine time is discrete like that also, so that observations are only possible at particular points in time.

Points take up no space. Infinitely many of them can be contained in a space without any length, area, volume, and the generalizations of these to higher dimensions.

With this model, it seems that a finite multiverse might exist representing all the possible branches of these observations.

It would guarantee that the model cannot even marginally approximate the possible outcomes of quantum mechanics.

A universe has conservation laws, but maybe there are similar laws that restrict the entire tree of universes.

Maybe. But we can't find them via the set of possible outcomes in quantum mechanics.

 

[picnic]

It would guarantee that the model cannot even marginally approximate the possible outcomes of quantum mechanics.

That's fine, but of course we can't know it is nothing until we know what it is. I was hoping that my description might remind somebody of a more developed form of this idea that I could read about.

Thanks for the replies, @LegionOnomaMoi . Much of what you said is over my head, but I learned a few things, and I don't want to beat this thread to death.

 

[picnic]

Why think about places to which you can never go?

Maybe we are in all these places simultaneously? Or maybe we are in none of these places?

I like the idea emotionally, because it means that all possibilities happen. There is no road not taken.

 

[Etritonakin]

Maybe we are in all these places simultaneously? Or maybe we are in none of these places?

I like the idea emotionally, because it means that all possibilities happen. There is no road not taken.

Even if it is not the case, thinking in those terms can help people make or choose the best road.