Tuesday, June 04, 2013

Infinity and beyond


In my last blog on More Monty I explored a few ideas around the meaning of probability and what that problem was telling us.  In doing so I used a few examples of infinity.  These on the surface look ok in their simplified manner and made the point well.  But the treatment of infinities is hard.  Firstly, it seems to be recognised that infinity is a concept and not a number and therefore the application of standard operators is not logical and leads one to some difficult situations.  I picked my words carefully in the last blog to avoid such issues.

But infinities are hard to deal with and it seems even more complicated in what they mean when it comes to probability.  For example, if there were infinite amount of doors (Monty Hall problem) then what would be the odds of picking the right door?

On the surface one would say 1/infinity.  Makes sense.  If you had 3 doors the odds would be 1/3.  4 doors 1/4 and so on.  But 1/infinity?  Its a number as close to 0 as can be without being 0.  But of course while we understand this concept it is also at the same time nonsense.  If we then start to look at infinities in probability then it would appear to get more tenuous.

If use of infinity with numerical operators in nonsense then calculation of odds with infinities is also nonsense.  1-infinity?  So the calculation of odds must also be nonsensical using infinities.  So how do we calculate odds in what could be an infinite universe?  It again may tell us something about probability.  Probability is inductive reasoning.  It tells us the outcome of past events and if we use that knowledge we can start to reduce randomness that would otherwise apply.

Back to dice.  If we have a 6 sided dice and roll it over a large sample range we would (assuming a perfect dice and proper rolling) get a relatively even spread of outcomes for each number on the dice.   True randomness will ensure that the outcome over a big enough period is even.  So we can use this knowledge how?  Well, an obvious statement is that when predicting the outcome of the dice we have one important piece of knowledge.  We know that we need to pick 1 to 6.  Obvious right but often overlooked.  If we had a random sized dice and we did not know its size then we would have a difficulty performing any form of reliable prediction using inductive reasoning.  Our performance outcome at a normal 6 sided dice will be on average 1/6 at picking the next number.  If the dice has random numbers then how would we perform?  How would one even calculate the odds?

The problem demonstrates that probability is all to do with knowledge.  For a better than random outcome from a random event we need to have some form of knowledge to reduce the odds from random to something better than random.  The odds of picking the next number on an x sided dice is not 1/x.  It is 1/x IF we know x.  Why?  Inductive reasoning is about patterns.  We look for past patterns with a view to improving out chances of avoiding randomness and uncertainty.  It is this reduction in uncertainty which has enabled us to evolve and survive as a race.  Inductive reasoning is not of course reliable and is a fallacy but in human evolution, while not perfect, has given us an edge over other species and indeed other members of our race in early days which no doubt enable evolution to lead us to where we are now. All humans reason using inductive reasoning with a view to reducing uncertainty.  Inductive reasoning over randomness is enough of an edge.

So what about infinities then?  The issue seems to be that odds are only calculable if we somehow have limits.  The dice shows this.  We can perform better than total uncertainty if we have a limit (knowledge of the number of sides of the dice and numbers on it).  Unbounded odds are simply not calculable and therefore performance will be random.  Probability is therefore a peculiar thing.  We can talk about the odds of a number 1 being rolled on a 6 sided dice as if it were linked to reality.  It is of course in a peculiar way.  It is simply saying that randomness dictates that if you know that it has 6 sides then you can have better chances of guessing the next number than if you did not know it had six sides.  These two concepts are key.  In infinite options there can be no meaningful calculation of odds.  Only when knowledge of boundaries (which infinity does not have in terms of upper limits) can probability become useful.

This might go some way to helping understand whether the universe is deterministic or not.  If determinism is true then all events are predictable (in theory) even if not in practice.  Therefore the probability of any event happening is finite and 1.  The debate is whether this in itself removes things like freewill.  If whatever event x is about to happen with a probability of 1 (determinism states it could not be otherwise) then we can see that to find this useful we would need two conditions to be true.  The first is that the event is certain (1/1) and the second is that we have knowledge of the boundaries to make any prediction better than random.  Of course this is where the irony kicks in.  If everything that occurs has a probability of 1/1 then we live in a world which is entirely predictable but at the same time due to its potentially infinite size and potential events totally unknowable.  So even certainty is useless in an unbounded universe even when determinism kicks in.  We have better chances with enormous but bounded odds and some knowledge than we do with certain outcomes and no knowledge.


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