Complexity and large numbers - relevance to Climate Change

HELL, Eternity, James Joyce, Large numbers and Climate Change Policy

A piece of writing that made a great impression on me many years ago in respect of the agonies of Hell (despite not believing in it) and how long one would have to suffer them for i.e. eternity is from James Joyce. See below

Quote by James Joyce: “What must it be, then, to bear the manifold tor...” | Goodreads

Essential to get across the level of suffering for sinners in Hell is the need to convey how long eternity is i.e infinity in a time sense, and I think James Joyce did that extremely well.

However unless you are of a mathematical bent which alas I am you probably won't have tried to estimate how long in years it would take the little bird to achieve its mindbogglingly difficult task. Using the wonders of modern software Excel and the web and some admittedly gross assumptions about "the number of leaves in forests" scales on fish etc one can come up with an estimate as below:

10.1E+324 YEARS that's 10.1 with 324 zeros after it

Now, that's a very, very big number and suffering the tortures of Hell for that long is beyond comprehension. However despite how mind bogglingly large that number is much larger numbers arise in real world problems involving combinations and permutations.

For comparison the age of the Earth is considered to be 4,540,000,000 (4.54 Billion Years) which in scientific notation is 4.54E+9 absolutely minute in comparison to 10.1 E324.

Permutation type problems 

For a quick recap /intro on permutations see 

http://home.avvanta.com/~math/permutations2.html

and to calculate the factorials used in the calculations of permutations especially very large ones see.. 

https://coolconversion.com/math/factorial/

So taking the number above "how long the bird  .. not even first second of eternity" you can find that that is only equivalent to about 178 factorial which contains 325 digits as below 

6235135397241908741680674639927586558582878361231153877729215864681715113389074517106770270394081618113953882713761626619212667044889245663364667862568799843977208366195524719896472807475423880975816549024219691598995799655217698833954618998814300242650289839922492033762220111298560000000000000000000000000000000000000000000

Relevance to Climate Change

Now stories about Hell, little birds and mountains of sand may or may not interest you in the slightest but I assure you they do have a direct relevance to many real world problems. There are according to Wikipedia currently 196 countries in the world. The contention of the mainstream on Climate Changes is all these countries will be negatively impacted to a greater or lesser extent by Global Warming (that's the thrust) and policies  must be adopted to deal with this. 

So if one believes in equality and fairness then the scrupulously right thing to do would be to analyse the needs of all countries and then try and identify an overall global climate change control policy that was in some  sense "the best" for all 196 countries. Now of course there will be quite a few alternatives to consider and compromises to be made - yes, yes you can probably see where I'm going 😀  but perhaps its worth following it through.. 

So now lets just consider not 196 but just three countries A,B,C assuming:

• the effectiveness of the policy is time dependent  i.e no point in padlocking the door after the horse has bolted 
• while its important that all countries implement the policy the impact globally will be different depending on the countries e.g its may be really important China stops increasing its build of coal fired power stations and as soon as possible, only slightly less so that India does so and the impact and urgency of  of Poland doing so less than either. 

To work out what is globally best one can draw up a table like that shown where we assign a value to the importance of  a specific country in implementing the policy and a weighting to the timeliness with which they implement it. We can then consider all possibilities and assign an overall Global benefit to each of the options to work out what it is best to do. In the example implementation of the policy should be BCA:

Possible Order of Implementation	Value of A  (weighting 1)	Value of B  (weighting 3)	Value of  C  ( weighting 2)	Global benefit
ABC	10	5	1	27
ACB	10	1	5	23
BAC	5	10	1	37
BCA	1	10	5	41
CAB	5	1	10	28
CBA	1	5	10	36

Now if we try and scale this up to the 196 countries of the World the table of all permutations to be assessed will have 196 factorial rows - you might try and attempt on your calculator or excel to work this out  and be surprised as to what happens. The answer is according to ref:

https://coolconversion.com/math/factorial/_196_

which has 336 digits in it as shown 

• 50801221108670467625027357853474485583272975249470269829299714310435905748001360370554013724211519571926262867104303166750125208816130922846164797282368228049534890346129156088948368782326391586029134561713739265719468698374988750170217611309676677779711031060019608283576803094698692188285748113739606947612227692134400000000000000000000000000000000000000000000000

This is so vast a number that even if you could identify simple metrics of importance and weightings for timeliness for each country then the amount of time it would take for the the current fastest supercomputer in the world, which ref Wikipedia, has a processing power of :

• 123 PetaFLOPS (Floating Point Operations per Second) or in long hand 123,000,000,000,000,000  

to even generate the number of scenarios to be assessed is almost beyond comprehension. Even for  40 countries generating  the scenarios required is a 40 factorial type calculation and using said supercomputer the time required would be be in the order of  :

•  4,627,032,271,967,450 times the age of the earth 

Ah!..  so thats why "Deep Thought" took so long answering "The Meaning of the Life Universe and Everything 

Concluding remarks 

Before I hear the well merited howls of "Andy that's crazy in practise for policy setting you would never need to be that rigorous - you could broadly categorise countries into a much smaller set of perhaps less than 10 major types making the problem vast simpler an still get pretty good assessment  - YES I fully accept that.  But and it's a very big but, in practise permutation type problems will still occur when trying to develop simple values of merit e.g for the importance of a given country implementing a specific policy as dozens or more conflicting factors will need to be considered.  

My point is that once you understand how quickly these sorts of permutation type problems become totally quantitatively intractable and cannot be solved without an almost inconceivable leap forward in our understanding, to implicitly claim you can solve them as one is in talking about a global climate change policy ( but see below) is akin to implying you are that hypothetical Omnipotent being "God". 

However good the scientists & policy makers supporting the mainstream climate change view are to consider they can develop even one global policy that's best "for the world" is shall we say "overstating it a little bit".. 

To set policy of course you can as they say "cut through the (crap)" by simply taking a view e.g that coal stations are bad so let's just shut them down - period. OR even simpler you can also simplify the task by claiming - "if we don't do something and right now we are going to die so basically fuck you and your over complicated analysis"

In genuinely accept the either of those simplifying approaches may actually be good approaches to take but their justification I think has little to do with science and a lot to do with the type of decisions made by politicians and resulting from referendums but lets not get onto that ..