Monday, February 15, 2010

Math part IV: Chaos Theory, Statistic and the Rest

In my last post I said that what I had presented about math could deal with known and unknown quantities, shape, position, size and mouvement of a thing in just about any dimension but that it could not yet deal with “undeterministic process”. That was slightly false, Calculus already hosted quite a lot of undeterministic equations. In fact, some equationi system have an infinite number of solution and as such are undeterministic. Does that means math can’t say anything about them? Of course not, it just mean that while math can give you the shape of the solutions it cannot give you a clear solution without more information. There are also equations that are very surprising and that gave rise to a field of math known as chaos theory.

You see we found that some system of equation in which 3 or more variable are dependant on each other where very (but I do mean very) sensible to initial condition. They are not stricly speaking undeterministic but small variation in initial condition can result in widely different answers. Throught the study of these equations and phenomenum we can study things we though purely random before, such as the weather and turbulance. In fact, Chaos theory explains easilly why we can’t pretty the weather accuratly over large period of time. As it happen the weather is a chaotic system where the influencing factor depend on each other and since we cannot known perfectly any of those factor at any point in time (there are always measurement error) the little errors make the solutive diverge more or less rapidly. Of course the less accurate your mesurement the faster it diverge and you find yourself away from your predictions.

Which will bring me to statistics; you know the lies, damn lies and statistics. Strangely people believe that statistics are falsehood and should not be trusted. The thing is that statistics are very easy to missuse with people that do not understand statistics. Take for exemple your usual pre-election polls. You have daily polls telling you who people want to vote for, with usually a 3-4% margin of error 19/20. How many of you understand what that means? The 3-4% margin of error most people guess means that each number might change from up to 3-4%, which is mostly correct. But what does that 19/20 actually means? Well, that what is called a 95% confidence level, a slightly low confidance level (for exemple, in science below a 99.9% we usually can’t say that we have measured a quantity, we only detect it’s presence), it means that depending on your sample size your pourcentage are spread out over a larger pourcentage that if you had a stronger measurement. If you couple this with the selection bias in phone surveys, relatively small number (you shouldn’t trust a survey with less then 1000 participant) and misleading questions you can explain strange discrepty between statistics and reality.

But enough explaining why statistic can be made wrong, lets talk about what statistics are. Well statistic is the study of probability and distribution, that means its studies the odd of something happening give some (or no) preexisting knowledge. It also give tools to understand how random (or semi random) event combine to provide a range of possibile result and the probability of theses result occuring. For exemple, lets say you threw 35 coins up in the air, what would be the odd of getting all of them showing face (~1/35 000 000 000) or any combinaison of the side you might want. You can also determine if two things might be related (a correlation) to each other and how strongly it is. You need statistics to understand quantum physics, thermodynamics and all that. For exemple, it’s impossible to follow the behavior of all particules in a gaz, however thanks to the distribution of velocity we can determine a lot (if not all) of the behavior of the gaz, all without ever knowning much about the details of the gaz particules.

Now, about the rest of math you get theory and system to deal with how groups are formed, how number and function are related to each other (group theory), theory to deal with order (order theory), how space behave (that topology), study of fractional dimension (fractal geometry), theory about how measurement are related to reality (measurement theory) and idea on how to analyse and represent system. What I presented in those four post barely scratch the surface of mathematics, I’ve failled to talk about vectors, complex analysist and trigonometry just to name those few concept but already you should see that math can be used to represent and characterize everything in reality and even much more. It use the rest of science to anchor itself in the universe but it allows us to explore even more complexe space and set up.

To finish by going back to the initial conversation with Yofed, what does blue+yellow give? Green, to a human, other creature would see other colors, or no color at all and the two photons (i.e remain blue and yellow) would remain independent until caught by a photodetector. Explaining love mathematically might sound amazingly dry and boring to some of you but here is the language equivalent of it (mostly because I do not have the neurochemicial knowledge to write the equation… not to mention that I’m pretty sure it’s way beyond our current understanding of the brain). So here I go, you have an initial release of neurochemical, from the stimulus of smelling, seeing, hearing basically relating to the other person, depending on the current chemestry of your brain (how the recent past was for you) this reaction might find itself amplyfied triggering neuron firing and identifying this person as someone interesting. You star forming attachement through short term memory, then depending on a billion factor from the environnement and within yourself the chemestry soup and electronic mess inside the labyrinth of neuron of your brain this attachement might grow stronger or weaker. As it grows you find yourself more and more attracted to this person, more in love, if a simillar process happen in the other brain you can say that you are both in love and if condition are right and the process remain or other process with simillar result replace them you can stay in love for a long time. The complexity of the process is amazing (well the complexity needed to make the human mind is just so beautiful) and anyone that would claim love is more then “just” chemical reaction and electrical response just doesn’t understand how much chemical reactionand electrical response are, and just how beautiful their complex interaction are. And this might lead to a future post about free will…

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