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…

Friday, February 12, 2010

XKCD: scientific romantism.. or not.



Don't worry I don't do graph like that, beside it's not like I would belive that correlation would mean causation anyway... still it's funny, in a shark with a party hat kinda way.

Math part III: Algebra and Calculus

So after numbers, arithmetic and geometry I get to the last part of mathematic that is taught in high school science: algebra. It is also the point where most people that already dislike math move up to hate and what they hell will that ever be usefull for. So what happen at this point, well most people will tell you that it’s the addition of letter to the number that just sound illogical and learning all those new “rules” to work with these letter. Now, you can probably guess I don’t agree, first off because I always found (elementary) algebra easy and second because I know that it is found everywhere and that almost everyone (even those that have not been taught algebra) use it everyday.

First off because, I think, the problem is how algebra is taught in school. In my opinion algebra is taugh way too late in the curiculum, it should be taught right at the beginning when we are learning the rules of arithmetics and numbers. You see, algebra introduce in math the concept of variables, i.e. unknown quantities. The problem is when you are first taught about this concept you have already been using math with unknown quantities when solving problem. So you have to relearn how to incorporate unknown in your math and the “special” rules you need to use to apply arithmetic on unknown quantities. Lots of duplicated work and lots of confusion from student as you have to basically rewrire the brain. Also, what we are taught is just the surface of algebra. So in the spirit of these post about math what can we do with algebra that we could not do before?

Well, a lot. You see algebra is not just the study of what happen when you replace number with variable, it is also the study of what happen when you change the operation and rules relatated to them and (more importantly, I think) how to apply mathematical operation on thing other then numbers. This last thing gives us chemical equations that give us the opportunity to understand what happen in chemical reaction, such as combustion. In fact, most of not all of science need algebra to be represented in comprehensive mathematical form. So basically, algebra add to the ability to representing physical and imaginary, manipulating these quantities and representing their shape, position and size, the ability to work with unknown or variable quantities, the ability to see how other type and set of manipulation would work and most importantly how to apply mathematical logic beyond numbers.

So with all that power, math must surely being done for right? Wrong, we now get to calculus, a part of math some of you might only have heard through horrified mention of college student having their first taste. So what is this dreaded discipline? Well calculus is slightly akind to geometry, in the way that while geometry deals with the shape and size of thing calculus deal with the movement and change of things. It involve such concept as the limit (seeing how as you approch a quantity another varies), fonction (describing a quantity in relation to others), derivative (describing the change in a fonction as it’s input change), integral (the reverse of the derivative and also the area under a fonction) and infinite series (numbers with recursive proprieties between themselve).

Lost of component that have an unparallel power, just about everything in the univers can be represented by a fonction, a derivative of a fonction or it’s integral (depending on what the phenomum involve). Most of advanced physics, engeenering, economic, all science in general involve calculus to a great degree. To continu with the language analogy, if arithmetics, numbers, and algebra form the “letters” of the language in our models of relality, geometry and calculus (and some would argue calculus alone) is the gramar, words and text structures. Throught this we can for exemple show how blue and yellow light form green light (it involve the transmission fonction of our eyes photoreceptor coupled with the wavelenght of the light we are seeing and how our brain intepret the resulting electro-chemical signals). Maths can be used to model much more complex phenomenum, such as the nuclear fusion in stars, chemical process involving complexe molecules (such sugars) and would appear at this point to be limited to predictable, deterministic process. In the next post I will talk about how maths deal with undeterministic process when I talk about statistic, chaos theory and probably the more advanced maths that I am only slightly famillar with (such as group theory).

Thursday, February 11, 2010

Maths part II: Arithmetic and geometry

So let get into another post about the beauty and power of math, this time I will talk about arithmetic and geometry. The first one because it is the next logical step after numbers, it’s the branch of mathematic everyone is the most famillar with and it is the oldest part of math, while the second countributed so much to our understanding of the universe early in our history that including it later would do it much disrespect.

Arithmetic in short the manipulation of number, allow us to work with many quantities and predict an outcome. For a simple example, what happen when someone has 4 stick and you remove 2 and 2/3 of one (your left with 1 and 1/3 of another). Arithmetic includes a lot of way to manipulate number, at the beginning we have addition and substraction (which are basically the same thing, one with positive integers and the another with negative). Next in the “simple” operation we have multiplication, division and exponentionation. These operations are mostly the same operation, exponentionation is repeted multiplicaiton of the same number, division is multiplication by a fraction. Now, multiplication are basically addition, so we could technically say that we have only one operation on number, the addition. However keep in mind that writing everything as addition is very time and space consumming. After the sample operation of addition we have more complex operation such as finding the roots of a number (square, cube or even higher power) the root of an equation is in short the exponentiation by a fractional number of another number, and also logarithm of number. The logrithmic operation is much more complex, it consist in finding the number that elevate another number to give the number we are taking the logarithm from. To be clearer, say we have log(10 000)=4, that means that 10 to the 4th power is 10 000 (the 10 is implied in the log without a base denominator). Now, this might seem to old very little power, but being able to predict relationship between quantities and answer question those relationship imply is a corner stone to our thinking.

Now to talk about another ancient field of math, geometry. Geometry is the study of shapes, sizes, positions and proprieties of space. At first, geometry was described for a plane following a set of axiom that forms a basic for logically dedicing a lot of theorems and from those theorems information about the nature of shapes, sizes, posititions and all. This form of geometry called Euclidian geometry (for the greek that formulated it as a set of axiom) is what most of you learned in high school and use most of the time to infere information about the spatial relation and distribution of things. If you combine this with cartesian coordinate you get a very powerful tool to move about and locate thing on the Earth (as long as they are not too far appart). But geometry doest not stop there, we have found through experiment and observation we have discovered that we can formulate other axiom that describ other type of space. Such as for exemple, if our universe was on a sphere, the proprieties of a sphere lead to some interesting phenomeon such as having square triangles with more then 1 90 degree angle, or that the shortest distance between two point is not found as a straight line but as the arc of a circle. Many other geometry are possible for space, in fact curved space is an observed thing in astrophysic notably around very dense and massive object. Oh, one last thing about the power of geometry, understanding shapes, size and position might seems like something that not really spectacular, however one should recall that the greek measured the size of the Moon, the diameter of the Earth, the distance between the Earth and the Moon and many other quantities with amazing precision using only Euclydian geometry principles!

Now, adding arithmetic to number gives humanity the power to predict relation between quantities and manipulate, trade and use number as a model of what is happening or will happen in simple terms. If you have knowledge of geometry too you get an in dept knowledge of the shape of the world you live in, the shape of the object around you, ways to relate to them and predict their positions. In short with just math that was availlable to the ancient greek you can understand and model quite a lot of the world. Your models are time consuming to use but still you have a phenomenal understanding. My next post will show you how math power can make those model faster to use, become more abstract to represent even unknown quantity and even study very small and very dynamic things while I talk about, algebra and calculus.

Wednesday, February 10, 2010

Math part 1: Numbers.

As I said in my last post this should be the first post about maths. Let me first give you a little history about why I’m doing these posts. Well it all started last time I met with Yofed on her visit to Quebec city. I made the claim that math could potentially explain and represent everything, so Yofed challenged me with the old explain love with math (quickly answered by simplistic 1+1=2 answer) and blue+yelow (answered with a not so simple answer of adding to wavelenght of blue and yellow to get green, which I quickly admited was wrong but that there was a way to add blue and yelow mathmatically to show that it would give green for our eyes). Now the conversation continued with a “you should make a post about the power of math.” So instead I choose to make many post about it!

So before I start I must say that I am not an historian of science, nor a mathematician. The ideas expressed below might not be completely acurate. Now that it’s out of the way, I’ll first start by writing about the numbers, the part of mathematics most people are arguably famillar with. Now the power of by themselve is only descriptive but we are deceived by their ubiquity into forgetting just how powerfull they are. Lets start at the beginning of number, the positive integers, you know the first number you know (1, 2, 3, …). Those number can represent almost all physical quantity, from the number of pencil, to the number of stars in the univers as long as it is something physical the positive integer can represent it.Now, you can not, with those nomber, easilly quantify lack of something. For that we extend to the integer set, adding the minus sign before a nombre give us a representation and quantification for the lack of something. Your missing 15$ so you have -15$, simple. Then we have the zero, strangely a relatively recent invention, that recent the absence of quantity, emptiness, it’s the very usefull quantification of nothing.

Not with the natural number set (integer +0) you can arguably represent any physical quantity, however you can’t easilly represent part of a whole. So we have another set (that include the natural numbers), that we call the rational set. The set includes the rational fraction, i.e fraction that can represent “physical” fraction of a thing. For exemple, 1/2, 1/4 and 1/10 would be these some of these fraction, they are constructed from any number that can be constructed from the ratio of two non-zero natural number. We now have a lot of descriptive power, however there are still quantity that cannot be represente, for exemple, the square root of 2. To represent these quantities we have another set greater then the rational that is call the irrational set. It includes the rational number as well as all those quantity that cannot be expressed as the ratio of two non-zero natural number. They describ slightly more esoteric quantity so as the lengh of the hypothenus of some triangle and some logarithm (more on those in a later post) they might also include very fondamental constant in natural such as pi and the euler number e (it’s still not clear if pi or e are member of the rational or irrational number).

Now with the irrational set we can pretty much describ every quantity you find in natural (in fact I’m pretty sure you can describ everything physical thing with them), however math descriptive power is not limited to nature, or our 4-dimentional univers. Lets imagine for exemple you want to find the square root of a negative number. Now, you know that two negative numbers squared give you a positive number while two positive numbers squared also give you a positive number. While it might seem unlikely to encounter such a thing in nature they are essential to solving many equation. So to represent those quantity that have no “real” manifestation we expend our knowledge toward another set of number called the complex (or imaginary) number. These numbers are very special because you need 2 real numbers to represent them, for exemple one such number is 3+2i. Now, what do we need them for? Well they are usefull in engineering, quantum physics, chaos theory, and I could not easilly (or shortly) explain how they are used and why. You’ll probably never have to use them if you’re not working in advanced scientific field, but you probably encounter device who work on principle needing such number to be understood.

So just from the number in math we can gather a lot of information about the world, quantitative information. In fact, number do not just give information about the world but they can give us information about world event imaginary or world without relation to our own. In a way if you try to consider math a language for describing models of reality and predicting reality from those model you could think of number as part of this language alphabet. It’s only a part of this alphabet because number are not the building block of the communication of mathematic, equation are. And while number are usefull to solve equation and to obtaint physical representaiton of that solution they are by themselve meaning less. In fact, I think you could say that numbers are what we would use to translate between the models and the “real world”. In the next post about math, I will talk about two of the oldest branch of mathematique, arithmetics and geometry.

State of the astrogeek

I haven’t posted on this blog for a while now, novembre 2009 if I believe the last edit notice from blogger or july if I believe my word document where I keep most of my recent post. So since Anyflower asked and because I’ve been hitching to post again, here is a short post on the state of the Astrogeek today, it should be followed by a long post (the first of a many) to asnwer yofed challenge about math from the last time we’ve met.

So where is the Astrogeek at now? Well I’ve finished the writing part of my thesis… I’ve had finished that in june 2009 anyway. At the moment I’ve sent my thesis for the first part of correction where one professor read the 431 pages of it and comment on what I should change to make it acceptable for the next step: the evaluation part. Hopefully very little will need to be changed and all the change will be cosmetics. On a more personal note, well I’m still single. I’ve meet a few girl but nothing came out of it. I’m looking for a job so if you’d know any one looking for a phd in physics send them my way. Well, so that’s mostly it, I’ll give more news as stuff happen I guess.

On an unrelated note: screw you word 2008 stop putting html/xml tags in my copy-paste!