Sorry for taking so long to respond, ive was in the middle of a complicated situation yesterday
Quote:
Originally Posted by Travelling fella
Ahhh math, something so pure and infinite.
untangible yet omnipresent.
not matter, not energy, just ideas meaning it's one of the highest expressions, and yet they can be used to meassure energy and matter.
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Its incredible that maths happens to be something that gets down to the deepest confines of philosophy with its "hey lets make stuff up as long as it doesn't contradicts itself" attitude.
In science we base our observations around reality but mathematics doesn't care and is independent of all this. Rather than mathematics being part of reality, it is reality itself that borrows stuff from mathematics.
Consider the natural numbers 1,2,3,4........
Now its easy to see how this fits with our existence. For example i own 1 television, 4 wooden chairs and there are 45678909876543 crumpled up papers with weird symbols written on them in my room
Say i extend this line to the integers ...-3 -2-1 0 1 2 3.... Well now the uses become trickier after all, what do the negatives actually mean? how can i have -2 potatoes? Sure you could say that i owe you 2 potatoes or that -19 mph is the same as 19 mph the other way but what is a negative in more physical terms? Scientists over a few centuries ago found one thing: charge.
F=Kq1q2/r^2 ect ect
So it seems that sometimes this completely made up system shows up in nature but is not necessarily a part of nature. The best example of this is with imaginary numbers. Gerolamo Cardano, Leonhard Euler and Descrates all had a field day with their curiosity when pressed to find answers such as x^2+1=0
x=±√-1
What number could possibly square to a negative if:
axa=a^2
-ax-a=a^2
Its impossible! no number in our real number line will ever square to a negative. So what did these people do? well rather than just leaving it as that, they decided to make up a new number line perpendicular to ours. Suddenly we have the answer to x^2+1=0 by saying that √-1= i

And so we get complex numbers which are made up of both real and imaginary parts.
But surely what is the point of this? imaginary numbers were just things made up by a few people about 500 years ago. How can this is any way reflect real life?
Well doing the Maclaurin series(i can expand on this for the lurkers out there):
f(x)=e^x= 1+x+(x^2)/2+(x^3)/6..........
g(x)=e^-x=1-x+(x^2)/2-(x^3)/6.......
e^x+e^-x=2+x^2+(x^4)/12.....
since i^2=-1 then
e^ix+e^-ix=2-x^2+(x^4)/12-(x^6)/360....
(e^ix+e^-ix)/2=1-(1/2)x^2+(x^4)/24-(x^6)/720....=cos(x)
(e^ix-e^-ix)/2i=sin(x)
Well hang on, so imaginary numbers are used in trigonometry? You mean that every time i throw a frisbee, imaginary numbers are somehow related?
Worse still if we look at quantum physics. Imaginary numbers pop up every 3 seconds there. Figures.
Its this sort of thing that has caused a world within a world in mathematics. Mathematicians themselves have become divided into 2 groups: the pure and the applied.
As the name implies the pure are to do with aspects of mathematics itself(eg: proving conjectures) whilst the applied are to do with using the mathematics in real world prolems(eg: fluid mechanics)
Pure
Analysis
Algebra(hard!!)
.....
Applied
Mechanics
Statistics
.....
Now as my teacher puts it, the pure mathematicians think that the applied are cretins who don't know how to use maths. In retrospect, the applied ones think that the pure are insane, spending their days living in an office like social outcasts.
Quote:
Originally Posted by Travelling fella
why are math infinite my friend? i know i'm asking something really hard but it'd be interesting to know what's your take on it
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There are several reasons. A big one pertains to a paradigm shift with mathematicians back early 20th century. You see just like scientists a century earlier there was a growing belief that perhaps everything about mathematics might one day be understood. People like Bertrand Russell were thinking that perhaps they could use set theory to organize mathematics into an understandable bundle. Whenever they found self references, they would avoid them by adding layer after layer of limitations making the foundations of mathematics look more like the sediments in the grand canyon.
Alfred Whitehead for example worked with Russell to produce a 360 or so page book proving that 1+1=2(dont believe me?
University of Michigan Historical Math Collection)
Even someone like David Hilbert, one of the most influential mathematicians of the time was into the idea of organizing everything in mathematics and rounding the edges off. It seemed like this indomitable mountain was about to be conquered. That is until Godel came along
Now Kurt Godel changed everything with his incompleteness theorems. The most relevant part of them being that even in a self-consistent system, the axioms cannot be proven. That it will be impossible to prove certain conjectures no matter how much time and effort we put into them.
To put this into perspective think of Fermats Last theorem. A problem thought up by an amateur took generations of mathematicians before it was actually proven. Neither Euler, Dirichlet nor Galois could crack it(though they made progress) and it wasn't until 357 years later that finally Andrew Wiles bought such a giant down to its knees.