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Originally Posted by Travelling fella
What is your goal with math?
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Over time i have come to conclude that i do maths for my own selfish ambitions. Sure, there is always the thought that one day i might pull a Dantzig(
Urban Legends Reference Pages: The Unsolvable Math Problem) but mostly i do it because its addictive.
I mean do you know the sensation of accomplishment you get after doing a particularly hard puzzle or a brain twister? Now imagine constantly having this feeling x1000, its like being on crack!
So to get back to the question, my goal with maths is simple: to become a fully pledged mathematician so that i can continue to do what i love doing(maths!). The good thing about this is that mathematics is so huge that i will be dead before i can have the chance to say "i've seen it all"
Quote:
Originally Posted by Travelling fella
where do you want to go (or be taken) by them?
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I don't really know but luckily i have more freedom of choice(career-wise)than it would appear.
You see, ironically, the people from a subject that is known for its perceived limited usefulness in the real life situations are sought after the most by employers. You could spend your youth studying maths and suddenly find yourself doing something like law or computing.
The explanation for this is that the nature of a maths course is different from all others. You see, to be accurate you don't learn things in maths in the same sense that you do for any other subject. What you do is learn how to solve problems which is something that seems to really impress employers. For example, you could hire an expert in a company with years of experience and all the know how associated with your job but when the unexpected comes,panic strikes. This is the point were problem solving skills become really useful.
Quote:
Originally Posted by Travelling fella
Do you think Math has a source? an starting point? where did they come from?
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Imo mathematics is something we make up and discover, questions about where they come from, where they start don't really make sense here. Its a really complicated thing to explain but basically we make the rules and everything falls from there(or gets built up to be more precise). There is no need to follow the rules and we can always make up new ones so long as we don't reach a contradiction. It may turn out that what we find applies to the real world or it may not since mathematics itself seems to exist in a platonic world that only partially overlaps with reality and the mind.
Its insane, you just define the beginning and then it just spontaneously explodes into something infinitely deep. Its sort of like how i can feel a white wall but the "whiteness" of the wall itself is intangible. Does the whiteness of a wall even exist?
Ill use the MIU puzzle(from one of my favourite books) to try to explain this:
The MIU system uses the 3 letters M,I,U. Lets call every line that we make which involves those letters strings ok?
examples:
MI
MU
MUUMUU
UIIUMIUUIMUIIUMIUUIMUIIU
Now lets make up some rules. Anything that we get using those rules we shall call theorems
Rule one: If you possess a string whose last letter is I, you can add a U at the end
example:
UMIUIII
turns into
UMIUIIIU
Rule two: Suppose you have Mx, then you may add Mxx to your collection
example:
MIIU
turns into
MIIUIIU
Rule three: If III appears in your string, then you may replace then with U
ie:
MIII
turns into
MU
Rule four:If UU occurs inside one of the strings, then you can drop it
ie:
MIUU=>MI
Now that we have the rules(theorems) we are going to have to arbitrarily decide a starting point which we shall call an axiom
Axiom: MI
That is to say that whenever we attempt this, we shall always start with MI
Now a question can be: can we get theorem MUIIU?
lets try it:
MI axiom
MII rule 2
MIIII rule 2
MIIIIU rule 1
MUIU rule 3
MUIUUIU rule 2
MUIIU rule 4
Yes we can so MUIIU is a theorem! See how we can build up something big by just defining a starting point and making some rules to follow?
Now try to see if we can get:
MU
Well, what does this have to do with the essence of mathematics? Truth is that all the laws of mathematics are simply continuations from simpler laws.
I'm sure that everyone has done or at least heard of Calculus. Saying that it is an essential field of mathematics is a big understatement. Using differentiation we can get velocities from distances or acceleration from velocities. I can do the reverse and integrate turning my accelerations into velocities or my curves into areas.
It seems that Calculus was invented(discovered?!?!?) by 2 people independently with their own notacion and all. Leibniz who had been working on it for the last 20 years and Newton who invented it in an afternoon while drinking a cup of tea to help his friend with an astronomy question(yes he was a genius but his fans always slowly come to the realization that he was a grade-A jerk). How did they do it? well they just expanded on simpler rules which are themselves products of ever simpler rules going all the way down to the axioms.
Now we are at the heart of all this: the axioms.
A dumb question to ask is if any of the axioms are right, after all they are just assumed to be right. The axioms themselves cannot be derived and only serve as a starting point but they can't be right or wrong either. They just "apply" or "don't apply".
Consider the axioms of euclidean geometry(named after the greek mathematician euclid):
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Originally Posted by wiki
- Any two points can be joined by a straight line.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are congruent.
- Parallel postulate. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
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They sound innocent enough. I can grab a pencil and a piece of paper and no matter where i put the points, i can always find a straight path to connect them. From this i can conclude for example that all triangles add up to pi radians or 180 degrees.
Of course this doesn't always happen.

The triangle to the right seems to have wonky lines but expressed in non-euclidean geometry, they are straight
For example imagine that you are in the north pole. You walk all the way down to panama and turn 90 degrees to the left. Walk to Africa, turn another 90 degrees to the left again and end up back where you started and find a right angle between the two points.
Hold on? 90+90+90=270>180 Shouldn't all the angles add up to 180?
Does this mean that Euclid was wrong? But if i make a triangle between my house, the neighbours house next door and the house at the end of the street, I find that they do in fact add up to 180. How can it be that they are both right and wrong simultaneously? Well its simply a matter of wether they apply or not. We arbitrarily set up the rules and the world immediately created will either implode in a puff of contradiction and self-reference or it might one day end up being useful
Quote:
Originally Posted by Travelling fella
What do you find the most fascinating about them?
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gee i don't know

. Maybe its the elegance of it or the complexity of it. I also seem to be uncontrollably attracted to hard things and ive yet to hear someone find something harder than maths.
I also enjoy the history of the people behind mathematics. They may not seem like it but the more famous ones have stories so dramatic, they put greek tragedies to shame