Greetings, wimps. I hope you have been practicing your logic. As every barbarian knows, practice and exercise are the key to building strong muscles, and strong muscles are big!
AND BIGGER IS BETTER!
Quite a few of you are so pathetically weak as to be frightened of numbers, the stepping stone to great mathematics. In this blog, I will slay your fears, or they will slay you. Either way, my Valkyrie is keeping my bed warm, and I do not intend to keep her waiting.
I shall introduce the runes required to properly understand what I will say. If you think this is somehow a sign of weakness, it is because you yourself are weak and I could crush you with just a FART!
ODIN IMPALED HIMSELF ON HIS SPEAR AND TOOK OUT ONE OF HIS EYES SO THAT WE MAY BE ABLE TO USE RUNES!
Behold $\mathbb{R}$, this rune, or symbol, represents the collection of all the so called "real numbers". Do not bother yourself with what other kinds of numbers there may be just yet, for the strain will cause your slow and painful death. Content yourself that any number you can concieve of at this moment is a real number, or as true barbarians would say: $x \in \mathbb{R}$, where $x$ is any number you can think of.
Recalling your basic logic (YES, LOGIC IS IMPORTANT IN MATHEMATICS YOU WHELP!), we shall start with statements assumed to be true. In mathematics, such statements are known as axioms. You may disagree with the choice of axioms, and you are free to select new ones. Indeed, mathematicians do this on a daily basis, else they starve. And in doing so, you are no longer discussing real numbers.
But we are barbarians, we please no-one but ourselves, such considerations are beneath us!
Even the most lowly weakling can comprehend the following:
In all that follows $a,b,c \in \mathbb{R}$
1. There exists an operation called addition, represented by $+$, such that for any $a,b$; $a+b = c$ for some $c$
2. $a+b = b+a$. This property means that the operation, in this case additon, is commutative.
3. $(a + b) + c$ = $a + (b +c)$. Such a property means that the operation is associative.
4. There exists a number zero, whose symbol is $0$ such that $a + 0 = a$ for all $a$ there exists $b$ such that $a + b = 0$. $b$ is known as the additive inverse of $a$, often written as $-a$ and vice versa.
You may now be having nightmares from your early school days. I laugh at your distress, and shall now give the axioms of multiplication.
5. For all $a,b$, $a \times b = c$ for some $c$
6. multiplication is both commutative and associative.
7. $a \times (b + c) = (a\times b) + (a\times c)$. That is to say multiplication is distributive over addition
8. There exists a number called one, represented by $1$, such that $a \times 1 = a$
and for any $a \neq 0$ there exists a $b$ such that $a \times b = 1$. $b$ is the multiplicative inverse of $a$, and vice versa.
Notice that the additive inverse of $a$ is equivalent to $-1 \times a = -a$
As a notational shorthand $a - b$ is taken to mean $a + (-b)$ and is sometimes called the difference of $a$ and $b$, or the subtraction of $b$ from $a$
These 8 axioms establish the real numbers as a(n algebriac) field. Do not confuse these as the places you'd usually go to when you want to pillage, for that would be stupid. When you have gained more muscle mass, the discussion of fields in mathematics may be tackled in fuller detail.
Notice now that we know that some numbers are larger than others, or at least that was what your least favourite teacher told you. We shall now define what we mean by order on $\mathbb{R}$
We define a subset, $P \subset \mathbb{R}$, called the positive set.
$P$ has the the following properties:
$a,b \in P \implies a+b \in P$
$a,b \in P \implies a \times b \in P$
Exactly one of the following is true:
$a \in P$ $a = 0$ $-a \in P$
We define an order, $<$.
$a < b$ if $b - a \in P$ Also $a \leq b$ if $a = b$ or $a < b$
Now we can define what we mean by an upperbound of some set $A \subset \mathbb{R}$: $b$ is an upperbound of $A$ if $x \leq b$ for all $x \in A$
One final axiom is now required:
If $A \subset \mathbb{R}$ and $A$ has an upperbound in $\mathbb{R}$, it also has a least upperbound in $\mathbb{R}$ In order words: if for all $x \in A$ there exists some $a$ such that $x \leq a$ then there exists a number b such that $ x \leq b \leq a$ for any upperbound $a$ of $P$
This axiom is known as the Completeness Axiom, and is required when more technical arguments are met.
Remember to practice, or you will forever remain weak and unloved!
Monday, January 31, 2011
Sunday, January 30, 2011
Logic Primer
As I was saying, logic is skill, and if you have skill you can certainly kill. I shall demonstrate to you the first simple techniques that will put you on track to being as great a barbarian as me! Unlikely as it may be.
What is logic?
Philosophers can spend their entire careers debating this question while I steal their lands, their women, burn their books and defecate over their reputation. A true barbarian would simply take logic to be a system of rules by which one can conclude that a statement is true given than some other statements are also true.
Now this is all a little high brow and pompous. Indeed I shall need to armwrestle Surtr after this blog in order to regain some of my barbarity, but without it you would be little more than an unarmed peasant facing a horde of angry Vikings!
Ahhh, good times. Hint: I am not a peasant.
Let us begin. I make my first statement, assumed to be true:
The blood of my enemies is red. (S1)
What statements can I conclude from it? Can I conclude that the people I pillage are weak?
NO!
Even though experience verifies it to be true, its conclusion is not possible given my initial statement (S1)
One statement which can be concluded from (S1) is:
The blood of my enemies is not blue (S2)
Why? Because if something is red, IT CANNOT BE BLUE! Notice, that from (S2) I cannot then conclude (S1), for if something is not blue, then it may be green, yellow, purple, cyan, magenta, grapefruit and so on, none of which are red.
Notice, that given any statement P implying any other statement Q, one may not necessarily then conclude that Q then implies P.
This is a fundamental error that many make, and it makes Olaf ANGRY! Many a barbarian have found themselves slain after showing this lack of basic technique.
I need to go breathe in some mead now.
Until next time, wimps!
What is logic?
Philosophers can spend their entire careers debating this question while I steal their lands, their women, burn their books and defecate over their reputation. A true barbarian would simply take logic to be a system of rules by which one can conclude that a statement is true given than some other statements are also true.
Now this is all a little high brow and pompous. Indeed I shall need to armwrestle Surtr after this blog in order to regain some of my barbarity, but without it you would be little more than an unarmed peasant facing a horde of angry Vikings!
Ahhh, good times. Hint: I am not a peasant.
Let us begin. I make my first statement, assumed to be true:
The blood of my enemies is red. (S1)
What statements can I conclude from it? Can I conclude that the people I pillage are weak?
NO!
Even though experience verifies it to be true, its conclusion is not possible given my initial statement (S1)
One statement which can be concluded from (S1) is:
The blood of my enemies is not blue (S2)
Why? Because if something is red, IT CANNOT BE BLUE! Notice, that from (S2) I cannot then conclude (S1), for if something is not blue, then it may be green, yellow, purple, cyan, magenta, grapefruit and so on, none of which are red.
Notice, that given any statement P implying any other statement Q, one may not necessarily then conclude that Q then implies P.
This is a fundamental error that many make, and it makes Olaf ANGRY! Many a barbarian have found themselves slain after showing this lack of basic technique.
I need to go breathe in some mead now.
Until next time, wimps!
Sound the Gjallarhorn
Behold and quake in fear, for Olaf is among you! Olaf is very disappointed in you all, Olaf will teach you about the world and how to be a great Barbarian!
Physics and Mathematics, like the axe and spear, are things no true Viking blooded barbarian would be caught without! Far better to be naked in the frozen north, hunting polar bears, than to be without them. Logic is the skill with which you wield your axe in a crazed bloodbath! Before you whelps can be trusted with weapons, I'd best make sure you don't accidentally decapitate yourself while puncturing your lungs with them even though it would be very amusing to see.
WAAAAAAGH
Physics and Mathematics, like the axe and spear, are things no true Viking blooded barbarian would be caught without! Far better to be naked in the frozen north, hunting polar bears, than to be without them. Logic is the skill with which you wield your axe in a crazed bloodbath! Before you whelps can be trusted with weapons, I'd best make sure you don't accidentally decapitate yourself while puncturing your lungs with them even though it would be very amusing to see.
WAAAAAAGH
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