Peter said:
"My position in this is that "absoluteness" is a useful, but entirely
mythical and unprovable, concept. It only exists in areas such as
mathematics, which is similarly 'un-real' and tautological. In fact, it
seems to me that the condition of absoluteness requires the tautological
condition. So, 1+1=2 is absolutely true, but what has been said? - a variety
of implications absolutely come from this, (2-1=1, etc.,) which we find very
useful, but essentially we are saying the same thing over and over.
I can't think of any other area where such absoluteness can definitely* be
said to exist."
And, in fact, one of the most powerful features of Mathematics is the
ability to change it around at will. Since axioms are so arbitrary, one can
make up as many mathematical systems as one likes. I owe the movement down
these thought lines to ZMM. Here is a post I have out on the newsgroups.
Please take a look at it.
Frozzbozz said:
>>Mathematics, from my
understanding is used to describe a problem we already understand so that
we can solve problems of that type in the future. But, if we want to
make something new, mathematics can't really describe it yet, and thus,
we are forced to use our own experience.<<
.
That is an interesting point. There are however, other ways to stretch
Mathematics to make it fit. One way is to remove it from any binding to
reality, and make it its own reality. When you do this, you can make
mathematics do whatever you wish.
Did Newton "discover" Calculus or did he "invent" Calculus?
What are the axioms of arithmetic? I have said this before, but I would
really like to discuss it more thoroughly.
One axiom of arithmetic is 0. Another axiom of arithmetic is infinity. Once
these axioms are understood, new axioms and new logic can be put into place
to create a new arithmetic. One can view an arithmetic as a world, if one
chooses to do so.
An arithmetic where there are two directions on a number line, one positive
and one negative describes a one-dimensional universe with an origin. 0 is
an origin. It isn't really a number, but an arbitrary point from which we
measure distance.
Infinity is another axiom of the present arithmetic. There is always a
bigger number, no matter how high you go.
Infinite divisibility is yet another axiom of present arithmetic. Right?
There is no such thing as a smallest number. Whatever number you can think
of, I can divide it by 2 and create a smaller number.
So what kind of universe does the present system describe? A one
dimensional, infinite universe with no smallest particle and movement
allowed in all directions (two) and an arbitrary origin from which those
directions point.
Let's write down the axioms:
1. 0 or an origin
2. Infinity-no largest number
3. Infinite divisibility-no smallest number
4. No division by 0-0 isn't a quantity-
5. Negative and positive directions.
6. Equal distance between units
There may be others. But let's just start with a couple of these, and
proceed to invent new arithmetics! It's easy! It's child's play! It's like
writing a fairy tale!
Let's eliminate 0 first off. If we do that, we eliminate the problem of any
number divided by 0 being undefined. We also create a smallest unit-the most
basic unit that makes up our universe is equivalent to this smallest number.
As soon as we do this, we solve Zeno's paradox of Achilles and the turtle.
The reason? At some point in this finite! divisibility, Achilles must begin
to move forward, and when he does-watch out turtle!
Could we eliminate negative numbers? Sure! Then we go from a line stretching
off in both directions, to a ray! How about this..let's only allow movement
in one direction! No subtraction! Woo hoo! I always hated subtraction!
Like pawns on a chess board, we can not ever take back any step forward.
Like time. Once it's done it's done.
So now what do we have? No 0, no negative numbers, no change of directions,
and a smallest number! How about a biggest number? Let's make one and close
up our universe. And once you get to the end, that's it, it's over.
Obviously any permutation of those axioms could be done. Here is another
interesting one-
Given a smallest number and a largest number
The smallest number=the largest number
0=360
The number "line" becomes a "circle"
The universe repeats itself forever!
The possibilities are as endless as the universes and axioms you can create.
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