“On the shoulders of giant numbers”
http://www.allergrootste.com/big/book/ch1/ch1_0.html
bigΨ
bigΨ.I. Number operations
Ψ.1. Natural repetition
chapter 1.0, edit 0.2.9
published: 2010-11-05
updated: 2011-12-30
§1.0.1. Addition?
1,2,3,4,5,6,7,8,9,10,
¿Where is Buddha when I am counting?
– Prabhâsadharma Roshi
20th century international Zen master
When you count you repeat a certain unit – a single entity.
Stars, spirals, sleep, sheep, sets, seconds, satyrs, smiles or sand grains –
all objects are subjects for counting,
as long as their individual units can be recognized.
Any two objects are separated by some space
and space can be measured in units.
These measures can be items for a comparative list, etcetera...
until one fails to find meaning and the item's separator is just
empty space .
In real life, except for quantity quality always matters.
It is perfectly possible to sell a quantity of pigs
in exchange for a selection of roots and a busty woman of bad reputation,
without knowing their exact number.
To belittle a people who only have use for counting numbers up to three
is no less barbaric.
The initial unit of arithmetic is one 1
and by repetition of 1.. you can count natural numbers.
Then the natural representation of zero
..is nothing!
As far as we know this makes nothing
the only ing
that is what it is in an expression
(where it is void).
0 = 1... {1#0} =
Don't think of addition as an operation on numbers.
Like zero
addition doesn't actually exist.
When we relate it to the first operation of multiplication,
addition can be defined as a
zeroth operation – the operation
without stars.
We add
all integers
in our notation for natural counting automatically,
by simply joining their units with 0* space in between
(where zero is ubiquitous).
The old-style plus + operator
is unmasked here as an empty star.
= 1...1... {1#a 1#b} = a+b
Peano's
successor
function S(n) = n+1
uniquely covers all numbers 1.. starting from n=0
by iteration of S, and thereby defines the set of natural numbers.
But it is preposterous to name individual integers with a function or a set.
Who would order S(S(0)) baked eggs?
Mathematicians are crazy, and they still have to count!
(Count the (depth of nested) brackets…)
We can understand a natural number as output
of a single character 1 forming (an initially unspecified)
series of ones until it stops (once).
An additional operation would be output 1..
that stops,
runs on and stops again.
Unit 1, the variable number (or number variable)
and star space 0* are the Godmothers of repetition
AKA ...
§1.0.2. Multiplication!!
There are two ways to define multiplication –
by repeated addition or by repeated substitution.
In our notation a*b the a
is repeated b times, which is a binary
operation on a pair (a,b) of numbers.
Use the single star * as operator
to multiply left operand a (called the item)
by right operand b (the iterator).
On the first line multiplication is a simple iteration
of a starting from
nothing.
We write this operation as a*b and not as b×a
(Question: "How many?" Answer: "6 items"),
because we prefer to position higher iterators to the right.
The second evaluation shows that multiplication can also be defined by
substitution
b {1:a} of the item a for each unit 1
in natural number b.
= 1... {1#b 1:a}
When we define multiplication a*b by the method of
meta-repetition
{a#b}
this seems to open a vicious cycle which obfuscates
the arithmetical fact,
namely that multiplication is repetition:
a series of items a of length b which in
star space 0* simply add up.
In our story we won't actually multiply many numbers,
but repetition in the form of iteration (another of its guises)
is used throughout in elaborate recursive schemes
and function enumerations (more of the same).
As soon as it's recognized any construct can and will be held accountable
(counted ;-)
Here multiplication is really a repetition of a repetition of units one
– a row of equal lengths, a discrete
rectangle.
Mathematicians like to hide series 1.. in number variables,
and focus on other types of signs.
Multiplication isn't necessarily the first operation after addition.
In the chapter on superpowers we'll
discuss
a function family Fa,
which takes minimal effort to define, where the operation
a... {a#b1} comes next.
Of course in such a scheme the so called identity element
a*1 = a does not exist.
And for practical purposes we'd dearly miss the commutative
property a*b = b*a of multiplication.
Without it algebra would have to be a Lie!
# Context of empty operations
Classic mathematics smokes the empty joint
ab {0^} of multiplication
and enumerates the single arrow a^b
of exponentiation
as the first case of the a^..b
superpowers.
Addition isn't necessarily covered by multiple arrow operators
^.. {^#c}
as the reduction rule at one ^ or at zero ^#0
drops operator counter c altogether.
An arrow operator
a+b
is not so simple to define.
To resolve an expression with good old-fashioned
adding
we write the
lower level
operation a+b
with a plus,
but by nature addition doesn't belong to the world
of arrow operators.
Most calculations we make rely on addition at a basic level,
but it is by no means clear how physical states and sizes can be taken together.
How do you add force to distance? It's impossible,
the qualities don't match.
Yet force measured in Newton multiplied by distance in meters amounts to energy
Nm = J in Joule.
Every quantity in nature is counted by some qualitative physical unit,
and only quantities of the same quality add up.
This makes us wonder, does there really exist such an arrow universe,
where joining the same qualitative units
a... {#b} = a^b
is dimensional in nature?
And if not, because deep down under in zero space
we live in a discrete world of integer addition,
what is it that qualifies as distance
or dimension?
According to a theory of
Gerard 't Hooft
information content
depends on the surface of a holographic sphere –
ultimately square Planck lengths.
Perhaps ours is actually a universe of stars
a... {#b 0*} = a*b
where the probabilistic walk of a quantum man
is to be measured in square root square foot.
In our experience multiplication is a messy business, in need of an operator.
Its classic notation ab without an operator sign,
obscures the arithmetical fact that we add numbers by simply
joining and counting them.
There isn't any place for primordial zero 0 in a world
of instant multiplication, where all is 1 at least.
In pure mathematics new constructs may arise from multiplication
that shy away from direct addition.
As such we propose the existence of a utopian
or infinite number line with a zero square
0*0 on its
inverse part.
