#
*“On the shoulders of giant numbers”*

http://www.iteror.org/big/book/ch1/ch1_0.html
big**Ψ**

http://www.iteror.org/big/book/ch1/ch1_0.html

# 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*).

`{1#n}`

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.

`{0`= a*

_{*}}^{..}b

`{*#0}`

`= 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(`

baked eggs?
Mathematicians are crazy, and they still have to count!
(Count the (depth of nested) brackets…)**S( 0)**)

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 `

of the item `{1:a}``a`

for each unit `1`

in natural number `b`

.

`{a#b 0`

_{*}}= 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.

`{0`= ψ^(-2) ≠ 00+0

_{^}}`{0`= 0

_{^}}