#
*“On the shoulders of giant numbers”*

http://www.iteror.org/big/book/ch1/ch1_7.html
big**Ψ**

http://www.iteror.org/big/book/ch1/ch1_7.html

## Ψ.1. Natural repetition

#####
chapter 1.7, *edit 0.2.9*

published: *2010-11-08*

updated: *2011-12-30*

### # 1.7. A new ancient world record number

Then again, on a single point

Buddhas may dwell for untold aeons

And as on one point, so on all points

For the same number of aeons...

– Avatamsaka Sutra 30.

#### §1.7.1. Measuring the asamkhyeya

To give the most definitive definition of an *asamkhyeya* –
the traditional Indian number *where infinity starts* –
we turn our eyes to the most grandiose work of buddhist literature. The
Flower Ornament Scripture
(Sanskrit: *Avatamsaka Sutra*, India, 1st-7th century AD),
in the West translated by the perspicacious linguist Thomas Cleary, hail!

In chapter 30
The Incalculable
(Sanskrit: *Asamkhyeyas*) a list is constructed
of the largest numbers in ancient history,
even surpassing the *asamkhyeya* itself, but
experts
disagree on the exact size of these number records.

The problem with the primary list of squares is
that the two earliest Chinese translators sum their exponents up differently,
resulting in somewhat different *incalculable* numbers.
To confuse the issue, Cleary takes his own course,
(but our definition of that foremost innumerable number

– the *asamkhyeya* – is most sacred :o)~

Where to begin?
The oldest translation by Buddhabhadra (c.420 AD) starts credibly by squaring a base
`100000` (Sanskrit: laksha).
Cleary, whose translation follows Shikshananda's (c.700 AD), places his induction base
`10^10` one square higher than Buddhabhadra,
but Shikshananda starts squaring only after he counts 100 lakshas `10^7`
(Sanskrit: koti).

Cleary knows perfectly well what a *koti* is,
but he reserves it for the start of a similar list [pp.1229-1230
FOS]
which results in the smaller incalculable

asamkhya
`10^(7×2^96)` of the *Gandavyuha* or Garland chapter.

To make our squaring formula work for Shikshananda,
think of his *koti* as the induction base
and consider his list one item shorter than usual, namely `n=103`
squares before it reaches the *uncountable* numbers.

From the choice of the base `s _{0}`

`= 10...`a general formula for the n-th square

`{0#a}``s`in the Sutra's list of numbers follows. We ourselves take the initial

_{n}*lakh*exponent

`5`from Buddhabhadra as our base. Bear in mind that the Indians (although inventors of the decimal system) assemble lengthy fantasy names instead.

*s _{0} = 10^a* &

*s*<=>

_{n+1}= s_{n}×s_{n}
*s _{n} = 10^(a×2^n)*
~ 2^2^

*(n+log(a×3.32)/log(2))*

_{ }

*= 10^(5×2^n)*~ 2^2^

*(n+4.05)*

`{a=5}`

The highest exponent `n` is the length of the list,
and the size of the *asamkhyeya* depends most on this.

But when to end?
Bhikshu
Jin Yong
(our intermediary source)
notes that the list of Buddhabhadra misses a term before it gets to the
*asamkhyeya*.
Thomas Cleary considers that Shikshananda,
who did his work for the Empress Wu, offers the most complete text.
But Cleary places his *incalculable* one large step
ahead
of Shikshananda.

Also Cleary and/or his printing devil were a bit reckless.
Their number list contains 10 cumulative miscalculations,
6 copying and 2 printing errors, so that Cleary's last sum at `n=103`
is barely accurate up to 4 decimal digits.

We have reason to assume the precise *asamkhyeya*
is given in the above formula by `s _{0}=10^5`
after

`104`steps. But because other constructions are also justifiable it is best to think of our number as an authoritative restoration – like an artwork is restored when the paint gets blurred over time.

To compare ours with other versions of the *asamkhyeya*
click here!

For two modern approximations of the *incalculable* number,
click here!

*10^(5×*2^*103) =
10 ^{50706024009129176059868128215040}*

~ 2^2^

*107.05*(Buddhabhadra skipped

`s`term)

_{100}
*10^(7×*2^*103) =
10 ^{70988433612780846483815379501056}*

~ 2^2^

*107.54*(Shikshananda past a laksha)

*10^(9.6×*2^*103) ~
10 ^{97689818800677646398555859804252.}*

= 2^2^

*108*(Asamkhyeya's absoluteya)

*10^(4.9×*2^*104) ~
10 ^{100000000000000000000000000000000} = 1E1E32*

~ 2^2^

*108.03*(Père Ubu's googolkhyeya)

*10^(5×*2^*104) =
10 ^{101412048018258352119736256430080}*

~ 2^2^

*108.05394894*(Novaloka's restoration)

*10^(10×*2^*104) =
10 ^{202824096036516704239472512860160}*

~ 2^2^

*109.05*(Cleary with errors corrected)

The question why `s _{104}`

`= 2^2^`*108.05*

should lie just out of reachin buddhism is of special interest.

A likely answer is that during the traditional practice of mantra meditation a monk keeps count of his prayers with rounds of

`108`beads on his rosary. Because past

`108`there is

*no counting*possible – round we go!

Our own

*asamkhyeya*and the modern and preciser estimates have a hidden property. For when the binary power tower exponent

`108`has just passed by, the great gate to Indian infinity is officially opened.

Could a mathematician in 3d century India have figured this out?
The answer is yes – counting the Archimedian
power laws
from index `0`

and using the obvious facts that `10^3` `~ 2^10`
and `2^5` `= 32` …beautifully!

George Joseph
quotes the *Anuyoga Dwara Sutra*
and affirms that both the power laws and specific logarithms of base two
(ardhacheda) where known in ancient
Jaina mathematics,
so this may very well have happened.

_{104}= 10^

*(10×*2^

*103)*< 10^

*(3×32/10×*2^

*103)*

_{ }= (10^3)^

*(1/10×*2^

*(5+103))*~ (2^

*10)*^

*(1/10×*2^

*108)*

_{ }<~ 2^2^

*108*

It's important to hide the absolute boundary power `108`
in another base `2`
than in the base `10` of the system used for writing numbers.
So there can be no crossover of numbers in between,
given that the scale is so large.

From all this we conclude that the number *asamkhyeya*
of the Avatamsaka Sutra is now properly restored.

With `2^60 TB` the *asamkhyeya* lies barely out of reach,
perhaps one day humanity will produce
such an amount of digital information.
When bit size is reduced to a single atom,
random numbers the size of this *asamkhyeya*
can be expressed in `2766`
metric ton of silicon
_{
<which fits inside a big house>}.

#### §1.7.2. Untold number records

The construction of Big numbers
by squaring
in the buddhist Avatamsaka Sutra had its precursor in the 1st? century
Anuyoga Dwara Sutra
of the Jains.
Still both sects seem to try to attain infinity with explicitly finite comparisons.
So infinity can be watched walking away incessantly
– as expected, after one Big number comes another…
Though the ancient Indians may have believed the purpose of these numbers was just to
ornament the uncountable

.

Now the realm of still *nameable* infinity in Buddhist mathematics
mentions ten names with their squares._{ }
Starting from the first *uncountable* number, the
asamkhyeya
`s _{104}`

*incalculable*, the Avatamsaka Sutra continues with a series of fourth powers to further name

`s`

_{106}*measureless*,

`s`

_{108}*boundless*,

`s`

_{110}*incomparable*,

`s`

_{112}*innumerable*,

`s`

_{114}*unaccountable*,

`s`

_{116}*unthinkable*,

`s`

_{118}*immeasurable*,

`s`

_{120}*unspeakable*,

`s`

_{122}*untold*and finally

`s`

_{123}*square untold*.

_{ }You can use our squaring formula to calculate the exponents of this group of

*asamkhyeyas*, for example the number unspeakable

`s`

_{120}`= 10^(5×2^120)`

`~ 2^2^`*124.05*

**square untold ***10^(5×2^123)*

will be the currently accepted ancient Indian record.By comparison, any number this size (but just one!) can be expressed, on a fine future day, by atom sized bits in a solid iron cube with sides measuring

`452`meter.

With the *untolds*
we've arrived at the end of the Sutra's long list of number names.
What follows may be called *unnameables* or *unmentionables*
and then the *uncallables* and *unlistables*
(all fine Buddhist paradoxes ;-)

On page 833 in Cleary's
FOS
a similar list reads * impure* instead of

*untold*, suggesting that a large enough

*quantity*can turn into a

*quality*. There the atomic and the astronomically large become interchangeable steps on the path to enlightening concentration – any

*order*in terms of size a passing stage in the discrimination of a living world.

Taking the group of *asamkhyeyas* (uncountable numbers)
as fuel for an enlightenment rocket that leaves our petty
little universe,
the Sutra chapter of which we've thus far studied the prose, concludes with a long poem.
Highly elevated concepts come into play, first still squares by nature,
then recursively expanding.
This poem is no less than a mystic mathematical formula that extends the
*asamkhyeyas* to a stairway of exponents `a^b^c^..`

or *power tower*.

The highest number in the
Anuyoga Dwara
is described by counting mustard seeds contained by cylinders
with recursively increasing radius.
The parallel passage in the poem in the Avatamsaka starts iterating
over the extremes of atoms in universes (*buddha-lands*)
and instants in ages of universes (Sanskrit: kalpas = aeons).

[Generally every item in] an

unspeakables[quantity] is filled with_{120}untoldsnumbers of_{122}unspeakables, and when this substitution is repeated for endless ages [to arbitrary depth] not a singleunspeakableatom can ever be completely explained. [vs.1]Now if

untoldbuddha-lands are reduced tounspeakableatoms in aninstant, where every atom containsuntoldlands... [1½iteration, subtotal

s*_{120}s^2 =_{122}10^(45×2^120)lands]

...and thiscontinuous reduction[recursion] moment to momentgoes on for untold aeons, then it's hard to tell the final number of lands or atoms... [vs.2-3]verses 1 ·· 3 · chapter 30 · FOS

It's hard… but let's give it a try! Fill in the known values
and let `m` be the number of moments in an aeon.
The larger than *untold* length of the continuous reduction will be dominant
and dwarf the number operation in the recursion step itself.
Also if `m` is less than *unspeakable* it's hardly significant,
as shown in the calculation below.

Following the general principle set out in verse 1 above, we may assume
that an aeon (of which there is an *untold* number) contains an
*unspeakable* `s _{120}` amount

`m`of instances (atoms of time).

*7*)^(

*s**m) ~ 2^(2^2^

_{122}*126.1**m)

`{m=s`~ 2^2^2^

_{120}}*126.4*

_{ }=

**(**(first new record step)

*10^(25×2^120)*)^^2
A power tower of almost `2^2^2^2^`

about equals the last number of atoms expressed in verse 3
(or the cumulative total of lands and atoms for that matter).*7*

Here we approximate Big numbers with binary power towers
`2^...`

*b*`{2^#c 7≤b<2^7}`
which is the notation we prefer for numbers that go *unnamed*,
but are actually *described* in the buddhist poem at hand.

In our own universe `m` is negligible (as time is running short),
but in higher Buddha worlds it reaches the size of the
asamkhyeyas.
This interpretation is based on
science! and
scripture!
but has no relevance for our
new record!

It's
historically
unclear what kind of *instant* is meant here
(Luk
notes `60` kshana
in a finger-snap at `75` snaps per minute),
but a lower bound can be given by modern physics,
as there are `1.855E42` *shortest instants* of
Planck Time
in a second, about `10^47` per day or `10^50` per year.

Define an *aeon* as the period that life (or consciousness)
exists in a universe.
Everyone will probably be dead when
star formation
stops after a hundred trillion years –
which sets a maximum of `10^17` days and `10^64` *moments*
for the aeon of our cosmos.
Scriptural explanation of the
data use
in the formula below.

In
chapter 31
of the Avatamsaka Sutra called Life Span
every aeon in the field of a Buddha equals a day (and night)
in a higher Buddha world.
So the number of moments `m _{r}` in a Buddha's aeon is

*increasing*exponentially against the number of reduction levels

`r = s`of buddha-lands or fields.

_{122}Our instant land formula calculates the total number

`f`of level

_{r}`r`fields issuing from a higher level

`r+1`field, where

`f`is the field of Shakyamuni Buddha, our stellar universe. The constant

_{0}`c`is the number of fields resulting from the reduction of all the atoms in a single buddha-land, which was fixed in the 2nd verse.

*= 10^(25×2^120)*~ 2^2^

*126*(untold unspeakables)

m

_{r}

*~ 10^(64+17×r)*~ 2^

*(212+55×r)*

f

_{r}= c^m

_{r}~ 2^2^

*(338+55×r)*(instant land formula)

The exact values of the physical coefficients in this
instant land
formula don't matter much, and the effect of the constant `c` is negligible.
Important is that when, as argued
above,
the aeon consists of `s _{120}` instants,
we can use

`m`to derive the level

_{r}`r ~ 4E35`of the Buddhas talking from the Avatamsaka platform (a well kept secret ;o)~

#### §1.7.3. Early evidence of tetration

The numbers that follow in the poem (in chapter 30 of the Avatamsaka Sutra)
are certainly larger, but exactly how large is uncertain.
It's a pity the mathematical theory of recursion
was never in the purview of the translators.
What makes the interpretation difficult is to establish the intended order of reasoning,
which is to begin with in Sanskrit poetry often the other way round,
and then obscured by Chinese grammar which is all too flexible.

Our focus must be on paragraph 4 of the poem in Thomas Cleary's literal version, reinterpreted.
For the latter part of the chapter there is a cornucopia of *multiplications*,
widening in a grand ornamented parable, but then lacking the pure speed
to race up a *power tower*, as we see here.

Iterating this way an

unspeakablenumbert=sof times [or worse:_{120}aeons],

while recursivelycounting aeons by these atoms[by their expanding number].verse 4 · chapter 30 · FOS

What this verse says is, take the
expression
from the previous verses as the first step of a formula
that continues to raise *powers* but now to arbitrary height.
Each step `t`

of this formula expresses a number of atoms,
which will be fed back the next step `t1`

into the coefficient for the number of aeons,
each time adding an exponent on top.

In reality it will cost you at least an aeon to take one such step,
which is why we chose the word times (to stay safe).
Finally, after counting an *unspeakable* number `s _{120}`
of recursive steps

`t`

the true *tetrational record*is set. From this height neither the value of

`m`nor the other coefficients so precisely defined in the past carry weight any more.

*7*)^(m*2^2^2^2^

*7*) ~ 2^2^2^2^2^

*7*

step t: 2^(2^2^

*7**2^...^

*7*)

`{2#t2}`~ 2^...^

*7*

`{2#t3}`

`~ 2^^`

*(t+5½)*(tetration)

record step

*s*:

_{120}**2^^(2^2^**~ 2^^2^^

*124.*)*5½*

The new record number is not just speculation,
for this second iteration is a minimal interpretation.
It is possible that two separate, consecutive recursions can be read in verse 4
(the 2nd verse on
page 892),
which would lead to even higher operations,
but we can't support it – the translation is inevitably confused
and the evidence is too thin.

So what we have here is probably the first description of a
*superpower* in history.
Despite their obviously obscure, perhaps even clumsy,
literary formulations we claim that the Indians achieved
the superpower operation of tetration already…
well before Celtic poets sang of CuChulinn
_{
<and the Ulster Bull>}.

The first iteration grinds worlds to atoms during a number of ages,
counted in the second iteration by the atoms left over from past (minor) ages.
This combination, of minor and major *kalpas*,
has the tetrational might to raise a power tower to
*unspeakable* height after stepping it up so many (major) times.

It doesn't matter if an extra exponent was added to the stairway
in the beginning – not even if we run up these stairs two steps at a time –
which settles the question whether to try to define
aeons
as in the Avatamsaka
chapter 31
or to strictly count the *unspeakable* moments of
chapter 30
what we do here.
The answer is that these stations have past and have no significant impact
on the resulting record number, which is thus established.

The new ancient world record number
is defined exactly and traditionally

as the great Indian
*unspeakable* tetration
10^^*(10^(5×2^120))*

of the Avatamsaka Sutra
thus far unknown in the history of mathematics.

What value the exponents of the power tower actually have is not so significant, given
that the iterator over the height of the tower is itself a large enough number.

The value of the generic `10`

could historically be put at
`10` (decimal base) or standardized as `2`

(binary),
without changing the result significantly.
In fact the constant `ê`

of the power whose derivative increases
equally fast, would be the natural choice for such an idealized exponent.

While these buddhist machinations remained hidden behind the proverbial oriental veil,
an absolute form of *infinity* was planted over from Neoplatonist
philosophy to the Christian world by St.
Augustine (354-430 AD).

The great Indian *unspeakable tetration* is still less than `4^^^3`

so in the context of true infinity it doesn't stand tall.
Strange, but the creators of the Avatamsaka Sutra
thought they were mapping the *uncountable*
all the way!

Enlightening beings called Universally Good, each praised for having

`2^^2`

virtues or more, will return to a point as small as the tip of a fine hair, and occupy it in^{227}unspeakablenumbers...

The same is true ofallpoints in the universe.verses 4 · 5 · chapter 30 · FOS

(^_^) The Buddha way appears mixed up, modular, with little large loopholes for buttonholes in the end (^o^).

Everybody's talkin' at me,
I don't hear a word they're sayin',
only the echoes of my mind.

People stop and stare,
I can't see the faces,
only the shadows of their eyes.

I'm goin' where the sun keep shinin'
through the pouring rain,
goin' where the weather suits my clothes.

Banking off on the northest wind,
sailing on a summer breeze,
skippin’ over the ocean like a stone.

And I won't let you leave my love behind.

– Everybody's Talkin'
Fred Neil

in the film:
Midnight Cowboy